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+The State of Human-centered NLP Technology for Fact-checking
+Anubrata Das∗, Houjiang Liu, Venelin Kovatchev and Matthew Lease
+aSchool of Information, The University of Texas at Austin, Austin, TX, USA
+A R T I C L E I N F O
+Keywords:
+Natural Language Processing
+Misinformation
+Disinformation
+Explainability
+Human-AI Teaming
+A B S T R A C T
+Misinformation threatens modern society by promoting distrust in science, changing narratives in
+public health, heightening social polarization, and disrupting democratic elections and financial
+markets, among a myriad of other societal harms. To address this, a growing cadre of professional
+fact-checkers and journalists provide high-quality investigations into purported facts. However,
+these largely manual efforts have struggled to match the enormous scale of the problem. In
+response, a growing body of Natural Language Processing (NLP) technologies have been
+proposed for more scalable fact-checking. Despite tremendous growth in such research, however,
+practical adoption of NLP technologies for fact-checking still remains in its infancy today.
+In this work, we review the capabilities and limitations of the current NLP technologies for
+fact-checking. Our particular focus is to further chart the design space for how these technologies
+can be harnessed and refined in order to better meet the needs of human fact-checkers. To do
+so, we review key aspects of NLP-based fact-checking: task formulation, dataset construction,
+modeling, and human-centered strategies, such as explainable models and human-in-the-loop
+approaches. Next, we review the efficacy of applying NLP-based fact-checking tools to assist
+human fact-checkers. We recommend that future research include collaboration with fact-checker
+stakeholders early on in NLP research, as well as incorporation of human-centered design
+practices in model development, in order to further guide technology development for human
+use and practical adoption. Finally, we advocate for more research on benchmark development
+supporting extrinsic evaluation of human-centered fact-checking technologies.
+1. Introduction
+Misinformation and related issues (disinformation, deceptive news, clickbait, rumours, and information credibility)
+increasingly threaten society. While concerns of misinformation existed since the early days of written text (Marcus,
+1992), with recent development of social media, the entry barrier for creating and spreading content has never been
+lower. Moreover, polarization online drives the spread of misinformation that in turn increases polarization (Cinelli,
+Pelicon, Mozetič, Quattrociocchi, Novak and Zollo, 2021a,b; Vicario, Quattrociocchi, Scala and Zollo, 2019). Braking
+such a vicious cycle would require addressing the problem of misinformation at its root.
+Fields such as journalism (Graves, 2018b; Graves and Amazeen, 2019; Neely-Sardon and Tignor, 2018) and
+archival studies (LeBeau, 2017) have extensively studied misinformation, and recent years have seen a significant
+growth in fact-checking initiatives to address this problem. Various organizations now focus on fact-checks (e.g.,
+PolitiFact, Snopes, FactCheck, First Draft, and Full Fact), and organizations such as the International Fact-Checking
+Network (IFCN)1 train and provide resources for independent fact-checkers and journalists to further scale expert
+fact-checking.
+While professional fact-checkers and journalists provide high-quality investigations of purported facts to inform
+the public, human effort struggles to match the global Internet scale of the problem. To address this, a growing
+body of research has investigated Natural Language Processing (NLP) to fully or partially automate fact-checking
+(Guo, Schlichtkrull and Vlachos, 2022; Nakov, Corney, Hasanain, Alam, Elsayed, Barrón-Cedeño, Papotti, Shaar and
+Da San Martino, 2021a; Zhou and Zafarani, 2020; Zeng, Abumansour and Zubiaga, 2021; Graves, 2018a). However,
+even state-of-the-art NLP technologies still cannot match human capabilities in many areas and remain insufficient
+to automate fact-checking in practice. Experts argue (Arnold, 2020; Nakov et al., 2021a) that fact-checking is a
+complex process and requires subjective judgement and expertise. While current NLP systems are increasingly better
+at addressing simple fact-checking tasks, identifying false claims that are contextual and beyond simple declarative
+∗Corresponding author
+ORCID(s): 0000-0002-5412-6149 (A. Das); 0000-0003-0983-6202 (H. Liu); 0000-0003-1259-1541 (V. Kovatchev);
+0000-0002-0056-2834 (M. Lease)
+1https://www.politifact.com/,
+https://www.snopes.com/,
+https://www.factcheck.org/,
+https://firstdraftnews.
+org/, https://fullfact.org/, and https://www.poynter.org/ifcn/, respectively.
+Anubrata Das et al.: Preprint submitted to Elsevier
+Page 1 of 30
+arXiv:2301.03056v1  [cs.CL]  8 Jan 2023
+
+The State of Human-centered NLP Technology for Fact-checking
+statements remains beyond the reach for fully automated systems (Chen, Sriram, Choi and Durrett, 2022; Fan, Piktus,
+Petroni, Wenzek, Saeidi, Vlachos, Bordes and Riedel, 2020). For example, claims buried in conversational systems,
+comment threads in social media community, and claims in multimedia contents are particularly challenging for
+automated systems. Additionally, most fact-checking practitioners desire NLP tools that are integrated into the existing
+fact-checking workflow and reduce latency (Nakov et al., 2021a; Graves, 2018b; Alam, Shaar, Dalvi, Sajjad, Nikolov,
+Mubarak, Da San Martino, Abdelali, Durrani, Darwish, Al-Homaid, Zaghouani, Caselli, Danoe, Stolk, Bruntink and
+Nakov, 2021b).
+In this literature review, we provide the reader with a comprehensive and holistic overview of the current state-
+of-the-art challenges and opportunities to more effectively leverage NLP technology in fact-checking. Our objectives
+in this work are twofold. First, we cover all aspects of the NLP pipeline for fact checking:
+task formulation, dataset
+construction, and modeling approaches. Second, we emphasize the human-centered approaches that seek to augment
+and accelerate human fact-checking, rather than supplant it. In contrast, prior literature reviews (Zeng et al., 2021;
+Guo et al., 2022; Oshikawa, Qian and Wang, 2020) either provide an overview of the existing approaches or capture
+the details of only a specific part of the fact-checking pipeline (Kotonya and Toni, 2020a; Hardalov, Arora, Nakov and
+Augenstein, 2021; Hanselowski, AvineshP.V., Schiller, Caspelherr, Chaudhuri, Meyer and Gurevych, 2018; Demartini,
+Mizzaro and Spina, 2020).
+Furthermore, we argue that it is important to extend the review of NLP technologies for fact-checking from
+modeling development to the area of Human-Computer Interaction (HCI) because technology design should reflect
+user needs so that its development can be better integrated in the real-world use context (Graves, 2018a; Lease, 2020;
+Kovatchev, Smith, Lee, Traynor, Aguilera and Devine, 2020; Nakov et al., 2021a; Micallef, Armacost, Memon and
+Patil, 2022; Juneja and Mitra, 2022). Specifically, we point the reader towards Section 7 where we propose concrete
+directions for future work.
+Current challenges are largely due to the relatively early stage of development of the automated fact-checking tech-
+nology. Specifically, current studies tend to adopt an intrinsic evaluation of components of the fact-checking pipeline
+rather than an end-to-end extrinsic evaluation of the entire fact-checking task. Moreover, component-wise accuracies
+may remain below the threshold required for practical adoption. Furthermore, while the research community’s focus
+on prediction accuracy has yielded laudable improvements, human factors (e.g., usability, intelligibility, trust) have
+garnered far less attention or progress yet are crucial for practical adoption.
+Such limitations have implications for future research. First, practical use of NLP technologies for fact-checking
+is likely to come from hybrid, human-in-the-loop approaches rather than full automation. Second, as the technology
+matures, end-to-end evaluation becomes increasingly important to ensure practical solutions are being developed to
+solve the real-world use-case. To this end, new benchmarks that facilitate the extrinsic evaluation of automated fact-
+checking applications in practical settings may help drive progress on solutions that can be adopted for use in the wild.
+Finally, to craft effective human-in-the-loop systems, more cross-cutting NLP and HCI integration could strengthen
+design of fact-checking tools, so that they are accurate, scalable, and usable in practice. Toward this end, it may
+be fruitful to collaborate more with stakeholders early on in NLP research and incorporate human-centered design
+practices in developing models.
+We have written this article with different audiences in mind. For researchers and fact-checkers who are new to
+automated fact-checking, this article provides a comprehensive overview of the problem. We discuss the challenges,
+the state-of-the-art capabilities, and the opportunities in the field, and we emphasize how machine learning and natural
+language processing can be used to combat disinformation. We recommend researchers new to this topic read the
+article in its entirety, following the logical structure of sections. Other readers who have more experience in the
+field may already be familiar with some of the concepts that we discuss. For them, this paper offers a novel human-
+centered perspective of automated fact checking and a discussion on how that perspective can affect system design,
+implementation, and evaluation. To facilitate the use of the paper by more experienced readers, we provide a quick
+overview of the content covered in each section.
+• Section 2 introduces the automated fact-checking pipeline. We provide an overview of the process for human
+fact-checkers and for automated solutions.
+• Section 3 discusses the task formulation: the goals and formal definitions of different sub-tasks in fact checking.
+• Section 4 describes the process of dataset construction, presents the most popular corpora for automated fact
+checking, and outlines some limitations of the data.
+Anubrata Das et al.: Preprint submitted to Elsevier
+Page 2 of 30
+
+The State of Human-centered NLP Technology for Fact-checking
+Figure 1: Fact-checking pipeline
+• Section 5 reviews approaches for automating fact checking. We discuss general NLP capabilities (Section 5.1),
+explainable approaches (Section 5.2), and human-in-the-loop (Section 5.3) approaches for fact-checking.
+• Section 6 surveys existing tools that apply NLP for fact-checking in a practical, real-world context. We argue
+that the human-centered perspective is necessary for the practical adoption of automated solutions.
+• Section 7 provides future research directions in the context of human-centered fact checking. We discuss the work
+division between human and AI for mixed-initiative fact-checking in Section 7.1. In Section 7.2 we propose a
+novel concept for measuring trust and a novel human-centered evaluation of NLP to assist fact-checkers.
+• We conclude our literature review with Section 8.
+2. Fact-Checking Pipeline
+The core idea behind automated fact-checking is enabling AI to reason over available information to determine the
+truthfulness of a claim. For successful automation, it is essential first to understand the complex process of journalistic
+fact-checking that involves human expertise along with skilled effort towards gathering evidence and synthesizing
+the evidence. Additional complexity comes from the need to process heterogeneous sources (e.g., information across
+various digital and non-digital sources). Data is also spread across different modalities such as images, videos, tables,
+graphs, among others. Moreover, there is a lack of tools that support effective and efficient fact-checking workflows
+(Graves, 2018a; Nakov et al., 2021a; Arnold, 2020).
+Graves (2017) breaks down the practical fact-checking mechanism for human fact-checkers into multiple steps
+such as a) identifying the claims to check, b) tracing false claims, c) consulting experts, and d) sharing the resulting
+fact-check. A growing body of AI literature — specifically in NLP — focuses on automating the fact-checking process.
+We synthesize several related surveys (Guo et al., 2022; Nakov et al., 2021a; Graves, 2018a; Zeng et al., 2021;
+Micallef et al., 2022) and distinguish four typical stages that constitute the automated fact-checking technology pipeline
+(illustrated in Figure 1). Note that the pipeline we describe below closely follows the structure of Guo et al. (2022),
+though the broader literature is also incorporated within these four stages:
+• Claim Detection, Checkworthiness, and Prioritization: Claim detection involves monitoring news and/or
+online information for potentially false content to fact-check. One must identify claims that are potentially
+falsifiable (e.g., purported facts rather subjective opinions) (Guo et al., 2022; Zeng et al., 2021). Moreover,
+because it is impractical to fact-check everything online given limited fact-checking resources (human or
+automated), fact checkers must prioritize what to fact-check (Arnold, 2020). NLP researchers have sought to
+inform such prioritization by automatically predicting the "checkworthiness" of claims (Nakov et al., 2021a).
+Additionally, to avoid repeated work, fact-checkers may consult existing fact-checking databases before judging
+the veracity of a new claim (claim matching (Zeng et al., 2021)). We see claim matching as a part of prioritizing
+claims, as fact-checkers would prioritize against checking such claims.
+• Evidence Retrieval: Once it is clear which claims to fact-check, the next step is to gather relevant, trustworthy
+evidence for assessing the claim (Guo et al., 2022; Zeng et al., 2021).
+• Veracity Prediction: Given the evidence, it is necessary to assess it to determine the veracity of the claim (Guo
+et al., 2022; Zeng et al., 2021).
+Anubrata Das et al.: Preprint submitted to Elsevier
+Page 3 of 30
+
+回田
+β
+R.m
+Claim detection,
+Evidence
+Veracity
+Checkworthiness,
+Explanation
+retrieval
+prediction
+and PrioritizationThe State of Human-centered NLP Technology for Fact-checking
+• Explanation: Finally, for human use, one must explain the fact-checking outcome via human-understandable
+justification for the model’s determination (Graves, 2018a; Kotonya and Toni, 2020a; Guo et al., 2022).
+In the subsequent sections, we discuss each of the tasks above in the context of existing NLP research in automated
+fact-checking. Some other steps (for example, detecting propaganda in text, click-bait detection) are also pertinent to
+fact-checking but do not directly fit into the stages described above. They are briefly discussed in Section 3.5.
+3. Task Formulation for Automated Fact-Checking
+Modern Natural Language Processing is largely-data driven. In this article, we distinguish task formulation
+(conceptual) vs. dataset construction (implementation activity, given the task definition). That said, the availability
+of a suitable dataset or the feasibility of constructing a new dataset can also bear on how tasks are formulated.
+3.1. Claim Detection, Checkworthiness, and Prioritization
+Fact-checkers and news organizations monitor information sources such as social media (Facebook, Twitter, Reddit,
+etc.), political campaigns and speeches, and public addresses from government officials on critical issues (Arnold,
+2020; Nakov et al., 2021a). Additional sources include tip-lines on end-to-end encrypted platforms (such as WhatsApp,
+Telegram, and Signal) (Kazemi, Garimella, Shahi, Gaffney and Hale, 2021b). The volume of information on various
+platforms makes it challenging to efficiently monitor all sources for misinformation. Zeng et al. (2021) define the claim
+detection step as identifying, filtering, and prioritizing claims.
+To identify claims, social media streams are often monitored for rumors (Guo et al., 2022). A rumour can be
+defined as a claim that is unverified and being circulated online (Zubiaga, Aker, Bontcheva, Liakata and Procter, 2018).
+Rumours are characterized by the subjectivity of the language and the reach of the content to the users (Qazvinian,
+Rosengren, Radev and Mei, 2011). Additionally, metadata related to virality, such as the number of shares (or retweets
+and re-posts), likes, or comments are also considered when identifying whether a post is a rumour (Zhang, Cao, Li,
+Sheng, Zhong and Shu, 2021a; Gorrell, Kochkina, Liakata, Aker, Zubiaga, Bontcheva and Derczynski, 2019). However,
+detecting rumours alone is not sufficient to decide whether a claim needs to be fact-checked.
+For each text of interest, the key questions fact-checking systems need to address include:
+1. Is there a claim to check?
+2. Does the claim contain verifiable information?
+3. Is the claim checkworthy?
+4. Has a trusted source already fact-checked the claim?
+Regarding the first criterion — is there a claim — one might ask whether the claim contains a purported fact or
+an opinion (Hassan, Arslan, Li and Tremayne, 2017a). For example, a statement such as “reggae is the most soulful
+genre of music” represents personal preference that is not checkable. In contrast, “<NAME> won a gold medal in the
+Olympics” is checkable by matching <NAME> to the list of all gold medal winners.
+Whether the claim contains verifiable information is more challenging. For example, if a claim can only be verified
+by private knowledge or personal experience that is not broadly accessible, then it cannot be checked (Konstantinovskiy,
+Price, Babakar and Zubiaga, 2021). For example, if someone claims to have eaten a certain food yesterday, it is probably
+impossible to verify beyond their personal testimony.
+As this example suggests, the question of whether the claim contains verifiable information depends in large part
+on what evidence is available for verification. This, in turn, may not be clear until after evidence retrieval is performed.
+In practice fact-checkers may perform some preliminary research, but mostly try to gauge checkworthiness only based
+on the claim itself.
+In addition, this consideration is only one of many that factors into deciding whether to check a claim. Even a claim
+that may appear to be unverifiable may still be of such great public interest that it is worth conducting the fact-check.
+Moreover, even if the fact-check is conducted and ultimately indeterminate (i.e., evidence does not exist either to verify
+or refute the claim), simply showing that a claim’s veracity cannot be determined may still be a valuable outcome.
+A claim is deemed checkworthy if a claim is of significant public interest or has the potential to cause harm (Nakov,
+Da San Martino, Elsayed, Barrón-Cedeño, Míguez, Shaar, Alam, Haouari, Hasanain, Babulkov et al., 2021b; Hassan,
+Li and Tremayne, 2015). For example, a claim related to the effect of a vaccine on the COVID-19 infection rate is more
+relevant to the public interest and hence more checkworthy than a claim about some philosopher’s favorite food.
+Anubrata Das et al.: Preprint submitted to Elsevier
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+
+The State of Human-centered NLP Technology for Fact-checking
+Claims, like memes, often appear several times and/or across multiple platforms (in the same form or with
+slight modification) (Leskovec, Backstrom and Kleinberg, 2009; Nakov et al., 2021a; Arnold, 2020). Fact-checking
+organizations maintain a growing database claims which have already been fact-checked. Thus, detected claims are
+compared against databases of already fact-checked claims by trusted organizations (Shaar, Martino, Babulkov and
+Nakov, 2020; Shaar, Alam, Da San Martino and Nakov, 2021a). Comparing new claims against such databases helps
+to avoid duplicating work on previously fact-checked claims. This step is also known as claim matching (Zeng et al.,
+2021).
+Reports from practitioners argue that if a claim is not checked within the first few hours, a late fact-check
+does not have much impact on changing the ongoing misinformation narrative (Nakov et al., 2021a; Arnold, 2020).
+Moreover, limited resources for fact-checking make it crucial for organizations to prioritize the claims to be checked
+(Borel, 2016). Claims can be prioritized based on their checkworthiness (Nakov, Da San Martino, Barrón-Cedeño,
+Zaghouani, Míguez, Alam, Caselli, Kutlu, Strub and Mandl, 2022; Nakov et al., 2021b). Nakov et al. (2022) note that
+checkworthiness is determined based on factors such as
+1. How urgently a claim needs to be checked?
+2. How much harm can a claim cause (Alam et al., 2021b; Alam, Dalvi, Shaar, Durrani, Mubarak, Nikolov,
+Da San Martino, Abdelali, Sajjad, Darwish and Nakov, 2021a; Shaar, Hasanain, Hamdan, Ali, Haouari, Nikolov,
+Kutlu, Kartal, Alam, Da San Martino, Barrón-Cedeño, Míguez, Beltrán, Elsayed and Nakov, 2021b)?
+3. Would the claim require attention from policy makers for addressing the underlying issue?
+Note that estimating harms is quite challenging, especially without first having a thorough understanding and measures
+of harm caused by misinformationNeumann, De-Arteaga and Fazelpour (2022).
+The spread of a claim on social media provides another potential signal for identifying public interest (Arnold,
+2020). In the spirit of doing “the greatest good for the greatest number”, viral claims might be prioritized highly because
+any false information in them has the potential to negatively impact a large number of people. On the other hand, since
+fairness considerations motivate equal protections for all people, we cannot serve only the majority at the expense of
+minority groups (Ekstrand, Das, Burke, Diaz et al., 2022; Neumann et al., 2022). Moreover, such minority groups may
+be more vulnerable, motivating greater protections, and may be disproportionately impacted by mis/disinformation
+(Guo et al., 2022). See Section 3.6 for additional discussion.
+3.2. Evidence Retrieval
+Some sub-tasks in automated fact-checking can be performed without the presence of explicit evidence. For
+example, the linguistic properties of the text can be used to determine whether it is machine-generated (Wang, 2017;
+Rashkin, Choi, Jang, Volkova and Choi, 2017). However, assessing assessing claim veracity without evidence is clearly
+more challenging (Schuster, Schuster, Shah and Barzilay, 2020).
+Provenance of a claim can also signal information quality; known unreliable source or distribution channels are
+often repeat offenders in spreading false information2. Such analysis of provenance can be further complicated when
+content is systematically propagated by multiple sources (twitter misinformation bots) (Jones, 2019).
+It is typically assumed that fact-checking requires gathering of reliable and trustworthy evidence that provides
+information to reason about the claim (Graves, 2018a; Li, Gao, Meng, Li, Su, Zhao, Fan and Han, 2016). In some
+cases, multiple aspects of a claim needs to be checked. A fact-checker would then decompose such a claim into distinct
+questions and gather relevant evidence for the question (Borel, 2016; Chen et al., 2022). From an information retrieval
+(IR) perspective, we can conceptualize each of those questions as an “information need” for which the fact-checker must
+formulate one or more queries to a search engine (Bendersky, Metzler and Croft, 2012) in order to retrieve necessary
+evidence.
+Evidence can be found across many modalities, including text, tables, knowledge graphs, images, and videos.
+Various metadata can also provide evidence and are sometimes required to assess the claim. Examples include context
+needed to disambiguate claim terms, or background of the individual or organization from whom the claim originated.
+Retrieving relevant evidence also depends on the following questions (Singh, Das, Li and Lease, 2021):
+1. Is there sufficient evidence available related to a claim?
+2. Is it accessible or available in the public domain?
+3. Is it in a format that can be read and processed?
+2https://disinformationindex.org/
+Anubrata Das et al.: Preprint submitted to Elsevier
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+The State of Human-centered NLP Technology for Fact-checking
+As noted earlier in Section 3.1, the preceding claim detection task involves assessing whether a claim contains verifiable
+information; this depends in part on what evidence exists to be retrieved, which is not actually known until evidence
+retrieval is performed. Having now reached this evidence retrieval step, we indeed discover whether sufficient evidence
+exists to support or refute the claim.
+Additionally, evidence should be trustworthy, reputable (Nguyen, Kharosekar, Lease and Wallace, 2018b; Lease,
+2018), and unbiased (Chen, Khashabi, Yin, Callison-Burch and Roth, 2019a).
+Once evidence is retrieved, stance detection assesses the degree to which the evidence supports or refutes the
+claim (Nguyen, Kharosekar, Krishnan, Krishnan, Tate, Wallace and Lease, 2018a; Ferreira and Vlachos, 2016; Popat,
+Mukherjee, Yates and Weikum, 2018). Stance detection is typically formulated as a classification task (or ordinal
+regression) over each piece of retrieved evidence. Note that some works formulate stance detection as an independent
+task (Hanselowski et al., 2018; Hardalov et al., 2021).
+3.3. Veracity Prediction
+Given a claim and gathered evidence, veracity prediction involves reasoning over the collected evidence and the
+claim. Veracity prediction can be formulated as a binary classification task (i.e., true vs. false) (Popat et al., 2018;
+Nakashole and Mitchell, 2014; Potthast, KIESELJ et al., 2018), or as a fine-grained, multi-class task following the
+journalistic fact-checking practices (Augenstein, Lioma, Wang, Lima, Hansen, Hansen and Simonsen, 2019; Shu,
+Mahudeswaran, Wang, Lee and Liu, 2020; Wang, 2017). In some cases, there may not be enough information available
+to determine the veracity of a claim (Thorne, Vlachos, Christodoulopoulos and Mittal, 2018).
+Note that fact-checking is potentially a recursive process because retrieved evidence may itself need to be fact-
+checked before it can be trusted and acted upon (Graves, 2018a). This is also consistent with broader educational
+practices in information literacy3 in which readers are similarly encouraged to evaluate the quality of information
+they consume. Such assessment of information reliability can naturally integrate with the veracity prediction task in
+factoring in the reliability of the evidence along with its stance (Nguyen et al., 2018b; Guo et al., 2022).
+3.4. Explaining Veracity Prediction
+While a social media platform might use automated veracity predictions in deciding whether to automatically block
+or demote content, the use of fact-checking technology often involves a human-in-the-loop, whether it is a platform
+moderator, a journalist, or an end-user. When we consider such human-centered use of fact-checking technologies,
+providing an automated veracity prediction without justifying the answer can cause a system to be ignored or distrusted,
+or even reinforce mistaken human beliefs in false claims (the “backfire effect” (Lewandowsky, Ecker, Seifert, Schwarz
+and Cook, 2012)). Explanations and justifications are especially important given the noticeable drop in performance of
+state-of-the-art NLP systems when facing adversarial examples (Kovatchev, Chatterjee, Govindarajan, Chen, Choi,
+Chronis, Das, Erk, Lease, Li et al., 2022). Consequently, automated fact-checking systems intended for human-
+consumption should seek to explain their veracity predictions in a similar manner to that of existing journalistic
+fact-checking practices (Uscinski, 2015). A brief point to make is that much of the explanation research has focused
+on explanations for researchers and engineers engaged in system development (types of explanations, methods of
+generating them, and evaluation regimens). In contrast,we emphasize here explanations for system users.
+Various types of explanations can be provided, such as through
+1. evidence attribution
+2. explaining the decision-making process for a fact-check
+3. summarizing the evidence
+4. case-based explanations
+Evidence attribution is the process of identifying evidence or a specific aspect of the evidence (such as paragraphs,
+sentences, or even tokens of interest) (Thorne et al., 2018; Popat et al., 2018; Shu, Cui, Wang, Lee and Liu, 2019; Lu
+and Li, 2020). Furthermore, the relative importance of the evidence can also justify the fact-checking outcome (Nguyen
+et al., 2018a). Alternatively, a set of rules or interactions to break down parts of the decision-making process can also
+serve as an explanation (Gad-Elrab, Stepanova, Urbani and Weikum, 2019; Nguyen et al., 2018b). Such formulation
+focuses more on how the evidence is processed to arrive at a decision. Explaining the veracity can also be formulated
+as a summarization problem over the gathered evidence to explain a fact-check (Atanasova, Simonsen, Lioma and
+Augenstein, 2020a; Kotonya and Toni, 2020b). Finally, case-based explanations can provide the user with similar,
+human-labeled instances (Das, Gupta, Kovatchev, Lease and Li, 2022).
+3https://en.wikipedia.org/wiki/Information_literacy
+Anubrata Das et al.: Preprint submitted to Elsevier
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+
+The State of Human-centered NLP Technology for Fact-checking
+3.5. Related Tasks
+In addition to tasks that are considered central to the automated fact-checking pipeline, some additional tasks bear
+mentioning as related and complementary to the fact-checking enterprise. Examples of such tasks include propaganda
+detection (Da San Martino, Cresci, Barrón-Cedeño, Yu, Pietro and Nakov, 2020), clickbait detection (Potthast, Köpsel,
+Stein and Hagen, 2016), and argument mining (Lawrence and Reed, 2020). Furthermore, some tasks can be formulated
+independent of the fact-checking pipeline and utilized later to improve individual fact-checking sub-tasks. For example,
+predicting the virality of social media content (Jain, Garg and Jain, 2021) can help improve claim detection and claim
+checkworthiness. Similarly, network analysis on fake news propagation (Shao, Ciampaglia, Flammini and Menczer,
+2016) can help in analyzing provenance.
+With an eye toward building more human-centered AI approaches, there are also some tasks that could be applied
+to help automate parts of the fact-checking process. For example, claim detection might be improved via an URL
+recommendation engine for content that might need fact-checking (Vo and Lee, 2018). Additionally, fact-checkers
+could benefit from a predicted score for claim difficulty (Singh et al., 2021). In terms of evidence retrieval and
+veracity prediction, one might generate fact-checking briefs to aid inexpert fact-checkers (Fan et al., 2020). Instead
+of summarizing the evidence in general (Section 3.4), one might instead summarize with the specific goal of decision
+support (Hsu and Tan, 2021).
+3.6. Key Challenges
+Most work in automated fact-checking has been done on veracity prediction, and to a lesser extent, on explanation
+generation. Recently, we have seen more attention directed towards claim detection and checkworthiness. In contrast,
+work on evidence retrieval remains less developed.
+Claim Detection Guo et al. (2022) points out several sources of biases in the claim check-worthiness task. Claims
+could be of variable interest to different social groups. Additionally, claims that might cause more harm to marginalized
+groups compared to the general population may not get enough attention. Ideally, models identifying check-worthiness
+need to overcome any possible disparate impact.
+Similar concerns appear in the report by Full Fact (Arnold, 2020). One of the criteria for selecting a claim for fact-
+checking across several organizations is “Could the claim threaten democratic processes or minority groups?” However,
+such criterion may be at odds with the concerns of virality. Fact-checking organizations often monitor virality metrics
+to decide which claims to fact-check (Arnold, 2020; Nakov et al., 2021a). Nevertheless, if a false claim is targeted
+towards an ethnic minority, such claims may not cross the virality thresholds.
+Prioritizing which claims to fact-check requires attention to various demographic traits: content creators, readers,
+and subject matter. Claim check-worthiness dataset design can thus benefit from consideration of demographics.
+Evidence Retrieval Evidence retrieval has been largely neglected in the automated fact-checking NLP literature. It
+is often assumed that evidence is already available, or, coarse-grained evidence is gathered from putting the claims into
+a search engine (Popat et al., 2018; Nguyen et al., 2018a). However, Hasanain and Elsayed (2022) show in their study
+that search engines optimized for relevance seldom retrieve evidence most useful for veracity prediction. Although
+retrieving credible information has been studied thoroughly in IR (Clarke, Rizvi, Smucker, Maistro and Zuccon,
+2020a), more work is needed that is focused on retrieving evidence for veracity assessment (Lease, 2018; Clarke,
+Rizvi, Smucker, Maistro and Zuccon, 2020b).
+Veracity Prediction and Explanation A critical challenge for automated systems is to reason over multiple sources
+of evidence while also taking source reputation into account. Additionally, explaining a complex reasoning process is
+a non-trivial task. The notion of model explanations itself is polysemous and evolving in general, not to mention in the
+context of fact checking. As explainable NLP develops, automated fact-checking tasks also need to evolve and provide
+explanations that are accessible to human stakeholders yet faithful to the underlying model. For example, case-based
+explanations are mostly unexplored in automated fact-checking, although working systems have been proposed for
+propaganda detection (Das et al., 2022).
+In many NLP tasks, such as machine translation or natural language inference, the goal is to build fully-automated,
+end-to-end solutions. However, in the context of fact-checking, state-of-the-art limitations suggest the need for humans-
+in-the-loop for the forseable future. Given this, automated tooling to support human fact-checkers is crucial. However,
+Anubrata Das et al.: Preprint submitted to Elsevier
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+
+The State of Human-centered NLP Technology for Fact-checking
+understanding the fact-checker needs and incorporating those needs in the task formulation has been largely absent
+from the automated fact-checking literature, with a few notable exceptions (Nakov et al., 2021a; Demartini et al., 2020).
+Future research could benefit from greater involvement of fact-checkers in the NLP research process and shifting goals
+from complete automation toward human support.
+4. Dataset Construction
+Corresponding to task formulation (Section 3), our presentation of fact-checking datasets is also organized around
+claims, evidence, veracity prediction, and explanation. Note that not all datasets have all of these components.
+4.1. Claim detection and claim check-worthiness
+Claim detection datasets typically contain claims and their sources (documents, social media streams, transcripts
+from political speeches) (Guo et al., 2022). One form of claim detection is identifying rumours on social media, where
+datasets are primarily constructed with text from Twitter (Zubiaga et al., 2018; Qazvinian et al., 2011) and Reddit
+(Gorrell et al., 2019; Lillie, Middelboe and Derczynski, 2019). Some works provide the claims in the context they
+appeared on social media (Zhang et al., 2021a; Ma, Gao, Mitra, Kwon, Jansen, Wong and Cha, 2016). However,
+several studies note that most claim detection datasets do not contain enough context. As the discussion of metadata
+in Section 3 suggests, broader context might include: social media reach, virality metrics, the origin of a claim, and
+relevant user data (i.e., who posted a claim, how influential they are online, etc.) (Arnold, 2020; Nakov et al., 2021a).
+Claim check-worthiness datasets (Nakov et al., 2021b; Shaar et al., 2021b; Barrón-Cedeño, Elsayed, Nakov,
+Da San Martino, Hasanain, Suwaileh, Haouari, Babulkov, Hamdan, Nikolov et al., 2020; Atanasova, Barrón-Cedeño,
+Elsayed, Suwaileh, Zaghouani, Kyuchukov, Da San Martino and Nakov, 2018; Konstantinovskiy et al., 2021; Hassan
+et al., 2015) filter claims from a source (similar to claim detection, sources include social media feeds and political
+debate transcripts, among others) by annotating claims based on the checkworthiness criteria (mentioned in the section
+3.1). Each claim is given a checkworthiness score to obtain a ranked list. Note that claim detection and checkworthiness
+datasets may be expert annotated (Hassan et al., 2015) or crowd annotated (Nakov et al., 2021b; Shaar et al., 2021b;
+Barrón-Cedeño et al., 2020; Atanasova et al., 2018; Konstantinovskiy et al., 2021)4.
+The datasets discussed above do not capture multi-modal datasets, and few do. One such dataset is r/Fakeddit
+(Nakamura, Levy and Wang, 2020). This dataset contains images and associated text content from Reddit as claims.
+Misinformation can also spread through multi-modal memes, and tasks such as Facebook (now Meta) Hateful Memes
+Challenge (Kiela, Firooz, Mohan, Goswami, Singh, Ringshia and Testuggine, 2020) for hate speech suggest what might
+be similarly done for misinformation detection.
+4.2. Evidence
+Early datasets in fact-checking provide metadata with claims as the only form of evidence. Such metadata include
+social media post properties, user information, publication date, source information (Wang, 2017; Potthast et al., 2018).
+As discussed earlier in Section 3.2, such metadata does not contain the world knowledge necessary to reason about a
+complex claim. To address the above limitations, recent datasets consider external evidence (Guo et al., 2022).
+Evidence is collected differently depending upon the problem setup. For artificial claims, evidence is often retrieved
+from a single source such as Wikipedia articles (Thorne et al., 2018; Jiang, Bordia, Zhong, Dognin, Singh and Bansal,
+2020; Schuster, Fisch and Barzilay, 2021). Domain limited evidence for real-world claims is collected from problem-
+specific sources, such as academic articles for scientific claims (Kotonya and Toni, 2020b; Wadden, Lin, Lo, Wang, van
+Zuylen, Cohan and Hajishirzi, 2020), or specific evidence listed in fact-checking websites (Vlachos and Riedel, 2014;
+Hanselowski, Stab, Schulz, Li and Gurevych, 2019). Open-domain evidence for real-world claims is usually collected
+from the web via search engines (Popat et al., 2018; Augenstein et al., 2019).
+Recently, there has been more work considering evidence beyond free text. Such formats include structured or semi-
+structured forms of evidence. Sources include knowledge bases for structured form of evidence (Shi and Weninger,
+2016) and semi-structured evidence from semi-structured knowledge bases (Vlachos and Riedel, 2015), tabular data
+(Chen et al., 2019a; Gupta, Mehta, Nokhiz and Srikumar, 2020), and tables within a document (Aly, Guo, Schlichtkrull,
+Thorne, Vlachos, Christodoulopoulos, Cocarascu and Mittal, 2021).
+Additionally, there are some retrieval-specific datasets that aim at retrieving credible information from search
+engines (Clarke et al., 2020b). However, such tasks don’t incorporate claim checking as an explicit task.
+4Some of these datasets, such as the CheckThat! datasets, are partially crowd and partially expert annotated.
+Anubrata Das et al.: Preprint submitted to Elsevier
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+The State of Human-centered NLP Technology for Fact-checking
+4.3. Veracity Prediction
+Evidence retrieval and veracity prediction datasets are usually constructed jointly. Note, in some cases, evidence
+may be absent from the datasets. Veracity prediction datasets usually do not deal with claim detection or claim
+checkworthiness tasks separately. Instead, such datasets contain a set of claims that are either artificially constructed
+or collected from the internet.
+Artificial claims in veracity prediction datasets are often limited in scope and constructed for natural language
+reasoning research (Aly et al., 2021; Thorne et al., 2018; Schuster et al., 2021; Jiang et al., 2020; Chen, Wang, Chen,
+Zhang, Wang, Li, Zhou and Wang, 2019b). For example, FEVER (Thorne et al., 2018) and HoVer (Jacovi and Goldberg,
+2021) obtain claims from Wikipedia pages. Some datasets also implement subject-predicate-object triplets for fact-
+checking against knowledge bases (Kim and Choi, 2020; Shi and Weninger, 2016).
+Fact-checking websites are popular sources for creating veracity prediction datasets based on real claims. Several
+datasets obtain claims from either a single website or collect claims from many such websites and collate them (Wang,
+2017; Hanselowski et al., 2018; Augenstein et al., 2019; Vlachos and Riedel, 2014). Note that such claims are inherently
+expert annotated. Other sources of claims are social media (Potthast et al., 2018; Shu, Sliva, Wang, Tang and Liu,
+2017), news outlets (Horne, Khedr and Adali, 2018; Gruppi, Horne and Adalı, 2021; Nørregaard, Horne and Adalı,
+2019), blogs, discussions in QA forums, or similar user-generated publishing platforms (Mihaylova, Nakov, Màrquez,
+Barrón-Cedeño, Mohtarami, Karadzhov and Glass, 2018).
+Additionally some fact-checking datasets target domain-specific problems such as scientific literature (Wadden
+et al., 2020), climate change (Diggelmann, Boyd-Graber, Bulian, Ciaramita and Leippold, 2020), and public health
+(Kotonya and Toni, 2020b). Most datasets are monolingual but recent effort have started to incorporate multi-lingual
+claims (Gupta and Srikumar, 2021; Barnabò, Siciliano, Castillo, Leonardi, Nakov, Da San Martino and Silvestri, 2022).
+Early datasets focus on a binary veracity prediction - true or false (Mihalcea and Strapparava, 2009). Recent datasets
+often adopt an ordinal veracity labeling scheme that mimics fact checkering websites (Vlachos and Riedel, 2014; Wang,
+2017; Augenstein et al., 2019). However, every fact-checking website has a different scale for veracity, so datasets
+that span across multiple websites come with a normalization problem. While some datasets do not normalize the
+labels (Augenstein et al., 2019), some normalize them post-hoc (Kotonya and Toni, 2020a; Gupta and Srikumar, 2021;
+Hanselowski et al., 2019).
+4.4. Explanation
+While an explanation is tied to veracity prediction, only a few datasets explicitly address the problem of explainable
+veracity prediction (Atanasova et al., 2020a; Kotonya and Toni, 2020b; Alhindi, Petridis and Muresan, 2018). Broadly
+in NLP, often parts of the input is highlighted to provide an explanation for the prediction. This form of explanations
+is known as extractive rationale (Zaidan, Eisner and Piatko, 2007; Kutlu, McDonnell, Elsayed and Lease, 2020).
+Incorporating the idea of the extractive rationale, some datasets include a sentence from the evidence along with
+the label (Thorne et al., 2018; Hanselowski et al., 2018; Wadden et al., 2020; Schuster et al., 2021). Although such
+datasets do not explicitly define evidence as a form of explanation in such cases, the line between evidence retrieval and
+explanation blurs if the evidence is the explanation. However, explanations are different from evidence in a few ways.
+Particularly, explanations need to be concise for user consumption, while evidence can be a collection of documents
+or long documents. Explanations are user sensitive. Consequently, evidence alone as a form of explanation might have
+some inherent assumption about the user that might not be understandable for different groups of users (e.g., experts
+vs. non-experts).
+4.5. Challenges
+Claims Checkworthiness datasets are highly imbalanced, i.e., the number of checkworthy claims are relatively low
+compared to non-checkworthy claims (Williams, Rodrigues and Novak, 2020). Datasets are also not generalizable
+due to their limited domain-specific context (Guo et al., 2022). Additionally, while existing datasets cover various
+languages such as English, Arabic, Spanish, Bulgarian, and Dutch, they are primarily monolingual.
+Consequently,
+building multilingual checkworthiness predictors is still challenging. Much of the data in check-worthiness datasets is
+not originally intended to be used in classification. The criteria used by different organizations when selecting which
+claims to check is often subjective and may not generalize outside of the particular organization.
+Some annotation practices can result in artifacts in the dataset. For example, artificially constructed false claims,
+such as a negation-based false claim in FEVER, can lead to artifacts in models (Schuster et al., 2021). Models do not
+generalize well beyond the dataset because they might overfit to the annotation schema (Bansal, Nushi, Kamar, Lasecki,
+Anubrata Das et al.: Preprint submitted to Elsevier
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+
+The State of Human-centered NLP Technology for Fact-checking
+Weld and Horvitz, 2019). One way to identify such blind spots is by using adversarial datasets for fact-checking. Such
+a setting is incorporated in FEVER 2.0 (Thorne and Vlachos, 2019).
+Datasets constructed for research may not always capture how fact-checkers work in practice. This leads to
+limitations in the algorithms built on them. For example, interviews with fact-checkers report that they tend to consider
+a combination of contents of the posts and associated virality metrics (indicating reach) during fact-checking (Arnold,
+2020). However, most fact-checking datasets do not include virality metrics.
+Evidence Retrieval Some datasets have been constructed by using a claim verbatim as a query and taking the
+top search results as evidence. However, some queries are better than others for retrieving desired information.
+Consequently, greater care might be taken in crafting effective queries or otherwise improving evidence retrieval such
+that resulting datasets are more likely to contain quality evidence for veracity prediction. Otherwise, poor quality
+evidence becomes a bottleneck for the quality of the models trained at the later stages in the fact-checking pipeline
+(Singh et al., 2021).
+Veracity Prediction A key challenge in veracity prediction datasets is that the labels are not homogeneous across
+fact-checking websites and normalizing might introduce noise.
+Explanation Some datasets include entire fact-checking articles as evidence and their summaries as the form of
+explanation (Atanasova et al., 2020a; Kotonya and Toni, 2020b). In such cases, “explanation” components assume an
+already available fact-checking article. Instead, providing abstractive summaries and explaining the reasoning process
+over the evidence would be more valuable.
+Data Generation Recent years have seen an increasing interest in the use of data generation and data augmentation
+for various NLP tasks (Liu, Swayamdipta, Smith and Choi, 2022; Hartvigsen, Gabriel, Palangi, Sap, Ray and Kamar,
+2022; Dhole, Gangal, Gehrmann, Gupta, Li, Mahamood, Mahendiran, Mille, Srivastava, Tan et al., 2021; Kovatchev,
+Smith, Lee and Devine, 2021). The use of synthetic data has not been extensively explored in the context of fact-
+checking.
+5. Automating Fact-checking
+NLP research in automated fact-checking has primarily focused on building models for different automated fact-
+checking tasks utilizing existing datasets. In the following section, we highlight the broad modeling strategies employed
+in the literature, with more detailed discussion related to explainable methods for automated fact-checking.
+5.1. General NLP Capabilities
+Claim Detection and Checkworthiness While claim detection is usually implemented as a classification task only,
+claim checkworthiness is typically implemented both as ranking (Nakov et al., 2021a) and classification task (Zeng
+et al., 2021). As discussed earlier in the task formulation Section (3.1), the broad task of claim detection can be
+broken down into sub-tasks of identifying claims, filtering duplicate claims, and prioritizing claims based on their
+checkworthiness. Another instance of identifying claims is detecting rumors in social media streams.
+Some early works in rumor detection focus on feature engineering from available metadata the text itself (Enayet
+and El-Beltagy, 2017; Aker, Derczynski and Bontcheva, 2017; Zhou, Jain, Phoha and Zafarani, 2020). More advanced
+methods for claim detection involve LSTM and other sequence models (Kochkina, Liakata and Augenstein, 2017).
+Such models are better at capturing the context of the text (Zubiaga, Liakata, Procter, Wong Sak Hoi and Tolmie,
+2016). Tree-LSTM (Ma, Gao and Wong, 2018) and Hierarchical attention networks (Guo, Cao, Zhang, Guo and Li,
+2018) capture the internal structure of the claim or the context in which the claim appears. Additionally, graph neural
+network approaches can capture the related social media activities along with the text (Monti, Frasca, Eynard, Mannion
+and Bronstein, 2019).
+Similarly, early works in claim-checkworthiness utilize support vector machines using textual features and rank
+the claims in terms of their priorities (Hassan et al., 2017a). For example, Konstantinovskiy et al. (2021) build a
+classification model for checkworthiness by collapsing the labels to checkable vs. non-checkable claim. They build
+a logistic regression model that uses word embeddings along with syntax based features (parts of speech tags, and
+named entities). Neural methods such as LSTM performed well in earlier checkworthiness shared tasks (Elsayed,
+Nakov, Barrón-Cedeño, Hasanain, Suwaileh, Da San Martino and Atanasova, 2019). Additionally, Atanasova, Nakov,
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+The State of Human-centered NLP Technology for Fact-checking
+Màrquez, Barrón-Cedeño, Karadzhov, Mihaylova, Mohtarami and Glass (2019b) show that capturing context helps
+with the checkworthiness task as well. Models such as RoBERTa obtained higher performance in the later edition of the
+CheckThat! shared task (Williams et al., 2020; Martinez-Rico, Martinez-Romo and Araujo, 2021) for English language
+claims. Fine-tuning such models for claim detection tasks has become more prevalent for claim checkworthiness in
+other languages as well (Hasanain and Elsayed, 2020; Williams et al., 2020).
+Filtering previously fact-checked claims is a relatively new task in this domain. Shaar et al. (2020) propose an
+approach using BERT and BM-25 to match claims against fact-checking databases for matching claims with existing
+databases. Additionally, fine-tuning RoBERTa on various fact-checking datasets resulted in high performance for
+identifying duplicate claims (Bouziane, Perrin, Cluzeau, Mardas and Sadeq, 2020). Furthermore, a combination of
+pretrained model Sentence-BERT and re-ranking with LambdaMART performed well for detecting previously fact-
+checked claims (Nakov et al., 2021b).
+Evidence Retrieval and Veracity Prediction Evidence retrieval and veracity prediction in the pipeline can be
+modeled sequentially or jointly. Similar to claim detection and checkworthiness models, early works use stylistic
+features and metadata to arrive at veracity prediction without external evidence (Wang, 2017; Rashkin et al., 2017).
+Models that include evidence retrieval often use commercial search APIs or some retrieval approach such as TF-IDF,
+and BM25 (Thorne et al., 2018). Similar to question-answering models, some works adopt a two-step approach. First
+a simpler model (TF-IDF or BM-25) is used at scale and then a more complex model is used for re-ranking after the
+initial pruning (Thorne et al., 2018; Nie, Wang and Bansal, 2019; Hanselowski et al., 2019). Additionally, document vs.
+passage retrieval, or 2-stage “telescoping” approaches, are adopted where the first stage is retrieving related documents
+and the second stage is to retrieve the relevant passage. Two stage approaches are useful for scaling up applications
+as the first stage is more efficient than the second stage. For domain specific evidence retrieval, using domain-bound
+word embeddings has been shown to be effective (Zeng et al., 2021).
+The IR task is not always a part of the process. Instead, it is often assumed that reliable evidence is already available.
+While this simplifies the fact-checking task so that researchers can focus on veracity prediction, in practice evidence
+retrieval is necessary and cannot be ignored. Moreover, in practice one must contend with noisy (non-relevant), low
+quality, and biased search results during inference.
+As discussed earlier in Section 3.3, assessing the reliability of gathered evidence may be necessary. If the evidence is
+assumed to be trustworthy, then it suffices to detect the stance of each piece of evidence and then aggregate (somehow)
+to induce veracity (e.g., perhaps assuming all evidence is equally important and trustworthy). However, often one
+must contend with evidence “in the wild” of questionable reliability, in which case assessing the quality (and bias) of
+evidence is an important precursor to using it in veracity prediction.
+Veracity prediction utilizes textual entailment for inferring veracity over either a single document as evidence
+or over multiple documents. Dagan, Dolan, Magnini and Roth (2010) define textual entailment as “deciding, given
+two text fragments, whether the meaning of one text is entailed (can be inferred) from another text.” Real-world
+applications often require reasoning over multiple documents (Augenstein et al., 2019; Kotonya and Toni, 2020b;
+Schuster et al., 2021). Reasoning over multiple documents can be done either by concatenation (Nie et al., 2019) or
+weighted aggregation (Nguyen et al., 2018b). Weighted aggregation virtually re-ranks the evidence considered to filter
+out the unreliable evidence (Ma, Gao, Joty and Wong, 2019; Pradeep, Ma, Nogueira and Lin, 2021). Some approaches
+also use Knowledge Bases as the central repository of all evidence (Shi and Weninger, 2016). However, evidence is
+only limited to what is available in the knowledge base (Guo et al., 2022; Zeng et al., 2021). Moreover, a fundamental
+limitation of knowledge bases is that not all knowledge fits easily into structured relations.
+Recent developments in large language models help extend the knowledge base approach. Fact-checking models
+can rely on pretrained models to provide evidence for veracity prediction (Lee, Li, Wang, Yih, Ma and Khabsa, 2020).
+However, this approach can encode biases present in the language model (Lee, Bang, Madotto and Fung, 2021).
+An alternative approach is to help fact-checkers with downstream tasks by processing evidence. An example of
+such work is generating summaries over available evidence using BERT (Fan et al., 2020).
+Limitations With the recent development of large, pre-trained language models and deep learning for NLP, we see
+a significant improvement across the fact-checking pipeline. With the introduction of FEVER (Thorne et al., 2018;
+Thorne, Vlachos, Cocarascu, Christodoulopoulos and Mittal, 2019; Aly et al., 2021) and CheckThat! (Nakov et al.,
+2021b) we have benchmarks for both artificial and real-life claim detection and verification models. However, even
+the state-of-the-art NLP models perform poorly on the benchmarks above. For example, the best performing model on
+Anubrata Das et al.: Preprint submitted to Elsevier
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+The State of Human-centered NLP Technology for Fact-checking
+FEVER 2018 shared task (Thorne et al., 2018) reports an accuracy of 0.675. Models perform worse on multi-modal
+shared task FEVEROUS (Aly et al., 2021): the best performing model reports 0.56 accuracy score6. Similarly, the best
+checkworthiness model only achieved an average precision of 0.65 for Arabic claims and 0.224 for English claims in
+the CheckThat! 2021 shared task for identifying checkworthiness in tweets (Nakov et al., 2021b). On the other hand,
+the best performing model for identifying check-worthy claims in debates reports 0.42 average precision. Surprisingly,
+Barrón-Cedeño et al. (2020), the top performing model for checkworthiness detection, report an average precision of
+0.806 (Williams et al., 2020). . For the fact-checking task of CheckThat! 2021 (Nakov et al., 2021b), the best performing
+model reports a 0.83 macro F1 score. However, the second-best model only reports a 0.50 F1 score. Given this striking
+gap in performance between the top system vs. others, it would be valuable for future work to benchmark systems on
+additional datasets in order to better assess the generality of these findings.
+It is not easy to make a direct comparison between different methods that are evaluated in different settings and
+with different datasets (Zeng et al., 2021). Moreover, the pipeline design of automated fact-checking creates potential
+bottlenecks, e.g., performance on the veracity prediction task on most datasets is dependent on the claim detection task
+performance or the quality of the evidence retrieved. Extensive benchmarks are required to incorporate all of the prior
+subtasks in the fact-checking pipeline systematically (Zeng et al., 2021).
+Much of AI research is faced with a fundamental trade-off between working with diverse formulations of a problem
+and standardized benchmarks for measuring progress. This trade-off also impacts automated fact-checking research.
+While there exist benchmarks such as FEVER and the CheckThat!, most models built on those benchmarks may
+not generalize well in a practical setting. Abstract and tractable formulations of a problem may help us develop
+technologies that facilitate practical adoption. However, practical adoption requires significant engineering effort
+beyond the research setting. Ideally, we would like to see automated fact-checking research continue to move toward
+increasingly realistic benchmarks while incorporating diverse formulations of the problem.
+5.2. Explainable Approaches
+Although the terms interpretability and explainability are often used interchangeably, and some times defined to be
+so (Molnar, 2020), we distinguish interpretability vs. explainability similar to (Kotonya and Toni, 2020a). Specifically,
+interpretability represents methods that provide direct insight into an AI system’s components (such as features and
+variables), often requiring some understanding of the specific to the algorithm, and often built for expert use cases
+such as model debugging. Explainability represents methods to understand an AI model without referring to the actual
+component of the systems. Note that, in the task formulation section, we have also talked about explaining veracity
+prediction. The goal of such explanation stems from fact-checker needs to help readers understand the fact-checking
+verdict. Therefore, explaining veracity prediction aligns more closely with explainability over interpretability. When
+the distinction between explainability vs. interpretability does not matter, we follow Vaughan and Wallach (2020) in
+adopting intelligibility (Vaughan and Wallach, 2020) as an umbrella term for both concepts.
+Sokol and Flach (2019) propose a desiderata for designing user experience for machine learning applications.
+Kotonya and Toni (2020a) extend them in the context of fact-checking and suggest eight properties of intelligibility:
+actionable, causal, coherent, context-full, interactive, unbiased or impartial, parsimonious, and chronological.
+Additionally, there are three dimensions specifically for explainable methods in NLP (Jacovi and Goldberg, 2020):
+1. Readability: are explanations clear?
+2. Plausibility: are explanations compelling or persuasive?
+3. Faithfulness: are explanations faithful to the model’s actual reasoning process?
+In comparison with the available intelligibility methods in NLP (Wiegreffe and Marasovic, 2021), only a few
+are applied to existing fact-checking works. Below, we highlight only commonly observed explainable fact-checking
+methods (also noted by Kotonya and Toni (2020a)).
+Attention-based Intelligibility Despite the debate about attention being a reliable intelligibility method (Jain and
+Wallace, 2019; Wiegreffe and Pinter, 2019; Serrano and Smith, 2019; Bibal, Cardon, Alfter, Wilkens, Wang, François
+and Watrin, 2022), it remains a popular method in existing deep neural network approaches in fact-checking. Attention-
+based explanations are provided in various forms:
+1. highlighting tokens in articles (Popat et al., 2018)
+5https://fever.ai/2018/task.html
+6https://fever.ai/task.html
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+The State of Human-centered NLP Technology for Fact-checking
+2. highlighting salient excerpts from evidence utilizing comments related to the post (Shu et al., 2019)
+3. n-gram extraction using self-attention (Yang, Pentyala, Mohseni, Du, Yuan, Linder, Ragan, Ji and Hu, 2019)
+4. attention from different sources other than the claim text itself, such as the source of tweets, retweet propagation,
+and retweeter properties (Lu and Li, 2020)
+Rule discovery as explanations Rule mining is a form of explanation prevalent in knowledge base systems (Gad-
+Elrab et al., 2019; Ahmadi, Lee, Papotti and Saeed, 2019). These explanations can be more comprehensive, but as noted
+in the previous section, not all statements can be fact-checked via knowledge-based methods due to limitations of the
+underlying knowledge-base itself. Some approaches provide general purpose rule mining in an attempt to address this
+limitation (Ahmadi, Truong, Dao, Ortona and Papotti, 2020).
+Summarization as explanations Both extractive and abstractive summaries can provide explanations for fact-
+checking. Atanasova et al. (2020a) provides natural language summaries to explain the fact-checking decision. They
+explore two different approaches - explanation generation and veracity prediction as separate tasks, and joint training
+of the both. Joint training performs worse than single training. Kotonya and Toni (2020b) combine abstractive and
+extractive approaches to provide a novel summarization approach. Brand, Roitero, Soprano, Rahimi and Demartini
+(2018) show jointly training prediction and explanation generation with encoder-decoder models such as BART (Lewis,
+Liu, Goyal, Ghazvininejad, Mohamed, Levy, Stoyanov and Zettlemoyer, 2020) results in explanations that help the
+crowd to perform better veracity assessment.
+Counterfactuals and adversarial methods Adversarial attacks on opaque models help to identify any blind-spots,
+biases and discover data artifacts in models (Ribeiro, Wu, Guestrin and Singh, 2020). Shared task FEVER 2.0 (Thorne
+et al., 2019) asked participants to devise methods for generating adversarial claims to identify weaknesses in the fact-
+checking methods. Natural language generation models such as GPT-3 can assist in formulating adversarial claims.
+More control over the generation can come from manipulating the input to natural language generation methods
+and constraining the generated text within original vocabulary (Niewiński, Pszona and Janicka, 2019). Atanasova,
+Wright and Augenstein (2020b) generate claims with n-grams inserted into the input text. Thorne and Vlachos (2019)
+experiment with several adversarial methods such as rule-based adversary, semantically equivalent adversarial rules
+(or SEARS) (Ribeiro, Singh and Guestrin, 2018), negation, and paraphrasing-based adversary. Adversarial attacks
+are evaluated based on the potency (correctness) of the example and reduction in system performance. While methods
+such as SEARS and paraphrasing hurt the system performance, hand-crafted adversarial examples have higher potency
+score.
+Interpretable methods (non-BlackBox) Some fact-checking works use a white-box or inherently interpretable
+model for fact-checking. Nguyen et al. (2018b,a) utilize a probabilistic graphical model and build an interactive
+interpretable model for fact-checking where users are allowed to directly override model decisions. Kotonya,
+Spooner, Magazzeni and Toni (2021) propose an interpretable graph neural network for interpretable fact-checking
+on FEVEROUS dataset (Aly et al., 2021).
+Limitations Intelligible methods in NLP and specifically within fact-checking are still in their infancy. Analysis of
+Kotonya and Toni (2020a) shows that most methods do not fulfill the desiderata mentioned earlier in this section.
+Specifically, they find that none of the existing models meet the criteria of being actionable, causal, and chronological.
+They also highlight that no existing method explicitly analyzes whether explanations are impartial. Some forms
+of explanations, such as rule-based triplets, are unbiased as they do not contain sentences or contain fragments of
+information (Kotonya and Toni, 2020a).
+Some explainable methods address a specific simplified formulation of the task. For example, Kotonya and Toni
+(2020b); Atanasova et al. (2020a) both assume that expert-written fact-checking articles already exist. They provide
+explanations as summaries of the fact-checking article. However, in practice, a fact-checking system would not have
+access to such an article for an unknown claim.
+In the case of automated fact-checking, most intelligible methods focus on explaining the outcome rather than
+describing the process to arrive at the outcome (Kotonya and Toni, 2020a). Moreover, all of the tasks in the fact-
+checking pipeline have not received equal attention for explainable methods. Kotonya and Toni (2020a) also argue
+that automatic fact-checking may benefit from explainable methods that provide insight into how outcomes of earlier
+sub-task in the fact-checking pipeline impact the outcome of later subtasks.
+Anubrata Das et al.: Preprint submitted to Elsevier
+Page 13 of 30
+
+The State of Human-centered NLP Technology for Fact-checking
+Most explainable NLP works evaluate explanation quality instead of explanation utility or faithfulness. Jacovi
+and Goldberg (2020) argue for a thorough faithfulness evaluation for explainable models. For example, even though
+attention-based explanations may provide quality explanations, they may not necessarily be faithful. Moreover, expla-
+nation utility requires separate evaluation by measuring whether explanations improve both i) human understanding
+of the model (Hase and Bansal, 2020) and ii) human effectiveness of the downstream task (Nakov et al., 2021a).
+Additionally, most intelligible methods establish only one-way communication from the model to humans. Instead,
+explanations might improve the model and human performance by establishing a bidirectional feedback loop.
+5.3. Human-in-the-loop Approaches
+Human-in-the-loop (HITL) approaches can help scale automated solutions while utilizing human intelligence for
+complex tasks. There are different ways of applying HITL methods, e.g., delegating sub-tasks to crowd workers
+(Demartini, Trushkowsky, Kraska, Franklin and Berkeley, 2013; Demartini, Difallah and Cudré-Mauroux, 2012;
+Sarasua, Simperl and Noy, 2012), active learning (Settles, 2009; Zhang, Lease and Wallace, 2017), interactive machine
+learning (Amershi, Cakmak, Knox and Kulesza, 2014; Joachims and Radlinski, 2007), and decision support systems
+where humans make the final decision based on model outcome and explanations (Zanzotto, 2019).
+While HITL approaches in artificial intelligence are prevalent, only a few recent works employ such approaches
+in fact-checking. HITL approaches are predominantly more present in the veracity prediction task than other parts of
+the pipeline. For example, Demartini et al. (2020) propose a HITL framework for combating online misinformation.
+However, they only consider hybrid approaches for two sub-tasks in the fact-checking pipeline: a) claim check-
+worthiness and b) truthfulness judgment (same as veracity prediction). Below, we discuss the existing HITL approaches
+by how the system leverages human effort for each sub-task in the fact-checking pipeline.
+Claim Detection, Checkworthiness, and Prioritization Social media streams are often monitored for rumors as
+a part of the claim detection task (Guo et al., 2022). Farinneya, Abdollah Pour, Hamidian and Diab (2021) apply an
+active learning-based approach at the claim detection stage for identifying rumors on social media. In-domain data is
+crucial for traditional supervised methods to perform well for rumor detection (Ahsan, Kumari and Sharma, 2019), but
+in real-world scenarios, sufficient in-domain labeled data may not be available in the early stages of development. A
+semi-supervised approach such as active learning is beneficial for achieving high performance with fewer data points.
+Empirical results shows that Tweet-BERT, along with the least confidence-based sample selection approach, performs
+on par with supervised approaches using far less labeled data (Farinneya et al., 2021).
+Similarly, Tschiatschek, Singla, Gomez Rodriguez, Merchant and Krause (2018) propose a HITL approach that
+aims to automatically aggregate user flags and recommend a small subset of the flagged content for expert fact-checking.
+Their Bayesian inference-based approach jointly learns to detect fake news and identify the accuracy of user flags over
+time. One strength of this approach is that the algorithm improves over time in identifying users’ flagging accuracy.
+Consequently, over time this algorithm’s performance improves. This approach is also robust against spammers. By
+running the model on publicly available Facebook data where a majority of the users are adversarial, experiments show
+that their algorithm still performs well.
+Duke’s Tech & Check team implemented HITL at the claim check-worthiness layer (Adair and Stencel, 2020). To
+avoid flagging false check-worthy claims, a human expert would sort claims detected by ClaimBuster (Hassan, Zhang,
+Arslan, Caraballo, Jimenez, Gawsane, Hasan, Joseph, Kulkarni, Nayak et al., 2017b), filter out the ones deemed more
+important for fact-checkers, and email them to several organizations. In essence, this approach helped fact-checkers
+prioritize the claims to check through an additional level of filtering. Currently, several published fact-checks on
+PolitiFact were first alerted by the emails from Tech & Check.
+Note that the CheckThat! (Nakov et al., 2021b; Shaar et al., 2021b; Barrón-Cedeño et al., 2020; Atanasova, Nakov,
+Karadzhov, Mohtarami and Da San Martino, 2019a) is a popular shared task for claim detection, check-worthiness, and
+prioritization tasks. However such shared tasks often have no submissions that employs HITL methodologies. Shared
+tasks for HITL approaches could encourage more solutions that can complement the limitations of model-only based
+approaches.
+Evidence Retrieval and Veracity Prediction Most work in HITL fact-checking caters to veracity prediction, and
+only a few consider evidence retrieval as a separate task. While there is a body of literature on HITL approaches in
+information retrieval (Chen and Jain, 2013; Demartini, 2015), we know of no work in that direction for fact-checking.
+Anubrata Das et al.: Preprint submitted to Elsevier
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+The State of Human-centered NLP Technology for Fact-checking
+Shabani, Charlesworth, Sokhn and Schuldt (2021) leverage HITL approaches for providing feedback about claim
+source, author, message, and spelling (SAMS). Annotators answer four yes/no questions about whether the article has
+a source, an author, a clear and unbiased message, and any spelling mistake. Furthermore, this work integrates the
+features provided by humans in a machine learning pipeline, which resulted in a 7.1% accuracy increase. However,
+the evaluation is performed on a small dataset with claims related to Covid-19. It is unclear if this approach would
+generalize outside of the domain. Moreover, further human effort can be reduced in this work by automating spell-
+check and grammar-check. SAMS could be quite limited in real life situations as most carefully crafted misinformation
+often looks like real news. Model generated fake news can successfully fool annotators (Zellers, Holtzman, Rashkin,
+Bisk, Farhadi, Roesner and Choi, 2020), and thus SAMS might also fail to flag such fake news.
+Qu, Barbera, Roitero, Mizzaro, Spina and Demartini (2021a) and Qu, Roitero, Mizzaro, Spina and Demartini
+(2021b) provide an understanding of how human and machine confidence scores can be leveraged to build HITL
+approaches for fact-checking. They consider explicit self-reported annotator confidence and compute implicit con-
+fidence based on standard deviation among ten crowd workers. Model confidence is obtained from bootstrapping
+(Efron and Tibshirani, 1985) ten different versions of the model and then computing standard deviation over the scores
+returned by the soft-max layer. Their evaluation shows that explicit crowd and model confidence are poor indicators
+of accurate classification decisions. Although the crowd and the model make different mistakes, there is no clear
+signal that confidence is related to accuracy. However, they show that implicit crowd confidence can be a useful
+signal for identifying when to engage experts to collect labels. A more recent study shows that a politically balanced
+crowd of ten is correlated with the average rating of three fact-checkers (Allen, Arechar, Pennycook and Rand, 2020).
+Gold, Kovatchev and Zesch (2019) also find that annotations by a crowd of ten correlate with the judgments of three
+annotators for textual entailment, which is utilized by veracity prediction models.
+A series of studies show that the crowd workers can reliably identify misinformation (Roitero, Soprano, Fan, Spina,
+Mizzaro and Demartini, 2020a; Roitero, Soprano, Portelli, Spina, Della Mea, Serra, Mizzaro and Demartini, 2020b;
+Soprano, Roitero, La Barbera, Ceolin, Spina, Mizzaro and Demartini, 2021). Furthermore, Roitero et al. (2020b) show
+that crowd workers not only can identify false claims but also can retrieve proper evidence to justify their annotation.
+One weakness of this study is that it only asks users to provide one URL as evidence. However, in practice, fact-
+checking might need reasoning over multiple sources of information. Although these studies do not propose novel
+HITL solutions, they provide sufficient empirical evidence and insights about where crowd workers can be engaged
+reliably in the fact-checking pipeline.
+Nguyen et al. (2018b) propose joint modeling of crowd annotations and machine learning to detect the veracity of
+textual claims. The key strength of the model is that it assumes all annotators can make mistakes, which is a possibility
+as fact-checking is a difficult task. Another strength is that this model allows users to import their knowledge into the
+system. Moreover, this HITL approach can collect on-demand stance labels from the crowd and incorporate them in
+veracity prediction. Empirical evaluation shows that this approach achieves strong predictive performance. A follow-up
+study provides an interactive HITL tool for fact-checking (Nguyen et al., 2018a).
+Nguyen, Weidlich, Yin, Zheng, Nguyen and Nguyen (2020) propose a HITL system to minimise user effort and cost.
+Users validate algorithmic predictions but do so at a minimal cost by only validating the most-beneficial predictions
+for improving the system. This system provides a guided interaction to the users and incrementally gets better as users
+engage with it.
+It is important to note that research on crowdsourcing veracity judgment is at an early stage. Different factors such
+as demographics, political leaning, criteria for determining the expertise of the assessors (Bhuiyan, Zhang, Sehat and
+Mitra, 2020), cognitive factors (Kaufman, Haupt and Dow, 2022), and even the rating scale (La Barbera, Roitero,
+Demartini, Mizzaro and Spina, 2020) led to different levels of alignment with expert ratings. Bhuiyan et al. (2020)
+outline research directions for designing better crowd processes specific to different types of misinformation for the
+successful utilization of crowd workers.
+Explaining Veracity Prediction HITL systems in fact-checking often use veracity explanations to correct model
+errors. As discussed earlier, Nguyen et al. (2018a) provides an interpretable model that allows users to impart their
+knowledge when the model is wrong. Empirical evaluation shows that users could impart their knowledge into the
+system. Similarly, Zhang, Rudra and Anand (2021b) propose a method that collects user feedback from explanations.
+Note that this method explains veracity prediction outcomes based on the evidence retrieved and their stance. Users
+provide feedback in terms of stance and relevance of the retrieved evidence. The proposed approach employs lifelong
+Anubrata Das et al.: Preprint submitted to Elsevier
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+The State of Human-centered NLP Technology for Fact-checking
+learning which enables the system to improve over time. Currently there is no empirical evaluation of this system to
+identify the effectiveness of this approach.
+Although natural language generation models are getting increasingly better (Radford, Wu, Child, Luan, Amodei,
+Sutskever et al., 2019), generating abstractive fact-checking explanations is still in its infancy (Kotonya and Toni,
+2020b). HITL methods could be leveraged to write reports justifying fact-checking explanations.
+Limitations After reviewing existing HITL approaches across different fact-checking tasks, we also list out several
+limitations as follow. First, some HITL approaches adopt several interpretable models to integrate human input, but
+the resulting models do not perform as well as the state-of-the-art deep learning models (Nguyen et al., 2018b,a).
+Farinneya et al. (2021) apply HITL approaches to scale up rumor detection from a limited amount of annotated data.
+Although it performs well to generalize the algorithm for a new topic in a few-shot manner, one of the weaknesses
+is that data from other domains or topics causes a high variance in model performance. Consequently, in-domain
+model performance might degrade when out-of-domain data is introduced in model training. This issue may hinder the
+model’s generalizability in practice, especially where a clear demarcation between topic domains may not be possible.
+More importantly, there is a lack of empirical studies on how to apply HITL approaches of fact-checking for
+practical adoption. Although HITL approaches provide a mechanism to engage human in the process of modeling
+development, several human factors, such as usability, intelligibility, and trust, become important to consider when
+applying this method in the real-world use case. Fact-checking is a time-sensitive task and requires expertise to process
+complex information over multiple sources (Graves, 2017). Fact-checkers and policy makers are often skeptical about
+any automated or semi-autoamted solutions as this type of research requires human creativity and expertise (Arnold,
+2020; Micallef et al., 2022). Therefore, more empirical evidence needs to be found to assess the effectiveness of
+applying different HITL approaches to automated fact-checking.
+6. Existing Tools for Fact-checking
+In the previous section, we reviewed the details of current NLP technologies for fact-checking. Subsequently,
+we extend our review of automated fact-checking to the HCI literature and discuss existing practices of applying fact-
+checking into real-world tools that assist human fact-checkers. In brief, there is a lack of holistic review of fact-checking
+tools from a human-centered perspective. Additionally, we found that the articulation of work between human labor and
+AI tools is still opaque in this field. Research questions include but are not limited to: 1) how can NLP tools facilitate
+human work in different fact-checking tasks? 2) how can we incorporate user needs and leverage human expertise to
+inform the design of automated fact-checking?
+In this section, we examine current real-world tools that apply NLP technologies in different stages of fact-checking
+and clarify the main use cases of these tools. We argue that more research concerning human factors for building
+automated fact-checking, such as user research, human-centered design, and usability studies, should be conducted to
+improve the practical adoption of automated fact-checking. These studies help us identify the design space of applying
+explainable and HITL approaches for real-world NLP technologies.
+6.1. Claim Detection and Prioritization
+The first step in claim detection is sourcing content to possibly check. On end-to-end encrypted platforms, such
+as WhatsApp, Telegram, and Signal, crowdsourcing-based tip-lines play a vital role in identifying suspicious content
+that is not otherwise accessible (Kazemi et al., 2021b). As another example, Check from Meedan 7, a tip-line service
+tool, also helps fact-checkers monitor fake news for in-house social media. User flagging of suspect content on social
+media platforms such as Facebook is also a valuable signal for identifying such content, and crowdsourcing initiatives
+like Twitter’s BirdWatch can further help triage and prioritize claims for further investigation.
+In the stage of finding and choosing claims to check, fact-checkers assess the fact-checking related quality of a
+claim and decide whether to fact-check it (Graves, 2017; Micallef et al., 2022). NLP models in claim detection, claim
+matching, and check-worthiness are useful to assist the above decision-making process. However, integrating them
+into real-world tools that help fact-checkers prioritize what to check requires more personalized effort. Graves (2018a)
+points out that it is important to design the aforementioned models to cater to fact-checker organizational interests,
+stakeholder needs, and changing news trends.
+7https://meedan.com/check
+Anubrata Das et al.: Preprint submitted to Elsevier
+Page 16 of 30
+
+The State of Human-centered NLP Technology for Fact-checking
+As one of the fact-checking qualities of a claim, checkability can be objectively analyzed by whether a claim
+contains one or more purported facts that can be verified (Section 3.1). Fact-checkers find it useful to apply models
+that identify checkable claims to their existing workflow because the model helps them filter irrelevant content and
+claims that are uncheckable when they are choosing claims to check (Arnold, 2020). ClaimBuster, one of the well-
+known claim detection tools, is built to find checkable claims from a large scale of text content (Hassan et al., 2017a).
+Claim detection can also be integrated into speech recognition tools to spot claims from live speech (Adair, 2020).
+Additionally, if a claim has already been fact-checked, fact-checkers can skip it and prioritize claims that have
+not been checked. As a relatively new NLP task, claim matching has been integrated into some current off-the-shelf
+search engines or fact-checking tools to help fact-checkers find previously fact-checked claims. For example, Google
+Fact Check Explorer8 can retrieve previously fact-checked claims by matching similar fact-check content to user input
+queries. Similarly, with Meedan’s Check, if users send a tip with fake news that has been previously fact-checked, the
+tool further helps fact-checkers retrieve the previous fact-check and send it to users.
+Whether or not to fact-check a claim depends on an organization’s goals and interests. Tools built for claim detection
+need to take such interests into account. For example, Full Fact developed a claim detection system that classifies
+claims into different categories, such as quantity, predictions, correlation or causation, personal experience, and laws
+or rules of operations (Konstantinovskiy et al., 2021). The claim categories are designed by their fact-checkers to cater
+to their needs of fact-checking UK political news in a live fact-checking situation. Identifying certain claims, such as
+quantity, correlation or causation, might be particularly useful for fact-checkers to evaluate the credibility of politician
+statements and claims. The system also helps tailor fact-checkers’ downstream tasks, such as fact-check assignments
+and automated verification for statistical claims (Nakov et al., 2021a).
+Fact-checkers also use social media monitoring tools to find claims to check, such as CrowdTangle, TweetDeck, and
+Facebook’s (unnamed) fact-checking tool, but those tools are not very effective to detect checkable claims. Some fact-
+checkers reported that only roughly 30% of claims flagged by Facebook’s fact-checking tool were actually checkable
+(Arnold, 2020). A low hanging fruit is to integrate claim detection models into these social media monitoring tools so
+that it is easier for fact-checkers to identify claims that are both viral and checkable. Additionally, these tools should
+enable fact-checkers to locate certain figures, institutions, or agencies according to their fact-checking interests and
+stakeholder needs so that these tools can better identify and prioritize truly check-worthy claims. An important question
+in implementing those systems is how to measure the virality of a claim and its change over time.
+It would also be useful to integrate veracity prediction into previous fact-checking tools because fact-checkers
+may pay the most attention to claims9 that are suspect and uncertain (since obviously true or false claims likely do
+not require a fact-check). However, information or data points that are used to give such predictions should also be
+provided to fact-checkers. If sources, evidence, propagation patterns, or other contextual information that models use
+to predict claim veracity can be explained clearly for fact-checkers, they can also triage these indicators to prioritize
+claims more holistically.
+6.2. Tools for Evidence Retrieval
+After finding and prioritizing which claim to check, fact-checkers investigate claims following three main activities:
+1) decomposing claims, 2) finding evidence, and 3) tracing the provenance of claims and their spread. Note that these
+three activities are intertwined with each other by using different information-seeking tools in the fact-checking process.
+Fact-checkers search for evidence by decomposing claims into sub questions. Evidence found while investigating a
+claim may further modify or add to the sub-questions (Singh et al., 2021). By iteratively investigating claims via
+online search, fact-checkers reconstruct the formation and the spread of a claim to assess its truth (Graves, 2017). In
+this section, we discuss the utility of existing information-seeking tools, including off-the-shelf search engines and
+domain-specific databases, that assist fact-checkers in each activity.
+Claim decomposition is not a specific activity that qualitative researchers have reported or analyzed in their fact-
+checking studies, but we can find more details from where fact-checking organizations describe their methodology10
+and how fact-checkers approach complex claims in their fact-checks11. Claim decomposition refers to how fact-
+checkers interpret ambiguous terms of a claim and set the fact-checking boundaries to find evidence. Decomposing
+8https://toolbox.google.com/factcheck/explorer
+9https://www.factcheck.org/our-process/
+10https://leadstories.com/how-we-work.html
+11https://www.factcheck.org/2021/10/oecd-data-conflict-with-bidens-educational-attainment-claim/ In this fact-
+check, fact-checkers decompose what President Biden mean by “advanced economies” and “young people”. The approach of defining these two
+terms directly influence their fact-checking results.
+Anubrata Das et al.: Preprint submitted to Elsevier
+Page 17 of 30
+
+The State of Human-centered NLP Technology for Fact-checking
+claims effectively requires sensitive curiosity and news judgments for fact-checkers that are cultivated through years
+of practice. Unfortunately, we are not aware of any existing tools that facilitate this process.
+Traditional methodology to decompose claims is to ask sub-questions. Recent NLP studies simulate this process
+by formulating it as a question-answering task (Fan et al., 2020; Chen et al., 2022). Researchers extract justifications
+from existing fact-checks and crowdsource sub-questions to decompose the claim. For automated-fact-checking, this
+NLP task might be very beneficial to improve the performance of evidence retrieval by auto-decomposing claims into
+smaller checkable queries (Chen et al., 2022). Although it is difficult for NLP to match the abilities of professional
+fact-checkers, it might help scale up the traditional, human fact-checking process. It could also help the public, new
+fact-checkers, or journalists to more effectively investigate complex claims and search for evidence.
+How fact-checkers find evidence is usually a domain-specific reporting process, contacting experts or looking for
+specific documents from reliable sources (Graves, 2017; Micallef et al., 2022). Instead of conducting random searches
+online, most fact-checkers include a list of reliable sources in which to look for evidence. Tools that are designed for
+searching domain datasets can also help fact-checkers to find evidence. For example, Li, Fang, Lou, Li and Zhang
+(2021) built an analytical search engine for retrieving the COVID-19 news data and summarizing it in an easy to
+digest, tabular format. The system can decompose analytical queries into structured entities and extract quantitative
+facts from news data. Furthermore, if evidence retrieval is accurate enough for in-domain datasets, the system can
+take a leap further to auto-verify domain-related claims. We provide more detailed use cases of veracity prediction in
+Section 6.3.
+Fact-checkers mainly use off-the-shelf search engines, such as Google, Bing, etc., to trace a claim’s origin from
+publicly accessible documents (Beers, Haughey, Melinda, Arif and Starbird, 2020; Arnold, 2020). Other digital
+datasets, such as LexisNexis and InternetArchive, are also useful for fact-checkers to trace claim origin. To capture the
+formation and change of a claim, search engines should not only filter unrelated content, but also retrieve both topically
+and evidentially relevant content. Hasanain and Elsayed (2021) report that most topically relevant pages retrieved from
+Google do not contain evidential information, such as statistics, quotes, entities, or other types of facts. Additionally,
+most built-in search engines in social media platforms, such as Twitter and Facebook, only filter “spreadable” content
+not “credible” content (Alsmadi, Alazzam and AlRamahi, 2021).
+Furthermore, these off-the-shelf search engines do not support multilingual search, so it is difficult for fact-checkers
+to trace claims if they are translated from other languages (Graves, 2017; Nakov et al., 2021a). NLP researchers have
+started to use multilingual embedding models to represent claim-related text in different languages and match existing
+fact-checks (Kazemi, Gaffney, Garimella and Hale, 2021a). This work not only helps fact-checkers find previously
+fact-checked claims more easily from other languages, but also to examine how the claim is transformed and reshaped
+by the media in different languages and socio-political contexts.
+6.3. Domain-specific Tools for Claim Verification
+As discussed in Sections 3.2 and 3.3, most verification prediction models are grounded on the collected evidence
+and the claim. To build an end-to-end claim verification system, NLP developers need to construct domain-specific
+datasets incorporating both claims and evidence. Different from complex claims that contain multiple arguments and
+require decomposition, claims that have simple linguistic structure with purported evidence or contain statistical facts
+can be automatically verified (Nakov et al., 2021a).
+Karagiannis, Saeed, Papotti and Trummer (2020) built CoronaCheck, a search engine that can directly verify Covid-
+19 related statistical claims by retrieving official data curated by experts (Dong, Du and Gardner, 2020). Full Fact
+(The Poynter Institute, 2021) also took a similar approach to verify statistical macroeconomic claims by retrieving
+evidence from UK parliamentary reports and national statistics. Additionally, Wadden et al. (2020) built a scientific
+claim verification pipeline by using abstracts that contain evidence to verify a given scientific claim.
+However, pitfalls still exist if fact-checkers use these domain-specific verification tools in practice. For example, the
+CoronaCheck tool cannot check the claim “The Delta variant causes more death than the Alpha variant” simply because
+the database does not contain fine-grained death statistics for Covid variants. Additionally, checking a statistical or
+scientific claim might only be a part of the process of checking a more complex claim, which requires fact-checkers
+to contextualize the veracity of previous statistical or scientific checks. In general, domain-specific tools are clearly
+valuable to use when available, though in practice they are often incomplete and insufficient on their own to check
+complex claims.
+Anubrata Das et al.: Preprint submitted to Elsevier
+Page 18 of 30
+
+The State of Human-centered NLP Technology for Fact-checking
+7. Discussion
+In this review, we have 1) horizontally outlined the research of applying NLP technologies for fact-checking from
+the beginning of task formulation to the end of tools adoption; as well as 2) vertically discussing the capabilities and
+limitations of NLP for each step of a fact-checking pipeline. We perceive a lack of research that bridges both to assist
+fact-checkers. Explainable and HITL approaches leverage both human and computational intelligence from a human-
+centered perspective, but there is a need to provide actionable guides to utilize both methods for designing useful
+fact-checking tools. In this section, we propose several research directions to explore the design space of applying
+NLP technologies to assist fact-checkers.
+7.1. Distributing Work between Human and AI for Mixed-initiative Fact-checking
+The practice of fact-checking has already become a type of complex and distributed computer-mediated work
+(Graves, 2018a). Although Graves (2017) breaks down a traditional journalist fact-checking pipeline into five steps,
+the real situation of fact-checking a claim is more complicated (Juneja and Mitra, 2022). Various AI tools are adopted
+dynamically and diversely by fact-checkers to complete different fact-checking tasks (Arnold, 2020; Beers et al., 2020;
+Micallef et al., 2022).
+Researchers and practitioners increasingly believe that future fact-checking should be a mixed-initiative practice
+in which humans perform specific tasks while machines take over others (Nguyen et al., 2018a; Lease, 2020; Nakov
+et al., 2021a). To embed such hybrid and dynamic human-machine collaborations into existing fact-checking workflow,
+the task arrangement between human and AI need to be articulated clearly by understanding the expected outcomes
+and criteria for each. Furthermore, designing a mixed-initiative tool for different fact-checking tasks requires a more
+fine-grained level of task definition for human and AI (Lease, 2018, 2020). In Section 5.3, we discuss several studies
+highlighting the role of humans in the fact-checking workflow, e.g., a) human experts select check-worthy claims from
+claim detection tools (Hassan et al., 2017b) and deliver them to fact-checkers, b) ask crowd workers to judge reliable
+claims sources (Shabani et al., 2021), or c) flag potential misinformation (Roitero et al., 2020b) to improve veracity
+prediction. All of the above human activities are examples of micro-tasks within a mixed-initiative fact-checking
+process.
+Prior work in crowdsourcing has shown that it is possible to effectively break down the academic research process
+and utilize crowd workers to partake in smaller research tasks (Vaish, Davis and Bernstein, 2015; Vaish, Gaikwad,
+Kovacs, Veit, Krishna, Arrieta Ibarra, Simoiu, Wilber, Belongie, Goel et al., 2017). Given this evidence, we can
+also break down sub-tasks of a traditional fact-checking process into more fine-grained tasks. Therefore, key research
+questions include: a) How can we design these micro-tasks to facilitate each sub-task of fact-checking, and b) What
+are the appropriate roles for human and AI in different micro-tasks?
+To effectively orchestrate human and AI work, researchers need to understand the respective roles of human and AI,
+and how they will interact with one another, because it will directly affect whether humans decide to take AI advice
+(Cimolino and Graham, 2022). Usually, if AI aims to assist high-stake decision-making tasks, such as recidivism
+prediction (Veale, Van Kleek and Binns, 2018) and medical treatments (Cai, Winter, Steiner, Wilcox and Terry, 2019),
+considerations of risk and trust will be important factors for people to adopt such AI assistants (Lai, Chen, Liao,
+Smith-Renner and Tan, 2021). In the context of fact-checking, if AI directly predicts the verdict of a claim, fact-
+checkers may be naturally skeptical about how the AI makes such a prediction (Arnold, 2020). On the other hand, if
+AI only helps to filter claims that are uncheckable, such as opinions and personal experience, fact-checkers may be
+more willing to use such automation with less concern about how AI achieves it. Deciding whether a claim is true or
+false is a high-stake decision-making task for fact-checkers, while filtering uncheckable claims is a less important but
+tedious task that fact-checkers want automation to help with. Therefore, the extent of human acceptance of AI varies
+according to how humans assess the task assigned to AI, resulting in different human factors, such as trust, transparency,
+and fairness. Researchers need to specify or decompose these human factors into different key variables that can be
+measured during the model development process. Given a deep understanding of the task relationship between human
+and AI, researchers can then ask further research questions on how to apply an explainable approach, or employ a
+HITL system vs. automated solutions, to conduct fact-checking. Here we list out several specific research topics that
+contain mixed-initiative tasks, including: a) assessing claim difficulty leveraging crowd workers, b) breaking down a
+claim into a multi-hop reasoning task and engaging the crowd to find information relevant to the sub-claims, and c)
+designing micro-tasks to parse a large number of documents retrieved by web search to identify sources that contain
+the evidence needed for veracity prediction.
+Anubrata Das et al.: Preprint submitted to Elsevier
+Page 19 of 30
+
+The State of Human-centered NLP Technology for Fact-checking
+7.2. Human-centered Evaluation of NLP Technology for Fact-Checkers
+We begin this section by proposing key metrics from human factors for evaluating systems (i.e., what to measure
+and how to measure them): accuracy, time, model understanding, and trust (Section 7.2.1). Following this, we further
+propose a template for an experimental protocol for human-centered evaluations in fact-checking (Section 7.2.2).
+7.2.1. Metrics
+Accuracy Most fact-checking user studies assume task accuracy as the primary user goal (Nguyen et al., 2018a;
+Mohseni, Yang, Pentyala, Du, Liu, Lupfer, Hu, Ji and Ragan, 2021). Whereas non-expert users (i.e., social media users
+or other form of content consumers) might be most interested in the veracity outcome along with justification, fact-
+checkers often want to use automation and manual effort interchangeably in their workflow (Arnold, 2020; Nakov et al.,
+2021a). Thus, we need a more fine-grained approach towards measuring accuracy beyond the final veracity prediction
+accuracy. For fine-grained accuracy evaluation, it is also crucial to capture fact-checker accuracy, particularly for the
+sub-tasks for which they use the fact-checking tool.
+With the assumption that “ground truth” exists for all of the sub-tasks in the fact-checking pipeline, accuracy can
+be computed by comparing user answers with the ground truth. Note that measuring sub-task level accuracy is trickier
+than end-to-end fact-checking accuracy. Sub-task level accuracy can be captured by conducting separate experiments
+for each sub-task. Suppose the point of interest is to understand user performance for detecting claim-checkworthiness.
+In that case, we will need to collect additional data specific to the claim-checkworthiness task.
+In some cases, it is possible to merge multiple sub-tasks for evaluation purposes. For example, Miranda, Nogueira,
+Mendes, Vlachos, Secker, Garrett, Mitchel and Marinho (2019) evaluate the effectiveness of their tool with journalists
+by capturing the following two key variables: a) the relevance of retrieved evidence, and b) the accuracy of the predicted
+stance. This method provides essential insight into evidence retrieval, stance detection, and the final fact-checking task.
+Depending on the tool, the exact detail of this metric will require specific changes according to tool affordances.
+Note that both time and accuracy measures need to control for claim properties. For example, if a claim has been
+previously fact-checked, it would take less time to fact-check such claims. On the other hand, a new claim that is more
+difficult to assess would require more time.
+Model Understanding Fact-checkers want to understand the tools they use. For example, Arnold (2020) pointed
+out that fact-checkers expressed a need for understanding CrowdTangle’s algorithm for detecting viral content on
+various social media platforms. Similarly, Nakov et al. (2021a) observed a need for increased system transparency in
+the fact-checking tools used by different organizations. Lease (2018) argues that transparency is equally important for
+non-expert users to understand the underlying system and make an informed judgement. Although this is not a key
+variable related to user performance, it is important for practical adoption.
+To measure understanding, users could be asked to self-report their level of understanding on a Likert-scale.
+However, simply asking participants if they understand the algorithm is not a sufficient metric. For example, it does not
+indicate whether participants will be able to simulate tool behavior (Hase and Bansal, 2020). We suggest the following
+steps for measuring model understanding based on prior work (Cheng, Wang, Zhang, O’Connell, Gray, Harper and
+Zhu, 2019).
+1. Decision Prediction. To capture users’ holistic understanding of a tool, users could be provided claims and
+asked the following: “What label would the tool assign to this claim?”
+2. Alternative Prediction. Capturing how changes in the input influence the output can also measure understand-
+ing, e.g., by asking users how the tool would assign a label to a claim when input parts are changed. Imagine a
+tool that showed the users the evidence it has considered to arrive at a veracity conclusion. Now, if certain pieces
+of evidence were swapped, how would that be reflected in the model prediction?
+Trust For practical adoption, trust in a fact-checking tool is crucial across all user groups. While model understanding
+is often positively correlated with trust, understanding alone may not suffice to establish trust. In this domain, fact-
+checkers and journalists may have less trust in algorithmic tools (Arnold, 2020). On the other hand, there is also the risk
+of over-trust, or users blindly following model predictions (Nguyen et al., 2018a; Mohseni et al., 2021). To maximize
+the tool effectiveness, we would want users to neither dismiss all model predictions out of hand (complete skepticism)
+nor blindly follow all model predictions (complete faith). Instead, it is important to calibrate user trust for the most
+effective tool usage. We suggest measuring a notion of calibrated trust (Lee and See, 2004): how often users abide by
+correct model decisions and override erroneous model decisions.
+Anubrata Das et al.: Preprint submitted to Elsevier
+Page 20 of 30
+
+The State of Human-centered NLP Technology for Fact-checking
+Both
+Model
+User
+Neither
+User Prediction
+Correct
+Incorrect
+Correct
+Incorrect
+Model 
+Prediction
+Figure 2: Confusion Matrix for User Predictions vs. Model Predictions with respect to ground truth (gold). We assume
+model predictions are provided to the user, who then decides whether to accept or reject the model prediction. The top-left
+quadrant (Both) covers cases where users correctly follow model predictions. The top-right quadrant (Model) denotes the
+cases where the model is correct but users mistakenly reject the model decisions. The bottom-left quadrant User denotes
+the cases where users correctly reject erroneous model predictions. The bottom-right quadrant Neither denotes the cases
+where users incorrectly accept erroneous model predictions. Quantifying user vs. model predictions in this manner enables
+measurement of calibrated trust: how often users abide by correct model decisions and override erroneous model decisions.
+To measure calibrated trust, we imagine a confusion matrix shown in Figure 2. The rows denote correct vs. incorrect
+model predictions while the columns denote correct vs. incorrect user predictions. A user who blindly followed all
+model predictions would have their behavior entirely captured by the main (primary) diagonal, whereas a user who
+skeptically rejected all model predictions would have their behavior captured entirely in the secondary diagonal. The
+ideal user’s behavior would be entirely captured in the first column: accepting all correct model predictions and rejecting
+all incorrect model predictions. To promote effective human-AI teaming, AI tools should assist their human users in
+developing strong calibrated trust to appropriately trust and distrust model predictions as each case merits.
+Beyond calibrated trust, one could also measure quantitative trust by adopting methodologies from the human-
+machine trust literature (Lee and Moray, 1992). For example, Cheng et al. (2019) adapted prior work into a 7-point
+Likert scale. A similar scale can be reused for evaluating trust in a fact-checking tool. For example, we can create five
+different Likert-scales to measure the agreement (or disagreement) of users with the following statements:
+• I understand the fact-checking tool.
+• I can predict how the tool will behave.
+• I have faith that the tool would be able to cope with the different fact-checking task.
+• I trust the decisions made by the tool.
+• I can count on the tool to provide reliable fact-checking decisions.
+Additional factors Individual differences among users might result in substantial variation in experimental outcomes.
+For example, varying technical literacy (Cheng et al., 2019), any prior knowledge about the claims, and users’ political
+leaning (Thornhill, Meeus, Peperkamp and Berendt, 2019) might influence user performance on the task while using
+fact-checking tools. Thus it is valuable to capture these factors in study design. For example:
+1. Technical Literacy: Users’ familiarity with popular technology tools (e.g, recommendation engines, spam
+detectors) and their programming experience (Cheng et al., 2019) as well as familiarity with existing fact-
+checking tools.
+2. Media Literacy: Users’ familiarity with 1) the fact-checking process, and 2) fact-checks from popular
+organizations such as PolitiFact and FactCheck.Org.
+3. Demographics: Users’ education level, gender, age, and political leaning.
+Quantitative measures alone are not sufficient as they do not capture certain nuances about how effectively a tool
+integrates into a fact-checker’s workflow. For example, even if users understand and trust the working principle of a
+Anubrata Das et al.: Preprint submitted to Elsevier
+Page 21 of 30
+
+The State of Human-centered NLP Technology for Fact-checking
+tool, it is unclear why they do so. Hence, users might be asked a few open-ended questions at the end of the study to
+gather qualitative insights. Such questions could include:
+1. Describe your understanding of the tool. Do any specific aspects of its design seem to assist or detract from your
+understanding of how it works?
+2. Why do you trust or not trust the tool?
+3. Would you use this tool beyond this study, and if so, in what capacity?
+7.2.2. Experimental Protocol
+One strategy to capture the aforementioned metrics is to design a mixed-methods study. Here we outline the
+template for such a study. Imagine the goal were to measure the user performance for fact-checking using a new tool
+(let’s call it tool A) compared to an existing tool (tool B). Fact-checking tasks in the real world might be influenced by
+user priors about the claims being checked. Thus, a within-subject study protocol may be more appropriate to account
+for such priors (Shi, Bhattacharya, Das, Lease and Gwizdka, 2022).
+1. Pre-task: Users would first be asked to fact-check a set of claims. To do so, first a user would be asked to leverage
+a pre-existing tool B at this stage. Tool B can be replaced with different baselines, depending on the particular
+use case, ranging from simple web-search by non-expert users to proprietary tools used by fact-checkers and
+journalists. Users would be asked to think aloud at this stage.
+2. Learning: At this stage users would familiarize themselves with the new tool (tool A). Users would need to
+fact-check a different set of claims from the first one. Ground truth would also accessible to the user to form a
+prior about what kind of mistakes a tool might make. Claims here would be selected at random to reflect tool
+capabilities. Moreover, tool performance metrics would be given to the users as additional information. Users
+would be encouraged to ask questions about the tool at this stage.
+3. Prediction: Users would now be asked to fact-check the same claims from step-1 above but this time they are
+asked to leverage the tool A. Users would be asked to think out loud through this stage. Users could simply guess
+the answers and achieve a high accuracy score. Thus, claims selected for stages (1) & (3) would be a balanced
+set of claims with an equal distribution of true positive, true negative, false positive, and false negative samples.
+This idea is adopted from prior work (Hase and Bansal, 2020).
+4. Post-task survey: Users would now be asked to take a small survey for capturing trust, understanding, technical
+literacy, media literacy, and demographic information.
+5. Post-task interview: Upon completion of these steps, users would be interviewed with open-ended questions to
+gather insights about their understanding and trust in the system.
+The measures and study protocol could be useful in the context of evaluating any new fact-checking system compared to
+an existing system or practices. Specifics might vary depending on the target user group and the tool’s intended purpose.
+Above we use the whole fact-checking pipeline to illustrate our experimental protocol. However this technique can be
+applied to other sub-tasks of automated fact checking, granted that we have the ground truth of the outcome for that
+sub-task. For example, let us assume a new claim detection tool has been proposed that takes claims from a tip-line
+(Kazemi et al., 2021b). Currently, fact-checkers use an existing claim-matching algorithm to filter out the already
+fact-checked claim. Now, if we replace tool B above with the existing claim-matching algorithm and tool A with the
+proposed claim detection tool, we can utilize the protocol mentioned above. In conclusion, one could evaluate how
+users perform for claim detection tasks using the new tool compared to the existing ones in terms of their accuracy,
+time, understanding, and trust.
+While we have proposed an ideal, extensive version of an evaluation protocol for evaluating new fact-checking tools,
+note that the actual protocol used in practice could be tailored according to the time required from the participants and
+the cost of conducting the experiment.
+8. Conclusion
+This review highlights the practices and development of the state-of-the-art in using NLP for automated fact-
+checking, emphasizing both the advances and the limitations of existing task formulation, dataset construction, and
+modeling approaches. We partially discuss existing practices of applying these NLP tasks into real-world tools that
+assist human fact-checkers. In recent years we have seen significant progress in automated fact-checking using NLP.
+Anubrata Das et al.: Preprint submitted to Elsevier
+Page 22 of 30
+
+The State of Human-centered NLP Technology for Fact-checking
+A broad range of tasks, datasets, and modeling approaches have been introduced in different parts of the fact-checking
+pipeline. Moreover, with recent developments in transformers and large language models, the model accuracy has
+improved across tasks. However, even state-of-the-art models on existing benchmarks — such as FEVER and CLEF!
+— may not yet be ready for practical adoption and deployment.
+To address these limitations, we advocate development of hybrid, HITL systems for fact-checking. As a starting
+point, we may wish to reorient the goals of existing NLP tasks from full automation towards decision support. In
+contrast with fully-automated systems, hybrid systems instead involve humans-in-the-loop and facilitate human-AI
+teaming (Bansal et al., 2019; Bansal, Wu, Zhou, Fok, Nushi, Kamar, Ribeiro and Weld, 2021b; Bansal, Nushi, Kamar,
+Horvitz and Weld, 2021a). Such use of hybrid systems can help a) scale-up human decision making; b) augment
+machine learning capabilities with human accuracy; and c) mitigate unintended consequences from machine errors.
+Additionally, we need new benchmarks and evaluation practices that can measure how automated and hybrid systems
+can improve downstream human accuracy (Smeros, Castillo and Aberer, 2021; Fan et al., 2020) and efficiency in
+fact-checking.
+Acknowledgements
+This research was supported in part by the Knight Foundation, the Micron Foundation, Wipro, and by Good
+Systems12, a UT Austin Grand Challenge to develop responsible AI technologies. The statements made herein are
+solely the opinions of the authors and do not reflect the views of the sponsoring agencies.
+References
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+Zubiaga, A., Liakata, M., Procter, R., Wong Sak Hoi, G., Tolmie, P., 2016. Analysing how people orient to and spread rumours in social media by
+looking at conversational threads. PloS one 11, e0150989.
+Anubrata Das et al.: Preprint submitted to Elsevier
+Page 30 of 30
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf,len=3274
+page_content='The State of Human-centered NLP Technology for Fact-checking  Anubrata Das∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Houjiang Liu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Venelin Kovatchev and Matthew Lease  aSchool of Information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' The University of Texas at Austin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Austin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' TX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' USA  A R T I C L E I N F O  Keywords:  Natural Language Processing  Misinformation  Disinformation  Explainability  Human-AI Teaming  A B S T R A C T  Misinformation threatens modern society by promoting distrust in science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' changing narratives in  public health,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' heightening social polarization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' and disrupting democratic elections and financial  markets,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' among a myriad of other societal harms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' To address this, a growing cadre of professional  fact-checkers and journalists provide high-quality investigations into purported facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' However,  these largely manual efforts have struggled to match the enormous scale of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' In  response, a growing body of Natural Language Processing (NLP) technologies have been  proposed for more scalable fact-checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Despite tremendous growth in such research, however,  practical adoption of NLP technologies for fact-checking still remains in its infancy today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  In this work, we review the capabilities and limitations of the current NLP technologies for  fact-checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Our particular focus is to further chart the design space for how these technologies  can be harnessed and refined in order to better meet the needs of human fact-checkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' To do  so, we review key aspects of NLP-based fact-checking: task formulation, dataset construction,  modeling, and human-centered strategies, such as explainable models and human-in-the-loop  approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Next, we review the efficacy of applying NLP-based fact-checking tools to assist  human fact-checkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' We recommend that future research include collaboration with fact-checker  stakeholders early on in NLP research, as well as incorporation of human-centered design  practices in model development, in order to further guide technology development for human  use and practical adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Finally, we advocate for more research on benchmark development  supporting extrinsic evaluation of human-centered fact-checking technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Introduction  Misinformation and related issues (disinformation, deceptive news, clickbait, rumours, and information credibility)  increasingly threaten society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' While concerns of misinformation existed since the early days of written text (Marcus,  1992), with recent development of social media, the entry barrier for creating and spreading content has never been  lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Moreover, polarization online drives the spread of misinformation that in turn increases polarization (Cinelli,  Pelicon, Mozetič, Quattrociocchi, Novak and Zollo, 2021a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Vicario, Quattrociocchi, Scala and Zollo, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Braking  such a vicious cycle would require addressing the problem of misinformation at its root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Fields such as journalism (Graves, 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Graves and Amazeen, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Neely-Sardon and Tignor, 2018) and  archival studies (LeBeau, 2017) have extensively studied misinformation, and recent years have seen a significant  growth in fact-checking initiatives to address this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Various organizations now focus on fact-checks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=',  PolitiFact, Snopes, FactCheck, First Draft, and Full Fact), and organizations such as the International Fact-Checking  Network (IFCN)1 train and provide resources for independent fact-checkers and journalists to further scale expert  fact-checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  While professional fact-checkers and journalists provide high-quality investigations of purported facts to inform  the public, human effort struggles to match the global Internet scale of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' To address this, a growing  body of research has investigated Natural Language Processing (NLP) to fully or partially automate fact-checking  (Guo, Schlichtkrull and Vlachos, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Nakov, Corney, Hasanain, Alam, Elsayed, Barrón-Cedeño, Papotti, Shaar and  Da San Martino, 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Zhou and Zafarani, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Zeng, Abumansour and Zubiaga, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Graves, 2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' However,  even state-of-the-art NLP technologies still cannot match human capabilities in many areas and remain insufficient  to automate fact-checking in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Experts argue (Arnold, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Nakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021a) that fact-checking is a  complex process and requires subjective judgement and expertise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' While current NLP systems are increasingly better  at addressing simple fact-checking tasks, identifying false claims that are contextual and beyond simple declarative  ∗Corresponding author  ORCID(s): 0000-0002-5412-6149 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Das);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' 0000-0003-0983-6202 (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Liu);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' 0000-0003-1259-1541 (V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Kovatchev);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  0000-0002-0056-2834 (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Lease)  1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='politifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='com/,  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='snopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='com/,  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='factcheck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='org/,  https://firstdraftnews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  org/, https://fullfact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='org/, and https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='poynter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='org/ifcn/, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Anubrata Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 1 of 30  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='03056v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='CL]  8 Jan 2023  The State of Human-centered NLP Technology for Fact-checking  statements remains beyond the reach for fully automated systems (Chen, Sriram, Choi and Durrett, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Fan, Piktus,  Petroni, Wenzek, Saeidi, Vlachos, Bordes and Riedel, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For example, claims buried in conversational systems,  comment threads in social media community, and claims in multimedia contents are particularly challenging for  automated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Additionally, most fact-checking practitioners desire NLP tools that are integrated into the existing  fact-checking workflow and reduce latency (Nakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Graves, 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Alam, Shaar, Dalvi, Sajjad, Nikolov,  Mubarak, Da San Martino, Abdelali, Durrani, Darwish, Al-Homaid, Zaghouani, Caselli, Danoe, Stolk, Bruntink and  Nakov, 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  In this literature review, we provide the reader with a comprehensive and holistic overview of the current state-  of-the-art challenges and opportunities to more effectively leverage NLP technology in fact-checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Our objectives  in this work are twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' First, we cover all aspects of the NLP pipeline for fact checking:  task formulation, dataset  construction, and modeling approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Second, we emphasize the human-centered approaches that seek to augment  and accelerate human fact-checking, rather than supplant it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' In contrast, prior literature reviews (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Oshikawa, Qian and Wang, 2020) either provide an overview of the existing approaches or capture  the details of only a specific part of the fact-checking pipeline (Kotonya and Toni, 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Hardalov, Arora, Nakov and  Augenstein, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Hanselowski, AvineshP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', Schiller, Caspelherr, Chaudhuri, Meyer and Gurevych, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Demartini,  Mizzaro and Spina, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Furthermore, we argue that it is important to extend the review of NLP technologies for fact-checking from  modeling development to the area of Human-Computer Interaction (HCI) because technology design should reflect  user needs so that its development can be better integrated in the real-world use context (Graves, 2018a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Lease, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Kovatchev, Smith, Lee, Traynor, Aguilera and Devine, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Nakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Micallef, Armacost, Memon and  Patil, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Juneja and Mitra, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Specifically, we point the reader towards Section 7 where we propose concrete  directions for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Current challenges are largely due to the relatively early stage of development of the automated fact-checking tech-  nology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Specifically, current studies tend to adopt an intrinsic evaluation of components of the fact-checking pipeline  rather than an end-to-end extrinsic evaluation of the entire fact-checking task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Moreover, component-wise accuracies  may remain below the threshold required for practical adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Furthermore, while the research community’s focus  on prediction accuracy has yielded laudable improvements, human factors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', usability, intelligibility, trust) have  garnered far less attention or progress yet are crucial for practical adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Such limitations have implications for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' First, practical use of NLP technologies for fact-checking  is likely to come from hybrid, human-in-the-loop approaches rather than full automation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Second, as the technology  matures, end-to-end evaluation becomes increasingly important to ensure practical solutions are being developed to  solve the real-world use-case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' To this end, new benchmarks that facilitate the extrinsic evaluation of automated fact-  checking applications in practical settings may help drive progress on solutions that can be adopted for use in the wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Finally, to craft effective human-in-the-loop systems, more cross-cutting NLP and HCI integration could strengthen  design of fact-checking tools, so that they are accurate, scalable, and usable in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Toward this end, it may  be fruitful to collaborate more with stakeholders early on in NLP research and incorporate human-centered design  practices in developing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  We have written this article with different audiences in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For researchers and fact-checkers who are new to  automated fact-checking, this article provides a comprehensive overview of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' We discuss the challenges,  the state-of-the-art capabilities, and the opportunities in the field, and we emphasize how machine learning and natural  language processing can be used to combat disinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' We recommend researchers new to this topic read the  article in its entirety, following the logical structure of sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Other readers who have more experience in the  field may already be familiar with some of the concepts that we discuss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For them, this paper offers a novel human-  centered perspective of automated fact checking and a discussion on how that perspective can affect system design,  implementation, and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' To facilitate the use of the paper by more experienced readers, we provide a quick  overview of the content covered in each section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Section 2 introduces the automated fact-checking pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' We provide an overview of the process for human  fact-checkers and for automated solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Section 3 discusses the task formulation: the goals and formal definitions of different sub-tasks in fact checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Section 4 describes the process of dataset construction, presents the most popular corpora for automated fact  checking, and outlines some limitations of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Anubrata Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 2 of 30  The State of Human-centered NLP Technology for Fact-checking  Figure 1: Fact-checking pipeline  Section 5 reviews approaches for automating fact checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' We discuss general NLP capabilities (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='1),  explainable approaches (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='2), and human-in-the-loop (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='3) approaches for fact-checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Section 6 surveys existing tools that apply NLP for fact-checking in a practical, real-world context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' We argue  that the human-centered perspective is necessary for the practical adoption of automated solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Section 7 provides future research directions in the context of human-centered fact checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' We discuss the work  division between human and AI for mixed-initiative fact-checking in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' In Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='2 we propose a  novel concept for measuring trust and a novel human-centered evaluation of NLP to assist fact-checkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  We conclude our literature review with Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Fact-Checking Pipeline  The core idea behind automated fact-checking is enabling AI to reason over available information to determine the  truthfulness of a claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For successful automation, it is essential first to understand the complex process of journalistic  fact-checking that involves human expertise along with skilled effort towards gathering evidence and synthesizing  the evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Additional complexity comes from the need to process heterogeneous sources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', information across  various digital and non-digital sources).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Data is also spread across different modalities such as images, videos, tables,  graphs, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Moreover, there is a lack of tools that support effective and efficient fact-checking workflows  (Graves, 2018a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Nakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Arnold, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Graves (2017) breaks down the practical fact-checking mechanism for human fact-checkers into multiple steps  such as a) identifying the claims to check, b) tracing false claims, c) consulting experts, and d) sharing the resulting  fact-check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' A growing body of AI literature — specifically in NLP — focuses on automating the fact-checking process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  We synthesize several related surveys (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Nakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Graves, 2018a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Micallef et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2022) and distinguish four typical stages that constitute the automated fact-checking technology pipeline  (illustrated in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Note that the pipeline we describe below closely follows the structure of Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' (2022),  though the broader literature is also incorporated within these four stages:  Claim Detection, Checkworthiness, and Prioritization: Claim detection involves monitoring news and/or  online information for potentially false content to fact-check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' One must identify claims that are potentially  falsifiable (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', purported facts rather subjective opinions) (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Moreover,  because it is impractical to fact-check everything online given limited fact-checking resources (human or  automated), fact checkers must prioritize what to fact-check (Arnold, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' NLP researchers have sought to  inform such prioritization by automatically predicting the "checkworthiness" of claims (Nakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Additionally, to avoid repeated work, fact-checkers may consult existing fact-checking databases before judging  the veracity of a new claim (claim matching (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' We see claim matching as a part of prioritizing  claims, as fact-checkers would prioritize against checking such claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Evidence Retrieval: Once it is clear which claims to fact-check, the next step is to gather relevant, trustworthy  evidence for assessing the claim (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Veracity Prediction: Given the evidence, it is necessary to assess it to determine the veracity of the claim (Guo  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Anubrata Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 3 of 30  回田  β  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='m  Claim detection,  Evidence  Veracity  Checkworthiness,  Explanation  retrieval  prediction  and PrioritizationThe State of Human-centered NLP Technology for Fact-checking  Explanation: Finally, for human use, one must explain the fact-checking outcome via human-understandable  justification for the model’s determination (Graves, 2018a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Kotonya and Toni, 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  In the subsequent sections, we discuss each of the tasks above in the context of existing NLP research in automated  fact-checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Some other steps (for example, detecting propaganda in text, click-bait detection) are also pertinent to  fact-checking but do not directly fit into the stages described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' They are briefly discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Task Formulation for Automated Fact-Checking  Modern Natural Language Processing is largely-data driven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' In this article, we distinguish task formulation  (conceptual) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' dataset construction (implementation activity, given the task definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' That said, the availability  of a suitable dataset or the feasibility of constructing a new dataset can also bear on how tasks are formulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Claim Detection, Checkworthiness, and Prioritization  Fact-checkers and news organizations monitor information sources such as social media (Facebook, Twitter, Reddit,  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' ), political campaigns and speeches, and public addresses from government officials on critical issues (Arnold,  2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Nakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Additional sources include tip-lines on end-to-end encrypted platforms (such as WhatsApp,  Telegram, and Signal) (Kazemi, Garimella, Shahi, Gaffney and Hale, 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' The volume of information on various  platforms makes it challenging to efficiently monitor all sources for misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' (2021) define the claim  detection step as identifying, filtering, and prioritizing claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  To identify claims, social media streams are often monitored for rumors (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' A rumour can be  defined as a claim that is unverified and being circulated online (Zubiaga, Aker, Bontcheva, Liakata and Procter, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Rumours are characterized by the subjectivity of the language and the reach of the content to the users (Qazvinian,  Rosengren, Radev and Mei, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Additionally, metadata related to virality, such as the number of shares (or retweets  and re-posts), likes, or comments are also considered when identifying whether a post is a rumour (Zhang, Cao, Li,  Sheng, Zhong and Shu, 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Gorrell, Kochkina, Liakata, Aker, Zubiaga, Bontcheva and Derczynski, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' However,  detecting rumours alone is not sufficient to decide whether a claim needs to be fact-checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  For each text of interest, the key questions fact-checking systems need to address include:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Is there a claim to check?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Does the claim contain verifiable information?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Is the claim checkworthy?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Has a trusted source already fact-checked the claim?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Regarding the first criterion — is there a claim — one might ask whether the claim contains a purported fact or  an opinion (Hassan, Arslan, Li and Tremayne, 2017a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For example, a statement such as “reggae is the most soulful  genre of music” represents personal preference that is not checkable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' In contrast, “<NAME> won a gold medal in the  Olympics” is checkable by matching <NAME> to the list of all gold medal winners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Whether the claim contains verifiable information is more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For example, if a claim can only be verified  by private knowledge or personal experience that is not broadly accessible, then it cannot be checked (Konstantinovskiy,  Price, Babakar and Zubiaga, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For example, if someone claims to have eaten a certain food yesterday, it is probably  impossible to verify beyond their personal testimony.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  As this example suggests, the question of whether the claim contains verifiable information depends in large part  on what evidence is available for verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' This, in turn, may not be clear until after evidence retrieval is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  In practice fact-checkers may perform some preliminary research, but mostly try to gauge checkworthiness only based  on the claim itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  In addition, this consideration is only one of many that factors into deciding whether to check a claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Even a claim  that may appear to be unverifiable may still be of such great public interest that it is worth conducting the fact-check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Moreover, even if the fact-check is conducted and ultimately indeterminate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', evidence does not exist either to verify  or refute the claim), simply showing that a claim’s veracity cannot be determined may still be a valuable outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  A claim is deemed checkworthy if a claim is of significant public interest or has the potential to cause harm (Nakov,  Da San Martino, Elsayed, Barrón-Cedeño, Míguez, Shaar, Alam, Haouari, Hasanain, Babulkov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Hassan,  Li and Tremayne, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For example, a claim related to the effect of a vaccine on the COVID-19 infection rate is more  relevant to the public interest and hence more checkworthy than a claim about some philosopher’s favorite food.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Anubrata Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 4 of 30  The State of Human-centered NLP Technology for Fact-checking  Claims, like memes, often appear several times and/or across multiple platforms (in the same form or with  slight modification) (Leskovec, Backstrom and Kleinberg, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Nakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Arnold, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Fact-checking  organizations maintain a growing database claims which have already been fact-checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Thus, detected claims are  compared against databases of already fact-checked claims by trusted organizations (Shaar, Martino, Babulkov and  Nakov, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Shaar, Alam, Da San Martino and Nakov, 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Comparing new claims against such databases helps  to avoid duplicating work on previously fact-checked claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' This step is also known as claim matching (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=',  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Reports from practitioners argue that if a claim is not checked within the first few hours, a late fact-check  does not have much impact on changing the ongoing misinformation narrative (Nakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Arnold, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Moreover, limited resources for fact-checking make it crucial for organizations to prioritize the claims to be checked  (Borel, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Claims can be prioritized based on their checkworthiness (Nakov, Da San Martino, Barrón-Cedeño,  Zaghouani, Míguez, Alam, Caselli, Kutlu, Strub and Mandl, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Nakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Nakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' (2022) note that  checkworthiness is determined based on factors such as  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' How urgently a claim needs to be checked?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' How much harm can a claim cause (Alam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Alam, Dalvi, Shaar, Durrani, Mubarak, Nikolov,  Da San Martino, Abdelali, Sajjad, Darwish and Nakov, 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Shaar, Hasanain, Hamdan, Ali, Haouari, Nikolov,  Kutlu, Kartal, Alam, Da San Martino, Barrón-Cedeño, Míguez, Beltrán, Elsayed and Nakov, 2021b)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Would the claim require attention from policy makers for addressing the underlying issue?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Note that estimating harms is quite challenging, especially without first having a thorough understanding and measures  of harm caused by misinformationNeumann, De-Arteaga and Fazelpour (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  The spread of a claim on social media provides another potential signal for identifying public interest (Arnold,  2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' In the spirit of doing “the greatest good for the greatest number”, viral claims might be prioritized highly because  any false information in them has the potential to negatively impact a large number of people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' On the other hand, since  fairness considerations motivate equal protections for all people, we cannot serve only the majority at the expense of  minority groups (Ekstrand, Das, Burke, Diaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Neumann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Moreover, such minority groups may  be more vulnerable, motivating greater protections, and may be disproportionately impacted by mis/disinformation  (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' See Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='6 for additional discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Evidence Retrieval  Some sub-tasks in automated fact-checking can be performed without the presence of explicit evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For  example, the linguistic properties of the text can be used to determine whether it is machine-generated (Wang, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Rashkin, Choi, Jang, Volkova and Choi, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' However, assessing assessing claim veracity without evidence is clearly  more challenging (Schuster, Schuster, Shah and Barzilay, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Provenance of a claim can also signal information quality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' known unreliable source or distribution channels are  often repeat offenders in spreading false information2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Such analysis of provenance can be further complicated when  content is systematically propagated by multiple sources (twitter misinformation bots) (Jones, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  It is typically assumed that fact-checking requires gathering of reliable and trustworthy evidence that provides  information to reason about the claim (Graves, 2018a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Li, Gao, Meng, Li, Su, Zhao, Fan and Han, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' In some  cases, multiple aspects of a claim needs to be checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' A fact-checker would then decompose such a claim into distinct  questions and gather relevant evidence for the question (Borel, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' From an information retrieval  (IR) perspective, we can conceptualize each of those questions as an “information need” for which the fact-checker must  formulate one or more queries to a search engine (Bendersky, Metzler and Croft, 2012) in order to retrieve necessary  evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Evidence can be found across many modalities, including text, tables, knowledge graphs, images, and videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Various metadata can also provide evidence and are sometimes required to assess the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Examples include context  needed to disambiguate claim terms, or background of the individual or organization from whom the claim originated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Retrieving relevant evidence also depends on the following questions (Singh, Das, Li and Lease, 2021):  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Is there sufficient evidence available related to a claim?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Is it accessible or available in the public domain?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Is it in a format that can be read and processed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  2https://disinformationindex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='org/  Anubrata Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 5 of 30  The State of Human-centered NLP Technology for Fact-checking  As noted earlier in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='1, the preceding claim detection task involves assessing whether a claim contains verifiable  information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' this depends in part on what evidence exists to be retrieved, which is not actually known until evidence  retrieval is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Having now reached this evidence retrieval step, we indeed discover whether sufficient evidence  exists to support or refute the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Additionally, evidence should be trustworthy, reputable (Nguyen, Kharosekar, Lease and Wallace, 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Lease,  2018), and unbiased (Chen, Khashabi, Yin, Callison-Burch and Roth, 2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Once evidence is retrieved, stance detection assesses the degree to which the evidence supports or refutes the  claim (Nguyen, Kharosekar, Krishnan, Krishnan, Tate, Wallace and Lease, 2018a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Ferreira and Vlachos, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Popat,  Mukherjee, Yates and Weikum, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Stance detection is typically formulated as a classification task (or ordinal  regression) over each piece of retrieved evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Note that some works formulate stance detection as an independent  task (Hanselowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Hardalov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Veracity Prediction  Given a claim and gathered evidence, veracity prediction involves reasoning over the collected evidence and the  claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Veracity prediction can be formulated as a binary classification task (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', true vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' false) (Popat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Nakashole and Mitchell, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Potthast, KIESELJ et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2018), or as a fine-grained, multi-class task following the  journalistic fact-checking practices (Augenstein, Lioma, Wang, Lima, Hansen, Hansen and Simonsen, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Shu,  Mahudeswaran, Wang, Lee and Liu, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Wang, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' In some cases, there may not be enough information available  to determine the veracity of a claim (Thorne, Vlachos, Christodoulopoulos and Mittal, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Note that fact-checking is potentially a recursive process because retrieved evidence may itself need to be fact-  checked before it can be trusted and acted upon (Graves, 2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' This is also consistent with broader educational  practices in information literacy3 in which readers are similarly encouraged to evaluate the quality of information  they consume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Such assessment of information reliability can naturally integrate with the veracity prediction task in  factoring in the reliability of the evidence along with its stance (Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Explaining Veracity Prediction  While a social media platform might use automated veracity predictions in deciding whether to automatically block  or demote content, the use of fact-checking technology often involves a human-in-the-loop, whether it is a platform  moderator, a journalist, or an end-user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' When we consider such human-centered use of fact-checking technologies,  providing an automated veracity prediction without justifying the answer can cause a system to be ignored or distrusted,  or even reinforce mistaken human beliefs in false claims (the “backfire effect” (Lewandowsky, Ecker, Seifert, Schwarz  and Cook, 2012)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Explanations and justifications are especially important given the noticeable drop in performance of  state-of-the-art NLP systems when facing adversarial examples (Kovatchev, Chatterjee, Govindarajan, Chen, Choi,  Chronis, Das, Erk, Lease, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Consequently, automated fact-checking systems intended for human-  consumption should seek to explain their veracity predictions in a similar manner to that of existing journalistic  fact-checking practices (Uscinski, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' A brief point to make is that much of the explanation research has focused  on explanations for researchers and engineers engaged in system development (types of explanations, methods of  generating them, and evaluation regimens).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' In contrast,we emphasize here explanations for system users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Various types of explanations can be provided, such as through  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' evidence attribution  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' explaining the decision-making process for a fact-check  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' summarizing the evidence  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' case-based explanations  Evidence attribution is the process of identifying evidence or a specific aspect of the evidence (such as paragraphs,  sentences, or even tokens of interest) (Thorne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Popat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Shu, Cui, Wang, Lee and Liu, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Lu  and Li, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Furthermore, the relative importance of the evidence can also justify the fact-checking outcome (Nguyen  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Alternatively, a set of rules or interactions to break down parts of the decision-making process can also  serve as an explanation (Gad-Elrab, Stepanova, Urbani and Weikum, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2018b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Such formulation  focuses more on how the evidence is processed to arrive at a decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Explaining the veracity can also be formulated  as a summarization problem over the gathered evidence to explain a fact-check (Atanasova, Simonsen, Lioma and  Augenstein, 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Kotonya and Toni, 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Finally, case-based explanations can provide the user with similar,  human-labeled instances (Das, Gupta, Kovatchev, Lease and Li, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  3https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='org/wiki/Information_literacy  Anubrata Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 6 of 30  The State of Human-centered NLP Technology for Fact-checking  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Related Tasks  In addition to tasks that are considered central to the automated fact-checking pipeline, some additional tasks bear  mentioning as related and complementary to the fact-checking enterprise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Examples of such tasks include propaganda  detection (Da San Martino, Cresci, Barrón-Cedeño, Yu, Pietro and Nakov, 2020), clickbait detection (Potthast, Köpsel,  Stein and Hagen, 2016), and argument mining (Lawrence and Reed, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Furthermore, some tasks can be formulated  independent of the fact-checking pipeline and utilized later to improve individual fact-checking sub-tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For example,  predicting the virality of social media content (Jain, Garg and Jain, 2021) can help improve claim detection and claim  checkworthiness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Similarly, network analysis on fake news propagation (Shao, Ciampaglia, Flammini and Menczer,  2016) can help in analyzing provenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  With an eye toward building more human-centered AI approaches, there are also some tasks that could be applied  to help automate parts of the fact-checking process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For example, claim detection might be improved via an URL  recommendation engine for content that might need fact-checking (Vo and Lee, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Additionally, fact-checkers  could benefit from a predicted score for claim difficulty (Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' In terms of evidence retrieval and  veracity prediction, one might generate fact-checking briefs to aid inexpert fact-checkers (Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Instead  of summarizing the evidence in general (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='4), one might instead summarize with the specific goal of decision  support (Hsu and Tan, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Key Challenges  Most work in automated fact-checking has been done on veracity prediction, and to a lesser extent, on explanation  generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Recently, we have seen more attention directed towards claim detection and checkworthiness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' In contrast,  work on evidence retrieval remains less developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Claim Detection Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' (2022) points out several sources of biases in the claim check-worthiness task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Claims  could be of variable interest to different social groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Additionally, claims that might cause more harm to marginalized  groups compared to the general population may not get enough attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Ideally, models identifying check-worthiness  need to overcome any possible disparate impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Similar concerns appear in the report by Full Fact (Arnold, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' One of the criteria for selecting a claim for fact-  checking across several organizations is “Could the claim threaten democratic processes or minority groups?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' However,  such criterion may be at odds with the concerns of virality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Fact-checking organizations often monitor virality metrics  to decide which claims to fact-check (Arnold, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Nakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Nevertheless, if a false claim is targeted  towards an ethnic minority, such claims may not cross the virality thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Prioritizing which claims to fact-check requires attention to various demographic traits: content creators, readers,  and subject matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Claim check-worthiness dataset design can thus benefit from consideration of demographics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Evidence Retrieval Evidence retrieval has been largely neglected in the automated fact-checking NLP literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' It  is often assumed that evidence is already available, or, coarse-grained evidence is gathered from putting the claims into  a search engine (Popat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' However, Hasanain and Elsayed (2022) show in their study  that search engines optimized for relevance seldom retrieve evidence most useful for veracity prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Although  retrieving credible information has been studied thoroughly in IR (Clarke, Rizvi, Smucker, Maistro and Zuccon,  2020a), more work is needed that is focused on retrieving evidence for veracity assessment (Lease, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Clarke,  Rizvi, Smucker, Maistro and Zuccon, 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Veracity Prediction and Explanation A critical challenge for automated systems is to reason over multiple sources  of evidence while also taking source reputation into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Additionally, explaining a complex reasoning process is  a non-trivial task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' The notion of model explanations itself is polysemous and evolving in general, not to mention in the  context of fact checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' As explainable NLP develops, automated fact-checking tasks also need to evolve and provide  explanations that are accessible to human stakeholders yet faithful to the underlying model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For example, case-based  explanations are mostly unexplored in automated fact-checking, although working systems have been proposed for  propaganda detection (Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  In many NLP tasks, such as machine translation or natural language inference, the goal is to build fully-automated,  end-to-end solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' However, in the context of fact-checking, state-of-the-art limitations suggest the need for humans-  in-the-loop for the forseable future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Given this, automated tooling to support human fact-checkers is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' However,  Anubrata Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 7 of 30  The State of Human-centered NLP Technology for Fact-checking  understanding the fact-checker needs and incorporating those needs in the task formulation has been largely absent  from the automated fact-checking literature, with a few notable exceptions (Nakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Demartini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Future research could benefit from greater involvement of fact-checkers in the NLP research process and shifting goals  from complete automation toward human support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Dataset Construction  Corresponding to task formulation (Section 3), our presentation of fact-checking datasets is also organized around  claims, evidence, veracity prediction, and explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Note that not all datasets have all of these components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Claim detection and claim check-worthiness  Claim detection datasets typically contain claims and their sources (documents, social media streams, transcripts  from political speeches) (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' One form of claim detection is identifying rumours on social media, where  datasets are primarily constructed with text from Twitter (Zubiaga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Qazvinian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2011) and Reddit  (Gorrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Lillie, Middelboe and Derczynski, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Some works provide the claims in the context they  appeared on social media (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Ma, Gao, Mitra, Kwon, Jansen, Wong and Cha, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' However,  several studies note that most claim detection datasets do not contain enough context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' As the discussion of metadata  in Section 3 suggests, broader context might include: social media reach, virality metrics, the origin of a claim, and  relevant user data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', who posted a claim, how influential they are online, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=') (Arnold, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Nakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Claim check-worthiness datasets (Nakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Shaar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Barrón-Cedeño, Elsayed, Nakov,  Da San Martino, Hasanain, Suwaileh, Haouari, Babulkov, Hamdan, Nikolov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Atanasova, Barrón-Cedeño,  Elsayed, Suwaileh, Zaghouani, Kyuchukov, Da San Martino and Nakov, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Konstantinovskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Hassan  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2015) filter claims from a source (similar to claim detection, sources include social media feeds and political  debate transcripts, among others) by annotating claims based on the checkworthiness criteria (mentioned in the section  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Each claim is given a checkworthiness score to obtain a ranked list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Note that claim detection and checkworthiness  datasets may be expert annotated (Hassan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2015) or crowd annotated (Nakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Shaar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Barrón-Cedeño et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Atanasova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Konstantinovskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  The datasets discussed above do not capture multi-modal datasets, and few do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' One such dataset is r/Fakeddit  (Nakamura, Levy and Wang, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' This dataset contains images and associated text content from Reddit as claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Misinformation can also spread through multi-modal memes, and tasks such as Facebook (now Meta) Hateful Memes  Challenge (Kiela, Firooz, Mohan, Goswami, Singh, Ringshia and Testuggine, 2020) for hate speech suggest what might  be similarly done for misinformation detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Evidence  Early datasets in fact-checking provide metadata with claims as the only form of evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Such metadata include  social media post properties, user information, publication date, source information (Wang, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Potthast et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  As discussed earlier in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='2, such metadata does not contain the world knowledge necessary to reason about a  complex claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' To address the above limitations, recent datasets consider external evidence (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Evidence is collected differently depending upon the problem setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For artificial claims, evidence is often retrieved  from a single source such as Wikipedia articles (Thorne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Jiang, Bordia, Zhong, Dognin, Singh and Bansal,  2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Schuster, Fisch and Barzilay, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Domain limited evidence for real-world claims is collected from problem-  specific sources, such as academic articles for scientific claims (Kotonya and Toni, 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Wadden, Lin, Lo, Wang, van  Zuylen, Cohan and Hajishirzi, 2020), or specific evidence listed in fact-checking websites (Vlachos and Riedel, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Hanselowski, Stab, Schulz, Li and Gurevych, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Open-domain evidence for real-world claims is usually collected  from the web via search engines (Popat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Augenstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Recently, there has been more work considering evidence beyond free text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Such formats include structured or semi-  structured forms of evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Sources include knowledge bases for structured form of evidence (Shi and Weninger,  2016) and semi-structured evidence from semi-structured knowledge bases (Vlachos and Riedel, 2015), tabular data  (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Gupta, Mehta, Nokhiz and Srikumar, 2020), and tables within a document (Aly, Guo, Schlichtkrull,  Thorne, Vlachos, Christodoulopoulos, Cocarascu and Mittal, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Additionally, there are some retrieval-specific datasets that aim at retrieving credible information from search  engines (Clarke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' However, such tasks don’t incorporate claim checking as an explicit task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  4Some of these datasets, such as the CheckThat!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' datasets, are partially crowd and partially expert annotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Anubrata Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 8 of 30  The State of Human-centered NLP Technology for Fact-checking  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Veracity Prediction  Evidence retrieval and veracity prediction datasets are usually constructed jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Note, in some cases, evidence  may be absent from the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Veracity prediction datasets usually do not deal with claim detection or claim  checkworthiness tasks separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Instead, such datasets contain a set of claims that are either artificially constructed  or collected from the internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Artificial claims in veracity prediction datasets are often limited in scope and constructed for natural language  reasoning research (Aly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Thorne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Schuster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Chen, Wang, Chen,  Zhang, Wang, Li, Zhou and Wang, 2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For example, FEVER (Thorne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2018) and HoVer (Jacovi and Goldberg,  2021) obtain claims from Wikipedia pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Some datasets also implement subject-predicate-object triplets for fact-  checking against knowledge bases (Kim and Choi, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Shi and Weninger, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Fact-checking websites are popular sources for creating veracity prediction datasets based on real claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Several  datasets obtain claims from either a single website or collect claims from many such websites and collate them (Wang,  2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Hanselowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Augenstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Vlachos and Riedel, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Note that such claims are inherently  expert annotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Other sources of claims are social media (Potthast et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Shu, Sliva, Wang, Tang and Liu,  2017), news outlets (Horne, Khedr and Adali, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Gruppi, Horne and Adalı, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Nørregaard, Horne and Adalı,  2019), blogs, discussions in QA forums, or similar user-generated publishing platforms (Mihaylova, Nakov, Màrquez,  Barrón-Cedeño, Mohtarami, Karadzhov and Glass, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Additionally some fact-checking datasets target domain-specific problems such as scientific literature (Wadden  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2020), climate change (Diggelmann, Boyd-Graber, Bulian, Ciaramita and Leippold, 2020), and public health  (Kotonya and Toni, 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Most datasets are monolingual but recent effort have started to incorporate multi-lingual  claims (Gupta and Srikumar, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Barnabò, Siciliano, Castillo, Leonardi, Nakov, Da San Martino and Silvestri, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Early datasets focus on a binary veracity prediction - true or false (Mihalcea and Strapparava, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Recent datasets  often adopt an ordinal veracity labeling scheme that mimics fact checkering websites (Vlachos and Riedel, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Wang,  2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Augenstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' However, every fact-checking website has a different scale for veracity, so datasets  that span across multiple websites come with a normalization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' While some datasets do not normalize the  labels (Augenstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2019), some normalize them post-hoc (Kotonya and Toni, 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Gupta and Srikumar, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Hanselowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Explanation  While an explanation is tied to veracity prediction, only a few datasets explicitly address the problem of explainable  veracity prediction (Atanasova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Kotonya and Toni, 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Alhindi, Petridis and Muresan, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Broadly  in NLP, often parts of the input is highlighted to provide an explanation for the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' This form of explanations  is known as extractive rationale (Zaidan, Eisner and Piatko, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Kutlu, McDonnell, Elsayed and Lease, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Incorporating the idea of the extractive rationale, some datasets include a sentence from the evidence along with  the label (Thorne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Hanselowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Wadden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Schuster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Although such  datasets do not explicitly define evidence as a form of explanation in such cases, the line between evidence retrieval and  explanation blurs if the evidence is the explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' However, explanations are different from evidence in a few ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Particularly, explanations need to be concise for user consumption, while evidence can be a collection of documents  or long documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Explanations are user sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Consequently, evidence alone as a form of explanation might have  some inherent assumption about the user that might not be understandable for different groups of users (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', experts  vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' non-experts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Challenges  Claims Checkworthiness datasets are highly imbalanced, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', the number of checkworthy claims are relatively low  compared to non-checkworthy claims (Williams, Rodrigues and Novak, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Datasets are also not generalizable  due to their limited domain-specific context (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Additionally, while existing datasets cover various  languages such as English, Arabic, Spanish, Bulgarian, and Dutch, they are primarily monolingual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Consequently,  building multilingual checkworthiness predictors is still challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Much of the data in check-worthiness datasets is  not originally intended to be used in classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' The criteria used by different organizations when selecting which  claims to check is often subjective and may not generalize outside of the particular organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Some annotation practices can result in artifacts in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For example, artificially constructed false claims,  such as a negation-based false claim in FEVER, can lead to artifacts in models (Schuster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Models do not  generalize well beyond the dataset because they might overfit to the annotation schema (Bansal, Nushi, Kamar, Lasecki,  Anubrata Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 9 of 30  The State of Human-centered NLP Technology for Fact-checking  Weld and Horvitz, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' One way to identify such blind spots is by using adversarial datasets for fact-checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Such  a setting is incorporated in FEVER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='0 (Thorne and Vlachos, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Datasets constructed for research may not always capture how fact-checkers work in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' This leads to  limitations in the algorithms built on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For example, interviews with fact-checkers report that they tend to consider  a combination of contents of the posts and associated virality metrics (indicating reach) during fact-checking (Arnold,  2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' However, most fact-checking datasets do not include virality metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Evidence Retrieval Some datasets have been constructed by using a claim verbatim as a query and taking the  top search results as evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' However, some queries are better than others for retrieving desired information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Consequently, greater care might be taken in crafting effective queries or otherwise improving evidence retrieval such  that resulting datasets are more likely to contain quality evidence for veracity prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Otherwise, poor quality  evidence becomes a bottleneck for the quality of the models trained at the later stages in the fact-checking pipeline  (Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Veracity Prediction A key challenge in veracity prediction datasets is that the labels are not homogeneous across  fact-checking websites and normalizing might introduce noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Explanation Some datasets include entire fact-checking articles as evidence and their summaries as the form of  explanation (Atanasova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Kotonya and Toni, 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' In such cases, “explanation” components assume an  already available fact-checking article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Instead, providing abstractive summaries and explaining the reasoning process  over the evidence would be more valuable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Data Generation Recent years have seen an increasing interest in the use of data generation and data augmentation  for various NLP tasks (Liu, Swayamdipta, Smith and Choi, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Hartvigsen, Gabriel, Palangi, Sap, Ray and Kamar,  2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Dhole, Gangal, Gehrmann, Gupta, Li, Mahamood, Mahendiran, Mille, Srivastava, Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Kovatchev,  Smith, Lee and Devine, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' The use of synthetic data has not been extensively explored in the context of fact-  checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Automating Fact-checking  NLP research in automated fact-checking has primarily focused on building models for different automated fact-  checking tasks utilizing existing datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' In the following section, we highlight the broad modeling strategies employed  in the literature, with more detailed discussion related to explainable methods for automated fact-checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' General NLP Capabilities  Claim Detection and Checkworthiness While claim detection is usually implemented as a classification task only,  claim checkworthiness is typically implemented both as ranking (Nakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021a) and classification task (Zeng  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' As discussed earlier in the task formulation Section (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='1), the broad task of claim detection can be  broken down into sub-tasks of identifying claims, filtering duplicate claims, and prioritizing claims based on their  checkworthiness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Another instance of identifying claims is detecting rumors in social media streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Some early works in rumor detection focus on feature engineering from available metadata the text itself (Enayet  and El-Beltagy, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Aker, Derczynski and Bontcheva, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Zhou, Jain, Phoha and Zafarani, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' More advanced  methods for claim detection involve LSTM and other sequence models (Kochkina, Liakata and Augenstein, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Such models are better at capturing the context of the text (Zubiaga, Liakata, Procter, Wong Sak Hoi and Tolmie,  2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Tree-LSTM (Ma, Gao and Wong, 2018) and Hierarchical attention networks (Guo, Cao, Zhang, Guo and Li,  2018) capture the internal structure of the claim or the context in which the claim appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Additionally, graph neural  network approaches can capture the related social media activities along with the text (Monti, Frasca, Eynard, Mannion  and Bronstein, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Similarly, early works in claim-checkworthiness utilize support vector machines using textual features and rank  the claims in terms of their priorities (Hassan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2017a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For example, Konstantinovskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' (2021) build a  classification model for checkworthiness by collapsing the labels to checkable vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' non-checkable claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' They build  a logistic regression model that uses word embeddings along with syntax based features (parts of speech tags, and  named entities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Neural methods such as LSTM performed well in earlier checkworthiness shared tasks (Elsayed,  Nakov, Barrón-Cedeño, Hasanain, Suwaileh, Da San Martino and Atanasova, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Additionally, Atanasova, Nakov,  Anubrata Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 10 of 30  The State of Human-centered NLP Technology for Fact-checking  Màrquez, Barrón-Cedeño, Karadzhov, Mihaylova, Mohtarami and Glass (2019b) show that capturing context helps  with the checkworthiness task as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Models such as RoBERTa obtained higher performance in the later edition of the  CheckThat!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' shared task (Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Martinez-Rico, Martinez-Romo and Araujo, 2021) for English language  claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Fine-tuning such models for claim detection tasks has become more prevalent for claim checkworthiness in  other languages as well (Hasanain and Elsayed, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Filtering previously fact-checked claims is a relatively new task in this domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Shaar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' (2020) propose an  approach using BERT and BM-25 to match claims against fact-checking databases for matching claims with existing  databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Additionally, fine-tuning RoBERTa on various fact-checking datasets resulted in high performance for  identifying duplicate claims (Bouziane, Perrin, Cluzeau, Mardas and Sadeq, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Furthermore, a combination of  pretrained model Sentence-BERT and re-ranking with LambdaMART performed well for detecting previously fact-  checked claims (Nakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Evidence Retrieval and Veracity Prediction Evidence retrieval and veracity prediction in the pipeline can be  modeled sequentially or jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Similar to claim detection and checkworthiness models, early works use stylistic  features and metadata to arrive at veracity prediction without external evidence (Wang, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Rashkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Models that include evidence retrieval often use commercial search APIs or some retrieval approach such as TF-IDF,  and BM25 (Thorne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Similar to question-answering models, some works adopt a two-step approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' First  a simpler model (TF-IDF or BM-25) is used at scale and then a more complex model is used for re-ranking after the  initial pruning (Thorne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Nie, Wang and Bansal, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Hanselowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Additionally, document vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  passage retrieval, or 2-stage “telescoping” approaches, are adopted where the first stage is retrieving related documents  and the second stage is to retrieve the relevant passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Two stage approaches are useful for scaling up applications  as the first stage is more efficient than the second stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For domain specific evidence retrieval, using domain-bound  word embeddings has been shown to be effective (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  The IR task is not always a part of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Instead, it is often assumed that reliable evidence is already available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  While this simplifies the fact-checking task so that researchers can focus on veracity prediction, in practice evidence  retrieval is necessary and cannot be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Moreover, in practice one must contend with noisy (non-relevant), low  quality, and biased search results during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  As discussed earlier in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='3, assessing the reliability of gathered evidence may be necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' If the evidence is  assumed to be trustworthy, then it suffices to detect the stance of each piece of evidence and then aggregate (somehow)  to induce veracity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', perhaps assuming all evidence is equally important and trustworthy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' However, often one  must contend with evidence “in the wild” of questionable reliability, in which case assessing the quality (and bias) of  evidence is an important precursor to using it in veracity prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Veracity prediction utilizes textual entailment for inferring veracity over either a single document as evidence  or over multiple documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Dagan, Dolan, Magnini and Roth (2010) define textual entailment as “deciding, given  two text fragments, whether the meaning of one text is entailed (can be inferred) from another text.” Real-world  applications often require reasoning over multiple documents (Augenstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Kotonya and Toni, 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Schuster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Reasoning over multiple documents can be done either by concatenation (Nie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2019) or  weighted aggregation (Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2018b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Weighted aggregation virtually re-ranks the evidence considered to filter  out the unreliable evidence (Ma, Gao, Joty and Wong, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Pradeep, Ma, Nogueira and Lin, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Some approaches  also use Knowledge Bases as the central repository of all evidence (Shi and Weninger, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' However, evidence is  only limited to what is available in the knowledge base (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Moreover, a fundamental  limitation of knowledge bases is that not all knowledge fits easily into structured relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Recent developments in large language models help extend the knowledge base approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Fact-checking models  can rely on pretrained models to provide evidence for veracity prediction (Lee, Li, Wang, Yih, Ma and Khabsa, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  However, this approach can encode biases present in the language model (Lee, Bang, Madotto and Fung, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  An alternative approach is to help fact-checkers with downstream tasks by processing evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' An example of  such work is generating summaries over available evidence using BERT (Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Limitations With the recent development of large, pre-trained language models and deep learning for NLP, we see  a significant improvement across the fact-checking pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' With the introduction of FEVER (Thorne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Thorne, Vlachos, Cocarascu, Christodoulopoulos and Mittal, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Aly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021) and CheckThat!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' (Nakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=',  2021b) we have benchmarks for both artificial and real-life claim detection and verification models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' However, even  the state-of-the-art NLP models perform poorly on the benchmarks above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For example, the best performing model on  Anubrata Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 11 of 30  The State of Human-centered NLP Technology for Fact-checking  FEVER 2018 shared task (Thorne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2018) reports an accuracy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='675.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Models perform worse on multi-modal  shared task FEVEROUS (Aly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021): the best performing model reports 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='56 accuracy score6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Similarly, the best  checkworthiness model only achieved an average precision of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='65 for Arabic claims and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='224 for English claims in  the CheckThat!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' 2021 shared task for identifying checkworthiness in tweets (Nakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' On the other hand,  the best performing model for identifying check-worthy claims in debates reports 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='42 average precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Surprisingly,  Barrón-Cedeño et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' (2020), the top performing model for checkworthiness detection, report an average precision of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='806 (Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For the fact-checking task of CheckThat!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' 2021 (Nakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021b), the best performing  model reports a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='83 macro F1 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' However, the second-best model only reports a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='50 F1 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Given this striking  gap in performance between the top system vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' others, it would be valuable for future work to benchmark systems on  additional datasets in order to better assess the generality of these findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  It is not easy to make a direct comparison between different methods that are evaluated in different settings and  with different datasets (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Moreover, the pipeline design of automated fact-checking creates potential  bottlenecks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', performance on the veracity prediction task on most datasets is dependent on the claim detection task  performance or the quality of the evidence retrieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Extensive benchmarks are required to incorporate all of the prior  subtasks in the fact-checking pipeline systematically (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Much of AI research is faced with a fundamental trade-off between working with diverse formulations of a problem  and standardized benchmarks for measuring progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' This trade-off also impacts automated fact-checking research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  While there exist benchmarks such as FEVER and the CheckThat!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', most models built on those benchmarks may  not generalize well in a practical setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Abstract and tractable formulations of a problem may help us develop  technologies that facilitate practical adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' However, practical adoption requires significant engineering effort  beyond the research setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Ideally, we would like to see automated fact-checking research continue to move toward  increasingly realistic benchmarks while incorporating diverse formulations of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Explainable Approaches  Although the terms interpretability and explainability are often used interchangeably, and some times defined to be  so (Molnar, 2020), we distinguish interpretability vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' explainability similar to (Kotonya and Toni, 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Specifically,  interpretability represents methods that provide direct insight into an AI system’s components (such as features and  variables), often requiring some understanding of the specific to the algorithm, and often built for expert use cases  such as model debugging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Explainability represents methods to understand an AI model without referring to the actual  component of the systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Note that, in the task formulation section, we have also talked about explaining veracity  prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' The goal of such explanation stems from fact-checker needs to help readers understand the fact-checking  verdict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Therefore, explaining veracity prediction aligns more closely with explainability over interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' When  the distinction between explainability vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' interpretability does not matter, we follow Vaughan and Wallach (2020) in  adopting intelligibility (Vaughan and Wallach, 2020) as an umbrella term for both concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Sokol and Flach (2019) propose a desiderata for designing user experience for machine learning applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Kotonya and Toni (2020a) extend them in the context of fact-checking and suggest eight properties of intelligibility:  actionable, causal, coherent, context-full, interactive, unbiased or impartial, parsimonious, and chronological.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Additionally, there are three dimensions specifically for explainable methods in NLP (Jacovi and Goldberg, 2020):  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Readability: are explanations clear?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Plausibility: are explanations compelling or persuasive?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Faithfulness: are explanations faithful to the model’s actual reasoning process?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  In comparison with the available intelligibility methods in NLP (Wiegreffe and Marasovic, 2021), only a few  are applied to existing fact-checking works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Below, we highlight only commonly observed explainable fact-checking  methods (also noted by Kotonya and Toni (2020a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Attention-based Intelligibility Despite the debate about attention being a reliable intelligibility method (Jain and  Wallace, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Wiegreffe and Pinter, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Serrano and Smith, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Bibal, Cardon, Alfter, Wilkens, Wang, François  and Watrin, 2022), it remains a popular method in existing deep neural network approaches in fact-checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Attention-  based explanations are provided in various forms:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' highlighting tokens in articles (Popat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2018)  5https://fever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='ai/2018/task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='html  6https://fever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='ai/task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='html  Anubrata Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 12 of 30  The State of Human-centered NLP Technology for Fact-checking  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' highlighting salient excerpts from evidence utilizing comments related to the post (Shu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2019)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' n-gram extraction using self-attention (Yang, Pentyala, Mohseni, Du, Yuan, Linder, Ragan, Ji and Hu, 2019)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' attention from different sources other than the claim text itself, such as the source of tweets, retweet propagation,  and retweeter properties (Lu and Li, 2020)  Rule discovery as explanations Rule mining is a form of explanation prevalent in knowledge base systems (Gad-  Elrab et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Ahmadi, Lee, Papotti and Saeed, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' These explanations can be more comprehensive, but as noted  in the previous section, not all statements can be fact-checked via knowledge-based methods due to limitations of the  underlying knowledge-base itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Some approaches provide general purpose rule mining in an attempt to address this  limitation (Ahmadi, Truong, Dao, Ortona and Papotti, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Summarization as explanations Both extractive and abstractive summaries can provide explanations for fact-  checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Atanasova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' (2020a) provides natural language summaries to explain the fact-checking decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' They  explore two different approaches - explanation generation and veracity prediction as separate tasks, and joint training  of the both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Joint training performs worse than single training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Kotonya and Toni (2020b) combine abstractive and  extractive approaches to provide a novel summarization approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Brand, Roitero, Soprano, Rahimi and Demartini  (2018) show jointly training prediction and explanation generation with encoder-decoder models such as BART (Lewis,  Liu, Goyal, Ghazvininejad, Mohamed, Levy, Stoyanov and Zettlemoyer, 2020) results in explanations that help the  crowd to perform better veracity assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Counterfactuals and adversarial methods Adversarial attacks on opaque models help to identify any blind-spots,  biases and discover data artifacts in models (Ribeiro, Wu, Guestrin and Singh, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Shared task FEVER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='0 (Thorne  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2019) asked participants to devise methods for generating adversarial claims to identify weaknesses in the fact-  checking methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Natural language generation models such as GPT-3 can assist in formulating adversarial claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  More control over the generation can come from manipulating the input to natural language generation methods  and constraining the generated text within original vocabulary (Niewiński, Pszona and Janicka, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Atanasova,  Wright and Augenstein (2020b) generate claims with n-grams inserted into the input text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Thorne and Vlachos (2019)  experiment with several adversarial methods such as rule-based adversary, semantically equivalent adversarial rules  (or SEARS) (Ribeiro, Singh and Guestrin, 2018), negation, and paraphrasing-based adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Adversarial attacks  are evaluated based on the potency (correctness) of the example and reduction in system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' While methods  such as SEARS and paraphrasing hurt the system performance, hand-crafted adversarial examples have higher potency  score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Interpretable methods (non-BlackBox) Some fact-checking works use a white-box or inherently interpretable  model for fact-checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' (2018b,a) utilize a probabilistic graphical model and build an interactive  interpretable model for fact-checking where users are allowed to directly override model decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Kotonya,  Spooner, Magazzeni and Toni (2021) propose an interpretable graph neural network for interpretable fact-checking  on FEVEROUS dataset (Aly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Limitations Intelligible methods in NLP and specifically within fact-checking are still in their infancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Analysis of  Kotonya and Toni (2020a) shows that most methods do not fulfill the desiderata mentioned earlier in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Specifically, they find that none of the existing models meet the criteria of being actionable, causal, and chronological.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  They also highlight that no existing method explicitly analyzes whether explanations are impartial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Some forms  of explanations, such as rule-based triplets, are unbiased as they do not contain sentences or contain fragments of  information (Kotonya and Toni, 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Some explainable methods address a specific simplified formulation of the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For example, Kotonya and Toni  (2020b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Atanasova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' (2020a) both assume that expert-written fact-checking articles already exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' They provide  explanations as summaries of the fact-checking article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' However, in practice, a fact-checking system would not have  access to such an article for an unknown claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  In the case of automated fact-checking, most intelligible methods focus on explaining the outcome rather than  describing the process to arrive at the outcome (Kotonya and Toni, 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Moreover, all of the tasks in the fact-  checking pipeline have not received equal attention for explainable methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Kotonya and Toni (2020a) also argue  that automatic fact-checking may benefit from explainable methods that provide insight into how outcomes of earlier  sub-task in the fact-checking pipeline impact the outcome of later subtasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Anubrata Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 13 of 30  The State of Human-centered NLP Technology for Fact-checking  Most explainable NLP works evaluate explanation quality instead of explanation utility or faithfulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Jacovi  and Goldberg (2020) argue for a thorough faithfulness evaluation for explainable models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For example, even though  attention-based explanations may provide quality explanations, they may not necessarily be faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Moreover, expla-  nation utility requires separate evaluation by measuring whether explanations improve both i) human understanding  of the model (Hase and Bansal, 2020) and ii) human effectiveness of the downstream task (Nakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Additionally, most intelligible methods establish only one-way communication from the model to humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Instead,  explanations might improve the model and human performance by establishing a bidirectional feedback loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Human-in-the-loop Approaches  Human-in-the-loop (HITL) approaches can help scale automated solutions while utilizing human intelligence for  complex tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' There are different ways of applying HITL methods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', delegating sub-tasks to crowd workers  (Demartini, Trushkowsky, Kraska, Franklin and Berkeley, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Demartini, Difallah and Cudré-Mauroux, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Sarasua, Simperl and Noy, 2012), active learning (Settles, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Zhang, Lease and Wallace, 2017), interactive machine  learning (Amershi, Cakmak, Knox and Kulesza, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Joachims and Radlinski, 2007), and decision support systems  where humans make the final decision based on model outcome and explanations (Zanzotto, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  While HITL approaches in artificial intelligence are prevalent, only a few recent works employ such approaches  in fact-checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' HITL approaches are predominantly more present in the veracity prediction task than other parts of  the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For example, Demartini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' (2020) propose a HITL framework for combating online misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  However, they only consider hybrid approaches for two sub-tasks in the fact-checking pipeline: a) claim check-  worthiness and b) truthfulness judgment (same as veracity prediction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Below, we discuss the existing HITL approaches  by how the system leverages human effort for each sub-task in the fact-checking pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Claim Detection, Checkworthiness, and Prioritization Social media streams are often monitored for rumors as  a part of the claim detection task (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Farinneya, Abdollah Pour, Hamidian and Diab (2021) apply an  active learning-based approach at the claim detection stage for identifying rumors on social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' In-domain data is  crucial for traditional supervised methods to perform well for rumor detection (Ahsan, Kumari and Sharma, 2019), but  in real-world scenarios, sufficient in-domain labeled data may not be available in the early stages of development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' A  semi-supervised approach such as active learning is beneficial for achieving high performance with fewer data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Empirical results shows that Tweet-BERT, along with the least confidence-based sample selection approach, performs  on par with supervised approaches using far less labeled data (Farinneya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Similarly, Tschiatschek, Singla, Gomez Rodriguez, Merchant and Krause (2018) propose a HITL approach that  aims to automatically aggregate user flags and recommend a small subset of the flagged content for expert fact-checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Their Bayesian inference-based approach jointly learns to detect fake news and identify the accuracy of user flags over  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' One strength of this approach is that the algorithm improves over time in identifying users’ flagging accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Consequently, over time this algorithm’s performance improves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' This approach is also robust against spammers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' By  running the model on publicly available Facebook data where a majority of the users are adversarial, experiments show  that their algorithm still performs well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Duke’s Tech & Check team implemented HITL at the claim check-worthiness layer (Adair and Stencel, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' To  avoid flagging false check-worthy claims, a human expert would sort claims detected by ClaimBuster (Hassan, Zhang,  Arslan, Caraballo, Jimenez, Gawsane, Hasan, Joseph, Kulkarni, Nayak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2017b), filter out the ones deemed more  important for fact-checkers, and email them to several organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' In essence, this approach helped fact-checkers  prioritize the claims to check through an additional level of filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Currently, several published fact-checks on  PolitiFact were first alerted by the emails from Tech & Check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Note that the CheckThat!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' (Nakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Shaar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Barrón-Cedeño et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Atanasova, Nakov,  Karadzhov, Mohtarami and Da San Martino, 2019a) is a popular shared task for claim detection, check-worthiness, and  prioritization tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' However such shared tasks often have no submissions that employs HITL methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Shared  tasks for HITL approaches could encourage more solutions that can complement the limitations of model-only based  approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Evidence Retrieval and Veracity Prediction Most work in HITL fact-checking caters to veracity prediction, and  only a few consider evidence retrieval as a separate task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' While there is a body of literature on HITL approaches in  information retrieval (Chen and Jain, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Demartini, 2015), we know of no work in that direction for fact-checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Anubrata Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 14 of 30  The State of Human-centered NLP Technology for Fact-checking  Shabani, Charlesworth, Sokhn and Schuldt (2021) leverage HITL approaches for providing feedback about claim  source, author, message, and spelling (SAMS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Annotators answer four yes/no questions about whether the article has  a source, an author, a clear and unbiased message, and any spelling mistake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Furthermore, this work integrates the  features provided by humans in a machine learning pipeline, which resulted in a 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='1% accuracy increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' However,  the evaluation is performed on a small dataset with claims related to Covid-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' It is unclear if this approach would  generalize outside of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Moreover, further human effort can be reduced in this work by automating spell-  check and grammar-check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' SAMS could be quite limited in real life situations as most carefully crafted misinformation  often looks like real news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Model generated fake news can successfully fool annotators (Zellers, Holtzman, Rashkin,  Bisk, Farhadi, Roesner and Choi, 2020), and thus SAMS might also fail to flag such fake news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Qu, Barbera, Roitero, Mizzaro, Spina and Demartini (2021a) and Qu, Roitero, Mizzaro, Spina and Demartini  (2021b) provide an understanding of how human and machine confidence scores can be leveraged to build HITL  approaches for fact-checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' They consider explicit self-reported annotator confidence and compute implicit con-  fidence based on standard deviation among ten crowd workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Model confidence is obtained from bootstrapping  (Efron and Tibshirani, 1985) ten different versions of the model and then computing standard deviation over the scores  returned by the soft-max layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Their evaluation shows that explicit crowd and model confidence are poor indicators  of accurate classification decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Although the crowd and the model make different mistakes, there is no clear  signal that confidence is related to accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' However, they show that implicit crowd confidence can be a useful  signal for identifying when to engage experts to collect labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' A more recent study shows that a politically balanced  crowd of ten is correlated with the average rating of three fact-checkers (Allen, Arechar, Pennycook and Rand, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Gold, Kovatchev and Zesch (2019) also find that annotations by a crowd of ten correlate with the judgments of three  annotators for textual entailment, which is utilized by veracity prediction models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  A series of studies show that the crowd workers can reliably identify misinformation (Roitero, Soprano, Fan, Spina,  Mizzaro and Demartini, 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Roitero, Soprano, Portelli, Spina, Della Mea, Serra, Mizzaro and Demartini, 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Soprano, Roitero, La Barbera, Ceolin, Spina, Mizzaro and Demartini, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Furthermore, Roitero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' (2020b) show  that crowd workers not only can identify false claims but also can retrieve proper evidence to justify their annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  One weakness of this study is that it only asks users to provide one URL as evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' However, in practice, fact-  checking might need reasoning over multiple sources of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Although these studies do not propose novel  HITL solutions, they provide sufficient empirical evidence and insights about where crowd workers can be engaged  reliably in the fact-checking pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' (2018b) propose joint modeling of crowd annotations and machine learning to detect the veracity of  textual claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' The key strength of the model is that it assumes all annotators can make mistakes, which is a possibility  as fact-checking is a difficult task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Another strength is that this model allows users to import their knowledge into the  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Moreover, this HITL approach can collect on-demand stance labels from the crowd and incorporate them in  veracity prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Empirical evaluation shows that this approach achieves strong predictive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' A follow-up  study provides an interactive HITL tool for fact-checking (Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Nguyen, Weidlich, Yin, Zheng, Nguyen and Nguyen (2020) propose a HITL system to minimise user effort and cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Users validate algorithmic predictions but do so at a minimal cost by only validating the most-beneficial predictions  for improving the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' This system provides a guided interaction to the users and incrementally gets better as users  engage with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  It is important to note that research on crowdsourcing veracity judgment is at an early stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Different factors such  as demographics, political leaning, criteria for determining the expertise of the assessors (Bhuiyan, Zhang, Sehat and  Mitra, 2020), cognitive factors (Kaufman, Haupt and Dow, 2022), and even the rating scale (La Barbera, Roitero,  Demartini, Mizzaro and Spina, 2020) led to different levels of alignment with expert ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Bhuiyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' (2020)  outline research directions for designing better crowd processes specific to different types of misinformation for the  successful utilization of crowd workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Explaining Veracity Prediction HITL systems in fact-checking often use veracity explanations to correct model  errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' As discussed earlier, Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' (2018a) provides an interpretable model that allows users to impart their  knowledge when the model is wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Empirical evaluation shows that users could impart their knowledge into the  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Similarly, Zhang, Rudra and Anand (2021b) propose a method that collects user feedback from explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Note that this method explains veracity prediction outcomes based on the evidence retrieved and their stance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Users  provide feedback in terms of stance and relevance of the retrieved evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' The proposed approach employs lifelong  Anubrata Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 15 of 30  The State of Human-centered NLP Technology for Fact-checking  learning which enables the system to improve over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Currently there is no empirical evaluation of this system to  identify the effectiveness of this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Although natural language generation models are getting increasingly better (Radford, Wu, Child, Luan, Amodei,  Sutskever et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2019), generating abstractive fact-checking explanations is still in its infancy (Kotonya and Toni,  2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' HITL methods could be leveraged to write reports justifying fact-checking explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Limitations After reviewing existing HITL approaches across different fact-checking tasks, we also list out several  limitations as follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' First, some HITL approaches adopt several interpretable models to integrate human input, but  the resulting models do not perform as well as the state-of-the-art deep learning models (Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2018b,a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Farinneya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' (2021) apply HITL approaches to scale up rumor detection from a limited amount of annotated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Although it performs well to generalize the algorithm for a new topic in a few-shot manner, one of the weaknesses  is that data from other domains or topics causes a high variance in model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Consequently, in-domain  model performance might degrade when out-of-domain data is introduced in model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' This issue may hinder the  model’s generalizability in practice, especially where a clear demarcation between topic domains may not be possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  More importantly, there is a lack of empirical studies on how to apply HITL approaches of fact-checking for  practical adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Although HITL approaches provide a mechanism to engage human in the process of modeling  development, several human factors, such as usability, intelligibility, and trust, become important to consider when  applying this method in the real-world use case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Fact-checking is a time-sensitive task and requires expertise to process  complex information over multiple sources (Graves, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Fact-checkers and policy makers are often skeptical about  any automated or semi-autoamted solutions as this type of research requires human creativity and expertise (Arnold,  2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Micallef et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Therefore, more empirical evidence needs to be found to assess the effectiveness of  applying different HITL approaches to automated fact-checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Existing Tools for Fact-checking  In the previous section, we reviewed the details of current NLP technologies for fact-checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Subsequently,  we extend our review of automated fact-checking to the HCI literature and discuss existing practices of applying fact-  checking into real-world tools that assist human fact-checkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' In brief, there is a lack of holistic review of fact-checking  tools from a human-centered perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Additionally, we found that the articulation of work between human labor and  AI tools is still opaque in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Research questions include but are not limited to: 1) how can NLP tools facilitate  human work in different fact-checking tasks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' 2) how can we incorporate user needs and leverage human expertise to  inform the design of automated fact-checking?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  In this section, we examine current real-world tools that apply NLP technologies in different stages of fact-checking  and clarify the main use cases of these tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' We argue that more research concerning human factors for building  automated fact-checking, such as user research, human-centered design, and usability studies, should be conducted to  improve the practical adoption of automated fact-checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' These studies help us identify the design space of applying  explainable and HITL approaches for real-world NLP technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Claim Detection and Prioritization  The first step in claim detection is sourcing content to possibly check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' On end-to-end encrypted platforms, such  as WhatsApp, Telegram, and Signal, crowdsourcing-based tip-lines play a vital role in identifying suspicious content  that is not otherwise accessible (Kazemi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' As another example, Check from Meedan 7, a tip-line service  tool, also helps fact-checkers monitor fake news for in-house social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' User flagging of suspect content on social  media platforms such as Facebook is also a valuable signal for identifying such content, and crowdsourcing initiatives  like Twitter’s BirdWatch can further help triage and prioritize claims for further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  In the stage of finding and choosing claims to check, fact-checkers assess the fact-checking related quality of a  claim and decide whether to fact-check it (Graves, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Micallef et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' NLP models in claim detection, claim  matching, and check-worthiness are useful to assist the above decision-making process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' However, integrating them  into real-world tools that help fact-checkers prioritize what to check requires more personalized effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Graves (2018a)  points out that it is important to design the aforementioned models to cater to fact-checker organizational interests,  stakeholder needs, and changing news trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  7https://meedan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='com/check  Anubrata Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 16 of 30  The State of Human-centered NLP Technology for Fact-checking  As one of the fact-checking qualities of a claim, checkability can be objectively analyzed by whether a claim  contains one or more purported facts that can be verified (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Fact-checkers find it useful to apply models  that identify checkable claims to their existing workflow because the model helps them filter irrelevant content and  claims that are uncheckable when they are choosing claims to check (Arnold, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' ClaimBuster, one of the well-  known claim detection tools, is built to find checkable claims from a large scale of text content (Hassan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2017a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Claim detection can also be integrated into speech recognition tools to spot claims from live speech (Adair, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Additionally, if a claim has already been fact-checked, fact-checkers can skip it and prioritize claims that have  not been checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' As a relatively new NLP task, claim matching has been integrated into some current off-the-shelf  search engines or fact-checking tools to help fact-checkers find previously fact-checked claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For example, Google  Fact Check Explorer8 can retrieve previously fact-checked claims by matching similar fact-check content to user input  queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Similarly, with Meedan’s Check, if users send a tip with fake news that has been previously fact-checked, the  tool further helps fact-checkers retrieve the previous fact-check and send it to users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Whether or not to fact-check a claim depends on an organization’s goals and interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Tools built for claim detection  need to take such interests into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For example, Full Fact developed a claim detection system that classifies  claims into different categories, such as quantity, predictions, correlation or causation, personal experience, and laws  or rules of operations (Konstantinovskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' The claim categories are designed by their fact-checkers to cater  to their needs of fact-checking UK political news in a live fact-checking situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Identifying certain claims, such as  quantity, correlation or causation, might be particularly useful for fact-checkers to evaluate the credibility of politician  statements and claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' The system also helps tailor fact-checkers’ downstream tasks, such as fact-check assignments  and automated verification for statistical claims (Nakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Fact-checkers also use social media monitoring tools to find claims to check, such as CrowdTangle, TweetDeck, and  Facebook’s (unnamed) fact-checking tool, but those tools are not very effective to detect checkable claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Some fact-  checkers reported that only roughly 30% of claims flagged by Facebook’s fact-checking tool were actually checkable  (Arnold, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' A low hanging fruit is to integrate claim detection models into these social media monitoring tools so  that it is easier for fact-checkers to identify claims that are both viral and checkable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Additionally, these tools should  enable fact-checkers to locate certain figures, institutions, or agencies according to their fact-checking interests and  stakeholder needs so that these tools can better identify and prioritize truly check-worthy claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' An important question  in implementing those systems is how to measure the virality of a claim and its change over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  It would also be useful to integrate veracity prediction into previous fact-checking tools because fact-checkers  may pay the most attention to claims9 that are suspect and uncertain (since obviously true or false claims likely do  not require a fact-check).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' However, information or data points that are used to give such predictions should also be  provided to fact-checkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' If sources, evidence, propagation patterns, or other contextual information that models use  to predict claim veracity can be explained clearly for fact-checkers, they can also triage these indicators to prioritize  claims more holistically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Tools for Evidence Retrieval  After finding and prioritizing which claim to check, fact-checkers investigate claims following three main activities:  1) decomposing claims, 2) finding evidence, and 3) tracing the provenance of claims and their spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Note that these  three activities are intertwined with each other by using different information-seeking tools in the fact-checking process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Fact-checkers search for evidence by decomposing claims into sub questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Evidence found while investigating a  claim may further modify or add to the sub-questions (Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' By iteratively investigating claims via  online search, fact-checkers reconstruct the formation and the spread of a claim to assess its truth (Graves, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' In  this section, we discuss the utility of existing information-seeking tools, including off-the-shelf search engines and  domain-specific databases, that assist fact-checkers in each activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Claim decomposition is not a specific activity that qualitative researchers have reported or analyzed in their fact-  checking studies, but we can find more details from where fact-checking organizations describe their methodology10  and how fact-checkers approach complex claims in their fact-checks11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Claim decomposition refers to how fact-  checkers interpret ambiguous terms of a claim and set the fact-checking boundaries to find evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Decomposing  8https://toolbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='com/factcheck/explorer  9https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='factcheck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='org/our-process/  10https://leadstories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='com/how-we-work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='html  11https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='factcheck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='org/2021/10/oecd-data-conflict-with-bidens-educational-attainment-claim/ In this fact-  check, fact-checkers decompose what President Biden mean by “advanced economies” and “young people”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' The approach of defining these two  terms directly influence their fact-checking results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Anubrata Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 17 of 30  The State of Human-centered NLP Technology for Fact-checking  claims effectively requires sensitive curiosity and news judgments for fact-checkers that are cultivated through years  of practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Unfortunately, we are not aware of any existing tools that facilitate this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Traditional methodology to decompose claims is to ask sub-questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Recent NLP studies simulate this process  by formulating it as a question-answering task (Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Researchers extract justifications  from existing fact-checks and crowdsource sub-questions to decompose the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For automated-fact-checking, this  NLP task might be very beneficial to improve the performance of evidence retrieval by auto-decomposing claims into  smaller checkable queries (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Although it is difficult for NLP to match the abilities of professional  fact-checkers, it might help scale up the traditional, human fact-checking process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' It could also help the public, new  fact-checkers, or journalists to more effectively investigate complex claims and search for evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  How fact-checkers find evidence is usually a domain-specific reporting process, contacting experts or looking for  specific documents from reliable sources (Graves, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Micallef et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Instead of conducting random searches  online, most fact-checkers include a list of reliable sources in which to look for evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Tools that are designed for  searching domain datasets can also help fact-checkers to find evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For example, Li, Fang, Lou, Li and Zhang  (2021) built an analytical search engine for retrieving the COVID-19 news data and summarizing it in an easy to  digest, tabular format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' The system can decompose analytical queries into structured entities and extract quantitative  facts from news data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Furthermore, if evidence retrieval is accurate enough for in-domain datasets, the system can  take a leap further to auto-verify domain-related claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' We provide more detailed use cases of veracity prediction in  Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Fact-checkers mainly use off-the-shelf search engines, such as Google, Bing, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', to trace a claim’s origin from  publicly accessible documents (Beers, Haughey, Melinda, Arif and Starbird, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Arnold, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Other digital  datasets, such as LexisNexis and InternetArchive, are also useful for fact-checkers to trace claim origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' To capture the  formation and change of a claim, search engines should not only filter unrelated content, but also retrieve both topically  and evidentially relevant content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Hasanain and Elsayed (2021) report that most topically relevant pages retrieved from  Google do not contain evidential information, such as statistics, quotes, entities, or other types of facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Additionally,  most built-in search engines in social media platforms, such as Twitter and Facebook, only filter “spreadable” content  not “credible” content (Alsmadi, Alazzam and AlRamahi, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Furthermore, these off-the-shelf search engines do not support multilingual search, so it is difficult for fact-checkers  to trace claims if they are translated from other languages (Graves, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Nakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' NLP researchers have  started to use multilingual embedding models to represent claim-related text in different languages and match existing  fact-checks (Kazemi, Gaffney, Garimella and Hale, 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' This work not only helps fact-checkers find previously  fact-checked claims more easily from other languages, but also to examine how the claim is transformed and reshaped  by the media in different languages and socio-political contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Domain-specific Tools for Claim Verification  As discussed in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='3, most verification prediction models are grounded on the collected evidence  and the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' To build an end-to-end claim verification system, NLP developers need to construct domain-specific  datasets incorporating both claims and evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Different from complex claims that contain multiple arguments and  require decomposition, claims that have simple linguistic structure with purported evidence or contain statistical facts  can be automatically verified (Nakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Karagiannis, Saeed, Papotti and Trummer (2020) built CoronaCheck, a search engine that can directly verify Covid-  19 related statistical claims by retrieving official data curated by experts (Dong, Du and Gardner, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Full Fact  (The Poynter Institute, 2021) also took a similar approach to verify statistical macroeconomic claims by retrieving  evidence from UK parliamentary reports and national statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Additionally, Wadden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' (2020) built a scientific  claim verification pipeline by using abstracts that contain evidence to verify a given scientific claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  However, pitfalls still exist if fact-checkers use these domain-specific verification tools in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For example, the  CoronaCheck tool cannot check the claim “The Delta variant causes more death than the Alpha variant” simply because  the database does not contain fine-grained death statistics for Covid variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Additionally, checking a statistical or  scientific claim might only be a part of the process of checking a more complex claim, which requires fact-checkers  to contextualize the veracity of previous statistical or scientific checks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' In general, domain-specific tools are clearly  valuable to use when available, though in practice they are often incomplete and insufficient on their own to check  complex claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Anubrata Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 18 of 30  The State of Human-centered NLP Technology for Fact-checking  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Discussion  In this review, we have 1) horizontally outlined the research of applying NLP technologies for fact-checking from  the beginning of task formulation to the end of tools adoption;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' as well as 2) vertically discussing the capabilities and  limitations of NLP for each step of a fact-checking pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' We perceive a lack of research that bridges both to assist  fact-checkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Explainable and HITL approaches leverage both human and computational intelligence from a human-  centered perspective, but there is a need to provide actionable guides to utilize both methods for designing useful  fact-checking tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' In this section, we propose several research directions to explore the design space of applying  NLP technologies to assist fact-checkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Distributing Work between Human and AI for Mixed-initiative Fact-checking  The practice of fact-checking has already become a type of complex and distributed computer-mediated work  (Graves, 2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Although Graves (2017) breaks down a traditional journalist fact-checking pipeline into five steps,  the real situation of fact-checking a claim is more complicated (Juneja and Mitra, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Various AI tools are adopted  dynamically and diversely by fact-checkers to complete different fact-checking tasks (Arnold, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Beers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Micallef et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Researchers and practitioners increasingly believe that future fact-checking should be a mixed-initiative practice  in which humans perform specific tasks while machines take over others (Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2018a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Lease, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Nakov  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' To embed such hybrid and dynamic human-machine collaborations into existing fact-checking workflow,  the task arrangement between human and AI need to be articulated clearly by understanding the expected outcomes  and criteria for each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Furthermore, designing a mixed-initiative tool for different fact-checking tasks requires a more  fine-grained level of task definition for human and AI (Lease, 2018, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='3, we discuss several studies  highlighting the role of humans in the fact-checking workflow, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', a) human experts select check-worthy claims from  claim detection tools (Hassan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2017b) and deliver them to fact-checkers, b) ask crowd workers to judge reliable  claims sources (Shabani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021), or c) flag potential misinformation (Roitero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2020b) to improve veracity  prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' All of the above human activities are examples of micro-tasks within a mixed-initiative fact-checking  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Prior work in crowdsourcing has shown that it is possible to effectively break down the academic research process  and utilize crowd workers to partake in smaller research tasks (Vaish, Davis and Bernstein, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Vaish, Gaikwad,  Kovacs, Veit, Krishna, Arrieta Ibarra, Simoiu, Wilber, Belongie, Goel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Given this evidence, we can  also break down sub-tasks of a traditional fact-checking process into more fine-grained tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Therefore, key research  questions include: a) How can we design these micro-tasks to facilitate each sub-task of fact-checking, and b) What  are the appropriate roles for human and AI in different micro-tasks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  To effectively orchestrate human and AI work, researchers need to understand the respective roles of human and AI,  and how they will interact with one another, because it will directly affect whether humans decide to take AI advice  (Cimolino and Graham, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Usually, if AI aims to assist high-stake decision-making tasks, such as recidivism  prediction (Veale, Van Kleek and Binns, 2018) and medical treatments (Cai, Winter, Steiner, Wilcox and Terry, 2019),  considerations of risk and trust will be important factors for people to adopt such AI assistants (Lai, Chen, Liao,  Smith-Renner and Tan, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' In the context of fact-checking, if AI directly predicts the verdict of a claim, fact-  checkers may be naturally skeptical about how the AI makes such a prediction (Arnold, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' On the other hand, if  AI only helps to filter claims that are uncheckable, such as opinions and personal experience, fact-checkers may be  more willing to use such automation with less concern about how AI achieves it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Deciding whether a claim is true or  false is a high-stake decision-making task for fact-checkers, while filtering uncheckable claims is a less important but  tedious task that fact-checkers want automation to help with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Therefore, the extent of human acceptance of AI varies  according to how humans assess the task assigned to AI, resulting in different human factors, such as trust, transparency,  and fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Researchers need to specify or decompose these human factors into different key variables that can be  measured during the model development process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Given a deep understanding of the task relationship between human  and AI, researchers can then ask further research questions on how to apply an explainable approach, or employ a  HITL system vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' automated solutions, to conduct fact-checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Here we list out several specific research topics that  contain mixed-initiative tasks, including: a) assessing claim difficulty leveraging crowd workers, b) breaking down a  claim into a multi-hop reasoning task and engaging the crowd to find information relevant to the sub-claims, and c)  designing micro-tasks to parse a large number of documents retrieved by web search to identify sources that contain  the evidence needed for veracity prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Anubrata Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 19 of 30  The State of Human-centered NLP Technology for Fact-checking  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Human-centered Evaluation of NLP Technology for Fact-Checkers  We begin this section by proposing key metrics from human factors for evaluating systems (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', what to measure  and how to measure them): accuracy, time, model understanding, and trust (Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Following this, we further  propose a template for an experimental protocol for human-centered evaluations in fact-checking (Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Metrics  Accuracy Most fact-checking user studies assume task accuracy as the primary user goal (Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2018a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Mohseni, Yang, Pentyala, Du, Liu, Lupfer, Hu, Ji and Ragan, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Whereas non-expert users (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', social media users  or other form of content consumers) might be most interested in the veracity outcome along with justification, fact-  checkers often want to use automation and manual effort interchangeably in their workflow (Arnold, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Nakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=',  2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Thus, we need a more fine-grained approach towards measuring accuracy beyond the final veracity prediction  accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For fine-grained accuracy evaluation, it is also crucial to capture fact-checker accuracy, particularly for the  sub-tasks for which they use the fact-checking tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  With the assumption that “ground truth” exists for all of the sub-tasks in the fact-checking pipeline, accuracy can  be computed by comparing user answers with the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Note that measuring sub-task level accuracy is trickier  than end-to-end fact-checking accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Sub-task level accuracy can be captured by conducting separate experiments  for each sub-task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Suppose the point of interest is to understand user performance for detecting claim-checkworthiness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  In that case, we will need to collect additional data specific to the claim-checkworthiness task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  In some cases, it is possible to merge multiple sub-tasks for evaluation purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For example, Miranda, Nogueira,  Mendes, Vlachos, Secker, Garrett, Mitchel and Marinho (2019) evaluate the effectiveness of their tool with journalists  by capturing the following two key variables: a) the relevance of retrieved evidence, and b) the accuracy of the predicted  stance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' This method provides essential insight into evidence retrieval, stance detection, and the final fact-checking task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Depending on the tool, the exact detail of this metric will require specific changes according to tool affordances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Note that both time and accuracy measures need to control for claim properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For example, if a claim has been  previously fact-checked, it would take less time to fact-check such claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' On the other hand, a new claim that is more  difficult to assess would require more time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Model Understanding Fact-checkers want to understand the tools they use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For example, Arnold (2020) pointed  out that fact-checkers expressed a need for understanding CrowdTangle’s algorithm for detecting viral content on  various social media platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Similarly, Nakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' (2021a) observed a need for increased system transparency in  the fact-checking tools used by different organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Lease (2018) argues that transparency is equally important for  non-expert users to understand the underlying system and make an informed judgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Although this is not a key  variable related to user performance, it is important for practical adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  To measure understanding, users could be asked to self-report their level of understanding on a Likert-scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  However, simply asking participants if they understand the algorithm is not a sufficient metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For example, it does not  indicate whether participants will be able to simulate tool behavior (Hase and Bansal, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' We suggest the following  steps for measuring model understanding based on prior work (Cheng, Wang, Zhang, O’Connell, Gray, Harper and  Zhu, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Decision Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' To capture users’ holistic understanding of a tool, users could be provided claims and  asked the following: “What label would the tool assign to this claim?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Alternative Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Capturing how changes in the input influence the output can also measure understand-  ing, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', by asking users how the tool would assign a label to a claim when input parts are changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Imagine a  tool that showed the users the evidence it has considered to arrive at a veracity conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Now, if certain pieces  of evidence were swapped, how would that be reflected in the model prediction?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Trust For practical adoption, trust in a fact-checking tool is crucial across all user groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' While model understanding  is often positively correlated with trust, understanding alone may not suffice to establish trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' In this domain, fact-  checkers and journalists may have less trust in algorithmic tools (Arnold, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' On the other hand, there is also the risk  of over-trust, or users blindly following model predictions (Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2018a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Mohseni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' To maximize  the tool effectiveness, we would want users to neither dismiss all model predictions out of hand (complete skepticism)  nor blindly follow all model predictions (complete faith).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Instead, it is important to calibrate user trust for the most  effective tool usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' We suggest measuring a notion of calibrated trust (Lee and See, 2004): how often users abide by  correct model decisions and override erroneous model decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Anubrata Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 20 of 30  The State of Human-centered NLP Technology for Fact-checking  Both  Model  User  Neither  User Prediction  Correct  Incorrect  Correct  Incorrect  Model  Prediction  Figure 2: Confusion Matrix for User Predictions vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Model Predictions with respect to ground truth (gold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' We assume  model predictions are provided to the user, who then decides whether to accept or reject the model prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' The top-left  quadrant (Both) covers cases where users correctly follow model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' The top-right quadrant (Model) denotes the  cases where the model is correct but users mistakenly reject the model decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' The bottom-left quadrant User denotes  the cases where users correctly reject erroneous model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' The bottom-right quadrant Neither denotes the cases  where users incorrectly accept erroneous model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Quantifying user vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' model predictions in this manner enables  measurement of calibrated trust: how often users abide by correct model decisions and override erroneous model decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  To measure calibrated trust, we imagine a confusion matrix shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' The rows denote correct vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' incorrect  model predictions while the columns denote correct vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' incorrect user predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' A user who blindly followed all  model predictions would have their behavior entirely captured by the main (primary) diagonal, whereas a user who  skeptically rejected all model predictions would have their behavior captured entirely in the secondary diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' The  ideal user’s behavior would be entirely captured in the first column: accepting all correct model predictions and rejecting  all incorrect model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' To promote effective human-AI teaming, AI tools should assist their human users in  developing strong calibrated trust to appropriately trust and distrust model predictions as each case merits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Beyond calibrated trust, one could also measure quantitative trust by adopting methodologies from the human-  machine trust literature (Lee and Moray, 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For example, Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' (2019) adapted prior work into a 7-point  Likert scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' A similar scale can be reused for evaluating trust in a fact-checking tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For example, we can create five  different Likert-scales to measure the agreement (or disagreement) of users with the following statements:  I understand the fact-checking tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  I can predict how the tool will behave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  I have faith that the tool would be able to cope with the different fact-checking task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  I trust the decisions made by the tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  I can count on the tool to provide reliable fact-checking decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Additional factors Individual differences among users might result in substantial variation in experimental outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  For example, varying technical literacy (Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2019), any prior knowledge about the claims, and users’ political  leaning (Thornhill, Meeus, Peperkamp and Berendt, 2019) might influence user performance on the task while using  fact-checking tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Thus it is valuable to capture these factors in study design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For example:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Technical Literacy: Users’ familiarity with popular technology tools (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='g, recommendation engines, spam  detectors) and their programming experience (Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2019) as well as familiarity with existing fact-  checking tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Media Literacy: Users’ familiarity with 1) the fact-checking process, and 2) fact-checks from popular  organizations such as PolitiFact and FactCheck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='Org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Demographics: Users’ education level, gender, age, and political leaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Quantitative measures alone are not sufficient as they do not capture certain nuances about how effectively a tool  integrates into a fact-checker’s workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For example, even if users understand and trust the working principle of a  Anubrata Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 21 of 30  The State of Human-centered NLP Technology for Fact-checking  tool, it is unclear why they do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Hence, users might be asked a few open-ended questions at the end of the study to  gather qualitative insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Such questions could include:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Describe your understanding of the tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Do any specific aspects of its design seem to assist or detract from your  understanding of how it works?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Why do you trust or not trust the tool?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Would you use this tool beyond this study, and if so, in what capacity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Experimental Protocol  One strategy to capture the aforementioned metrics is to design a mixed-methods study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Here we outline the  template for such a study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Imagine the goal were to measure the user performance for fact-checking using a new tool  (let’s call it tool A) compared to an existing tool (tool B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Fact-checking tasks in the real world might be influenced by  user priors about the claims being checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Thus, a within-subject study protocol may be more appropriate to account  for such priors (Shi, Bhattacharya, Das, Lease and Gwizdka, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Pre-task: Users would first be asked to fact-check a set of claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' To do so, first a user would be asked to leverage  a pre-existing tool B at this stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Tool B can be replaced with different baselines, depending on the particular  use case, ranging from simple web-search by non-expert users to proprietary tools used by fact-checkers and  journalists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Users would be asked to think aloud at this stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Learning: At this stage users would familiarize themselves with the new tool (tool A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Users would need to  fact-check a different set of claims from the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Ground truth would also accessible to the user to form a  prior about what kind of mistakes a tool might make.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Claims here would be selected at random to reflect tool  capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Moreover, tool performance metrics would be given to the users as additional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Users  would be encouraged to ask questions about the tool at this stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Prediction: Users would now be asked to fact-check the same claims from step-1 above but this time they are  asked to leverage the tool A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Users would be asked to think out loud through this stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Users could simply guess  the answers and achieve a high accuracy score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Thus, claims selected for stages (1) & (3) would be a balanced  set of claims with an equal distribution of true positive, true negative, false positive, and false negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  This idea is adopted from prior work (Hase and Bansal, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Post-task survey: Users would now be asked to take a small survey for capturing trust, understanding, technical  literacy, media literacy, and demographic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Post-task interview: Upon completion of these steps, users would be interviewed with open-ended questions to  gather insights about their understanding and trust in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  The measures and study protocol could be useful in the context of evaluating any new fact-checking system compared to  an existing system or practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Specifics might vary depending on the target user group and the tool’s intended purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Above we use the whole fact-checking pipeline to illustrate our experimental protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' However this technique can be  applied to other sub-tasks of automated fact checking, granted that we have the ground truth of the outcome for that  sub-task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' For example, let us assume a new claim detection tool has been proposed that takes claims from a tip-line  (Kazemi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Currently, fact-checkers use an existing claim-matching algorithm to filter out the already  fact-checked claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Now, if we replace tool B above with the existing claim-matching algorithm and tool A with the  proposed claim detection tool, we can utilize the protocol mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' In conclusion, one could evaluate how  users perform for claim detection tasks using the new tool compared to the existing ones in terms of their accuracy,  time, understanding, and trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  While we have proposed an ideal, extensive version of an evaluation protocol for evaluating new fact-checking tools,  note that the actual protocol used in practice could be tailored according to the time required from the participants and  the cost of conducting the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Conclusion  This review highlights the practices and development of the state-of-the-art in using NLP for automated fact-  checking, emphasizing both the advances and the limitations of existing task formulation, dataset construction, and  modeling approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' We partially discuss existing practices of applying these NLP tasks into real-world tools that  assist human fact-checkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' In recent years we have seen significant progress in automated fact-checking using NLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Anubrata Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 22 of 30  The State of Human-centered NLP Technology for Fact-checking  A broad range of tasks, datasets, and modeling approaches have been introduced in different parts of the fact-checking  pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Moreover, with recent developments in transformers and large language models, the model accuracy has  improved across tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' However, even state-of-the-art models on existing benchmarks — such as FEVER and CLEF!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  — may not yet be ready for practical adoption and deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  To address these limitations, we advocate development of hybrid, HITL systems for fact-checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' As a starting  point, we may wish to reorient the goals of existing NLP tasks from full automation towards decision support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' In  contrast with fully-automated systems, hybrid systems instead involve humans-in-the-loop and facilitate human-AI  teaming (Bansal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Bansal, Wu, Zhou, Fok, Nushi, Kamar, Ribeiro and Weld, 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Bansal, Nushi, Kamar,  Horvitz and Weld, 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Such use of hybrid systems can help a) scale-up human decision making;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' b) augment  machine learning capabilities with human accuracy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' and c) mitigate unintended consequences from machine errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Additionally, we need new benchmarks and evaluation practices that can measure how automated and hybrid systems  can improve downstream human accuracy (Smeros, Castillo and Aberer, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=', 2020) and efficiency in  fact-checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  Acknowledgements  This research was supported in part by the Knight Foundation, the Micron Foundation, Wipro, and by Good  Systems12, a UT Austin Grand Challenge to develop responsible AI technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' The statements made herein are  solely the opinions of the authors and do not reflect the views of the sponsoring agencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
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+page_content='  Atanasova, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
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+page_content=' lab on automatic identification and verification of political claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' task 1: Check-worthiness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content='  arXiv preprint  arXiv:1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
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+page_content='edu/  Anubrata Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 23 of 30  The State of Human-centered NLP Technology for Fact-checking  Atanasova, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
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+page_content=' CLEF (Working Notes) 2380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE1T4oBgHgl3EQfRwNN/content/2301.03056v1.pdf'}
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+Multivariate Regression via Enhanced
+Response Envelope: Envelope Regularization
+and Double Descent
+Oh-Ran Kwon and Hui Zou
+School of Statistics, University of Minnesota
+Abstract
+The envelope model provides substantial efficiency gains over the standard multi-
+variate linear regression by identifying the material part of the response to the model
+and by excluding the immaterial part. In this paper, we propose the enhanced response
+envelope by incorporating a novel envelope regularization term in its formulation. It is
+shown that the enhanced response envelope can yield better prediction risk than the
+original envelope estimator. The enhanced response envelope naturally handles high-
+dimensional data for which the original response envelope is not serviceable without
+necessary remedies. In an asymptotic high-dimensional regime where the ratio of the
+number of predictors over the number of samples converges to a non-zero constant, we
+characterize the risk function and reveal an interesting double descent phenomenon for
+the first time for the envelope model. A simulation study confirms our main theoret-
+ical findings. Simulations and real data applications demonstrate that the enhanced
+response envelope does have significantly improved prediction performance over the
+original envelope method.
+Keywords: Double descent, Envelope model, High-dimension asymptotics, Prediction, Reg-
+ularization
+1
+arXiv:2301.04625v1  [stat.ME]  11 Jan 2023
+
+1
+Introduction
+The envelope model first introduced by Cook et al. (2010) is a modern approach to estimat-
+ing an unknown regression coefficient matrix β ∈ Rr×p in multivariate linear regression of
+the response vector y ∈ Rr on the predictors x ∈ Rp. It was shown by Cook et al. (2010) that
+the envelope estimator of β results in substantial efficiency gains relative to the standard
+maximum likelihood estimator of β. The gains arise by identifying the part of the response
+vector that is material to the regression and by excluding the immaterial part in the estima-
+tion. The original envelope model has been later extended to the envelope model based on
+excluding immaterial parts of the predictors to the regression by Cook et al. (2013). Cook
+et al. (2013) then established the connection between the latter envelope model and partial
+least squares, providing a statistical understanding of partial least squares algorithms.
+The success of the envelope models and their theories motivated some authors to propose
+new envelope models by applying or extending the core idea of envelope modeling to various
+statistical models. The two most common are the response envelope models and the predictor
+envelope models. The response envelope models (predictor envelope models) achieve estima-
+tion and prediction gains by eliminating the variability arising from the immaterial part of
+the responses (predictors) that is invariant to the changes in the predictors (responses). Pa-
+pers on response envelope models include the original envelope model (Cook et al., 2010), the
+partial envelope model (Su and Cook, 2011), the scaled response envelope model (Cook and
+Su, 2013), the reduced-rank envelope model (Cook et al., 2015), the sparse envelope model
+(Su et al., 2016), the Bayesian envelope model (Khare et al., 2017), the tensor response enve-
+lope model (Li and Zhang, 2017), the envelope model for matrix variate regression (Ding and
+Cook, 2018), and the spatial envelope model for spatially correlated data (Rekabdarkolaee
+et al., 2020). Papers on predictor envelope models include the envelope model for predictor
+reduction (Cook et al., 2013), the envelope model for generalized linear models and Cox’s
+proportional hazard model (Cook and Zhang, 2015a), the scaled predictor envelope model
+(Cook and Su, 2016), the envelope quantile regression model (Ding et al., 2020), the envelope
+model for the censored quantile regression (Zhao et al., 2022), tensor envelope partial least
+squares regression (Zhang and Li, 2017), and envelope-based sparse partial least squares
+regression (Zhu and Su, 2020). For a comprehensive review of the envelope models, readers
+2
+
+are referred to Cook (2018).
+High-dimensional data have become common in many fields. It is only natural to consider
+the performance of the envelope model under high dimensions. The likelihood-based method
+to estimate β under both the response/predictor envelope model is not serviceable for high-
+dimensional data because the likelihood-based method requires the inversion of the sample
+covariance matrix of predictors. Hence, one has to find effective ways to mitigate this issue.
+For the predictor envelope model, its connection to partial least squares provides one solution.
+Partial least squares (De Jong, 1993) can be used for estimating β for the predictor envelope
+model (Cook et al., 2013). The partial least squares algorithm is an iterative moment-based
+algorithm involving the sample covariance of predictors and the sample covariance between
+the response vector and predictors, which does not require inversion of the sample covariance
+matrix of predictors. In addition, the algorithm provides the root-n consistent estimator of β
+in the predictor envelope model with the number of predictors fixed (Chun and Kele¸s, 2010;
+Cook et al., 2013) and can yield accurate prediction in the asymptotic high-dimensional
+regime when the response is univariate (Cook and Forzani, 2019). Motivated by this, Zhu and
+Su (2020) introduced envelope-based sparse partial least squares and showed the consistency
+of the estimator for the sparse predictor envelope model. Zhang and Li (2017) proposed a
+tensor envelope partial least squares algorithm, which provides the consistent estimator for
+the tensor predictor envelope model. Another way to apply predictor envelope models for
+high-dimensional data is by selecting the principal components of predictors and then using
+likelihood-based estimation on the principal components. This simple remedy is adapted by
+Rimal et al. (2019) to compare the prediction performance of the likelihood-based predictor
+envelope method, principal component regression, and partial least squares regression for
+high-dimensional data. Their extensive numerical study showed that this simple remedy
+produced better prediction performance than principal component regression and partial
+least squares regression. The impact of high dimensions is more severe for the response
+envelope. There is far less work on making the response envelope model serviceable for high-
+dimensional data. The Bayesian approach for the response envelope model (Khare et al.,
+2017) can handle high-dimensional data. The sparse envelope model (Su et al., 2016) which
+performs variable selection on the responses can handle data with the sample size smaller
+3
+
+than the number of responses, but still requires the number of predictors smaller than the
+number of sample size.
+In this paper, we propose the enhanced response envelope for high-dimensional data by
+incorporating a novel envelope regularization term in its formulation. The envelope regu-
+larization term respects the fundamental idea of the original envelope model by considering
+the presence of the material and immaterial parts of the response in the model. The en-
+hancements are twofold. First, our enhanced response envelope estimator can handle both
+low- and high-dimensional data, while the original envelope estimator can only handle low-
+dimensional data where the sample size n is smaller than the number of predictors p. From
+the connection between the original envelope estimator and the enhanced response envelope
+estimator in low-dimension, we extend the definition of the original envelope estimator to
+high-dimensional data by considering the limiting case of the enhanced response envelope es-
+timator with a vanishing regularization parameter; see the discussion in Section 2.3. Second,
+we prove that the enhanced response envelope can reduce the prediction risk relative to the
+original envelope for all values of n and p. Moreover, we study the asymptotics of the predic-
+tion risk for the original envelope estimator and the enhanced response envelope estimator
+when both n, p → ∞ and their ratio converges to a nonzero constant p/n → γ ∈ (0, ∞).
+This kind of asymptotic regime has been considered in high-dimensional machine learning
+theory (El Karoui, 2018; Dobriban and Wager, 2018; Liang and Rakhlin, 2020; Hastie et al.,
+2022) for analyzing the behavior of prediction risk of certain predictive models. We derive an
+interesting asymptotic prediction risk curve for the envelope estimator. The risk increases as
+γ increases, and then decreases after γ > 1. This phenomenon is known as the double descent
+phenomenon in the machine learning literature. Although the double descent phenomenon
+has been observed for neural networks and ridgeless regression (Belkin et al., 2019; Hastie
+et al., 2022), this is the first time that such a phenomenon is shown for the envelope models.
+The rest of the paper is organized as follows. We review the original envelope model
+and the corresponding envelope estimator in Section 2.1. In Section 2.2, we introduce a
+new regularization term called the envelope regularization based on which we propose the
+enhanced response envelope in section 2.3. The enhanced response envelope estimator nat-
+urally provides a definition for the envelope estimator when p > n. Section 2.4 describes
+4
+
+how to implement this new method in practice. In Section 3.1, we prove that the enhanced
+response envelope can yield better prediction risk than the original envelope for any (n, p)
+pair. Considering n, p → ∞ and p/n → γ ∈ (0, ∞), we derive the limiting prediction risk
+result of the original envelope and the enhanced response envelope in Section 3.2. This result
+along with our simulation study in Section 4 verify the double descent phenomenon. Real
+data analyses are presented in Section 5. Proofs of theorems are provided in Appendix A.
+2
+Enhanced response envelope
+2.1
+Review of envelope model
+Envelope model
+Let us begin with the classical multivariate linear regression model of a
+response vector y ∈ Rr given a predictor vector x ∈ Rp:
+y = βx + ε, ε ∼ N(0, Σ),
+(1)
+where ε is the error vector with a positive definite Σ and independent to x. β ∈ Rr×p is
+an unknown matrix of regression coefficients and x ∼ Px where Px is a distribution on Rp
+such that E(x) = 0 and Cov(x) = Σx. We omit an intercept by assuming E(y) = 0 for easy
+communication.
+The envelope model allows for the possibility that there is a part of the response vector
+that is unaffected by changes in the predictor vector. Specifically, let E ⊆ Rr be a subspace
+such that for all x1 and x2,
+(i) QEy|(x = x1) ∼ QEy|(x = x2) and (ii) PEy ⊥⊥ QEy|x,
+(2)
+where PE is the projection onto E and QE = I − PE. Condition (i) states that the marginal
+distribution of QEy is invariant to changes in x. Condition (ii) says that QEy does not
+affect PEy if x is provided. Conditions together imply that PE includes the relevant depen-
+dency information of y on x (the material part) while QE is the irrelevant information (the
+immaterial part).
+Let B = span(β). The conditions in (2) hold if and only if
+span(β) = B ⊆ E and Σ = PEΣPE + QEΣQE.
+(3)
+5
+
+The definition of an envelope introduced by Cook et al. (2007, 2010) formalizes the smallest
+subspace satisfying the conditions in (2) using the equivalence relation of (2) and (3). The
+envelope is defined as the intersection of all subspaces E satisfying (3) and is denoted by
+EΣ,B, Σ-envelope of B.
+The envelope model arises by parameterizing the multivariate linear model in terms of
+the envelope EΣ,B. The parameterization is as follows. Let u = dim(EΣ,B), Γ ∈ Rr×u be any
+semi-orthogonal basis matrix for EΣ,B, and Γ0 ∈ Rr×(r−u) is any semi-orthogonal basis matrix
+for the orthogonal complement of EΣ,B. Then the multivariate linear model can be written
+as
+y = Γηx + ε, ε ∼ N(0, ΓΩΓT + Γ0Ω0ΓT
+0 ),
+(4)
+where β = Γη with η ∈ Ru×p, and Ω ∈ Rr×r and Ω0 ∈ R(r−u)×(r−u) are symmetric positive
+definite matrices. Model (4) is called the envelope model.
+Envelope estimator The parameters in the envelope model are estimated by maximizing
+the likelihood function from model (4). Assume that p+r < n and u is the dimension u of the
+envelope. SX = n−1XTX, SY = n−1YTY, SY,X = n−1YTX, and SY|X = SY−SY,XS−1
+X SX,Y,
+where Y ∈ Rn×r has rows yT
+i and X ∈ Rn×p has rows xT
+i .
+The envelope estimator of β is determined as
+ˆEΣ,B = span{arg
+min
+G∈Gr(r,u)(log |GTSY|XG| + log |GTS−1
+Y G|)},
+(5)
+where Gr(r, u) = {G ∈ Rr×u : G is a semi-orthogonal matrix}. Define ˆΓ as any semi-
+orthogonal basis matrix for ˆEΣ,B and let ˆΓ0 be any semi-orthogonal basis matrix for the
+orthogonal complement of ˆEΣ,B. The estimator of β is given by
+ˆβ = ˆΓˆΓTSY,XS−1
+X ,
+(6)
+and Σ is estimated by ˆΣ = ˆΓ ˆΩˆΓ + ˆΓT
+0 ˆΩ0ˆΓ0 where
+ˆΩ = ˆΓTSY|XˆΓ,
+ˆΩ0 = ˆΓT
+0 SY ˆΓ0,
+(7)
+2.2
+Envelope regularization
+In this section, we introduce the envelope regularization term that respects the fundamental
+idea in the envelope model by considering the presence of material and immaterial parts,
+6
+
+PEΣ,By and QEΣ,By, in the regression.
+We define the envelope regularization term as
+ρ(η, Ω) = tr(ηTΩ−1η).
+(8)
+The envelope model distinguishes between PEΣ,By and QEΣ,By in the estimation process.
+The log-likelihood function of the envelope model is decomposed into two log-likelihood
+functions. One is the log-likelihood function for the multivariate regression of ΓTy on x,
+ΓTy = ηx+ΓTε where ΓTε ∼ N(0, Ω). The other is the log-likelihood function for the zero-
+mean model of ΓT
+0 y, ΓT
+0 y = ΓT
+0 ε where ΓT
+0 ε ∼ N(0, Ω0). The envelope regularization term
+(8) is the function of η and Ω, the parameters in the likelihood for the material part of the
+envelope model. The envelope regularization term (8) can be seen as imposing the Frobenius
+norm regularization on the coefficient after standardizing the material part of the regression
+to have uncorrelated errors, Ω−1/2ΓTy = Ω−1/2ηx + Ω−1/2ΓTε where Ω−1/2ΓTε ∼ N(0, I).
+We emphasize that the envelope regularization is different from the ridge regularization.
+While the ridge regularization ∥β∥2
+F is the quadratic function of β, the envelope regulariza-
+tion is not because the components of Ω are not fixed values. The envelope regularization is
+the function of both η and Ω, and thus is optimized over η and Ω simultaneously, as shown
+in the next subsection.
+2.3
+The proposed estimator
+We only assume that r ≤ n but p is allowed to be bigger than n. The log-likelihood function
+under the envelope model (4) is
+Lu(η, EΣ,B, Ω, Ω0) = − (nr/2) log(2π) − (n/2) log |ΓΩΓT + Γ0Ω0ΓT
+0 |
+− (1/2)
+n
+�
+i=1
+(yi − Γηxi)T(ΓΩΓT + Γ0Ω0ΓT
+0 )−1(yi − Γηxi).
+By incorporating the envelope regularization term ρ given in the last subsection, we propose
+the following enhanced response envelope estimator via penalized maximum likelihood:
+arg max{Lu(η, EΣ,B, Ω, Ω0) − n
+2λ · ρ(η, Ω)},
+(9)
+where λ > 0 serves as a regularization parameter.
+7
+
+Let SX = n−1XTX, SY = n−1YTY, SY,X = n−1YTX, Sλ
+X = SX + λI and Sλ
+Y|X =
+SY − SY,X(Sλ
+X)−1SX,Y. After some basic calculations, (9) can be expressed as
+ˆEΣ,B(λ) = span{arg
+min
+G∈Gr(r,u)(log |GTSλ
+Y|XG| + log |GTS−1
+Y G|)},
+(10)
+where Gr(r, u) = {G ∈ Rr×u : G is a semi-orthogonal matrix}. Let ˆΓλ be any semi-orthogonal
+basis matrix for ˆEΣ,B(λ) and ˆΓ0,λ be any semi-orthogonal basis matrix for the orthogonal
+complement of ˆEΣ,B(λ). The enhanced envelope estimator of β is given by
+ˆβ(λ) = ˆΓλˆΓT
+λSY,X(Sλ
+X)−1
+(11)
+and Σ is estimated by ˆΣ(λ) = ˆΓλ ˆΩ(λ)ˆΓλ + ˆΓT
+0,λ ˆΩ0(λ)ˆΓ0,λ where
+ˆΩ(λ) = ˆΓT
+λSλ
+Y|XˆΓλ,
+ˆΩ0(λ) = ˆΓT
+0,λSY ˆΓ0,λ,
+(12)
+The enhanced response envelope estimator can naturally handle the case where p ≥ n−r,
+while the original envelope estimator (5) does not. Motivated by the definition of ridgeless
+regression (Hastie et al., 2022), we can consider taking the limit of the enhanced response
+envelope estimator with λ → 0+:
+ˆEΣ,B = span{arg
+min
+G∈Gr(r,u)( lim
+λ→0+ log |GTSλ
+Y|XG| + log |GTS−1
+Y G|)},
+ˆβ = lim
+λ→0+ ˆβ(λ)
+(13)
+We take (13) as the definition of envelope estimator. Obviously, when p < n−r, this extended
+definition recovers the original envelope estimator (5). This definition enables the use of the
+envelope estimator when p ≥ n−r, without altering the definition of the original envelope
+estimator (5) when p < n−r. In practice, we implement (13) by computing the enhanced
+response envelope estimator (10) with a very small value of λ such as 10−8.
+As the enhanced response envelope estimator (9) has flexibility on λ, the enhanced re-
+sponse envelope estimator with an appropriate choice of λ can yield better prediction risk
+compared to the envelope estimator, which is discussed in Section 3. We discuss the Grass-
+mannian manifold optimization required in (10) in the next subsection.
+2.4
+Implementation
+Suppose that the dimension u is specified and λ is given. Our proposed estimator ˆEΣ,B(λ)
+for EΣ(B) requires the optimization over the Grassmannian G(u, r). Such a computation
+8
+
+problem exists for the original envelope model as well. So far, the best-known algorithm for
+solving envelope models is the algorithm introduced by Cook et al. (2016). Thus, we employ
+their algorithm to compute ˆEΣ,B(λ) in (10). Note that we standardize X so that each column
+has a mean of 0 and a standard deviation of 1 before fitting any model.
+In practice, the tuning parameter λ and the dimension u of the envelope are unknown. We
+use the cross-validation method to choose (u, λ). For the original envelope, u can be selected
+by using AIC, BIC, LRT or cross-validation. BIC and LRT may be preferred as shown by
+simulations in Su and Cook (2013). Because the enhanced response envelope model has an
+additional tuning parameter λ, we propose to use cross-validation to find the best tuning
+parameter combination of u and λ.
+We have implemented the enhanced response envelope method in R and the code is
+available upon request.
+3
+Theory
+In this section, we show that the enhanced response envelope can reduce the prediction risk
+over the envelope for any (n, p) pair. We then consider the asymptotic setting when n, p → ∞
+p/n → γ ∈ (0, ∞). This asymptotic regime has been considered in the literature (El Karoui,
+2018; Dobriban and Wager, 2018; Liang and Rakhlin, 2020; Hastie et al., 2022) for analyzing
+the behavior of prediction risk of certain predictive models.
+In our discussion, we consider the case where EΣ(B) is known, which has been assumed
+in the existing envelope papers to understand the core mechanism of envelope methodologies
+(Cook et al., 2013; Cook and Zhang, 2015a,b).
+3.1
+Reduction in prediction risk
+Consider a test point xnew ∼ Px. For an estimator ˆβ, we define the prediction risk as
+R( ˆβ|X) = E[∥ ˆβxnew − βxnew∥2|X].
+Note that this definition has the bias-variance decomposition,
+R( ˆβ|X) = ∥bias(vec( ˆβ)|X)∥2 + tr{Var(vec( ˆβ)|X)}.
+9
+
+Let Γ be a semi-orthogonal basis matrix for EΣ,B. Following the discussion in Section 2.3,
+we take (13) as the definition of the envelope estimator ˆβΓ . The prediction risk of ˆβΓ is
+R( ˆβΓ|X) = vecT(β)[ΠXΣxΠX ⊗ Ir]vec(β)
+�
+��
+�
+bias2
++ tr(Ω)
+n
+tr(S+
+XΣx)
+�
+��
+�
+var
+,
+where ΠX = Ip − S+
+XSX.
+The prediction risk of the enhanced response envelope estimator ˆβΓ(λ) is
+R( ˆβΓ(λ)|X) = E[∥ ˆβΓ(λ)xnew − βxnew∥2|X]
+= λ2vecT(β)[(SX + λI)−1Σx(SX + λI)−1 ⊗ Ir]vec(β)
+�
+��
+�
+bias2
++ tr(Ω)
+n
+tr(ΣxSX(SX + λI)−2)
+�
+��
+�
+var
+.
+(14)
+Theorem 1 shows that using the envelope regularization always improves the prediction
+risk of the envelope model.
+Theorem 1. There always exists a λ > 0 such that R( ˆβΓ(λ)|X) < R( ˆβΓ|X).
+3.2
+Limiting prediction risk and double descent phenomenon
+The asymptotics of the envelope model are well-established in the case where n diverges
+while p is fixed (Cook et al., 2010), while not in a high-dimensional asymptotic setup. In
+this section, we examine the limiting risk of both the enhanced response envelope estimator
+and the envelope estimator in the high-dimensional asymptotic regime where n, p → ∞ with
+p/n → γ ∈ (0, ∞). The number of response variables r is fixed. This kind of asymptotic
+regime has been considered in high-dimensional machine learning theory (El Karoui, 2018;
+Dobriban and Wager, 2018; Liang and Rakhlin, 2020; Hastie et al., 2022) for analyzing the
+behavior of prediction risk of certain predictive models.
+Let x = Σ1/2
+x x∗, where E(x∗) = 0 and Cov(x∗) = Ip. Then the envelope model (4) of y
+on x can be expressed as the envelope model of y on x∗:
+y = Γηx + ε = Γη∗x∗ + ε,
+where η∗ = ηΣ1/2 and ε ∼ N(0, ΓΩΓT + Γ0Ω0ΓT
+0 ). We take advantage of the invariance
+property of the envelope model in the analysis. Considering the envelope on (y, x∗) amounts
+to assuming the covariance of the predictor is Ip.
+10
+
+0
+2
+4
+6
+8
+0
+5
+10
+15
+20
+25
+γ
+Limiting prediction risk
+Envelope
+Enhanced response envelope
+Figure 1: The limiting prediction risks of the enhanced response envelope with λ∗ =
+tr(Ω)γ/c2 (gray solid line) and the envelope (black solid line), illustrating Theorem 2 when
+tr(Ω) = 10 and tr(βTβ) = 10.
+Theorem 2. Assume that x has a bounded 4th moment and that tr(ηTη) = c2 for all n, p.
+Then as n, p → ∞, such that p/n → γ ∈ (0, ∞), almost surely,
+R( ˆβΓ|X) →
+�
+�
+�
+�
+�
+tr(Ω)
+γ
+1−γ
+for γ < 1
+c2(1 − 1
+γ) + tr(Ω)
+1
+γ−1
+for γ > 1,
+and
+R( ˆβΓ(λ∗)|X) → tr(Ω)γm(−λ∗),
+where λ∗ = tr(Ω)γ/c2 and m(z) =
+1−γ−z−√
+(1−γ−z)2−4γz
+(2γz)
+.
+Figure 1 visualizes the limiting prediction risk curves in Theorem 2. It plots the limiting
+risks of envelope (black solid line) and the enhanced response envelope with λ∗ = tr(Ω)γ/c2
+(dark-gray solid line), when tr(Ω) = 10 and tr(ηTη) = 10.
+We have four remarks from Theorem 2. The limiting risk of envelope increases before
+γ = 1 and then decreases after γ = 1. The double descent phenomenon has been observed in
+popular methods such as neural networks, kernel machines and ridgeless regression (Belkin
+et al., 2019; Hastie et al., 2022), but this is the first time that such a result is established
+11
+
+in the envelope literature. Second, the enhanced response envelope estimator always has a
+better asymptotic prediction risk than the envelope estimator (for any c2, tr(Ω), and γ).
+Third, in Theorem 1, we show the existence of a λ that gives a smaller prediction risk of
+the enhanced response envelope than the envelope estimator. In an asymptotic regime, we
+specify such a λ value: λ∗ = tr(Ω)γ/c2. Lastly, the gap between two limiting prediction risks,
+limn,p→∞ R( ˆβΓ|X) and n,p→∞R( ˆβΓ(λ∗)|X), increases as γ increases from 0 to 1. It is easy to
+see as
+1
+1−γ > m(−λ∗), 0 < γ < 1.
+4
+Simulation
+In this section, we use simulations to compare the performance of the enhanced response
+envelope estimator and the envelope estimator in terms of the prediction risk, E[∥ ˆβxnew −
+βxnew∥2|X] = tr[( ˆβ − β)Cov(xnew)( ˆβ − β)T]. In addition, we use simulations to have a
+numeric illustration of the double descent phenomenon to confirm the asymptotic theory.
+We consider a setting where yi ∈ R3 is generated from the model
+yi = βxi + εi, εi ∼ N(0, Σ), i = 1, . . . , n,
+and xi ∈ Rp is generated independently from xi ∼ N(0, Σx(ρ)) where (i, j)th element of
+Σx(ρ) ∈ Rp×p is ρ|i−j|. The covariance matrix Σ is set using three orthonormal vectors and
+has eigenvalues 10, 8 and 2. The columns of Γ are the second and third eigenvectors of Σ.
+Each component of ˜η ∈ R2×p is generated from the standard normal distribution. We then
+set η =
+√
+10 · ˜η/∥˜η∥F. In this setting, tr(ηTη) = 10, tr(Ω) = 10, and tr(Ω0) = 10. We
+assume that dim(EΣ,B) = 2 is known.
+Prediction risk comparison
+In this simulation, we try different combinations of n, p
+and ρ where n ∈ {50, 90, 200, 500}, p/n ∈ {0.1, 0.8, 1.2} and ρ ∈ {0, 0.8}. We compare the
+prediction risk of the enhanced response envelope estimator to three different estimators: the
+envelope estimator, multivariate linear regression, and multivariate ridge regression.
+For the enhanced response envelope and the multivariate ridge regression, we perform
+ten-fold cross-validation on simulated data to select λ among equally spaced 100 candidate
+λ-values in the scale of logarithm base 10. We compute the envelope estimator for data
+12
+
+with n ≤ p−r by taking a very small value of λ = 10−8 in the enhanced response envelope
+estimator. We fit multivariate regression model to n < p data by taking a tiny value of
+λ = 10−8 in the multivariate ridge regression. We then calculate the prediction risk. This
+process is repeated 100 times.
+In Table 1, we report the prediction risk averaged over 100 replications. First, we see that
+the prediction risks from the enhanced response envelope are consistently smaller than the
+envelope, as indicated in Theorem 1. Second, the enhanced response envelope consistently
+gives smaller prediction risks compared to the multivariate ridge regression. When u = r, the
+enhanced response envelope model reduces to the multivariate ridge regression. Therefore,
+the prediction risk of the enhanced envelope model can be smaller than that of multivariate
+ridge regression as long as tr(Ω0) > 0.
+Double descent confirmation
+This simulation is designed to support Theorem 2 and
+to illustrate the double descent phenomenon in the envelope model. We set n ∈ {200, 2000}
+and ρ = 0. p/n varies from 0.1 to 8. We compute the envelope and the enhanced response
+envelope with setting λ∗ = tr(Ω)p/(nc2) = p/n on simulated data. We then calculate the
+prediction risk for each estimator. Again, we fit n ≤ p−r data to the envelope estimator by
+taking a very small value of λ = 10−8 in the enhanced response envelope estimator.
+Figure 2 displays the prediction risks from n = 2000 with various p values. The gray
+rectangle points denote the prediction risk for the enhanced response envelope estimator. The
+black triangle points are the prediction risk for the envelope estimator. We see a fascinating
+double descent prediction risk curve for the envelope model, as discussed in Theorem 2. Also,
+the enhanced response envelope gives a smaller prediction risk across the entire range of p/n.
+Figure 3 plots the prediction risk curves from n = 200. We see that Figure 3 exhibits the
+same messages for the much smaller sample size. Although Theorem 2 is established when
+considering EΣ,B is known, we did not use this information in the actual estimation in the
+simulation study, yet the core message of Theorem 2 is confirmed by the simulation.
+13
+
+n
+p
+Enhanced
+envelope
+Envelope
+Multivariate
+linear reg
+Multivariate
+ridge reg
+Example 1: p/n = 0.1, ρ = 0
+50
+5
+1.31 (0.11)
+1.40 (0.12)
+2.39 (0.17)
+2.04 (0.12)
+90
+9
+1.24 (0.08)
+1.41 (0.10)
+2.33 (0.13)
+1.92 (0.09)
+200
+20
+1.16 (0.04)
+1.26 (0.05)
+2.31 (0.05)
+1.93 (0.04)
+500
+50
+1.06 (0.03)
+1.18 (0.03)
+2.28 (0.04)
+1.85 (0.04)
+Example 2: p/n = 0.8, ρ = 0
+50
+40
+6.73 (0.18)
+60.89 (5.80)
+104.45 (7.09)
+7.16 (0.11)
+90
+72
+6.44 (0.14)
+55.10 (2.93)
+94.81 (3.24)
+7.05 (0.08)
+200
+160
+5.86 (0.10)
+42.50 (0.99)
+81.33 (1.63)
+6.91 (0.06)
+500
+400
+5.67 (0.04)
+40.61 (0.85)
+79.17 (1.11)
+6.89 (0.03)
+Example 3: p/n = 1.2, ρ = 0
+50
+60
+8.02 (0.23)
+33.70 (1.33)
+93.79 (3.83)
+8.08 (0.11)
+90
+108
+7.58 (0.13)
+41.01 (1.36)
+94.38 (3.60)
+7.98 (0.07)
+200
+240
+7.02 (0.07)
+47.91 (1.18)
+99.94 (2.83)
+7.82 (0.04)
+500
+600
+6.78 (0.04)
+50.43 (0.91)
+103.33 (1.55)
+7.75 (0.03)
+Example 4: p/n = 0.1, ρ = 0.8
+50
+5
+1.76 (0.11)
+1.98 (0.19)
+2.39 (0.17)
+1.84 (0.07)
+90
+9
+1.02 (0.05)
+1.40 (0.08)
+2.33 (0.13)
+1.45 (0.06)
+200
+20
+0.90 (0.03)
+1.30 (0.04)
+2.31 (0.05)
+1.31 (0.03)
+500
+50
+0.78 (0.02)
+1.19 (0.03)
+2.28 (0.04)
+1.22 (0.02)
+Example 5: p/n = 0.8, ρ = 0.8
+50
+40
+4.16 (0.17)
+62.50 (6.34)
+104.45 (7.09)
+4.76 (0.12)
+90
+72
+3.78 (0.15)
+55.14 (2.85)
+94.81 (3.24)
+4.63 (0.10)
+200
+160
+3.32 (0.05)
+42.40 (0.99)
+81.33 (1.63)
+4.28 (0.05)
+500
+400
+3.09 (0.03)
+40.69 (0.87)
+79.17 (1.11)
+4.05 (0.03)
+Example 6: p/n = 1.2, ρ = 0.8
+50
+60
+5.24 (0.23)
+36.43 (1.12)
+104.17 (4.37)
+5.80 (0.14)
+90
+108
+4.41 (0.12)
+44.84 (1.68)
+103.01 (4.03)
+5.16 (0.09)
+200
+240
+4.05 (0.07)
+51.46 (1.30)
+109.34 (3.17)
+4.98 (0.06)
+500
+600
+3.86 (0.03)
+54.21 (0.98)
+112.62 (1.71)
+4.82 (0.03)
+Table 1: Prediction risk, averaged over 100 replications. The standard error is given in paren-
+theses. For n ≤ p−r data, we compute the envelope by taking a very small value of λ = 10−8
+in the enhanced response envelope; see the definition of the envelope estimator (13) in Sec-
+tion 2.3. For n < p data, we fit the multivariate regression model by taking a tiny value of
+λ = 10−8 in the multivariate ridge regression.
+14
+
+0
+2
+4
+6
+8
+0
+5
+10
+15
+20
+25
+p/n
+Prediction risk
+Envelope
+Enhanced response envelope
+Figure 2: Prediction risk of the envelope and the enhanced response envelope with λ∗ =
+tr(Ω)p/(nc2), when n = 2000 and p varies. For n ≤ p−r data, we fit the envelope by taking
+a very small value of λ = 10−8 in the enhanced response envelope estimator; see the definition
+of the envelope estimator (13) in Section 2.3.
+5
+Real data
+In this section, we use two real datasets to illustrate the enhanced response envelope esti-
+mator. We use air pollution data in which the number of samples is bigger than the number
+of predictors (n > p) in the next subsection. In Subsection 5.2, we analyze near-infrared
+spectroscopy data in which the number of predictors is much bigger than the number of
+predictors (p ≫ n).
+We compare the prediction performance of the enhanced response envelope estimator to
+the envelope estimator, multivariate regression, and multivariate ridge regression.
+5.1
+Air pollution data
+The air pollution data are available and obtained directly from Table 1.5 of Johnson et al.
+(2002). The response vector y ∈ R5 consists of atmospheric concentrations of CO, NO, NO2,
+O3, and HC, recorded at noon in the Los Angeles area on 42 different days. The two predictors
+15
+
+0
+2
+4
+6
+8
+0
+5
+10
+15
+20
+25
+p/n
+Prediction risk
+Envelope
+Enhanced response envelope
+Figure 3: Prediction risk of the envelope and the enhanced response envelope with λ∗ =
+tr(Ω)p/(nc2), when n = 200 and p varies. For n ≤ p−r data, we fit the envelope by taking a
+very small value of λ = 10−8 in the enhanced response envelope estimator; see the definition
+of the envelope estimator (13) in Section 2.3.
+are wind speed and solar radiation. This data were analyzed in Cook (2018) to illustrate the
+effectiveness of the original envelope model compared to the standard multivariate regression
+model. They showed that the asymptotic standard errors of estimated components of β
+from the envelope model are significantly reduced compared to those from the standard
+multivariate regression model. We use the data to predict atmospheric concentrations from
+wind speed and solar radiation and compare the prediction performance of the enhanced
+response envelope estimator to the envelope estimator, the standard multivariate regression,
+and multivariate ridge regression.
+To compare the prediction performance, we borrow the nested cross validation idea
+(Wang and Zou, 2021; Bates et al., 2021), in which an inner cross-validation is performed
+to tune a model and an outer cross-validation is performed to provide a prediction error of
+the tuned model. We adopt the leave-one-out cross-validation (LOOCV) procedure for the
+outer loop because the LOOCV error is an unbiased estimator of the generalization error
+of the tuned model and is shown to have nice performance compared to other methods for
+16
+
+Enhanced
+envelope
+Envelope
+Multivariate
+linear reg
+Multivariate
+ridge reg
+Error
+8.859
+8.951
+9.192
+9.124
+Table 2: Air pollution data: prediction error of the enhanced response envelope method,
+the original envelope method, the multivariate linear regression, and the multivariate ridge
+regression.
+estimating generalization errors (Wang and Zou, 2021).
+We take the ith observation out from the data and set the remaining n−1 observations
+as the training set to fit and tune models. We standardize X of the training set so that each
+column has a mean of 0 and a standard deviation of 1. We perform ten-fold cross-validation
+to select (u, λ) from a fine grid of u ∈ {0, . . . , 5} and 20 equally spaced candidate λ-values
+in the scale of logarithm base 10 for the enhanced response envelope. For the envelope,
+we perform ten-fold cross-validation to choose u from {0, . . . , 5}. For the multivariate ridge
+model, ten-fold cross-validation is performed to select λ from 20 equally spaced λ-values in
+the scale of logarithm base 10. The ith observation we take out at the beginning is set as
+the test set. We standardize xi of the test set using the mean and standard deviation of the
+training data. We then calculate the squared prediction error, ∥yi − ˆβ(−i)xi∥2
+2/r, where ˆβ(−i)
+is the estimated regression coefficient derived from the training set. We repeat this process
+for i = 1, . . . , n and report �n
+i=1 ∥yi − ˆβ(−i)xi∥2
+2/(nr) in Table 2. We see that the enhanced
+response envelope estimator gives the smallest prediction error among all competitors.
+5.2
+Near-infrared spectroscopy data of fresh cattle manure
+Near-infrared spectroscopy data of cattle manure were collected by Gog´e et al. (2021). The
+data are available in the Data INRAE Repository at https://doi.org/10.15454/JIGO8R. This
+data contain 73 cattle manure samples that were analyzed by near-infrared spectroscopy
+using a NIRFlex device. Near-infrared spectra were recorded every 2 nm from 1100 to 2498
+nm on fresh homogenized samples. In addition, the cattle manure samples were analyzed
+for three chemical properties: the amount of dry matter, magnesium oxide, and potassium
+17
+
+Enhanced
+envelope
+Envelope
+Multivariate
+linear reg
+Multivariate
+ridge reg
+Error
+0.437
+0.460
+0.692
+0.492
+Table 3: Near-infrared spectroscopy data: prediction error from the enhanced response enve-
+lope method, the envelope method, the multivariate linear regression, and the multivariate
+ridge regression. We compute the envelope estimator by taking a very small value of λ = 10−8
+in the enhanced response envelope estimator; see the definition of the envelope estimator
+(13) in Section 2.3. We fit the multivariate regression model by taking a very small value of
+λ = 10−8 in the multivariate ridge regression.
+oxide. We use the data of 62 cattle manure samples which have no missing values. We
+standardize each chemical property to have a sample mean of 0 and a standard deviation of
+1. In our analysis, we consider the multivariate linear model, where xi ∈ R700 is the vector
+of near-infrared spectroscopy measurements and yi ∈ R3 is the vector of three chemical
+measurements to predict the three chemical properties from the absorbance spectra.
+In Table 3, we report the prediction error which is calculated using the same procedure
+described in the previous subsection, except that u is chosen from {0, . . . , 3}. Again, We see
+that the enhanced response envelope estimator has the smallest prediction error among all
+competitors.
+6
+Discussion
+In this paper, we have developed a novel envelope regularization function which is used to
+define the enhanced envelope estimator. We have shown that the enhanced envelope estimator
+is indeed better than the un-regularized envelope estimator in prediction. The asymptotic
+analysis of the risk function of envelope reveals, for the first time in the envelope literature,
+an interesting double descent phenomenon. The numeric examples in this work also suggest
+that the enhanced response envelope estimator is a promising new tool for multivariate
+regression.
+18
+
+Although this paper is focused on the case where the number of responses (r) is less
+than the number of samples and the number of predictors, it is interesting to consider the
+case when r → ∞ in ultrahigh-dimensional problems. Su et al. (2016) studied the response
+envelope for r → ∞ but p is fixed. When both p, r > n and diverge, there are additional
+technical issues to be addressed. For example, we may need another penalty term to handle
+the issues caused by the large r in the model. This direction of research will be investigated
+in a separate paper.
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+Statistics, 48, 161–182.
+A
+Proofs of Theorems
+A.1
+Proof of Theorem 1
+Note that
+R( ˆβΓ(λ)|X) = λ2tr(β(SX + λI)−1Σx(SX + λI)−1βT) + tr(Ω)
+n
+tr(ΣxSX(SX + λI)−2).
+Therefore, we have
+∂
+∂λR( ˆβΓ(λ)|X)
+= 2λ · tr(βSX(SX + λI)−2Σx(SX + λI)−1βT) − 2tr(Ω)
+n
+tr(ΣxSX(SX + λI)−3)
+≤
+p
+�
+i=1
+�
+2λ · σi(βTβ) − 2tr(Ω)
+n
+�
+σi(ΣxSX(SX + λI)−3),
+where σi(M) denotes the i-th largest eigenvalue of M. The inequality above comes from Von
+Neumann’s trace inequality.
+22
+
+Since
+∂
+∂λR( ˆβΓ(λ)|X) < 0 if λ < tr(Ω)/(nσ1
+�
+βTβ)
+�
+, R( ˆβΓ(λ)|X) is a monotonically
+decreasing function if 0 ≤ λ ≤ tr(Ω)/(nσ1
+�
+βTβ)
+�
+. Therefore, we have
+R( ˆβΓ(λ)|X) < tr(Ω)
+n
+tr(ΣxS+
+X),
+when 0 < λ < tr(Ω)/(nσ1
+�
+βTβ)
+�
+. Since
+tr(Ω)
+n
+tr(ΣxS+
+X) ≤ R( ˆβΓ|X),
+we prove the theorem.
+A.2
+Proof of Theorem 2
+Our analyses of limiting prediction risk follow that of Hastie et al. (2022).
+As Σx = I,
+R( ˆβΓ|X) = vecT(β)[ΠX ⊗ Ir]vec(β) + tr(Ω)
+n
+tr(S+
+X),
+R( ˆβΓ(λ)|X) = λ2tr(β(SX + λI)−2βT) + tr(Ω)
+n
+tr(SX(SX + λI)−2),
+where ΠX = Ip − S+
+XSX.
+A.2.1
+Proof for envelope estimator when γ < 1
+Let us consider the case where p/n → γ ∈ (0, 1). From Theorem 1 of Bai and Yin (2008),
+σmin(SX) ≥ (1 − √γ)2/2 and σmax(SX) ≤ 2(1 + √γ)2 almost surely for all sufficiently large
+n. Therefore, in this case, SX is invertible and the bias term of R( ˆβΓ|X) is 0, almost surely.
+The variance term of R( ˆβΓ|X) is
+tr(Ω)
+n
+tr(S+
+X) = p · tr(Ω)
+n
+� 1
+sdFSX(s),
+where FSX(s) is the spectral measure of SX. By the Marchenko-Pastur theorem, which says
+that FSX → Fγ, and the Portmanteau theorem,
+� 2(1+√γ)2/
+(1−√γ)2/2
+1
+sdFSX(s) →
+� 2(1+√γ)2/
+(1−√γ)2/2
+1
+sdFγ(s) =
+� 1
+sdFγ(s).
+23
+
+The equality is because the support of Fγ is [(1 − √γ)2, (1 + √γ)2]. We can also remove the
+upper and lower limits of integration on the left-hand side by Theorem 1 of Bai and Yin
+(2008). Thus, combining above results, we arrive at
+R( ˆβΓ|X) → γ · tr(Ω)
+� 1
+sdFγ(s).
+The Stieltjes transformation of Fγ is given by
+m(z) =
+�
+1
+s − zdFγ(s) = (1 − γ − z) −
+�
+(1 − γ − z)2 − 4γz)
+2γz
+,
+for any real z < 0. By taking the limit z → 0−, the proof is completed.
+A.2.2
+Proof for envelope estimator when γ > 1
+The variance term of R( ˆβΓ|X) is
+tr(Ω)
+n
+tr(S+
+X) = tr(Ω)
+n
+tr((XXT/n)+) = tr(Ω)
+p
+tr((XXT/p)+).
+Considering n/p → τ = 1/γ < 1, by the same arguments from the proof above, we conclude
+that
+tr(Ω)
+n
+tr(S+
+X) → tr(Ω)
+1
+γ − 1.
+Let β = [bT
+1 . . . bT
+r ]. The bias term is
+vecT(β)[ΠX ⊗ Ir]vec(β) =
+r
+�
+i=1
+bT
+i ΠXbi =
+r
+�
+i=1
+lim
+z→0+ zbT
+i (SX + zI)−1bi.
+From Theorem 1 of Bai et al. (2007), we have that
+zbT
+i (SX + zI)−1bi → z
+�
+1
+s + zFγ(s) = z∥bi∥2m(−z) a.s.,
+for any i = 1, . . . , r. We further have that
+r
+�
+i=1
+zbT
+i (SX + zI)−1bi → zc2m(−z) a.s.
+By the Arzela-Ascoli theorem and the Moore-Osgood theorem, we exchange limits and
+arrive at
+lim
+z→0+
+r
+�
+i=1
+zbT
+i (SX + zI)−1bi → c2 lim
+z→0+ zm(−z) = c2(1 − 1/γ) a.s.
+Combining the variance and the bias terms, we complete the proof.
+24
+
+A.2.3
+Proof for enhanced envelope estimator
+We use the similar techniques from the envelope estimator for both variance and bias terms.
+The variance term of R( ˆβΓ(λ)) becomes
+tr(Ω)
+n
+tr(SX(SX + λI)−2) → γtr(Ω)
+�
+s
+(s + λ)2Fγ(s).
+Let gn,λ(η) = λ · tr(β(SX + λ(1 + η)I)−1βT), η ∈ [−1/2, 1/2]. The bias term of R( ˆβΓ(λ))
+is
+λ2tr(β(SX + λI)−2βT) = − ∂
+∂ηgn(λ, 0).
+Because
+gn,λ(η) → λc2m(−λ(1 + η)) = λc2
+�
+1
+s + λ(1 + η)dFγ(s),
+and derivative and limit are exchangeable, we have that
+λ2tr(β(SX + λI)−2βT) → λ2c2
+�
+1
+(s + λ)2dFγ(s).
+We can conclude that,
+R( ˆβΓ(λ)) →
+� λ2c2 + s · γtr(Ω)
+(s + λ)2
+Fγ(s).
+The right-hand side is minimized at λ∗ = γtr(Ω)/c2. In such case, the right-hand side
+becomes γtr(Ω) · m(−λ∗).
+25
+
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+Bipartite unique-neighbour expanders via Ramanujan graphs
+Ron Asherov and Irit Dinur∗
+Weizmann Institute, Rehovot, Israel
+Abstract
+We construct an infinite family of bounded-degree bipartite unique-neighbour expander graphs with
+arbitrarily unbalanced sides. Although weaker than the lossless expanders constructed by Capalbo et
+al., our construction is simpler and may be closer to be implementable in practice due to the smaller
+constants. We construct these graphs by composing bipartite Ramanujan graphs with a fixed-size gadget
+in a way that generalizes the construction of unique neighbour expanders by Alon and Capalbo. For
+the analysis of our construction we prove a strong upper bound on average degrees in small induced
+subgraphs of bipartite Ramanujan graphs. Our bound generalizes Kahale’s average degree bound to
+bipartite Ramanujan graphs, and may be of independent interest. Surprisingly, our bound strongly relies
+on the exact Ramanujan-ness of the graph and is not known to hold for nearly-Ramanujan graphs.
+1
+Introduction
+An infinite family Gn = (Ln ⊔ Rn, En) of (c, d)-biregular graphs with |Ln| + |Rn| → ∞ is called a unique
+neighbour expander family if there exists δ > 0 such that for every n and every set of left side vertices S ⊆ Ln
+of size |S| ≤ δ|Ln| there exists a unique neighbour of S in Gn, namely a vertex in Rn that is connected to
+exactly one vertex in S. We only require that sets of left vertices have unique neighbours, and arbitrarily
+small right side sets may have no unique neighbour.
+Alon and Capalbo [AC02] construct several explicit families of unique neighbour expanders, via an elegant
+composition of a Ramanujan graph and a gadget. They construct three families of general (non-bipartite)
+graphs in which all small sets have unique neighbours, and one family of slightly unbalanced bipartite graphs
+where small sets on the left have unique neighbors on the right. In their construction the left side is 22/21
+times bigger than the right side. The more imbalanced the graph, the harder it is for small left hand side
+sets to expand into the right hand side. Capalbo et. al. [Cap+02] construct arbitrarily unbalanced bipar-
+tite graphs that are lossless expanders, a notion strictly stronger than unique neighbour expansion. Their
+construction is based on a sequence of somewhat involved composition steps using randomness conductors.
+Our main theorem is an efficient construction of an infinite family of bipartite unique neighbour expanders
+for any constant imbalance α, and any sufficiently large left-regularity degrees of a specific form:
+Theorem 1. There is a function ˆq : N × R → N such that for every integer c0 > 5 and real number α > 1,
+if q > ˆq(c0, α) is a prime power and αc0(q + 1) is an integer, then there is a polynomial-time construction of
+an infinite family of (c0(q + 1), αc0(q + 1))-biregular unique neighbour expanders.
+The theorem is proven in Section 6.2, and provides a way to compute ˆq(c0, α). Here are some computed
+values of ˆq(c0, α) for several values of c0, α.
+∗Irit Dinur acknowledges support by ERC grant 772839 and ISF grant 2073/21.
+1
+arXiv:2301.03072v1  [math.CO]  8 Jan 2023
+
+c0
+α
+ˆq(c0, α)
+10
+2
+18907
+35
+2
+1492
+100
+100
+136051
+100
+1.01
+1135
+Notice that ˆq(co, α) increases with α, reflecting the fact that constructions with larger α (namely, more
+imbalanced sides) are harder to come by, and require larger degrees.
+The construction uses an infinite family of bipartite Ramanujan graphs, namely graphs whose non-
+trivial spectrum is contained in the spectrum of the (c, d)-biregular tree (see Preliminaries for details). We
+construct the unique neighbour expander family by taking a family of bipartite Ramanujan graphs and
+combining them with a fixed size graph (“gadget”), with a good unique neighbour property (small sets have
+unique neighbours), whose existence is shown via the probabilistic method (Lemma 11). The combination
+is done as follows. We first place a copy of the gadget for every right side vertex of the Ramanujan graph.
+The vertex is replaced by the right side of the gadget, and its neighbours are identified with the left side of
+the gadget. The gadget is used to route the neighbours of each left side vertex in the Ramanujan graph to
+its neighbours in the product graph.
+Expansion in the product graph comes from unique neighbour expansion of the gadget together with
+low degree vertices in the Ramanujan graph. Sufficiently low degree vertices are guaranteed to exist thanks
+to the following (new) bound on the average degree of induced subgraphs of bipartite Ramanujan graphs,
+which may be of independent interest.
+Theorem 2. Let G = (L ⊔ R, E) be a (c, d)-biregular Ramanujan graph, and let ε > 0. Then there exists
+δ > 0, that depends only on ε, c, d, such that for every S ⊂ L of size |S| ≤ δ|L|, the set N(S) ⊆ R of the
+neighbours of S satisfies
+c|S|
+|N(S)| ≤ 1 + (1 + ε)
+�
+d − 1
+c − 1 .
+The theorem shows that every small set on the left side admits neighbours on the right side with low degree
+in the induced subgraph. The proof involves recursive analysis of non-backtracking paths. Interestingly, the
+recursion has a nice solution only when the graph is Ramanujan. It is unclear whether this method can be
+extended to “nearly-Ramanujan” graphs.
+Combining the average degree upper bound with the gadget, the low-degree right-side vertices in the
+Ramanujan graph imply a small set of left-side vertices in the gadget; this set will have a unique neighbour
+in the gadget, which gives (via Lemma 12) a unique neighbour in the constructed graph.
+Even though Ramanujan graphs are the best spectral expanders one can hope for, an efficient construc-
+tion of Ramanujan graphs (be them bipartite or not) does not immediately imply that we can construct
+unique neighbour expanders. In the d-regular case, Kahale shows ([Kah95, Thm 5.2]) that there are nearly-
+Ramanujan graphs with expansion at most d/2, which is not enough for unique neighbour expansion. In fact,
+recently Kamber and Kaufman [KK22] proved that some Ramanujan graphs strongly fail to have unique
+neighbour expansion, by giving explicit constructions of arbitrarily small sets that do not admit a unique
+neighbour.
+As mentioned, the graph product we define requires a fixed size gadget, whose proof of existence is not
+constructive. In principle, such a gadget could be found by exhaustive search since we are working in a
+constant size search space. The gadget’s size in our construction is at least cubic in q, so exhaustive search
+is impractical for even small values of q. Unfortunately we know of no efficient construction of a gadget with
+the required parameters. It is possible that the graph sampling method present in [AK19] can be used to
+construct fixed size gadgets more efficiently.
+The rest of this work is organized as follows. In Section 2 we survey some of the uses of unique neighbour
+expanders, and mention known constructions of such graphs. Section 3 provides basic definitions and results.
+Our main technical tool, that asserts the low induced degree in bipartite Ramanujan graph, is stated and
+proven in Section 4. We prove the existence of a fixed-size gadget with good unique neighbour expansion
+2
+
+properties in Section 5. In Section 6 we define the way we use the Ramanujan graphs and the gadget to
+construct bipartite unique neighbour expanders, and by that prove Theorem 1.
+2
+Related work
+One of the prominent uses of bipartite expanders in general and bipartite unique neighbour expanders in
+particular, and the motivation for this work, is the construction of error correcting codes. The works of
+Tanner [Tan81] and later Sipser and Spielman [SS96] construct linear error correcting codes C(B, C0) from
+a bipartite graph B and a smaller linear code C0. It is shown that under some assumptions on the code C0
+and the expansion properties of the bipartite graph B, the resulting code has good distance. This gives a
+way to take a family of graphs and transform it into a family of codes. Our work describes a construction
+that, in a sense, goes the other way around: given two bipartite graphs, B and B0, we view B0 as a parity
+check graph1 of the base code C0, and B plays the role of the underlying graph of a Tanner code C(B, C0).
+Our output graph is just the parity check graph of C(B, C0). We give full details of this graph product in
+Section 6.1.
+In [DSW06; BV09] it is shown that codes constructed on top of unique neighbour expanders are weakly
+smooth and can be used to construct robustly testable codes. But the uses of unique neighbour expanders are
+not limited to error correcting codes: for example, such graphs may be used in the context of non-blocking
+networks, where it is required to connect several input-output terminals via paths in a non-intersecting
+fashion. Arora et al. [ALM96] use graphs with expansion beyond the d/2 barrier to establish the existence
+of unique neighbours in the graph, which are useful in finding input-output paths in the online settings.
+Roughly speaking, when routing a set of input-output pairs, the algorithm can use all unique neighbours
+freely since they are guaranteed not to interfere with any other paths.
+Pippenger [Pip93] uses explicit
+constructions of spectral expanders in order to solve a similar problem, in the case where the route planning
+is computed locally. There the spectral expansion of a graph is proven to imply a combinatorial expansion,
+in a similar way to our Theorem 2.
+Another use for unique neigbhour expanders is for load-balancing problems, such as the token distribution
+problem described in [PU89], and the similar pebble distribution problem, briefly discussed in [AC02]. In
+the latter, pebbles are placed arbitrarily on vertices of a graph, and need to be distributed via edges of the
+graph such that no vertex has more than one pebble. Given that the total number of pebbles is small and
+that the graph has the unique neighbour property, we have an efficient parallel algorithm for redistributing
+the pebbles.
+Alon and Capalbo [AC02] construct several families of unique neighbour expanders, one of them is a
+family of bipartite graphs whose left side is 22/21 times bigger than the right side. Similar to the construction
+presented at this work, each graph in the constructed family is a combination of a Ramanujan graph and a
+fixed graph. These graphs are not (bi-)regular but their degrees are bounded by a constant. Becker [Bec16]
+uses a different family of 8-regular Ramanujan graphs in order to construct a family of (non-bipartite) unique
+neighbour expanders, with the additional property that each graph in the family is a Cayley graph.
+A different approach to constructing bipartite graphs uses randomness conductors. Randomness conduc-
+tors are functions that receive a bitstring with some entropy (according to some measure of entropy), and a
+uniformly random bitstring, and output a bitstring, with certain guarantees on its entropy. Some conduc-
+tors can be constructed explicitly via a spectral method, and Capalbo et al. [Cap+02] combine them in a
+zig-zag-like fashion in order to construct an infinite family of bipartite lossless expanders, namely bipartite
+graphs with fixed left-regularity c where small enough sets contained in the left side have at least c(1 − ε)
+neighbours on the right side. These graphs are trivially unique neighbour expanders, since a simple counting
+argument shows that if a set expands by a factor of more than c/2, then it has unique neighbours.
+1This is a bipartite graph whose incidence structure is given by the parity check matrix.
+3
+
+3
+Preliminaries
+3.1
+Expander graphs
+In this work we deal with undirected graphs, that may contain multiple edges between two vertices, but do
+not contain self-loops. For a graph G and a subset of its vertices S we denote by NG(S) the neighbourhood
+of S, namely all vertices adjacent to some vertex in S. When the graph in discussion is obvious, we may
+omit it and write N(S). We say that v is a unique neighbour of S if there is a unique u ∈ S that is adjacent
+to v.
+Let (Gn) be a series of graphs with the number of vertices growing to infinity. There are several well
+studied notions of expansion in graph families; we note some of them.
+1. Vertex expansion. (Gn) is a (δ, α)-vertex expander if for every n and any subset S ⊆ VGn, if |S| ≤ δ|VGN |
+we have that |NGN (S)| ≥ α|S|.
+2. Edge expansion. (Gn) is a (δ, α)-edge expander if for every n and any subset S ⊆ VGn, if |S| ≤ δ|VGN |
+we have that at least an α-fraction of the edges with one endpoint in S have their other endpoint
+outside of S.
+3. Spectral expansion. Assume that (Gn) are all d-regular, and let An be the adjacency operator associated
+with Gn, so An is indexed by vertices of Gn and (An)uv counts how many edges there are between
+u and v in Gn. Let λ1 ≥ . . . ≥ λVn be its spectrum. It can be seen that λ1 = d. Then (Gn) is a
+λ-spectral expander if for all n and i ̸= 1 we have |λi| ≤ λ.
+4. Unique neighbour expansion. (Gn) is a δ-unique neighbour expander if for every n, any subset S ⊆ VGn
+of size at most δ|VGN | has a unique neighbour.
+These definitions apply to bipartite graphs Gn = (Ln ⊔ Rn, En) as well, with the exception that we usually
+consider sets contained in the left side only, and require that Ln/Rn is a constant, normally greater than
+1. In this case we note that edge expansion is meaningless (since all edges leaving the left side enter the
+right side), and if a bipartite graph is (c, d)-biregular, namely if all left-side vertices have degree c and all
+right-side vertices have degree d, then the largest eigenvalue of the associated adjacency operator is
+√
+cd.
+It can be seen that for d-regular graphs, the best spectral expansion we can hope for is α = 2
+√
+d − 1.
+These graphs are known as Ramanujan graphs.
+3.2
+Bipartite Ramanujan graphs
+Ramanujan graphs have the best spectral gap [Nil91], and their non-trivial eigenvalues are contained in the
+spectrum of the infinite d-regular tree Td. Similarly, in the bipartite case, Biregular Ramanujan graphs are
+defined via their relation to the infinite biregular trees: the infinite (c, d)-biregular tree Tc,d, for d > c, has
+the spectrum
+λ ∈ spec(Tc,d) ⇔ |λ| ∈ {0} ∪
+�√
+d − 1 −
+√
+c − 1,
+√
+d − 1 +
+√
+c − 1
+�
+(see, e.g., [GM88], [LS96].) We therefore say that a finite (c, d)-biregular graph is bipartite Ramanujan if its
+nontrivial eigenvalues lie in this set. That means that every eigenvalue λ of a bipartite Ramanujan graph
+belongs to one of these classes:
+1. Trivial: λ = ±
+√
+cd, with eigenvectors fixed on either sides, or λ = 0;
+2. λ ∈ [
+√
+d − 1 − √c − 1,
+√
+d − 1 + √c − 1] are the nontrivial positive eigenvalues;
+3. λ ∈ [−√c − 1 −
+√
+d − 1, √c − 1 −
+√
+d − 1] are the nontrivial negative eigenvalues. Note that since the
+graph is bipartite, λ is an eigenvalue if and only if −λ is an eigenvalue.
+4
+
+By an extension of the Alon-Boppana bound, given in [FL96], this is the best spectral gap we can hope for,
+at least as far as upper bounds for |λ| are concerned. We note that unlike the d-regular case, we require a
+lower bound to |λ| too, which is essential for our proof.
+While there is a vast literature on the construction of d-regular Ramanujan graph (most prominently
+[LPS88] and [Mar88]), less is known about bipartite Ramanujan graphs. In 2014 Marcus et al. [MSS13]
+proved the existence of biregular graphs with one-sided spectral graphs that resemble the Ramanujan bounds:
+these graphs satisfy the one-sided inequality only, namely |λ| ≤
+√
+d − 1 + √c − 1 for every nontrivial eigen-
+value λ. Gribinski et al. [GM21] showed a polynomial-time construction of such graphs, for every degrees
+(d, kd) for any integers d, k. These graphs do not suffice for our analysis, since we make explicit use of the
+lower bound |λ| ≥
+√
+d − 1 − √c − 1 too.
+In 2021 Brito et al. [BDH22] proved that a random biregular graph is nearly Ramanujan with high
+probability. Interestingly, and unlike other works in this field, our proof strongly relies on the graph to be
+exactly Ramanujan, so we cannot use those constructions either.
+We use an explicit construction of bipartite Ramanujan graphs (with both bounds on non-trivial eigen-
+values) given by Ballantine et al.:
+Theorem 3 ([Bal+15]). For every prime power q, there exists an explicit construction of a (q + 1, q3 + 1)-
+biregular Ramanujan graph.
+4
+Vertex expansion in biregular Ramanujan graphs
+Our main technical tool is the following theorem showing that bipartite Ramanujan graphs exhibit excellent
+left-to-right expansion. We restate the theorem for convenience.
+Theorem 2. Let G = (L ⊔ R, E) be a (c, d)-biregular Ramanujan graph, and let ε > 0. Then there exists
+δ > 0, that depends only on ε, c, d, such that for every S ⊂ L of size |S| ≤ δ|L|, the set N(S) ⊆ R of the
+neighbours of S satisfies
+c|S|
+|N(S)| ≤ 1 + (1 + ε)
+�
+d − 1
+c − 1 .
+We note that the quantity on the left hand side of the inequality can be interpreted as follows. Look
+at the bipartite graph induced by taking the vertices S on the left and N(S) on the right. Since every left
+vertex has c outgoing edges, the total number of edges in the induced subgraph is c|S|. This means that
+the expression on the left hand side of the inequality is exactly the average degree of the right side of the
+induced subgraph. Interestingly, the bound in this theorem is strictly stronger than what we would get from
+just applying the expander mixing lemma which amounts to
+c|S|
+|N(S)| ≤ (1 + ε) ·
+�
+1 + d − 1
+c − 1 + 2
+�
+d − 1
+c − 1
+�
+.
+See Claim 4 for details. The fact that we improve upon the expander mixing lemma is perhaps not surprising
+since our analysis is based on enumerating non-backtracking paths, and not just on magnitude of the second
+largest eigenvalue. We also use lower bounds on the magnitude of all nontrivial eigenvalues, whereas the
+expander mixing lemma uses just upper bounds.
+4.1
+Comparison to known bounds
+As noted above, Theorem 2 is an improvement of the bound that the expander mixing lemma gives in similar
+settings, which only uses the one-sided inequality |λ| ≤
+√
+d − 1 + √c − 1. For reference, we state and prove
+the expander mixing lemma for bipartite Ramanujan graphs.
+5
+
+Claim 4 (Expander mixing lemma for bipartite Ramanujan graphs). Let G = (L⊔R, E) be a (c, d)-biregular
+Ramanujan graph, and let ε > 0. Then there exists δ > 0 such that for every S ⊆ L of size |S| ≤ δ|L|, the
+neighbourhood of S satisfies
+c|S|
+|N(S)| ≤ (1 + ε)
+�
+1 + d − 1
+c − 1 + 2
+√
+d − 1
+√c − 1
+�
+.
+Proof. The expander mixing lemma for biregular graphs says that for every S ⊆ L, T ⊆ R we have
+����
+|e(S, T)|
+|E|
+− |S|
+|L| · |T|
+|R|
+���� ≤
+λ
+√
+cd
+�
+|S|
+|L| · |T|
+|R|
+where λ is the second largest eigenvalue of G (see, e.g., [Hae95]). It is clarified that we consider the spectrum
+of G as an adjacency operator, so the largest eigenvalue is
+√
+cd.
+Picking T = N(S) means all edges coming out from S are in the cut, namely |e(S, T)| = c|S|. Plugging
+that in gives
+����
+c|S|
+c|L| − |S|
+|L| · |N(S)|
+|R|
+���� ≤
+λ
+√
+cd
+�
+|S|
+|L| · |N(S)|
+|R|
+.
+Multiplying both sides by |L|
+|S| gives
+����1 − |N(S)|
+|R|
+���� ≤
+λ
+√
+cd
+�
+|S|
+|L| · |N(S)|
+|R|
+· |L|
+|S| =
+λ
+√
+cd
+�
+|N(S)|
+|R|
+· |L|
+|S| =
+λ
+√
+cd
+�
+|N(S)|
+|S|
+·
+�
+d
+c = λ
+c
+�
+|N(S)|
+|S|
+(1)
+where we also used the fact that |E| = c|L| = d|R|.
+Let us assume that |S| = α|L|. Then we can upper bound |N(S)| by
+|N(S)| ≤ c|S| = αc|L| = αd|R|
+and so we have
+1 − |N(S)|
+|R|
+≥ 1 − dα|R|
+|R|
+= 1 − dα.
+We square (1) and plug in the last inequality to get
+(1 − dα)2 ≤ λ2
+c · |N(S)|
+c|S| .
+Recall that G is bipartite Ramanujan, so |λ| ≤
+√
+d − 1 + √c − 1. Use that and rearrange:
+c|S|
+|N(S)| ≤ λ2
+c (1 − dα)−2
+≤ d − 1 + c − 1 + 2
+√
+d − 1√c − 1
+c
+(1 − dα)−2
+≤ d − 1 + c − 1 + 2
+√
+d − 1√c − 1
+c − 1
+(1 − dα)−2
+=
+�
+1 + d − 1
+c − 1 + 2
+√
+d − 1
+√c − 1
+�
+(1 − dα)−2.
+The claim is proven by noting that there is some δ > 0 such that (1 − dα)−2 ≤ 1 + ε for every α < δ, namely
+whenever |S| ≤ δ|L|.
+6
+
+Kahale proved ([Kah95, Thm 4.2]) that in d-regular Ramanujan graphs (not necessarily bipartite), small
+induced subgraphs have average degree at most 1 +
+√
+d − 1. Interestingly, this result can be deduced almost
+immediately from Theorem 2. This is due to the following lemma, proven in Appendix A, which asserts that
+the edge-vertex incidence graph (see [SS96]) of a d-regular Ramanujan graph is a (2, d)-biregular Ramanujan
+graph:
+Lemma 5. Let G be a d-regular Ramanujan graph, and G′ its edge-vertex incidence graph. Then G′ is a
+(2, d)-biregular Ramanujan graph.
+We state and prove Kahale’s bound, but we will not use it in our construction.
+Corollary 6. Let G = (VG, EG) be a d-regular Ramanujan graph, and let ε > 0. Then there exists δ > 0
+such that for every induced subgraph S with at most δ|VG| vertices, the average degree of S is at most
+dS := 2|ES|
+|VS| ≤ 1 + (1 + ε)
+√
+d − 1.
+Proof. Let G = (VG, EG) be a d-regular Ramanujan graph and ε > 0. We define G′ = (LG′ ⊔ RG′, EG′) as
+the edge-vertex incidence graph, namely LG′ = EG, RG′ = VG, and for every edge e = {u, v} in G we have
+the two edges {e, u} and {e, v} in G′. Since the degree of every vertex in G is d, and since every edge has
+two endpoints, we have that G′ is a (2, d)-biregular graph. Lemma 5 asserts that G′ is Ramanujan in the
+bipartite sense. By Theorem 2, there exists δ > 0 such that if T ⊆ LG′ is of size at most δ|LG′|, then
+2|T|
+|NG′(T)| ≤ 1 + (1 + ε)
+√
+d − 1.
+A subgraph S = (VS, ES) of G satisfies that ES is a subset of left-side vertices in G′, VS is a subset of
+right-side vertices in G′, and VS = NG′(ES) (because if an edge is in the subgraph then both of its endpoints
+are in the subgraph, and we assume that the subgraph does not contain an isolated vertex). Therefore, if ES
+is sufficiently small, namely if |ES| ≤ δ|LG′| = δ|EG|, then by Theorem 2 the average degree of NG′(ES) = VS
+is bounded by 1 + (1 + ε)
+√
+d − 1.
+We add that if we wish to find a bound the number of vertices, we note that |ES| ≤ d
+2|VS|. So every
+induced subgraph with no more than
+2
+dδ|EG| = δ|VG| vertices will satisfy the required average degree
+bound.
+4.2
+Proof of Theorem 2
+Theorem 2 is proven by enumerating non-backtracking paths. A non-backtracking path of length ℓ is a
+sequence of edges ((s(ei), t(ei)))ℓ
+i=1 such that for every i, t(ei) = s(ei+1) and s(ei) ̸= t(ei+1).
+For a bipartite graph G and a subset S of left side vertices we define Mℓ(S) to be the number of all non-
+backtracking paths whose all left-side vertices are in S, and Mℓ(S, G) to be the number of non-backtracking
+paths whose first and last left-side vertices are in S. Clearly Mℓ(S) ≤ Mℓ(S, G), as paths of the latter type
+may leave S ⊔N(S) (before re-entering S at the last step). We use a lower bound on Mℓ(S) due to [Kam19]:
+Lemma 7. For every undirected bipartite graph G = (LG ⊔ RG, EG) and integer l it holds that
+Mℓ(LG) ≥ |EG|
+��
+( ¯dL − 1)( ¯dR − 1)
+�ℓ−1
+where ¯dL, ¯dR are the average degrees of the left and right sides of G respectively.
+We state and prove an upper bound on Mℓ(S, G):
+7
+
+Lemma 8. Let G be a (c, d)-biregular Ramanujan graph with n vertices on the left side, and S a subset of
+the left side. Then for every integer ℓ:
+M2ℓ(S, G) ≤ |S|
+�
+(2 +
+√
+d − 1)ℓ + 2
+�
+(c − 1)ℓ/2(d − 1)ℓ/2
+provided that S is small enough:
+|S|(c − 1)ℓ/2(d − 1)ℓ/2 ≤ n.
+(2)
+Before proving the upper bound, we show how these bounds can be combined to obtain Theorem 2.
+Proof of Theorem 2. Let ℓ be an integer to be determined later, S ⊆ L a sufficiently small subset (where
+sufficiently smalls means (2)). Denote by N(S) ⊆ R the neighbours of S. The subgraph induced on S ∪N(S)
+has c|S| edges, with left degrees all c and average right degree ¯dR =
+c|S|
+|N(S)|.
+Chaining the inequalities in Lemma 7 and Lemma 8, we have
+c|S|
+�
+(c − 1)( ¯dR − 1)
+� 2ℓ−1
+2
+≤ M2ℓ(S) ≤ M2ℓ(S, VG) ≤ |S| ·
+�
+(2 +
+√
+d − 1)ℓ + 2
+�
+· (c − 1)ℓ/2(d − 1)ℓ/2.
+Simplifying, we get,
+c(c − 1)ℓ− 1
+2 ( ¯dR − 1)ℓ− 1
+2 ≤
+�
+(2 +
+√
+d − 1)ℓ + 2
+�
+· (c − 1)ℓ/2 · (d − 1)ℓ/2
+( ¯dR − 1)ℓ− 1
+2 ≤
+�
+(2 +
+√
+d − 1)ℓ + 2
+� √c − 1
+c
+��
+d − 1
+c − 1
+�ℓ
+¯dR − 1 ≤
+�
+�
+�
+�
+�
+(2 +
+√
+d − 1)ℓ + 2
+� √c − 1
+� ¯d − 1
+c
+�
+��
+�
+⋆
+�
+�
+�
+�
+1/ℓ
+·
+�
+d − 1
+c − 1
+Since ¯d ≤ d, we have that ⋆ = O(ℓ), so ⋆1/ℓ = O(1), hence for a fixed ε > 0 there exists a constant ℓ (that
+depends only on ε, c, d) such that ⋆1/ℓ ≤ 1 + ε; this ℓ determines, via inequality (2), a fixed δ such that
+whenever |S| ≤ δn we have
+¯dR ≤ 1 + (1 + ε)
+�
+d − 1
+c − 1 .
+We proceed to prove Lemma 8.
+For a bipartite graph G = (LG ⊔ RG, EG) and an integer ℓ, we define ALL
+ℓ
+, ALR
+ℓ
+, ARL
+ℓ
+, ARR
+ℓ
+as operators
+corresponding to non-backtracking paths of length ℓ, i.e.
+ALL
+ℓ
+: L2(LG) → L2(LG)
+,
+(ALL
+ℓ
+f)(x) =
+�
+(e1,...,eℓ),t(eℓ)=x,s(e1),t(eℓ)∈LG
+f(s(e1))
+with the summation over all non-backtracking paths of length ℓ, and similarly for the other operators.
+Let M be the operator corresponding to a single step from the right side G to the left side of G, namely
+M has |RG| rows and |LG| columns, with Muv counting the number of edges between u ∈ RG and v ∈ LG
+in G. Then the following recursive formulae hold for every integer ℓ > 1:
+M ⊤ALL
+ℓ
+= ARL
+ℓ+1 + (d − 1)ARL
+ℓ−1
+M ⊤ALR
+ℓ
+= ARR
+ℓ+1 + (d − 1)ARR
+ℓ−1
+MARL
+ℓ
+= ALL
+ℓ+1 + (c − 1)ALL
+ℓ−1
+MARR
+ℓ
+= ALR
+ℓ+1 + (c − 1)ALR
+ℓ−1
+8
+
+The first formula is explained as follows.
+Every non-backtracking path from R to L of length ℓ + 1 is
+composed of a non-backtracking path from L to L of length ℓ plus an extra step (that’s the M ⊤ALL
+ℓ
+factor.)
+The opposite is true, except for paths counted in M ⊤ALL
+ℓ
+that do backtrack, namely those made of a non-
+backtracking path of length ℓ − 1, and walking back and forth along the same edge. There are d − 1 ways to
+choose that edge (since it cannot be the one that was last in the path of length ℓ − 1, otherwise it wouldn’t
+be counted in M ⊤ALL
+ℓ
+), so we need to subtract (d − 1)ARL
+ℓ−1. The rest of the equations are explained in an
+analog way.
+Due to symmetry we have:
+(ALL
+ℓ
+)⊤ = ALL
+ℓ
+,
+(ARR
+ℓ
+)⊤ = ARR
+ℓ
+,
+(ALR
+ℓ
+)⊤ = ARL
+ℓ
+And since the graph is bipartite we have:
+ALR
+2ℓ = 0
+,
+ARL
+2ℓ = 0
+ALL
+2ℓ+1 = 0
+,
+ARR
+2ℓ+1 = 0
+These equations yield a recursive formula for ALL
+ℓ
+, with the following initial conditions:
+ALL
+2
+= MM ⊤ − cI
+ALL
+4
+= MM ⊤ALL
+2
+− (c − 1 + d − 1)ALL
+2
+− c(d − 1)I
+MM ⊤ALL
+ℓ
+= ALL
+ℓ+2 + ((c − 1) + (d − 1))ALL
+ℓ
++ (c − 1)(d − 1)ALL
+ℓ−2
+,
+∀ℓ ≥ 4
+(3)
+The following lemma, proven in Appendix A, suggests a way to find a non-recursive formula for ALL
+2ℓ , given
+such linear recursive relations with fixed coefficients.
+Lemma 9. Let (xn) be a series defined via a second order linear recurrence with fixed coefficients A, B ∈ C:
+xn = Axn−1 + Bxn−2
+Assume λ1 ̸= λ2 are (real or complex) roots of the characteristic polynomial λ2 − Aλ − B. Then there are
+α, β ∈ C, that depend on the initial conditions x0, x1, such that
+xn = αλn
+1 + βλn
+2
+for every n ≥ 0.
+If the characteristic polynomial has a single root λ of multiplicity 2, then there are α, β ∈ C such that
+xn = αλn + βnλn
+for every n ≥ 0.
+We use the lemma to bound the eigenvalues of ALL
+2ℓ given bounds on the spectrum of the biregular graph.
+Lemma 10. Let G be a (c, d)-biregular graph. Then there is a sequence of polynomials with integer coeffi-
+cients (pℓ(x)) such that for every eigenpair (λ, v) of G, pℓ(λ2) is an eigenvalue of ALL
+2ℓ , and moreover, for
+every λ ∈ R, if
+|λ| ∈ {0} ∪ [
+√
+d − 1 −
+√
+c − 1,
+√
+d − 1 +
+√
+c − 1]
+(4)
+then
+|pℓ(λ2)| ≤ (2 +
+√
+d − 1)ℓ(c − 1)ℓ/2(d − 1)ℓ/2.
+(5)
+9
+
+Proof. The recursive formulae proven above (3) suggest that there is a series of polynomials pn(x) with
+integer coefficients such that ALL
+2n = pn(MM ⊤). Note that the graph’s adjacency matrix is
+AG =
+� 0
+M
+M ⊤
+0
+�
+And so, if (λ, v) is an eigenpair of G, then (λ2, v) is an eigenpair of
+A2
+G =
+�MM ⊤
+0
+0
+M ⊤M
+�
+.
+This shows that pℓ(λ2) is an eigenvalue of ALL
+2ℓ whenever λ is an eigenvalue of G. The converse is also true.
+The formulae (3) can be transformed so as to convey that pn(x) satisfies these equations:
+p1(x) = x − c
+,
+p2(x) = x2 + (2 − 2c − d)x + c(c − 1)
+xpn(x) = pn+1(x) + (c − 1 + d − 1)pn(x) + (c − 1)(d − 1)pn−1(x)
+for all n > 1. Setting n = 1 gives an equation involving p0(x), p1(x), p2(x). We can solve this equation for
+p0(x) and get a simpler description of the initial conditions:
+p0(x) =
+c
+c − 1
+,
+p1(x) = x − c
+(6)
+xpn(x) = pn+1(x) + (c − 1 + d − 1)pn(x) + (c − 1)(d − 1)pn−1(x)
+(7)
+for all n > 0.
+We fix some t that satisfies (4), namely such that
+|t| ∈ {0} ∪ [
+√
+d − 1 −
+√
+c − 1,
+√
+d − 1 +
+√
+c − 1].
+We first deal with the case where |t| ∈ (
+√
+d − 1 − √c − 1,
+√
+d − 1 + √c − 1), and later we will consider the
+edge cases where t is one of the endpoints of the segment or 0. Let us write x = t2. We have that for this
+fixed x, the series (pn(x))n satisfies a second order linear recurrence with fixed coefficients. Using Lemma 9,
+we conclude that there are functions α(x), λ1(x), β(x), λ2(x) that depend only on x, c and d, such that
+pn(x) = α(x)(λ1(x))n + β(x)(λ2(x))n
+(8)
+for every n.
+In order to find λ1, λ2 we solve for λ the characteristic polynomial, namely the following quadratic
+equation derived from (7):
+xλ = λ2 + (c − 1 + d − 1)λ + (c − 1)(d − 1)
+To obtain
+λ1,2(x) = x − (c − 1) − (d − 1) ±
+�
+∆(x)
+2
+where
+∆(x) = x2 − 2x((c − 1) + (d − 1)) + (c − d)2.
+(9)
+Using the initial values for p0(x), p1(x) from (6), and plugging back into (8) we get the equations
+c
+c − 1 = α(x)(λ1(x))0 + β(x)(λ2(x))0 = α(x) + β(x)
+x − c = α(x)(λ1(x))1 + β(x)(λ2(x))1 = α(x)λ1(x) + β(x)λ2(x)
+10
+
+whose solution is
+α(x) = (c − 1)x − (c − 1)2 − (c − 1) + (c − 1)(d − 1) + (c − 1)
+�
+∆(x) − x + d − 1 +
+�
+∆(x)
+2(c − 1)
+�
+∆(x)
+β(x) =
+c
+c − 1 − α(x).
+We check when ∆(x) = 0 by solving (9) for x:
+x1,2 = 2((c − 1) + (d − 1)) ±
+�
+4(c − 1 + d − 1)2 − 4(c − d)2
+2
+= (c − 1 + d − 1) ±
+�
+(c + d)2 − 4(c + d) + 4 − (c − d)2
+= (c − 1 + d − 1) ±
+�
+c2 + 2cd + d2 − 4c − 4d + 4 − c2 + 2cd − d2
+= (c − 1 + d − 1) ±
+√
+4cd − 4c − 4d + 4
+= (c − 1 + d − 1) ± 2
+√
+c − 1
+√
+d − 1
+= (
+√
+d − 1 ±
+√
+c − 1)2
+We see that ∆(x) is quadratic in x and has roots at (
+√
+d − 1 ± √c − 1)2. This gives a nice factorization of
+∆(x):
+∆(x) = x2 − 2x((c − 1) + (d − 1)) + (c − d)2
+=
+�
+x −
+�√
+d − 1 +
+√
+c − 1
+�2� �
+x −
+�√
+d − 1 −
+√
+c − 1
+�2�
+Recall that for the x we fixed we have √x = t ∈ (
+√
+d − 1 − √c − 1,
+√
+d − 1 + √c − 1), so the first term in the
+product is negative and the second term is positive, so ∆ < 0, and so λ1,2 are complex numbers (conjugate
+to one another), with magnitude
+|λ1,2|2 = (x − (c − 1) − (d − 1))2 − ∆(x)
+4
+= x2 − 2x((c − 1) + (d − 1)) + (c − 1 + d − 1)2 − (x2 − 2x((c − 1) + (d − 1)) + (c − d)2)
+4
+= (c + d − 2)2 − (c − d)2
+4
+= (c − 1)(d − 1)
+(10)
+A very similar calculation shows that α, β are conjugates with magnitude
+|α|2 = |β|2 =
+x(x − cd)
+∆(x) · (c − 1)
+This finishes the proof for all such x’s:
+|pℓ(x)| = |α(x)λℓ
+1 + β(x)λℓ
+2| ≤ |α(x)λℓ
+1| + |β(x)λℓ
+2|
+= |α(x)||λ1|ℓ + |β(x)||λ2|ℓ
+= 2
+�
+x(x − cd)
+∆(x) · (c − 1)(c − 1)ℓ/2(d − 1)ℓ/2
+We keep in mind that x is fixed, so the expression is smaller than (2 +
+√
+d − 1) · ℓ · (c − 1)ℓ/2(d − 1)ℓ/2 for
+large enough ℓ.
+We are left with the cases x = t2 for t = 0,
+√
+d − 1 ± √c − 1:
+11
+
+1. t = 0. We use the same methods and find that the characteristic polynomial is
+λ2 + (c − 1 + d − 1)λ + (c − 1)(d − 1)
+whose roots are
+λ1 = −(c − 1)
+,
+λ2 = −(d − 1).
+Using the initial conditions (p0(0) = c/(c − 1), p1(0) = −c) we obtain
+α(0) =
+c
+c − 1
+,
+β(0) = 0
+and using the fact that c < d we get
+|pℓ(0)| = |α(0)λℓ
+1 + β(0)λℓ
+2|
+=
+c
+c − 1(c − 1)ℓ
+< 2l(c − 1)ℓ/2(c − 1)ℓ/2
+< 2l(c − 1)ℓ/2(d − 1)ℓ/2.
+2. t =
+√
+d − 1 + √c − 1. Then x = t2 = (
+√
+d − 1 + √c − 1)2 = d − 1 + c − 1 + 2
+√
+d − 1√c − 1, and the
+characteristic polynomial has a single root of multiplicity 2, namely
+λ = x − (c − 1) − (d − 1)
+2
+=
+√
+d − 1
+√
+c − 1.
+The solution, therefore, takes the form
+pn(x) = (α(x) + nβ(x))(c − 1)n/2(d − 1)n/2.
+Using the initial values we get
+α(x) =
+c
+c − 1
+,
+β(x) =
+x − c
+√
+d − 1√c − 1 −
+c
+c − 1 = 2 +
+d − 2
+√
+d − 1√c − 1 −
+c
+c − 1.
+1 <
+c
+c−1 ≤ 2 so β(x) ≤
+√
+d − 1 + 1, and in total we get
+|pℓ(x)| = |α(x) + ℓβ(x)|(c − 1)ℓ/2(d − 1)ℓ/2
+≤
+�����
+1
+ℓ ·
+c
+c − 1
+���� + |β(x)|
+�
+ℓ(c − 1)ℓ/2(d − 1)ℓ/2
+≤
+�
+2 +
+√
+d − 1
+�
+ℓ(c − 1)ℓ/2(d − 1)ℓ/2
+For sufficiently large ℓ.
+3. t =
+√
+d − 1 − √c − 1. We get x = t2 = d − 1 + c − 1 − 2
+√
+d − 1√c − 1, and the rest follows the same
+calculations as in the previous case.
+Bounds on the spectrum of ALL
+2ℓ give bounds on the number of non-backtracking paths completely con-
+tained in a small set, hence gives Lemma 8.
+12
+
+Proof of Lemma 8. Recall that M2ℓ(S, G) counts the number of non-backtracking paths of length 2ℓ that
+start and end in S, so by the definition of the ALL
+n
+operatore, we have M2ℓ(S, G) = ⟨ALL
+2ℓ 1S, 1S⟩.
+We note that ALL
+2ℓ 1L = c(c − 1)ℓ−1(d − 1)ℓ1L, because every non-backtracking path starting at a given
+vertex is made of picking the first left-to-right edge (we have c such edges to pick from), and then alternating
+between picking any of the d or c edges adjacent to the current vertex, except for the edge we picked to get
+to it.
+Write 1S =
+|S|
+n 1L + r, with r ⊥ 1L, and ∥r∥2
+2 ≤ ∥1S∥2
+2 = |S|. Since the graph is Ramanujan, the
+nontrivial eigenvalues in its spectral decomposition have their absolute value in the set {0} ∪ [
+√
+d − 1 −
+√c − 1,
+√
+d − 1 + √c − 1].
+We only care about the nontrivial eigenvalues because r ⊥ 1L, hence in the
+writing of r in the orthogonal basis made of eigenvectors, only eigenvectors with nontrivial eigenvalues
+appear. We use Lemma 10 to get
+⟨ALL
+2ℓ r, r⟩ ≤ (2 +
+√
+d − 1)ℓ(c − 1)ℓ/2(d − 1)ℓ/2 · ∥r∥2
+2 .
+Combine everything to get
+M2ℓ(S, G) = ⟨ALL
+2ℓ 1S, 1S⟩ =
+�
+ALL
+2ℓ
+|S|
+n 1L + r, |S|
+n 1L + r
+�
+= |S|2
+n2 ⟨ALL
+2ℓ 1L, 1L⟩ + ⟨ALL
+2ℓ r, r⟩
+= |S|2
+n2 · c(c − 1)ℓ−1(d − 1)ℓ⟨1L, 1L⟩ + ⟨ALL
+2ℓ r, r⟩
+≤ |S|2
+n c(c − 1)ℓ−1(d − 1)ℓ + (2 +
+√
+d − 1)ℓ(c − 1)ℓ/2(d − 1)ℓ/2 ∥r∥2
+2
+≤ |S|
+�|S| · c · (c − 1)ℓ/2(d − 1)ℓ/2
+n(c − 1)
++ (2 +
+√
+d − 1)ℓ
+�
+(c − 1)ℓ/2(d − 1)ℓ/2
+≤ |S|
+�
+c
+c − 1 + (2 +
+√
+d − 1)ℓ
+�
+(c − 1)ℓ/2(d − 1)ℓ/2
+≤ |S|
+�
+(2 +
+√
+d − 1)ℓ + 2
+�
+(c − 1)ℓ/2(d − 1)ℓ/2
+5
+Random gadget
+In this section we prove the existence of bipartite graphs such that every small set of left-side vertices has
+a unique neighbour on the right side. We draw a random biregular graph from a similar distribution as in
+[Pip77], and use techniques similiar to [Vad+12, Thm 4.4].
+Lemma 11. For every integers L, R, c, d with Lc = Rd, L > R, c > 3, if k is an integer that satisfies the
+inequality
+k
+c−3
+2
+≤
+1
+2Le ·
+� R
+3ec
+� c−1
+2
+then there is a (c, d)-biregular graph with sides [L] and [R] such that every set of left vertices of size at most
+k has a unique neighbour.
+We draw a random (c, d)-biregular graph in the following way: fix L vertices on the left side and R
+vertices on the right side (cL = dR), write c copies of each left-side vertex and d copies of each right-side
+vertex, and connect them via a uniformly random matching. That is, pick a uniformly random permutation
+π : L × [c] → R × [d], and for every u ∈ L, v ∈ R, i ∈ [c], j ∈ [d], if π(u, i) = (v, j), then add (u, v) as an edge.
+Note that we allow multiple edges between two vertices (if there are several i, j satisfying π(u, i) = (v, j)).
+13
+
+Let G be a random bipartite graph with L vertices on the left side and R vertices on the right side drawn
+from said distribution. Let A be a subset of left vertices of size k. We note that if A expands by at least
+(c + 1)/2, then, by a simple counting argument, A has a unique neighbour. It is therefore sufficient to find
+the probability that A expands by at least (c + 1)/2.
+Let us fix an arbitary ordering of the ck edges leaving A, and denote it e1, . . . , eck. We say that ei is a
+repeat if it touches a previously covered vertex, that is, if its right endpoint is contained in the set of right
+endpoints of the set e1, . . . , ei−1. We note that if A does not expand by at least (c + 1)/2, then, again by a
+simple counting argument, there are at least (c − 1)k/2 repeats. This is because the number of repeats and
+the size of the set of the neighbours of A add up to the number of edges leaving A, namely ck.
+We note that for every i, ei is a repeat if it touches one of i − 1 or less previously covered vertices. This
+means that Pr[ei is a repeat] ≤ i−1
+R < ck
+R . Moreover, if we condition on the event that some of the first i − 1
+edges are also repeats, then the probability that ei is a repeat may only decrease, since it means that there
+are less “forbidden” endpoints. We conclude that for every set of l edges:
+Pr[ei1, . . . , eil are repeats] =
+l�
+j=1
+Pr[eij is a repeat | ei1, . . . eij−1 are repeats] <
+�ck
+R
+�l
+.
+If A expands too little, then there are many repeats. We can use it to bound the probability that A has
+no unqiue neighbour:
+Pr[A has no unique neighbour] ≤ Pr[A expands by < (c + 1)/2]
+≤ Pr[there are at least (c − 1)k/2 repeats]
+≤
+�
+i1,...,i(c−1)k/2∈(
+ck
+(c−1)k/2)
+Pr[{ei1, . . . , ei(c−1)k/2} are repeats]
+<
+� ck
+c−1
+2 k
+�
+·
+�ck
+R
+� c−1
+2 k
+And by a union bound over the possible choices of A:
+Pr[∃ “bad” A of size k] ≤
+�L
+k
+�
+· Pr[A expands by < (c + 1)/2]
+≤
+�L
+k
+�
+·
+� ck
+c−1
+2 k
+�
+·
+�ck
+R
+� c−1
+2 k
+≤
+�Le
+k
+�k
+·
+� cke
+c−1
+2 k
+� c−1
+2 k
+·
+�ck
+R
+� c−1
+2 k
+=
+�
+Le
+k ·
+� 2ce
+c − 1 · ck
+R
+� c−1
+2 �k
+≤
+�
+Le
+k ·
+�3eck
+R
+� c−1
+2 �k
+(11)
+Where the last inequality follows from assuming that c ≥ 3 so
+2c
+c−1 ≤ 3.
+We are now ready to prove Lemma 11.
+Proof of Lemma 11. Let us draw a (c, d)-biregular graph G = ([L] ⊔ [R], E) from the distribution described
+above. Let k be an integer satsifying (11). Using a union bound and the inequality in (11), we have (where
+14
+
+probability is taken over the choice of G):
+Pr[∃ “bad” A ⊆ [L] of size ≤ k] =
+k
+�
+a=1
+Pr[∃ “bad” A ⊆ [L] of size a]
+≤
+k
+�
+a=1
+�
+Le
+a ·
+�3eca
+R
+� c−1
+2 �a
+<
+∞
+�
+a=1
+�
+Le
+k ·
+�3eck
+R
+� c−1
+2 �a
+=
+∞
+�
+a=1
+�
+k
+c−1
+3
+· Le ·
+�3ec
+R
+� c−1
+2 �a
+≤
+∞
+�
+a=1
+�1
+2
+�a
+< 1.
+We see that with strictly positive probability, a random graph has no “bad” subsets of size ≤ k, hence there
+exists a graph with the desired unique neighbour property.
+6
+Construction
+6.1
+Routed product definition
+Let us begin with a brief coding theory motivation. An error-correcting code is often given via an m × n
+parity check matrix H, so that C = Ker H ⊆ {0, 1}n. The matrix H can be visualized as a bipartite graph,
+called the parity check graph, with n left and m right vertices, and an edge i ∼ j whenever H(j, i) ̸= 0. A
+Tanner code is defined given a bipartite graph B and a base code C0 = Ker H0 [Tan81]. One way to view
+the routed product is through the point of view of codes. Consider the parity check graph B0 of H0 and
+define the routed product of B and B0 to be simply the parity check graph of the Tanner code C(B, C0).
+Here is a more detailed and combinatorial definition of the routed product without mention of codes.
+Let G = (L ⊔ R, E) be a (c, d)-biregular graph and G0 = (L0 ⊔ R0, E0) a (c0, d0)-biregular graph.
+We
+think of G as a big graph (in practice, an infinite family of Ramanujan graphs), and G0 as a fixed size graph
+(gadget). Assume that |L0| = d, and let us think of the edges of G as a function E : R × [d] → L which
+maps a right side vertex v and an index i to the ith neighbour of v in G.
+We can define the routed product graph G′ = G ◦ G0 as the bipartite graph whose left side is L, right
+side is the cartesian product R × R0, and the set of edges is
+E′ = {(E(v, i), (v, j)) : v ∈ R, i ∈ [d], j ∈ [R0], (i, j) ∈ E0}.
+That is, we write R0 copies of each vertex in R, and every right side vertex v in the big graph G and an
+edge (i, j) in the small gadget G0 gives an edge between the ith neighbour of v in G, and the jth vertex of
+the copy of G0 assigned to v in G′. Otherwise put, we use G0 to route every edge of the big graph G to c0
+edges in the product graph G′.
+More precisely, for every v ∈ R, the bipartite subgraph of G′ whose left side is NG(v) and right side
+is (v, ·) is isomorphic to G0. This means that, roughly speaking, unique neighbours are inherited from the
+small graph to the product graph:
+Lemma 12. Let S ⊆ L, v ∈ NG(S). Define S′ = {i : E(v, i) ∈ S} ⊆ [d] as the indexed neighbours of v in
+S. If S′, as a set of vertices in the gadget G0, has a unique neighbour j ∈ R0 in G0, then (v, j) is a unique
+neighbour of S in the product graph G′.
+The proof is immediate while staring at Fig. 1, but for the sake of completion it is given in Appendix A.
+15
+
+Figure 1: An example of a bipartite graph G (dashed, red), a small gadget G0 (dotted, green), and the
+routed product G′ = G ◦ G0 (solid, blue). The set S ⊆ L has a neighbour v ∈ R, and so S is associated with
+a set S′ of left side vertices of the copy of G0 associated with v. Since (i′, j) is the only edge connecting j
+to S′ in G0, we have that (v, j) is a unique neighbour of S in G′.
+16
+
+S6.2
+Proof of Theorem 1
+Let q be a prime power, c0 an integer, and α > 1. Assume that αc0(q + 1) is an integer. We construct an
+infinite family of (c0(q + 1), αc0(q + 1))-biregular graphs with the unique neighbour property under some
+assumptions specified below.
+Denote c = q + 1 and d = q3 + 1. By Theorem 3 there is an efficient construction of an infinite family
+of (c, d)-biregular Ramanujan graphs (Gn). Let G0 = (L0 ⊔ R0, E0) be a gadget: a c0-left-regular bipartite
+graph with |L0| = d = q3 + 1 vertices on the left side and R0 vertices on the right side, such that every
+left-side set of sufficiently small size admits a unique neighbour on the right side, where “sufficiently small”
+here means the bound given in Lemma 11. For the constructed graph to have the left side α times bigger
+than the right side, we set R0 =
+d
+αc =
+q3+1
+α(q+1).
+We define G′
+n = Gn ◦ G0 as the routed product of Gn and G0. For the rest of this (short) proof let us
+suppress n from the notation, for convenience.
+Let ε < 1
+q. By Theorem 2, there exists δ > 0 such that for every S ⊆ L of size at most δ|L|, the “average
+right degree” ¯dS, namely the average of the degrees of vertices in NG(S) in the induced subgraph S ⊔NG(S),
+is bounded:
+¯dS :=
+c|S|
+|NG(S)| ≤ 1 + (1 + ε)
+�
+d − 1
+c − 1 .
+We show that such S has a unique neighbour in G′.
+We note that d−1
+c−1 = q2, so since ε < 1
+q we have a vertex v ∈ R of “degree” at most q + 1 in G, that is, the
+set S′ ⊆ [d] of v’s neighbours in S is of size at most q + 1. By Lemma 12, if S′, as a set of left-side vertices
+in G0, has a unique neighbour j in G0, then our original set S has a unique neighbour (v, j) in G.
+It remains to choose the parameters in a way that all left-side sets of size at most q + 1 have a unique
+neighbour in G0. By Lemma 11, we need to have:
+(q + 1)
+c0−3
+2
+≤
+1
+2(q3 + 1)e ·
+�
+�
+q3+1
+α(q+1)
+3ec0
+�
+�
+c0−1
+2
+.
+(12)
+The LHS is O(q
+c0−3
+2 ) and RHS is Θ(qc0−4), so if c0 > 5 then for sufficiently large q the construction gives a
+unique neighbour expander. That is, there exists some ˆq(c0, α) such that if q > ˆq then (12) holds, hence we
+constructed a bipartite unique neighbour expander as promised in Theorem 1.
+7
+Future work
+The main pitfall of our approach is the non-constructive nature of the gadget. Theoretically since the gadget
+has constant size this is no issue. However, exhaustive search is impractical even for small values of q. This
+is because the gadget’s size is cubic in q so the search space is of size exponential in q3. A natural question
+would be whether it is possible to construct such a gadget in an efficient way, since that would lead to the
+whole unique neighbour expander family to be constructible in practice. For the bipartite Ramanujan family
+chosen in our work (the one by Ballantine et al. [Bal+15]) we ask the following.
+Question 13. For which prime power q and real number α ≥ 1 can one construct efficiently a biregular
+graph with left side q3 + 1, right side
+q3+1
+α(q+1), such that every left side set of size at most q + 1 has a unique
+neighbour?
+We note that the fixed size graph given in [AC02, Lemma 4.3] is a good gadget (for α = 22/21 and the
+edge-vertex incidence graphs of a 44-regular Ramanujan graph family), and indeed these graphs can be used
+to construct bipartite unique neighbour expanders.
+Since we prove that a random gadget is, with non-negligble probability, good for our construction, it
+may be interesting to construct such gadget by simply drawing random gadgets and testing whether they
+are good. Since drawing is simple, we are left with the task of testing. We therefore ask:
+17
+
+Question 14. Given a bipartite graph, can one efficiently find the smallest nonempty set of left-side vertices
+that has no unique neighbours?
+We currently know of no better way than just enumerating all left-side sets, which is exponential in the
+size of the graph, hence impractical. We refer to [AK19] for an interesting approach to testing expansion of
+random graphs.
+The methods presented in this work are not limited to the (q+1, q3 +1)-biregular Ramanujan family. We
+can therefore ask the question the other way around – find a gadget (by sampling or any other way), and see
+whether we can efficiently construct a bipartite Ramanujan family that will make it work, i.e. that would
+allow us to rewrite the proof of Theorem 1. This emphasizes the well-known natural question of constructing
+Ramanujan graphs with arbitrary degrees, specifically in the bipartite and biregular setting,
+Question 15. For which integers c < d can one construct efficiently an infinite family of (c, d)-biregular
+Ramanujan graphs?
+We note that our construction is far from “right-side unique neighbour expansion,” as the complete right
+side of a single gadget is a constant-size set with no unique neighbours on the left. We wonder whether it is
+possible to construct a bipartite graph where all small size sets (be them contained in either sides, or both)
+have unique neighbours.
+18
+
+References
+[AC02]
+Noga Alon and Michael Capalbo. “Explicit unique-neighbor expanders”. In: The 43rd Annual
+IEEE Symposium on Foundations of Computer Science, 2002. Proceedings. IEEE. 2002, pp. 73–
+79.
+[AK19]
+Benny Applebaum and Eliran Kachlon. “Sampling graphs without forbidden subgraphs and un-
+balanced expanders with negligible error”. In: 2019 IEEE 60th Annual Symposium on Foundations
+of Computer Science (FOCS). IEEE. 2019, pp. 171–179.
+[ALM96]
+Sanjeev Arora, Frank Thomson Leighton, and Bruce M Maggs. “On-line algorithms for path
+selection in a nonblocking network”. In: SIAM Journal on Computing 25.3 (1996), pp. 600–625.
+[Bal+15]
+Cristina Ballantine, Brooke Feigon, Radhika Ganapathy, Janne Kool, Kathrin Maurischat, and
+Amy Wooding. “Explicit construction of Ramanujan bigraphs”. In: Women in numbers europe.
+Springer, 2015, pp. 1–16.
+[BDH22]
+Gerandy Brito, Ioana Dumitriu, and Kameron Decker Harris. “Spectral gap in random bipartite
+biregular graphs and applications”. In: Combinatorics, Probability and Computing 31.2 (2022),
+pp. 229–267.
+[Bec16]
+Oren Becker. “Symmetric unique neighbor expanders and good LDPC codes”. In: Discrete Applied
+Mathematics 211 (2016), pp. 211–216. issn: 0166-218X. doi: https://doi.org/10.1016/
+j.dam.2016.04.022. url: https://www.sciencedirect.com/science/article/pii/
+S0166218X16301810.
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+Eli Ben-Sasson and Michael Viderman. “Tensor products of weakly smooth codes are robust”.
+In: Theory of Computing 5.1 (2009), pp. 239–255.
+[Cap+02]
+Michael Capalbo, Omer Reingold, Salil Vadhan, and Avi Wigderson. “Randomness conductors
+and constant-degree expansion beyond the degree/2 barrier”. In: Proceedings of the 34th Annual
+ACM Symposium on Theory of Computing. 2002, pp. 659–668.
+[DSW06]
+Irit Dinur, Madhu Sudan, and Avi Wigderson. “Robust local testability of tensor products of
+LDPC codes”. In: Approximation, Randomization, and Combinatorial Optimization. Algorithms
+and Techniques. Springer, 2006, pp. 304–315.
+[FL96]
+Keqin Feng and Wen-Ch’ing Winnie Li. “Spectra of hypergraphs and applications”. In: Journal
+of number theory 60.1 (1996), pp. 1–22.
+[GM21]
+Aurelien Gribinski and Adam W Marcus. “Existence and polynomial time construction of bireg-
+ular, bipartite Ramanujan graphs of all degrees”. In: arXiv preprint arXiv:2108.02534 (2021).
+[GM88]
+Chris D Godsil and Bojan Mohar. “Walk generating functions and spectral measures of infinite
+graphs”. In: Linear Algebra and its Applications 107 (1988), pp. 191–206.
+[Hae95]
+Willem H Haemers. “Interlacing eigenvalues and graphs”. In: Linear Algebra and its applications
+226 (1995), pp. 593–616.
+[Kah95]
+Nabil Kahale. “Eigenvalues and expansion of regular graphs”. In: Journal of the ACM (JACM)
+42.5 (1995), pp. 1091–1106.
+[Kam19]
+Amitay Kamber. Lp Expander Graphs. 2019. arXiv: 1609.04433 [math.CO].
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+Amitay Kamber and Tali Kaufman. “Combinatorics via closed orbits: number theoretic Ramanu-
+jan graphs are not unique neighbor expanders”. In: Proceedings of the 54th Annual ACM SIGACT
+Symposium on Theory of Computing. 2022, pp. 426–435.
+[LPS88]
+Alexander Lubotzky, Ralph Phillips, and Peter Sarnak. “Ramanujan graphs”. In: Combinatorica
+8.3 (1988), pp. 261–277.
+[LS96]
+Wen-Ch’ing Winnie Li and Patrick Solé. “Spectra of Regular Graphs and Hypergraphs and
+Orthogonal Polynomials”. In: European Journal of Combinatorics 17.5 (1996), pp. 461–477. doi:
+https://doi.org/10.1006/eujc.1996.0040.
+19
+
+[Mar88]
+G. A. Margulis. “Explicit group-theoretical constructions of combinatorial schemes and their
+application to the design of expanders and concentrators”. In: Problemy peredachi informatsii
+24.1 (1988), pp. 51–60.
+[MSS13]
+Adam Marcus, Daniel A Spielman, and Nikhil Srivastava. “Interlacing families I: Bipartite Ra-
+manujan graphs of all degrees”. In: 2013 IEEE 54th Annual Symposium on Foundations of com-
+puter science. IEEE. 2013, pp. 529–537.
+[Nil91]
+Alon Nilli. “On the second eigenvalue of a graph”. In: Discrete Mathematics 91.2 (1991), pp. 207–
+210.
+[Pip77]
+Nicholas Pippenger. “Superconcentrators”. In: SIAM Journal on Computing 6.2 (1977), pp. 298–
+304.
+[Pip93]
+Nicholas Pippenger. “Self-routing superconcentrators”. In: Proceedings of the twenty-fifth annual
+ACM symposium on Theory of Computing. 1993, pp. 355–361.
+[PU89]
+David Peleg and Eli Upfal. “The token distribution problem”. In: SIAM journal on computing
+18.2 (1989), pp. 229–243.
+[SS96]
+Michael Sipser and Daniel A Spielman. “Expander codes”. In: IEEE transactions on Information
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+[Tan81]
+R Tanner. “A recursive approach to low complexity codes”. In: IEEE Transactions on information
+theory 27.5 (1981), pp. 533–547.
+[Vad+12]
+Salil P Vadhan et al. “Pseudorandomness”. In: Foundations and Trends in Theoretical Computer
+Science 7.1–3 (2012).
+20
+
+8
+Appendix A
+We restate and prove the lemmas we used throughout the work.
+Lemma 5. Let G be a d-regular Ramanujan graph, and G′ its edge-vertex incidence graph. Then G′ is a
+(2, d)-biregular Ramanujan graph.
+Proof. Let G = (V, E) a d-regular Ramanujan graph. The adjacency matrix of G′ is A =
+� 0
+M
+M ⊤
+0
+�
+where
+M has |E| rows, each containing two 1’s, and |V | columns, each containing d 1’s. Let v be an eigenvector of
+A with eigenvalue λ; then v is an eigenvector of A2 with eigenvalue λ2. We note that
+A2 =
+�
+MM ⊤
+0
+0
+M ⊤M
+�
+so it suffices to consider the spectrum of M ⊤M, which is essentially the operator corresponding to a walk
+from a vertex of G to an edge that touches it and back to one of its endpoints (possibly the same vertex we
+started at).
+For every v ∈ V , there are d ways to walk from it to an edge and then back to v; all other legal paths
+correspond to picking an edge touching v. We conclude that M ⊤M = dI + A, so every eigenvalue λ of G′
+satisfies λ2 = d + σ where σ is an eigenvalue of G.
+The lemma is proven by noting that |σ| ≤ 2
+√
+d − 1 (since G is Ramanujan), so
+d − 2
+√
+d − 1 ≤ λ2 ≤ d + 2
+√
+d − 1
+The terms on the extreme sides of the inequality can be verified to be (
+√
+d − 1 ± 1)2 so we get |λ| ∈
+[
+√
+d − 1 − 1,
+√
+d − 1 + 1], as needed (recall that in G′ the left-regularity is c = 2 so √c − 1 = 1).
+Lemma 9. Let (xn) be a series defined via a second order linear recurrence with fixed coefficients A, B ∈ C:
+xn = Axn−1 + Bxn−2
+Assume λ1 ̸= λ2 are (real or complex) roots of the characteristic polynomial λ2 − Aλ − B. Then there are
+α, β ∈ C, that depend on the initial conditions x0, x1, such that
+xn = αλn
+1 + βλn
+2
+for every n ≥ 0.
+If the characteristic polynomial has a single root λ of multiplicity 2, then there are α, β ∈ C such that
+xn = αλn + βnλn
+for every n ≥ 0.
+Proof. We note that for every n ≥ 2 we have
+� xn
+xn−1
+�
+=
+�Axn−1 + Bxn−2
+xn−1
+�
+=
+�A
+B
+1
+0
+� �xn−1
+xn−2
+�
+Denote the 2 × 2 matrix by D, so by induction,
+� xn
+xn−1
+�
+= Dn
+�x1
+x0
+�
+Let us diagonalize D. The characteristic polyonmial is
+pD(λ) = det(λI − D) =
+����
+λ − A
+−B
+−1
+λ
+���� = λ(λ − A) − B = λ2 − Aλ − B
+21
+
+If pD(λ) has two distinct roots λ1, λ2, then the matrix is diagonalizable; that means that there exists a 2 × 2
+matrix M such that D = M · diag{λ1, λ2} · M −1. We get:
+� xn
+xn−1
+�
+= M
+�λ1
+0
+0
+λ2
+�n
+M −1
+�x1
+x0
+�
+= M
+�λn
+1
+0
+0
+λn
+2
+�
+M −1
+�x1
+x0
+�
+We can compute M, M −1 explicitly, multiple the matrices and get α, β ∈ C such that xn = αλn
+1 + βλn
+2 as
+required.
+Otherwise, if pD(λ) has a single root λ of multiplicity 2, then we can find its Jordan form, i.e. find M
+such that
+D = M
+�λ
+1
+0
+λ
+�
+M −1
+Dn = M
+�λ
+1
+0
+λ
+�n
+M −1 = M
+�λn
+nλn−1
+0
+λn
+�
+M −1
+Where the last equality follows from a simple induction.
+Similarly, we get
+� xn
+xn−1
+�
+= M
+�λ
+1
+0
+λ
+�n
+M −1
+�x1
+x0
+�
+= M
+�λn
+nλn−1
+0
+λn
+�
+M −1
+�x1
+x0
+�
+And again we can find α, β ∈ C as required.
+For the following lemma we remind that G = (L ⊔ R, E) is a (c, d)-biregular graph, G0 = (L0 ⊔ R0, E0) is
+a (c0, d0)-biregular graph, and G′ = G ◦ G0 is the routed product of G and G0. Recall that the edges of G′
+are (E(v, i), (v, j)) when v ∈ R is a right side vertex of G, i ∈ [d], E(v, i) is the ith neighbour of v according
+to G, and (i, j) ∈ E0.
+Lemma 12. Let S ⊆ L, v ∈ NG(S). Define S′ = {i : E(v, i) ∈ S} ⊆ [d] as the indexed neighbours of v in
+S. If S′, as a set of vertices in the gadget G0, has a unique neighbour j ∈ R0 in G0, then (v, j) is a unique
+neighbour of S in the product graph G′.
+Proof. Assume that i′ ∈ S′ is the unique neighbour of j in G0. By the definition of the routed product
+we have that (E(v, i′), (v, j)) is an edge in G. Since i′ ∈ S′ we have that E(v, i′) ∈ S, so indeed (v, j) is
+a neighbour of S in G′. It is therefore remaining to show that it is unique, i.e. that E(v, i′) is the only
+neighbour of (v, j) in S.
+The neighbours of (v, j) in G are E(v, i) for every i such that (i, j) ∈ E0. If E(v, i) ∈ S, then by the
+definition of S′ we have that i ∈ S′, so i is a neighbour of j in E0. But we know that j is a unique neighbour
+of S′ in E0, so we must have that i = i′, and indeed (v, j) is a unique neighbour of S in G′.
+22
+
diff --git a/5dE1T4oBgHgl3EQfTAM3/content/tmp_files/load_file.txt b/5dE1T4oBgHgl3EQfTAM3/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..fbbec8c623b2bfeca53a5adaa04564aefebad0ed
--- /dev/null
+++ b/5dE1T4oBgHgl3EQfTAM3/content/tmp_files/load_file.txt
@@ -0,0 +1,612 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf,len=611
+page_content='Bipartite unique-neighbour expanders via Ramanujan graphs  Ron Asherov and Irit Dinur∗  Weizmann Institute, Rehovot, Israel  Abstract  We construct an infinite family of bounded-degree bipartite unique-neighbour expander graphs with  arbitrarily unbalanced sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Although weaker than the lossless expanders constructed by Capalbo et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=', our construction is simpler and may be closer to be implementable in practice due to the smaller  constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' We construct these graphs by composing bipartite Ramanujan graphs with a fixed-size gadget  in a way that generalizes the construction of unique neighbour expanders by Alon and Capalbo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' For  the analysis of our construction we prove a strong upper bound on average degrees in small induced  subgraphs of bipartite Ramanujan graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Our bound generalizes Kahale’s average degree bound to  bipartite Ramanujan graphs, and may be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Surprisingly, our bound strongly relies  on the exact Ramanujan-ness of the graph and is not known to hold for nearly-Ramanujan graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  1  Introduction  An infinite family Gn = (Ln ⊔ Rn, En) of (c, d)-biregular graphs with |Ln| + |Rn| → ∞ is called a unique  neighbour expander family if there exists δ > 0 such that for every n and every set of left side vertices S ⊆ Ln  of size |S| ≤ δ|Ln| there exists a unique neighbour of S in Gn, namely a vertex in Rn that is connected to  exactly one vertex in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' We only require that sets of left vertices have unique neighbours, and arbitrarily  small right side sets may have no unique neighbour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Alon and Capalbo [AC02] construct several explicit families of unique neighbour expanders, via an elegant  composition of a Ramanujan graph and a gadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' They construct three families of general (non-bipartite)  graphs in which all small sets have unique neighbours, and one family of slightly unbalanced bipartite graphs  where small sets on the left have unique neighbors on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In their construction the left side is 22/21  times bigger than the right side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' The more imbalanced the graph, the harder it is for small left hand side  sets to expand into the right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Capalbo et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' [Cap+02] construct arbitrarily unbalanced bipar-  tite graphs that are lossless expanders, a notion strictly stronger than unique neighbour expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Their  construction is based on a sequence of somewhat involved composition steps using randomness conductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Our main theorem is an efficient construction of an infinite family of bipartite unique neighbour expanders  for any constant imbalance α, and any sufficiently large left-regularity degrees of a specific form:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' There is a function ˆq : N × R → N such that for every integer c0 > 5 and real number α > 1,  if q > ˆq(c0, α) is a prime power and αc0(q + 1) is an integer, then there is a polynomial-time construction of  an infinite family of (c0(q + 1), αc0(q + 1))-biregular unique neighbour expanders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  The theorem is proven in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='2, and provides a way to compute ˆq(c0, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Here are some computed  values of ˆq(c0, α) for several values of c0, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  ∗Irit Dinur acknowledges support by ERC grant 772839 and ISF grant 2073/21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='03072v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='CO]  8 Jan 2023  c0  α  ˆq(c0, α)  10  2  18907  35  2  1492  100  100  136051  100  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='01  1135  Notice that ˆq(co, α) increases with α, reflecting the fact that constructions with larger α (namely, more  imbalanced sides) are harder to come by, and require larger degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  The construction uses an infinite family of bipartite Ramanujan graphs, namely graphs whose non-  trivial spectrum is contained in the spectrum of the (c, d)-biregular tree (see Preliminaries for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' We  construct the unique neighbour expander family by taking a family of bipartite Ramanujan graphs and  combining them with a fixed size graph (“gadget”), with a good unique neighbour property (small sets have  unique neighbours), whose existence is shown via the probabilistic method (Lemma 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' The combination  is done as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' We first place a copy of the gadget for every right side vertex of the Ramanujan graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  The vertex is replaced by the right side of the gadget, and its neighbours are identified with the left side of  the gadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' The gadget is used to route the neighbours of each left side vertex in the Ramanujan graph to  its neighbours in the product graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Expansion in the product graph comes from unique neighbour expansion of the gadget together with  low degree vertices in the Ramanujan graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Sufficiently low degree vertices are guaranteed to exist thanks  to the following (new) bound on the average degree of induced subgraphs of bipartite Ramanujan graphs,  which may be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Let G = (L ⊔ R, E) be a (c, d)-biregular Ramanujan graph, and let ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Then there exists  δ > 0, that depends only on ε, c, d, such that for every S ⊂ L of size |S| ≤ δ|L|, the set N(S) ⊆ R of the  neighbours of S satisfies  c|S|  |N(S)| ≤ 1 + (1 + ε)  �  d − 1  c − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  The theorem shows that every small set on the left side admits neighbours on the right side with low degree  in the induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' The proof involves recursive analysis of non-backtracking paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Interestingly, the  recursion has a nice solution only when the graph is Ramanujan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' It is unclear whether this method can be  extended to “nearly-Ramanujan” graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Combining the average degree upper bound with the gadget, the low-degree right-side vertices in the  Ramanujan graph imply a small set of left-side vertices in the gadget;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' this set will have a unique neighbour  in the gadget, which gives (via Lemma 12) a unique neighbour in the constructed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Even though Ramanujan graphs are the best spectral expanders one can hope for, an efficient construc-  tion of Ramanujan graphs (be them bipartite or not) does not immediately imply that we can construct  unique neighbour expanders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In the d-regular case, Kahale shows ([Kah95, Thm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='2]) that there are nearly-  Ramanujan graphs with expansion at most d/2, which is not enough for unique neighbour expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In fact,  recently Kamber and Kaufman [KK22] proved that some Ramanujan graphs strongly fail to have unique  neighbour expansion, by giving explicit constructions of arbitrarily small sets that do not admit a unique  neighbour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  As mentioned, the graph product we define requires a fixed size gadget, whose proof of existence is not  constructive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In principle, such a gadget could be found by exhaustive search since we are working in a  constant size search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' The gadget’s size in our construction is at least cubic in q, so exhaustive search  is impractical for even small values of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Unfortunately we know of no efficient construction of a gadget with  the required parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' It is possible that the graph sampling method present in [AK19] can be used to  construct fixed size gadgets more efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  The rest of this work is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In Section 2 we survey some of the uses of unique neighbour  expanders, and mention known constructions of such graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Section 3 provides basic definitions and results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Our main technical tool, that asserts the low induced degree in bipartite Ramanujan graph, is stated and  proven in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' We prove the existence of a fixed-size gadget with good unique neighbour expansion  2  properties in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In Section 6 we define the way we use the Ramanujan graphs and the gadget to  construct bipartite unique neighbour expanders, and by that prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  2  Related work  One of the prominent uses of bipartite expanders in general and bipartite unique neighbour expanders in  particular, and the motivation for this work, is the construction of error correcting codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' The works of  Tanner [Tan81] and later Sipser and Spielman [SS96] construct linear error correcting codes C(B, C0) from  a bipartite graph B and a smaller linear code C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' It is shown that under some assumptions on the code C0  and the expansion properties of the bipartite graph B, the resulting code has good distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' This gives a  way to take a family of graphs and transform it into a family of codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Our work describes a construction  that, in a sense, goes the other way around: given two bipartite graphs, B and B0, we view B0 as a parity  check graph1 of the base code C0, and B plays the role of the underlying graph of a Tanner code C(B, C0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Our output graph is just the parity check graph of C(B, C0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' We give full details of this graph product in  Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  In [DSW06;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' BV09] it is shown that codes constructed on top of unique neighbour expanders are weakly  smooth and can be used to construct robustly testable codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' But the uses of unique neighbour expanders are  not limited to error correcting codes: for example, such graphs may be used in the context of non-blocking  networks, where it is required to connect several input-output terminals via paths in a non-intersecting  fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Arora et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' [ALM96] use graphs with expansion beyond the d/2 barrier to establish the existence  of unique neighbours in the graph, which are useful in finding input-output paths in the online settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Roughly speaking, when routing a set of input-output pairs, the algorithm can use all unique neighbours  freely since they are guaranteed not to interfere with any other paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Pippenger [Pip93] uses explicit  constructions of spectral expanders in order to solve a similar problem, in the case where the route planning  is computed locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' There the spectral expansion of a graph is proven to imply a combinatorial expansion,  in a similar way to our Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Another use for unique neigbhour expanders is for load-balancing problems, such as the token distribution  problem described in [PU89], and the similar pebble distribution problem, briefly discussed in [AC02].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In  the latter, pebbles are placed arbitrarily on vertices of a graph, and need to be distributed via edges of the  graph such that no vertex has more than one pebble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Given that the total number of pebbles is small and  that the graph has the unique neighbour property, we have an efficient parallel algorithm for redistributing  the pebbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Alon and Capalbo [AC02] construct several families of unique neighbour expanders, one of them is a  family of bipartite graphs whose left side is 22/21 times bigger than the right side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Similar to the construction  presented at this work, each graph in the constructed family is a combination of a Ramanujan graph and a  fixed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' These graphs are not (bi-)regular but their degrees are bounded by a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Becker [Bec16]  uses a different family of 8-regular Ramanujan graphs in order to construct a family of (non-bipartite) unique  neighbour expanders, with the additional property that each graph in the family is a Cayley graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  A different approach to constructing bipartite graphs uses randomness conductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Randomness conduc-  tors are functions that receive a bitstring with some entropy (according to some measure of entropy), and a  uniformly random bitstring, and output a bitstring, with certain guarantees on its entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Some conduc-  tors can be constructed explicitly via a spectral method, and Capalbo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' [Cap+02] combine them in a  zig-zag-like fashion in order to construct an infinite family of bipartite lossless expanders, namely bipartite  graphs with fixed left-regularity c where small enough sets contained in the left side have at least c(1 − ε)  neighbours on the right side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' These graphs are trivially unique neighbour expanders, since a simple counting  argument shows that if a set expands by a factor of more than c/2, then it has unique neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  1This is a bipartite graph whose incidence structure is given by the parity check matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  3  3  Preliminaries  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='1  Expander graphs  In this work we deal with undirected graphs, that may contain multiple edges between two vertices, but do  not contain self-loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' For a graph G and a subset of its vertices S we denote by NG(S) the neighbourhood  of S, namely all vertices adjacent to some vertex in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' When the graph in discussion is obvious, we may  omit it and write N(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' We say that v is a unique neighbour of S if there is a unique u ∈ S that is adjacent  to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Let (Gn) be a series of graphs with the number of vertices growing to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' There are several well  studied notions of expansion in graph families;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' we note some of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Vertex expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' (Gn) is a (δ, α)-vertex expander if for every n and any subset S ⊆ VGn, if |S| ≤ δ|VGN |  we have that |NGN (S)| ≥ α|S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Edge expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' (Gn) is a (δ, α)-edge expander if for every n and any subset S ⊆ VGn, if |S| ≤ δ|VGN |  we have that at least an α-fraction of the edges with one endpoint in S have their other endpoint  outside of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Spectral expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Assume that (Gn) are all d-regular, and let An be the adjacency operator associated  with Gn, so An is indexed by vertices of Gn and (An)uv counts how many edges there are between  u and v in Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Let λ1 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' ≥ λVn be its spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' It can be seen that λ1 = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Then (Gn) is a  λ-spectral expander if for all n and i ̸= 1 we have |λi| ≤ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Unique neighbour expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' (Gn) is a δ-unique neighbour expander if for every n, any subset S ⊆ VGn  of size at most δ|VGN | has a unique neighbour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  These definitions apply to bipartite graphs Gn = (Ln ⊔ Rn, En) as well, with the exception that we usually  consider sets contained in the left side only, and require that Ln/Rn is a constant, normally greater than  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In this case we note that edge expansion is meaningless (since all edges leaving the left side enter the  right side), and if a bipartite graph is (c, d)-biregular, namely if all left-side vertices have degree c and all  right-side vertices have degree d, then the largest eigenvalue of the associated adjacency operator is  √  cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  It can be seen that for d-regular graphs, the best spectral expansion we can hope for is α = 2  √  d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  These graphs are known as Ramanujan graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='2  Bipartite Ramanujan graphs  Ramanujan graphs have the best spectral gap [Nil91], and their non-trivial eigenvalues are contained in the  spectrum of the infinite d-regular tree Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Similarly, in the bipartite case, Biregular Ramanujan graphs are  defined via their relation to the infinite biregular trees: the infinite (c, d)-biregular tree Tc,d, for d > c, has  the spectrum  λ ∈ spec(Tc,d) ⇔ |λ| ∈ {0} ∪  �√  d − 1 −  √  c − 1,  √  d − 1 +  √  c − 1  �  (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=', [GM88], [LS96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=') We therefore say that a finite (c, d)-biregular graph is bipartite Ramanujan if its  nontrivial eigenvalues lie in this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' That means that every eigenvalue λ of a bipartite Ramanujan graph  belongs to one of these classes:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Trivial: λ = ±  √  cd, with eigenvectors fixed on either sides, or λ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' λ ∈ [  √  d − 1 − √c − 1,  √  d − 1 + √c − 1] are the nontrivial positive eigenvalues;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' λ ∈ [−√c − 1 −  √  d − 1, √c − 1 −  √  d − 1] are the nontrivial negative eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Note that since the  graph is bipartite, λ is an eigenvalue if and only if −λ is an eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  4  By an extension of the Alon-Boppana bound, given in [FL96], this is the best spectral gap we can hope for,  at least as far as upper bounds for |λ| are concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' We note that unlike the d-regular case, we require a  lower bound to |λ| too, which is essential for our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  While there is a vast literature on the construction of d-regular Ramanujan graph (most prominently  [LPS88] and [Mar88]), less is known about bipartite Ramanujan graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In 2014 Marcus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' [MSS13]  proved the existence of biregular graphs with one-sided spectral graphs that resemble the Ramanujan bounds:  these graphs satisfy the one-sided inequality only, namely |λ| ≤  √  d − 1 + √c − 1 for every nontrivial eigen-  value λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Gribinski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' [GM21] showed a polynomial-time construction of such graphs, for every degrees  (d, kd) for any integers d, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' These graphs do not suffice for our analysis, since we make explicit use of the  lower bound |λ| ≥  √  d − 1 − √c − 1 too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  In 2021 Brito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' [BDH22] proved that a random biregular graph is nearly Ramanujan with high  probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Interestingly, and unlike other works in this field, our proof strongly relies on the graph to be  exactly Ramanujan, so we cannot use those constructions either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  We use an explicit construction of bipartite Ramanujan graphs (with both bounds on non-trivial eigen-  values) given by Ballantine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' :  Theorem 3 ([Bal+15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' For every prime power q, there exists an explicit construction of a (q + 1, q3 + 1)-  biregular Ramanujan graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  4  Vertex expansion in biregular Ramanujan graphs  Our main technical tool is the following theorem showing that bipartite Ramanujan graphs exhibit excellent  left-to-right expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' We restate the theorem for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Let G = (L ⊔ R, E) be a (c, d)-biregular Ramanujan graph, and let ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Then there exists  δ > 0, that depends only on ε, c, d, such that for every S ⊂ L of size |S| ≤ δ|L|, the set N(S) ⊆ R of the  neighbours of S satisfies  c|S|  |N(S)| ≤ 1 + (1 + ε)  �  d − 1  c − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  We note that the quantity on the left hand side of the inequality can be interpreted as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Look  at the bipartite graph induced by taking the vertices S on the left and N(S) on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Since every left  vertex has c outgoing edges, the total number of edges in the induced subgraph is c|S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' This means that  the expression on the left hand side of the inequality is exactly the average degree of the right side of the  induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Interestingly, the bound in this theorem is strictly stronger than what we would get from  just applying the expander mixing lemma which amounts to  c|S|  |N(S)| ≤ (1 + ε) ·  �  1 + d − 1  c − 1 + 2  �  d − 1  c − 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  See Claim 4 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' The fact that we improve upon the expander mixing lemma is perhaps not surprising  since our analysis is based on enumerating non-backtracking paths, and not just on magnitude of the second  largest eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' We also use lower bounds on the magnitude of all nontrivial eigenvalues, whereas the  expander mixing lemma uses just upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='1  Comparison to known bounds  As noted above, Theorem 2 is an improvement of the bound that the expander mixing lemma gives in similar  settings, which only uses the one-sided inequality |λ| ≤  √  d − 1 + √c − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' For reference, we state and prove  the expander mixing lemma for bipartite Ramanujan graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  5  Claim 4 (Expander mixing lemma for bipartite Ramanujan graphs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Let G = (L⊔R, E) be a (c, d)-biregular  Ramanujan graph, and let ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Then there exists δ > 0 such that for every S ⊆ L of size |S| ≤ δ|L|, the  neighbourhood of S satisfies  c|S|  |N(S)| ≤ (1 + ε)  �  1 + d − 1  c − 1 + 2  √  d − 1  √c − 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' The expander mixing lemma for biregular graphs says that for every S ⊆ L, T ⊆ R we have  ����  |e(S, T)|  |E|  − |S|  |L| · |T|  |R|  ���� ≤  λ  √  cd  �  |S|  |L| · |T|  |R|  where λ is the second largest eigenvalue of G (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=', [Hae95]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' It is clarified that we consider the spectrum  of G as an adjacency operator, so the largest eigenvalue is  √  cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Picking T = N(S) means all edges coming out from S are in the cut, namely |e(S, T)| = c|S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Plugging  that in gives  ����  c|S|  c|L| − |S|  |L| · |N(S)|  |R|  ���� ≤  λ  √  cd  �  |S|  |L| · |N(S)|  |R|  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Multiplying both sides by |L|  |S| gives  ����1 − |N(S)|  |R|  ���� ≤  λ  √  cd  �  |S|  |L| · |N(S)|  |R|  |L|  |S| =  λ  √  cd  �  |N(S)|  |R|  |L|  |S| =  λ  √  cd  �  |N(S)|  |S| �  d  c = λ  c  �  |N(S)|  |S|  (1)  where we also used the fact that |E| = c|L| = d|R|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Let us assume that |S| = α|L|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Then we can upper bound |N(S)| by  |N(S)| ≤ c|S| = αc|L| = αd|R|  and so we have  1 − |N(S)|  |R|  ≥ 1 − dα|R|  |R|  = 1 − dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  We square (1) and plug in the last inequality to get  (1 − dα)2 ≤ λ2  c · |N(S)|  c|S| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Recall that G is bipartite Ramanujan, so |λ| ≤  √  d − 1 + √c − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Use that and rearrange:  c|S|  |N(S)| ≤ λ2  c (1 − dα)−2  ≤ d − 1 + c − 1 + 2  √  d − 1√c − 1  c  (1 − dα)−2  ≤ d − 1 + c − 1 + 2  √  d − 1√c − 1  c − 1  (1 − dα)−2  =  �  1 + d − 1  c − 1 + 2  √  d − 1  √c − 1  �  (1 − dα)−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  The claim is proven by noting that there is some δ > 0 such that (1 − dα)−2 ≤ 1 + ε for every α < δ, namely  whenever |S| ≤ δ|L|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  6  Kahale proved ([Kah95, Thm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='2]) that in d-regular Ramanujan graphs (not necessarily bipartite), small  induced subgraphs have average degree at most 1 +  √  d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Interestingly, this result can be deduced almost  immediately from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' This is due to the following lemma, proven in Appendix A, which asserts that  the edge-vertex incidence graph (see [SS96]) of a d-regular Ramanujan graph is a (2, d)-biregular Ramanujan  graph:  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Let G be a d-regular Ramanujan graph, and G′ its edge-vertex incidence graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Then G′ is a  (2, d)-biregular Ramanujan graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  We state and prove Kahale’s bound, but we will not use it in our construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Let G = (VG, EG) be a d-regular Ramanujan graph, and let ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Then there exists δ > 0  such that for every induced subgraph S with at most δ|VG| vertices, the average degree of S is at most  dS := 2|ES|  |VS| ≤ 1 + (1 + ε)  √  d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Let G = (VG, EG) be a d-regular Ramanujan graph and ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' We define G′ = (LG′ ⊔ RG′, EG′) as  the edge-vertex incidence graph, namely LG′ = EG, RG′ = VG, and for every edge e = {u, v} in G we have  the two edges {e, u} and {e, v} in G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Since the degree of every vertex in G is d, and since every edge has  two endpoints, we have that G′ is a (2, d)-biregular graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Lemma 5 asserts that G′ is Ramanujan in the  bipartite sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' By Theorem 2, there exists δ > 0 such that if T ⊆ LG′ is of size at most δ|LG′|, then  2|T|  |NG′(T)| ≤ 1 + (1 + ε)  √  d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  A subgraph S = (VS, ES) of G satisfies that ES is a subset of left-side vertices in G′, VS is a subset of  right-side vertices in G′, and VS = NG′(ES) (because if an edge is in the subgraph then both of its endpoints  are in the subgraph, and we assume that the subgraph does not contain an isolated vertex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Therefore, if ES  is sufficiently small, namely if |ES| ≤ δ|LG′| = δ|EG|, then by Theorem 2 the average degree of NG′(ES) = VS  is bounded by 1 + (1 + ε)  √  d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  We add that if we wish to find a bound the number of vertices, we note that |ES| ≤ d  2|VS|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' So every  induced subgraph with no more than  2  dδ|EG| = δ|VG| vertices will satisfy the required average degree  bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='2  Proof of Theorem 2  Theorem 2 is proven by enumerating non-backtracking paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' A non-backtracking path of length ℓ is a  sequence of edges ((s(ei), t(ei)))ℓ  i=1 such that for every i, t(ei) = s(ei+1) and s(ei) ̸= t(ei+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  For a bipartite graph G and a subset S of left side vertices we define Mℓ(S) to be the number of all non-  backtracking paths whose all left-side vertices are in S, and Mℓ(S, G) to be the number of non-backtracking  paths whose first and last left-side vertices are in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Clearly Mℓ(S) ≤ Mℓ(S, G), as paths of the latter type  may leave S ⊔N(S) (before re-entering S at the last step).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' We use a lower bound on Mℓ(S) due to [Kam19]:  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' For every undirected bipartite graph G = (LG ⊔ RG, EG) and integer l it holds that  Mℓ(LG) ≥ |EG|  ��  ( ¯dL − 1)( ¯dR − 1)  �ℓ−1  where ¯dL, ¯dR are the average degrees of the left and right sides of G respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  We state and prove an upper bound on Mℓ(S, G):  7  Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Let G be a (c, d)-biregular Ramanujan graph with n vertices on the left side, and S a subset of  the left side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Then for every integer ℓ:  M2ℓ(S, G) ≤ |S|  �  (2 +  √  d − 1)ℓ + 2  �  (c − 1)ℓ/2(d − 1)ℓ/2  provided that S is small enough:  |S|(c − 1)ℓ/2(d − 1)ℓ/2 ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  (2)  Before proving the upper bound, we show how these bounds can be combined to obtain Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Let ℓ be an integer to be determined later, S ⊆ L a sufficiently small subset (where  sufficiently smalls means (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Denote by N(S) ⊆ R the neighbours of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' The subgraph induced on S ∪N(S)  has c|S| edges, with left degrees all c and average right degree ¯dR =  c|S|  |N(S)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Chaining the inequalities in Lemma 7 and Lemma 8, we have  c|S|  �  (c − 1)( ¯dR − 1)  � 2ℓ−1  2  ≤ M2ℓ(S) ≤ M2ℓ(S, VG) ≤ |S| ·  �  (2 +  √  d − 1)ℓ + 2  �  (c − 1)ℓ/2(d − 1)ℓ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Simplifying, we get,  c(c − 1)ℓ− 1  2 ( ¯dR − 1)ℓ− 1  2 ≤  �  (2 +  √  d − 1)ℓ + 2  �  (c − 1)ℓ/2 · (d − 1)ℓ/2  ( ¯dR − 1)ℓ− 1  2 ≤  �  (2 +  √  d − 1)ℓ + 2  � √c − 1  c  ��  d − 1  c − 1  �ℓ  ¯dR − 1 ≤  �  �  �  �  �  (2 +  √  d − 1)ℓ + 2  � √c − 1  � ¯d − 1  c  �  ��  �  ⋆  �  �  �  �  1/ℓ �  d − 1  c − 1  Since ¯d ≤ d, we have that ⋆ = O(ℓ), so ⋆1/ℓ = O(1), hence for a fixed ε > 0 there exists a constant ℓ (that  depends only on ε, c, d) such that ⋆1/ℓ ≤ 1 + ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' this ℓ determines, via inequality (2), a fixed δ such that  whenever |S| ≤ δn we have  ¯dR ≤ 1 + (1 + ε)  �  d − 1  c − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  We proceed to prove Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  For a bipartite graph G = (LG ⊔ RG, EG) and an integer ℓ, we define ALL  ℓ  , ALR  ℓ  , ARL  ℓ  , ARR  ℓ  as operators  corresponding to non-backtracking paths of length ℓ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  ALL  ℓ  : L2(LG) → L2(LG)  ,  (ALL  ℓ  f)(x) =  �  (e1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=',eℓ),t(eℓ)=x,s(e1),t(eℓ)∈LG  f(s(e1))  with the summation over all non-backtracking paths of length ℓ, and similarly for the other operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Let M be the operator corresponding to a single step from the right side G to the left side of G, namely  M has |RG| rows and |LG| columns, with Muv counting the number of edges between u ∈ RG and v ∈ LG  in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Then the following recursive formulae hold for every integer ℓ > 1:  M ⊤ALL  ℓ  = ARL  ℓ+1 + (d − 1)ARL  ℓ−1  M ⊤ALR  ℓ  = ARR  ℓ+1 + (d − 1)ARR  ℓ−1  MARL  ℓ  = ALL  ℓ+1 + (c − 1)ALL  ℓ−1  MARR  ℓ  = ALR  ℓ+1 + (c − 1)ALR  ℓ−1  8  The first formula is explained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Every non-backtracking path from R to L of length ℓ + 1 is  composed of a non-backtracking path from L to L of length ℓ plus an extra step (that’s the M ⊤ALL  ℓ  factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=')  The opposite is true, except for paths counted in M ⊤ALL  ℓ  that do backtrack, namely those made of a non-  backtracking path of length ℓ − 1, and walking back and forth along the same edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' There are d − 1 ways to  choose that edge (since it cannot be the one that was last in the path of length ℓ − 1, otherwise it wouldn’t  be counted in M ⊤ALL  ℓ  ), so we need to subtract (d − 1)ARL  ℓ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' The rest of the equations are explained in an  analog way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Due to symmetry we have:  (ALL  ℓ  )⊤ = ALL  ℓ  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  (ARR  ℓ  )⊤ = ARR  ℓ  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  (ALR  ℓ  )⊤ = ARL  ℓ  And since the graph is bipartite we have:  ALR  2ℓ = 0  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  ARL  2ℓ = 0  ALL  2ℓ+1 = 0  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  ARR  2ℓ+1 = 0  These equations yield a recursive formula for ALL  ℓ  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' with the following initial conditions:  ALL  2  = MM ⊤ − cI  ALL  4  = MM ⊤ALL  2  − (c − 1 + d − 1)ALL  2  − c(d − 1)I  MM ⊤ALL  ℓ  = ALL  ℓ+2 + ((c − 1) + (d − 1))ALL  ℓ  + (c − 1)(d − 1)ALL  ℓ−2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  ∀ℓ ≥ 4  (3)  The following lemma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' proven in Appendix A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' suggests a way to find a non-recursive formula for ALL  2ℓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' given  such linear recursive relations with fixed coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Let (xn) be a series defined via a second order linear recurrence with fixed coefficients A, B ∈ C:  xn = Axn−1 + Bxn−2  Assume λ1 ̸= λ2 are (real or complex) roots of the characteristic polynomial λ2 − Aλ − B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Then there are  α, β ∈ C, that depend on the initial conditions x0, x1, such that  xn = αλn  1 + βλn  2  for every n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  If the characteristic polynomial has a single root λ of multiplicity 2, then there are α, β ∈ C such that  xn = αλn + βnλn  for every n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  We use the lemma to bound the eigenvalues of ALL  2ℓ given bounds on the spectrum of the biregular graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Let G be a (c, d)-biregular graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Then there is a sequence of polynomials with integer coeffi-  cients (pℓ(x)) such that for every eigenpair (λ, v) of G, pℓ(λ2) is an eigenvalue of ALL  2ℓ , and moreover, for  every λ ∈ R, if  |λ| ∈ {0} ∪ [  √  d − 1 −  √  c − 1,  √  d − 1 +  √  c − 1]  (4)  then  |pℓ(λ2)| ≤ (2 +  √  d − 1)ℓ(c − 1)ℓ/2(d − 1)ℓ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  (5)  9  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' The recursive formulae proven above (3) suggest that there is a series of polynomials pn(x) with  integer coefficients such that ALL  2n = pn(MM ⊤).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Note that the graph’s adjacency matrix is  AG =  � 0  M  M ⊤  0  �  And so, if (λ, v) is an eigenpair of G, then (λ2, v) is an eigenpair of  A2  G =  �MM ⊤  0  0  M ⊤M  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  This shows that pℓ(λ2) is an eigenvalue of ALL  2ℓ whenever λ is an eigenvalue of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' The converse is also true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  The formulae (3) can be transformed so as to convey that pn(x) satisfies these equations:  p1(x) = x − c  ,  p2(x) = x2 + (2 − 2c − d)x + c(c − 1)  xpn(x) = pn+1(x) + (c − 1 + d − 1)pn(x) + (c − 1)(d − 1)pn−1(x)  for all n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Setting n = 1 gives an equation involving p0(x), p1(x), p2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' We can solve this equation for  p0(x) and get a simpler description of the initial conditions:  p0(x) =  c  c − 1  ,  p1(x) = x − c  (6)  xpn(x) = pn+1(x) + (c − 1 + d − 1)pn(x) + (c − 1)(d − 1)pn−1(x)  (7)  for all n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  We fix some t that satisfies (4), namely such that  |t| ∈ {0} ∪ [  √  d − 1 −  √  c − 1,  √  d − 1 +  √  c − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  We first deal with the case where |t| ∈ (  √  d − 1 − √c − 1,  √  d − 1 + √c − 1), and later we will consider the  edge cases where t is one of the endpoints of the segment or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Let us write x = t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' We have that for this  fixed x, the series (pn(x))n satisfies a second order linear recurrence with fixed coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Using Lemma 9,  we conclude that there are functions α(x), λ1(x), β(x), λ2(x) that depend only on x, c and d, such that  pn(x) = α(x)(λ1(x))n + β(x)(λ2(x))n  (8)  for every n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  In order to find λ1, λ2 we solve for λ the characteristic polynomial, namely the following quadratic  equation derived from (7):  xλ = λ2 + (c − 1 + d − 1)λ + (c − 1)(d − 1)  To obtain  λ1,2(x) = x − (c − 1) − (d − 1) ±  �  ∆(x)  2  where  ∆(x) = x2 − 2x((c − 1) + (d − 1)) + (c − d)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  (9)  Using the initial values for p0(x), p1(x) from (6), and plugging back into (8) we get the equations  c  c − 1 = α(x)(λ1(x))0 + β(x)(λ2(x))0 = α(x) + β(x)  x − c = α(x)(λ1(x))1 + β(x)(λ2(x))1 = α(x)λ1(x) + β(x)λ2(x)  10  whose solution is  α(x) = (c − 1)x − (c − 1)2 − (c − 1) + (c − 1)(d − 1) + (c − 1)  �  ∆(x) − x + d − 1 +  �  ∆(x)  2(c − 1)  �  ∆(x)  β(x) =  c  c − 1 − α(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  We check when ∆(x) = 0 by solving (9) for x:  x1,2 = 2((c − 1) + (d − 1)) ±  �  4(c − 1 + d − 1)2 − 4(c − d)2  2  = (c − 1 + d − 1) ±  �  (c + d)2 − 4(c + d) + 4 − (c − d)2  = (c − 1 + d − 1) ±  �  c2 + 2cd + d2 − 4c − 4d + 4 − c2 + 2cd − d2  = (c − 1 + d − 1) ±  √  4cd − 4c − 4d + 4  = (c − 1 + d − 1) ± 2  √  c − 1  √  d − 1  = (  √  d − 1 ±  √  c − 1)2  We see that ∆(x) is quadratic in x and has roots at (  √  d − 1 ± √c − 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' This gives a nice factorization of  ∆(x):  ∆(x) = x2 − 2x((c − 1) + (d − 1)) + (c − d)2  =  �  x −  �√  d − 1 +  √  c − 1  �2� �  x −  �√  d − 1 −  √  c − 1  �2�  Recall that for the x we fixed we have √x = t ∈ (  √  d − 1 − √c − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  √  d − 1 + √c − 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' so the first term in the  product is negative and the second term is positive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' so ∆ < 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' and so λ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='2 are complex numbers (conjugate  to one another),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' with magnitude  |λ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='2|2 = (x − (c − 1) − (d − 1))2 − ∆(x)  4  = x2 − 2x((c − 1) + (d − 1)) + (c − 1 + d − 1)2 − (x2 − 2x((c − 1) + (d − 1)) + (c − d)2)  4  = (c + d − 2)2 − (c − d)2  4  = (c − 1)(d − 1)  (10)  A very similar calculation shows that α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' β are conjugates with magnitude  |α|2 = |β|2 =  x(x − cd)  ∆(x) · (c − 1)  This finishes the proof for all such x’s:  |pℓ(x)| = |α(x)λℓ  1 + β(x)λℓ  2| ≤ |α(x)λℓ  1| + |β(x)λℓ  2|  = |α(x)||λ1|ℓ + |β(x)||λ2|ℓ  = 2  �  x(x − cd)  ∆(x) · (c − 1)(c − 1)ℓ/2(d − 1)ℓ/2  We keep in mind that x is fixed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' so the expression is smaller than (2 +  √  d − 1) · ℓ · (c − 1)ℓ/2(d − 1)ℓ/2 for  large enough ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  We are left with the cases x = t2 for t = 0,  √  d − 1 ± √c − 1:  11  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' We use the same methods and find that the characteristic polynomial is  λ2 + (c − 1 + d − 1)λ + (c − 1)(d − 1)  whose roots are  λ1 = −(c − 1)  ,  λ2 = −(d − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Using the initial conditions (p0(0) = c/(c − 1), p1(0) = −c) we obtain  α(0) =  c  c − 1  ,  β(0) = 0  and using the fact that c < d we get  |pℓ(0)| = |α(0)λℓ  1 + β(0)λℓ  2|  =  c  c − 1(c − 1)ℓ  < 2l(c − 1)ℓ/2(c − 1)ℓ/2  < 2l(c − 1)ℓ/2(d − 1)ℓ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' t =  √  d − 1 + √c − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Then x = t2 = (  √  d − 1 + √c − 1)2 = d − 1 + c − 1 + 2  √  d − 1√c − 1, and the  characteristic polynomial has a single root of multiplicity 2, namely  λ = x − (c − 1) − (d − 1)  2  =  √  d − 1  √  c − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  The solution, therefore, takes the form  pn(x) = (α(x) + nβ(x))(c − 1)n/2(d − 1)n/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Using the initial values we get  α(x) =  c  c − 1  ,  β(x) =  x − c  √  d − 1√c − 1 −  c  c − 1 = 2 +  d − 2  √  d − 1√c − 1 −  c  c − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  1 <  c  c−1 ≤ 2 so β(x) ≤  √  d − 1 + 1, and in total we get  |pℓ(x)| = |α(x) + ℓβ(x)|(c − 1)ℓ/2(d − 1)ℓ/2  ≤  �����  1  ℓ ·  c  c − 1  ���� + |β(x)|  �  ℓ(c − 1)ℓ/2(d − 1)ℓ/2  ≤  �  2 +  √  d − 1  �  ℓ(c − 1)ℓ/2(d − 1)ℓ/2  For sufficiently large ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' t =  √  d − 1 − √c − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' We get x = t2 = d − 1 + c − 1 − 2  √  d − 1√c − 1, and the rest follows the same  calculations as in the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Bounds on the spectrum of ALL  2ℓ give bounds on the number of non-backtracking paths completely con-  tained in a small set, hence gives Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  12  Proof of Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Recall that M2ℓ(S, G) counts the number of non-backtracking paths of length 2ℓ that  start and end in S, so by the definition of the ALL  n  operatore, we have M2ℓ(S, G) = ⟨ALL  2ℓ 1S, 1S⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  We note that ALL  2ℓ 1L = c(c − 1)ℓ−1(d − 1)ℓ1L, because every non-backtracking path starting at a given  vertex is made of picking the first left-to-right edge (we have c such edges to pick from), and then alternating  between picking any of the d or c edges adjacent to the current vertex, except for the edge we picked to get  to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Write 1S =  |S|  n 1L + r, with r ⊥ 1L, and ∥r∥2  2 ≤ ∥1S∥2  2 = |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Since the graph is Ramanujan, the  nontrivial eigenvalues in its spectral decomposition have their absolute value in the set {0} ∪ [  √  d − 1 −  √c − 1,  √  d − 1 + √c − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  We only care about the nontrivial eigenvalues because r ⊥ 1L, hence in the  writing of r in the orthogonal basis made of eigenvectors, only eigenvectors with nontrivial eigenvalues  appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' We use Lemma 10 to get  ⟨ALL  2ℓ r, r⟩ ≤ (2 +  √  d − 1)ℓ(c − 1)ℓ/2(d − 1)ℓ/2 · ∥r∥2  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Combine everything to get  M2ℓ(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' G) = ⟨ALL  2ℓ 1S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' 1S⟩ =  �  ALL  2ℓ  |S|  n 1L + r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' |S|  n 1L + r  �  = |S|2  n2 ⟨ALL  2ℓ 1L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' 1L⟩ + ⟨ALL  2ℓ r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' r⟩  = |S|2  n2 · c(c − 1)ℓ−1(d − 1)ℓ⟨1L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' 1L⟩ + ⟨ALL  2ℓ r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' r⟩  ≤ |S|2  n c(c − 1)ℓ−1(d − 1)ℓ + (2 +  √  d − 1)ℓ(c − 1)ℓ/2(d − 1)ℓ/2 ∥r∥2  2  ≤ |S|  �|S| · c · (c − 1)ℓ/2(d − 1)ℓ/2  n(c − 1)  + (2 +  √  d − 1)ℓ  �  (c − 1)ℓ/2(d − 1)ℓ/2  ≤ |S|  �  c  c − 1 + (2 +  √  d − 1)ℓ  �  (c − 1)ℓ/2(d − 1)ℓ/2  ≤ |S|  �  (2 +  √  d − 1)ℓ + 2  �  (c − 1)ℓ/2(d − 1)ℓ/2  5  Random gadget  In this section we prove the existence of bipartite graphs such that every small set of left-side vertices has  a unique neighbour on the right side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' We draw a random biregular graph from a similar distribution as in  [Pip77], and use techniques similiar to [Vad+12, Thm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' For every integers L, R, c, d with Lc = Rd, L > R, c > 3, if k is an integer that satisfies the  inequality  k  c−3  2  ≤  1  2Le ·  � R  3ec  � c−1  2  then there is a (c, d)-biregular graph with sides [L] and [R] such that every set of left vertices of size at most  k has a unique neighbour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  We draw a random (c, d)-biregular graph in the following way: fix L vertices on the left side and R  vertices on the right side (cL = dR), write c copies of each left-side vertex and d copies of each right-side  vertex, and connect them via a uniformly random matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' That is, pick a uniformly random permutation  π : L × [c] → R × [d], and for every u ∈ L, v ∈ R, i ∈ [c], j ∈ [d], if π(u, i) = (v, j), then add (u, v) as an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Note that we allow multiple edges between two vertices (if there are several i, j satisfying π(u, i) = (v, j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  13  Let G be a random bipartite graph with L vertices on the left side and R vertices on the right side drawn  from said distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Let A be a subset of left vertices of size k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' We note that if A expands by at least  (c + 1)/2, then, by a simple counting argument, A has a unique neighbour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' It is therefore sufficient to find  the probability that A expands by at least (c + 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Let us fix an arbitary ordering of the ck edges leaving A, and denote it e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' , eck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' We say that ei is a  repeat if it touches a previously covered vertex, that is, if its right endpoint is contained in the set of right  endpoints of the set e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' , ei−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' We note that if A does not expand by at least (c + 1)/2, then, again by a  simple counting argument, there are at least (c − 1)k/2 repeats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' This is because the number of repeats and  the size of the set of the neighbours of A add up to the number of edges leaving A, namely ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  We note that for every i, ei is a repeat if it touches one of i − 1 or less previously covered vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' This  means that Pr[ei is a repeat] ≤ i−1  R < ck  R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Moreover, if we condition on the event that some of the first i − 1  edges are also repeats, then the probability that ei is a repeat may only decrease, since it means that there  are less “forbidden” endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' We conclude that for every set of l edges:  Pr[ei1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' , eil are repeats] =  l�  j=1  Pr[eij is a repeat | ei1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' eij−1 are repeats] <  �ck  R  �l  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  If A expands too little, then there are many repeats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' We can use it to bound the probability that A has  no unqiue neighbour:  Pr[A has no unique neighbour] ≤ Pr[A expands by < (c + 1)/2]  ≤ Pr[there are at least (c − 1)k/2 repeats]  ≤  �  i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=',i(c−1)k/2∈(  ck  (c−1)k/2)  Pr[{ei1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' , ei(c−1)k/2} are repeats]  <  � ck  c−1  2 k  � �ck  R  � c−1  2 k  And by a union bound over the possible choices of A:  Pr[∃ “bad” A of size k] ≤  �L  k  �  Pr[A expands by < (c + 1)/2]  ≤  �L  k  � � ck  c−1  2 k  � �ck  R  � c−1  2 k  ≤  �Le  k  �k � cke  c−1  2 k  � c−1  2 k �ck  R  � c−1  2 k  =  �  Le  k ·  � 2ce  c − 1 · ck  R  � c−1  2 �k  ≤  �  Le  k ·  �3eck  R  � c−1  2 �k  (11)  Where the last inequality follows from assuming that c ≥ 3 so  2c  c−1 ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  We are now ready to prove Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Proof of Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Let us draw a (c, d)-biregular graph G = ([L] ⊔ [R], E) from the distribution described  above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Let k be an integer satsifying (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Using a union bound and the inequality in (11), we have (where  14  probability is taken over the choice of G):  Pr[∃ “bad” A ⊆ [L] of size ≤ k] =  k  �  a=1  Pr[∃ “bad” A ⊆ [L] of size a]  ≤  k  �  a=1  �  Le  a ·  �3eca  R  � c−1  2 �a  <  ∞  �  a=1  �  Le  k ·  �3eck  R  � c−1  2 �a  =  ∞  �  a=1  �  k  c−1  3  Le ·  �3ec  R  � c−1  2 �a  ≤  ∞  �  a=1  �1  2  �a  < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  We see that with strictly positive probability, a random graph has no “bad” subsets of size ≤ k, hence there  exists a graph with the desired unique neighbour property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  6  Construction  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='1  Routed product definition  Let us begin with a brief coding theory motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' An error-correcting code is often given via an m × n  parity check matrix H, so that C = Ker H ⊆ {0, 1}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' The matrix H can be visualized as a bipartite graph,  called the parity check graph, with n left and m right vertices, and an edge i ∼ j whenever H(j, i) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' A  Tanner code is defined given a bipartite graph B and a base code C0 = Ker H0 [Tan81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' One way to view  the routed product is through the point of view of codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Consider the parity check graph B0 of H0 and  define the routed product of B and B0 to be simply the parity check graph of the Tanner code C(B, C0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Here is a more detailed and combinatorial definition of the routed product without mention of codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Let G = (L ⊔ R, E) be a (c, d)-biregular graph and G0 = (L0 ⊔ R0, E0) a (c0, d0)-biregular graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  We  think of G as a big graph (in practice, an infinite family of Ramanujan graphs), and G0 as a fixed size graph  (gadget).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Assume that |L0| = d, and let us think of the edges of G as a function E : R × [d] → L which  maps a right side vertex v and an index i to the ith neighbour of v in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  We can define the routed product graph G′ = G ◦ G0 as the bipartite graph whose left side is L, right  side is the cartesian product R × R0, and the set of edges is  E′ = {(E(v, i), (v, j)) : v ∈ R, i ∈ [d], j ∈ [R0], (i, j) ∈ E0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  That is, we write R0 copies of each vertex in R, and every right side vertex v in the big graph G and an  edge (i, j) in the small gadget G0 gives an edge between the ith neighbour of v in G, and the jth vertex of  the copy of G0 assigned to v in G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Otherwise put, we use G0 to route every edge of the big graph G to c0  edges in the product graph G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  More precisely, for every v ∈ R, the bipartite subgraph of G′ whose left side is NG(v) and right side  is (v, ·) is isomorphic to G0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' This means that, roughly speaking, unique neighbours are inherited from the  small graph to the product graph:  Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Let S ⊆ L, v ∈ NG(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Define S′ = {i : E(v, i) ∈ S} ⊆ [d] as the indexed neighbours of v in  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' If S′, as a set of vertices in the gadget G0, has a unique neighbour j ∈ R0 in G0, then (v, j) is a unique  neighbour of S in the product graph G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  The proof is immediate while staring at Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' 1, but for the sake of completion it is given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  15  Figure 1: An example of a bipartite graph G (dashed, red), a small gadget G0 (dotted, green), and the  routed product G′ = G ◦ G0 (solid, blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' The set S ⊆ L has a neighbour v ∈ R, and so S is associated with  a set S′ of left side vertices of the copy of G0 associated with v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Since (i′, j) is the only edge connecting j  to S′ in G0, we have that (v, j) is a unique neighbour of S in G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  16  S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='2  Proof of Theorem 1  Let q be a prime power, c0 an integer, and α > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Assume that αc0(q + 1) is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' We construct an  infinite family of (c0(q + 1), αc0(q + 1))-biregular graphs with the unique neighbour property under some  assumptions specified below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Denote c = q + 1 and d = q3 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' By Theorem 3 there is an efficient construction of an infinite family  of (c, d)-biregular Ramanujan graphs (Gn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Let G0 = (L0 ⊔ R0, E0) be a gadget: a c0-left-regular bipartite  graph with |L0| = d = q3 + 1 vertices on the left side and R0 vertices on the right side, such that every  left-side set of sufficiently small size admits a unique neighbour on the right side, where “sufficiently small”  here means the bound given in Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' For the constructed graph to have the left side α times bigger  than the right side, we set R0 =  d  αc =  q3+1  α(q+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  We define G′  n = Gn ◦ G0 as the routed product of Gn and G0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' For the rest of this (short) proof let us  suppress n from the notation, for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Let ε < 1  q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' By Theorem 2, there exists δ > 0 such that for every S ⊆ L of size at most δ|L|, the “average  right degree” ¯dS, namely the average of the degrees of vertices in NG(S) in the induced subgraph S ⊔NG(S),  is bounded:  ¯dS :=  c|S|  |NG(S)| ≤ 1 + (1 + ε)  �  d − 1  c − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  We show that such S has a unique neighbour in G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  We note that d−1  c−1 = q2, so since ε < 1  q we have a vertex v ∈ R of “degree” at most q + 1 in G, that is, the  set S′ ⊆ [d] of v’s neighbours in S is of size at most q + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' By Lemma 12, if S′, as a set of left-side vertices  in G0, has a unique neighbour j in G0, then our original set S has a unique neighbour (v, j) in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  It remains to choose the parameters in a way that all left-side sets of size at most q + 1 have a unique  neighbour in G0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' By Lemma 11, we need to have:  (q + 1)  c0−3  2  ≤  1  2(q3 + 1)e ·  �  �  q3+1  α(q+1)  3ec0  �  �  c0−1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  (12)  The LHS is O(q  c0−3  2 ) and RHS is Θ(qc0−4), so if c0 > 5 then for sufficiently large q the construction gives a  unique neighbour expander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' That is, there exists some ˆq(c0, α) such that if q > ˆq then (12) holds, hence we  constructed a bipartite unique neighbour expander as promised in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  7  Future work  The main pitfall of our approach is the non-constructive nature of the gadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Theoretically since the gadget  has constant size this is no issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' However, exhaustive search is impractical even for small values of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' This  is because the gadget’s size is cubic in q so the search space is of size exponential in q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' A natural question  would be whether it is possible to construct such a gadget in an efficient way, since that would lead to the  whole unique neighbour expander family to be constructible in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' For the bipartite Ramanujan family  chosen in our work (the one by Ballantine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' [Bal+15]) we ask the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Question 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' For which prime power q and real number α ≥ 1 can one construct efficiently a biregular  graph with left side q3 + 1, right side  q3+1  α(q+1), such that every left side set of size at most q + 1 has a unique  neighbour?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  We note that the fixed size graph given in [AC02, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='3] is a good gadget (for α = 22/21 and the  edge-vertex incidence graphs of a 44-regular Ramanujan graph family), and indeed these graphs can be used  to construct bipartite unique neighbour expanders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Since we prove that a random gadget is, with non-negligble probability, good for our construction, it  may be interesting to construct such gadget by simply drawing random gadgets and testing whether they  are good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Since drawing is simple, we are left with the task of testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' We therefore ask:  17  Question 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Given a bipartite graph, can one efficiently find the smallest nonempty set of left-side vertices  that has no unique neighbours?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  We currently know of no better way than just enumerating all left-side sets, which is exponential in the  size of the graph, hence impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' We refer to [AK19] for an interesting approach to testing expansion of  random graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  The methods presented in this work are not limited to the (q+1, q3 +1)-biregular Ramanujan family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' We  can therefore ask the question the other way around – find a gadget (by sampling or any other way), and see  whether we can efficiently construct a bipartite Ramanujan family that will make it work, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' that would  allow us to rewrite the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' This emphasizes the well-known natural question of constructing  Ramanujan graphs with arbitrary degrees, specifically in the bipartite and biregular setting,  Question 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' For which integers c < d can one construct efficiently an infinite family of (c, d)-biregular  Ramanujan graphs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  We note that our construction is far from “right-side unique neighbour expansion,” as the complete right  side of a single gadget is a constant-size set with no unique neighbours on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' We wonder whether it is  possible to construct a bipartite graph where all small size sets (be them contained in either sides, or both)  have unique neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  18  References  [AC02]  Noga Alon and Michael Capalbo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' “Explicit unique-neighbor expanders”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In: The 43rd Annual  IEEE Symposium on Foundations of Computer Science, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' 2002, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' 73–  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  [AK19]  Benny Applebaum and Eliran Kachlon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' “Sampling graphs without forbidden subgraphs and un-  balanced expanders with negligible error”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In: 2019 IEEE 60th Annual Symposium on Foundations  of Computer Science (FOCS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
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+page_content='  [ALM96]  Sanjeev Arora, Frank Thomson Leighton, and Bruce M Maggs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' “On-line algorithms for path  selection in a nonblocking network”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In: SIAM Journal on Computing 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='3 (1996), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
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+page_content='  [Bal+15]  Cristina Ballantine, Brooke Feigon, Radhika Ganapathy, Janne Kool, Kathrin Maurischat, and  Amy Wooding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' “Explicit construction of Ramanujan bigraphs”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In: Women in numbers europe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Springer, 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' 1–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  [BDH22]  Gerandy Brito, Ioana Dumitriu, and Kameron Decker Harris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' “Spectral gap in random bipartite  biregular graphs and applications”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In: Combinatorics, Probability and Computing 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='2 (2022),  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
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+page_content='  [Bec16]  Oren Becker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' “Symmetric unique neighbor expanders and good LDPC codes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In: Discrete Applied  Mathematics 211 (2016), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' 211–216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' issn: 0166-218X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='1016/  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='dam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
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+page_content='022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' url: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='com/science/article/pii/  S0166218X16301810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  [BV09]  Eli Ben-Sasson and Michael Viderman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' “Tensor products of weakly smooth codes are robust”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  In: Theory of Computing 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
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+page_content='  [Cap+02]  Michael Capalbo, Omer Reingold, Salil Vadhan, and Avi Wigderson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' “Randomness conductors  and constant-degree expansion beyond the degree/2 barrier”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In: Proceedings of the 34th Annual  ACM Symposium on Theory of Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' 2002, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
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+page_content='  [DSW06]  Irit Dinur, Madhu Sudan, and Avi Wigderson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' “Robust local testability of tensor products of  LDPC codes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In: Approximation, Randomization, and Combinatorial Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Algorithms  and Techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
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+page_content=' “Spectra of hypergraphs and applications”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In: Journal  of number theory 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
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+page_content=' “Existence and polynomial time construction of bireg-  ular, bipartite Ramanujan graphs of all degrees”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In: arXiv preprint arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
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+page_content=' “Walk generating functions and spectral measures of infinite  graphs”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
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+page_content=' “Interlacing eigenvalues and graphs”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In: Linear Algebra and its applications  226 (1995), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
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+page_content='  [Kah95]  Nabil Kahale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' “Eigenvalues and expansion of regular graphs”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In: Journal of the ACM (JACM)  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
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+page_content='  [Kam19]  Amitay Kamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Lp Expander Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' arXiv: 1609.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='04433 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  [KK22]  Amitay Kamber and Tali Kaufman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' “Combinatorics via closed orbits: number theoretic Ramanu-  jan graphs are not unique neighbor expanders”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In: Proceedings of the 54th Annual ACM SIGACT  Symposium on Theory of Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
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+page_content='  [LPS88]  Alexander Lubotzky, Ralph Phillips, and Peter Sarnak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' “Ramanujan graphs”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In: Combinatorica  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='3 (1988), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
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+page_content='  [LS96]  Wen-Ch’ing Winnie Li and Patrick Solé.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' “Spectra of Regular Graphs and Hypergraphs and  Orthogonal Polynomials”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In: European Journal of Combinatorics 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='5 (1996), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' 461–477.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' doi:  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='1006/eujc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='0040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  19  [Mar88]  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Margulis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' “Explicit group-theoretical constructions of combinatorial schemes and their  application to the design of expanders and concentrators”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In: Problemy peredachi informatsii  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='1 (1988), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' 51–60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  [MSS13]  Adam Marcus, Daniel A Spielman, and Nikhil Srivastava.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' “Interlacing families I: Bipartite Ra-  manujan graphs of all degrees”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In: 2013 IEEE 54th Annual Symposium on Foundations of com-  puter science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' 529–537.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  [Nil91]  Alon Nilli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' “On the second eigenvalue of a graph”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In: Discrete Mathematics 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='2 (1991), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' 207–  210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  [Pip77]  Nicholas Pippenger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' “Superconcentrators”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In: SIAM Journal on Computing 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='2 (1977), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' 298–  304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  [Pip93]  Nicholas Pippenger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' “Self-routing superconcentrators”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In: Proceedings of the twenty-fifth annual  ACM symposium on Theory of Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' 1993, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' 355–361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  [PU89]  David Peleg and Eli Upfal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' “The token distribution problem”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In: SIAM journal on computing  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='2 (1989), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' 229–243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  [SS96]  Michael Sipser and Daniel A Spielman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' “Expander codes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In: IEEE transactions on Information  Theory 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='6 (1996), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' 1710–1722.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  [Tan81]  R Tanner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' “A recursive approach to low complexity codes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In: IEEE Transactions on information  theory 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='5 (1981), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' 533–547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  [Vad+12]  Salil P Vadhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' “Pseudorandomness”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' In: Foundations and Trends in Theoretical Computer  Science 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='1–3 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  20  8  Appendix A  We restate and prove the lemmas we used throughout the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Let G be a d-regular Ramanujan graph, and G′ its edge-vertex incidence graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Then G′ is a  (2, d)-biregular Ramanujan graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Let G = (V, E) a d-regular Ramanujan graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' The adjacency matrix of G′ is A =  � 0  M  M ⊤  0  �  where  M has |E| rows, each containing two 1’s, and |V | columns, each containing d 1’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Let v be an eigenvector of  A with eigenvalue λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' then v is an eigenvector of A2 with eigenvalue λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' We note that  A2 =  �  MM ⊤  0  0  M ⊤M  �  so it suffices to consider the spectrum of M ⊤M, which is essentially the operator corresponding to a walk  from a vertex of G to an edge that touches it and back to one of its endpoints (possibly the same vertex we  started at).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  For every v ∈ V , there are d ways to walk from it to an edge and then back to v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' all other legal paths  correspond to picking an edge touching v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' We conclude that M ⊤M = dI + A, so every eigenvalue λ of G′  satisfies λ2 = d + σ where σ is an eigenvalue of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  The lemma is proven by noting that |σ| ≤ 2  √  d − 1 (since G is Ramanujan), so  d − 2  √  d − 1 ≤ λ2 ≤ d + 2  √  d − 1  The terms on the extreme sides of the inequality can be verified to be (  √  d − 1 ± 1)2 so we get |λ| ∈  [  √  d − 1 − 1,  √  d − 1 + 1], as needed (recall that in G′ the left-regularity is c = 2 so √c − 1 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Let (xn) be a series defined via a second order linear recurrence with fixed coefficients A, B ∈ C:  xn = Axn−1 + Bxn−2  Assume λ1 ̸= λ2 are (real or complex) roots of the characteristic polynomial λ2 − Aλ − B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Then there are  α, β ∈ C, that depend on the initial conditions x0, x1, such that  xn = αλn  1 + βλn  2  for every n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  If the characteristic polynomial has a single root λ of multiplicity 2, then there are α, β ∈ C such that  xn = αλn + βnλn  for every n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' We note that for every n ≥ 2 we have  � xn  xn−1  �  =  �Axn−1 + Bxn−2  xn−1  �  =  �A  B  1  0  � �xn−1  xn−2  �  Denote the 2 × 2 matrix by D, so by induction,  � xn  xn−1  �  = Dn  �x1  x0  �  Let us diagonalize D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' The characteristic polyonmial is  pD(λ) = det(λI − D) =  ����  λ − A  −B  −1  λ  ���� = λ(λ − A) − B = λ2 − Aλ − B  21  If pD(λ) has two distinct roots λ1, λ2, then the matrix is diagonalizable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' that means that there exists a 2 × 2  matrix M such that D = M · diag{λ1, λ2} · M −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' We get:  � xn  xn−1  �  = M  �λ1  0  0  λ2  �n  M −1  �x1  x0  �  = M  �λn  1  0  0  λn  2  �  M −1  �x1  x0  �  We can compute M, M −1 explicitly, multiple the matrices and get α, β ∈ C such that xn = αλn  1 + βλn  2 as  required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Otherwise, if pD(λ) has a single root λ of multiplicity 2, then we can find its Jordan form, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' find M  such that  D = M  �λ  1  0  λ  �  M −1  Dn = M  �λ  1  0  λ  �n  M −1 = M  �λn  nλn−1  0  λn  �  M −1  Where the last equality follows from a simple induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Similarly, we get  � xn  xn−1  �  = M  �λ  1  0  λ  �n  M −1  �x1  x0  �  = M  �λn  nλn−1  0  λn  �  M −1  �x1  x0  �  And again we can find α, β ∈ C as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  For the following lemma we remind that G = (L ⊔ R, E) is a (c, d)-biregular graph, G0 = (L0 ⊔ R0, E0) is  a (c0, d0)-biregular graph, and G′ = G ◦ G0 is the routed product of G and G0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Recall that the edges of G′  are (E(v, i), (v, j)) when v ∈ R is a right side vertex of G, i ∈ [d], E(v, i) is the ith neighbour of v according  to G, and (i, j) ∈ E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Let S ⊆ L, v ∈ NG(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Define S′ = {i : E(v, i) ∈ S} ⊆ [d] as the indexed neighbours of v in  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' If S′, as a set of vertices in the gadget G0, has a unique neighbour j ∈ R0 in G0, then (v, j) is a unique  neighbour of S in the product graph G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Assume that i′ ∈ S′ is the unique neighbour of j in G0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' By the definition of the routed product  we have that (E(v, i′), (v, j)) is an edge in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' Since i′ ∈ S′ we have that E(v, i′) ∈ S, so indeed (v, j) is  a neighbour of S in G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' It is therefore remaining to show that it is unique, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' that E(v, i′) is the only  neighbour of (v, j) in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  The neighbours of (v, j) in G are E(v, i) for every i such that (i, j) ∈ E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' If E(v, i) ∈ S, then by the  definition of S′ we have that i ∈ S′, so i is a neighbour of j in E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content=' But we know that j is a unique neighbour  of S′ in E0, so we must have that i = i′, and indeed (v, j) is a unique neighbour of S in G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
+page_content='  22' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfTAM3/content/2301.03072v1.pdf'}
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+arXiv:2301.03983v1  [cs.IT]  10 Jan 2023
+On the Performance of Dual RIS-assisted V2I
+Communication under Nakagami-m Fading
+Mohd Hamza Naim Shaikh, Khaled Rabie◦, Xingwang Li#, Theodoros Tsiftsis†, and Galymzhan Nauryzbayev
+School of Engineering and Digital Sciences, Nazarbayev University, Nur-Sultan City, 010000, Kazakhstan
+◦Department of Engineering, Manchester Metropolitan University, Manchester, M15 6BH, UK
+#School of Physics and Electronic Information Engineering, Henan Polytechnic University, Jiaozuo 454000, China
+†Department of Informatics & Telecommunications, University of Thessaly, Greece;
+†School of Intelligent Systems Science and Engineering, Jinan University, China
+Email: {hamza.shaikh, galymzhan.nauryzbayev}@nu.edu.kz, ◦k.rabie@mmu.ac.uk,
+#lixingwang@hpu.edu.cn, †tsiftsis@ieee.org
+Abstract—Vehicle-to-everything (V2X) connectivity in 5G-and-
+beyond communication networks supports the futuristic intelligent
+transportation system (ITS) by allowing vehicles to intelligently
+connect with everything. The advent of reconfigurable intelligent
+surfaces (RISs) has led to realizing the true potential of V2X
+communication. In this work, we propose a dual RIS-based
+vehicle-to-infrastructure (V2I) communication scheme. Following
+that, the performance of the proposed communication scheme
+is evaluated in terms of deriving the closed-form expressions
+for outage probability, spectral efficiency and energy efficiency.
+Finally, the analytical findings are corroborated with simulations
+which illustrate the superiority of the RIS-assisted vehicular
+networks.
+Keywords— Reconfigurable intelligent surface (RIS), dual RIS,
+energy efficiency, spectral efficiency, vehicular communication.
+I. INTRODUCTION
+As a key enabler for intelligent transportation systems (ITSs),
+vehicle-to-everything (V2X) communication has sparked a re-
+newed interest in the research community. V2X encompasses
+a wide range of wireless technologies such as vehicle-to-
+pedestrian (V2P), vehicle-to-infrastructure (V2I), and vehicle-
+to-vehicle (V2V). Additionally, it also includes the vehicu-
+lar communications with vulnerable road users (VRUs), grid
+(V2G), network (V2N) and cloud (V2C) [1]. The V2X com-
+munications will be a critical component of the futuristic
+connected and self-driving cars, envisioned and enabled by
+the sixth-generation (6G) wireless technologies. Furthermore,
+the V2X communications will also enhance and transform
+the quality-of-service (QoS) in terms of unparalleled user
+experience, ultra-high road safety and air quality improvement.
+In addition, a slew of advanced applications will also be
+supported like platooning, trajectory alignments, exchanging
+sensor data and high precision maps, and so on [2]. Thanks to
+the enhanced capabilities of 6G, vehicles will receive accurate
+safety information, intelligent traffic management support, and
+innovative infotainment features. Thus, the 6G services will be
+used to create a fully automated, autonomous, and ubiquitously
+connected vehicular network [3].
+Recently, reconfigurable intelligent surfaces (RISs) have
+emerged as a breakthrough technology that offers a great deal
+in terms of wireless communication [4]. Inherently, RIS is a
+software-defined artificial structure made up of a large number
+of scattering passive elements, termed as reflecting units (RUs).
+These RUs are capable to adjust the electromagnetic (EM)
+properties of a reflected wave that is incident on them. Thus,
+RIS can use not only this ability to boost the received signal’s
+power, but also the capability to create an additional reflective
+link to mitigate the impact of blockages. With the large number
+of RUs, RISs are particularly known to have large spectral and
+energy efficiency [5]. As a result, RIS may be used to improve
+the quality of vehicular communication through establishing
+a low-cost, highly energy efficient indirect line-of-sight (LoS)
+communications [6].
+In [7], the authors investigated the outage performance for
+RIS-assisted vehicular communication networks. Likewise, the
+secrecy outage performance of RIS-aided vehicular communi-
+cations has been studied in [8]. RISs were also investigated for
+detecting VRUs such as cyclists, pedestrians and wheelchair
+users [9]. Specifically, the authors utilized RISs for enhancing
+the radar visibility for VRUs. Further, in [10], the authors
+provided a optimization framework for resource allocation
+in the RIS-aided vehicular communications. Specifically, they
+jointly optimized the power allocation, RIS reflection coeffi-
+cients and spectrum allocation for different QoS requirements
+of the V2V and V2I communication links. Likewise, in [11],
+the authors discussed a system model where RSU leverages RIS
+to connect the dark zones, i.e., areas blocked due by obstacles.
+Moreover, a comprehensive overview on the recent advances
+in 6G vehicular networks was provided in [12, 13], where the
+authors also described various open challenges and possible
+research directions.
+Motivated by the above, in this work, we investigate the
+performance of a dual RIS-assisted V2I communication net-
+work scenario. Specifically, the proposed scenario considers the
+uplink transmission where the vehicle is communicating with
+the base station. To enhance the communication capabilities, the
+vehicle is supported through two RISs which create a virtual
+line-of-sight (LoS) link, which, otherwise, was inherently non-
+LoS (NLoS). The major contributions can be summarized as
+• Explicitly, we invoked the central limit theorem (CLT) to
+characterize the received signal-to-noise ratio (SNR) for
+
+Vehicle-to-Vehicle (V2V)
+Vehicle-to-Infrastructure (V2I)
+Fig. 1. Schematic for the considered dual RIS-aided V2I communication.
+the proposed dual RIS case. Further, based on this, we
+derived the closed-form expression for outage probability.
+• Further, we derived the closed-form expressions for the
+upper and lower bounds of SE and EE of the proposed
+dual RIS-assisted V2I communication scenario.
+• Finally, as a performance benchmark, the proposed sce-
+nario is compared with the single RIS-assisted V2I com-
+munication and with RIS V2I communication. The results
+show the superiority of the proposed scenario of dual RIS-
+assisted V2I over the single RIS-assisted V2I communi-
+cation case.
+II. SYSTEM MODEL
+As illustrated in Fig. 1, in this work, we consider a V2I
+communication model, wherein the vehicular user (V) tries to
+communicate with a nearby base station (BS). Apart from the
+direct cellular link, a reflected path through RISs is considered
+to support this uplink transmission. In particular, we consider
+a dual RIS-assisted uplink V2I transmission with two RISs,
+one each placed near V and BS both, respectively. For the two
+RISs, the number of RUs is assumed to be M1 and M2 for
+RIS-1 and RIS-2, respectively, while keeping the total number
+of RUs unchanged, i.e., M1+M2 = N, where N is the number
+of RUs in large RIS for the single RIS scenario, which is the
+benchmark for comparison. Thus, based on RIS, the following
+scenarios are considered in this work
+• Dual RIS-assisted Transmission (DRAT): In DRAT, the
+transmission takes place only through the two RISs and
+the reflected link, as shown in Fig. 1.
+• Single RIS-assisted Transmission (SRAT): In SRAT, the
+transmission takes place through single large RIS which
+is placed near to BS.
+• Direct Cellular Transmission (DCT): In DCT, V commu-
+nicates with BS directly without utilizing RISs. Thus, the
+transmission is inherently NLoS and experiences a higher
+pathloss exponent. This would also serve as the baseline
+scheme for the performance comparison of the above two
+cases.
+A. Channel Model
+The channels between V-to-RIS-1 and RIS-2-to-BS can
+be modeled as deterministic LOS channels as the distances
+are small and the probability of having LoS is very high.
+However, the distance between RIS-1 and RIS-2 is large and
+thus the small scale fading for the channel between the ith
+element of RIS-1 and the jth element of RIS-2, denoted by
+hRR
+ij , is modeled through Nakagami-m fading. Hence, for
+i = {1, 2, . . ., M1} and j = {1, 2, . . ., M2}. Further, the
+distances related to the V-to-RIS-1, RIS-1-to-RIS-2 and RIS-2-
+to-BS links are represented by d1, dRR and d2, respectively.
+B. Received Signal Model
+The received base-band signal at BS, denoted by r, for the
+dual RIS-aided transmission case can be expressed as
+r =
+�
+B Pt
+��M1
+i=1
+�M2
+j=1 ejφ(1)
+i hRR
+ij ejφ(2)
+j
+�
+s + No,
+(1)
+where Pt is the transmit power constraint at V, B is the distance-
+dependent pathloss, s ∼ CN (0, 1) is the transmitted symbol,
+and No ∼ CN
+�
+0, σ2�
+is the additive white Gaussian noise
+(AWGN). Further, φ1 and φ2 are the phase of the V-to-RIS1
+and RIS2-to-BS channels. Further, for a link distance d, B at
+the carrier frequency of 3 GHz can be given by [14]
+B(d) [dB] =
+�
+−37.5 − 22 log10(d/1 m)
+if LOS,
+−35.1 − 36.7 log10(d/1 m)
+if NLOS.
+(2)
+Likewise, instantaneous SNR at BS can be formulated as
+γ =
+����
+�M1
+i=1
+�M2
+j=1 δije
+j
+�
+φ(1)
+i
++φ(2)
+j
+−ϕij
+�����
+2
+B Pt
+σ2
+,
+(3)
+where δij and ϕij denote the amplitude and phase of hRR
+ij .
+1) RIS Reflection Parameters: Now, SNR at BS can be
+maximized through adjusting the phase at RISs to cancel
+the resultant phase, i.e., φ(1)
+i
++ φ(2)
+j
+− ϕij = 0, for i =
+{1, 2, . . ., M1} and j = {1, 2, . . ., M2}. Thus, by substituting
+ϕij = φ(1)
+i
++ φ(2)
+j , ∀i, j, the received signal power at BS can
+be maximized. Consequently, maximized SNR corresponding
+to the optimal phase can be given as
+γmax =
+����M1
+i=1
+�M2
+j=1 δij
+���
+2
+B Pt
+σ2
+= A2B Pt
+σ2
+= A2 B ¯γ,
+(4)
+where A2 =
+���
+�M1
+i=1
+�M2
+j=1 δij
+���
+2
+is the cascaded channel gain
+provided by RISs, and ¯γ = Pt/σ2 is transmit SNR.
+Likewise, proceeding in the similar way, for the SRAT
+scenario, maximized SNR at BS can be given as1
+ˆγmax =
+��N
+i=1 βi
+�2
+¯γ = B2¯γ,
+(5)
+where βi is the amplitude of a channel between RIS and
+V, denoted by hRU
+i
+, i.e., hRU
+i
+= βie−jϕi, and B2 is the
+corresponding channel gain provided by single RIS.
+1For the SRAT scenario, the analysis is similar. Thus, the detailed description
+is omitted for the sake of brevity. In particular, for SRAT, large RIS with N
+RUs is present near BS, where N = M1 + M2. Likewise, the RIS-to-BS link
+is also modeled as Nakagami-m fading with the rest of the parameters being
+the same, as in DRAT, like transmit power constraint at V, etc.
+
+III. PERFORMANCE ANALYSIS
+This section initially evaluates SNR for the dual RIS-aided
+V2I scenario. Utilizing the SNR expressions formulated earlier,
+the outage probability, SE and EE are derived.
+A. Statistical Characterization of the Dual RIS Channel Gain
+Now utilizing CLT for M
+≫ 1, A = �M1
+i=1
+�M2
+j=1 δij
+can be approximated through a Gaussian distribution, i.e.,
+A ∼ N(µy, σ2
+y) [15], with a probability density function (PDF)
+given by
+fA(y) =
+
+
+
+1
+√
+2πσ2
+A exp
+�
+−(y−µA)2
+2σ2
+A
+�
+,
+if y > 0,
+0,
+if y = 0,
+(6)
+where µA = �M1
+i=1
+�M2
+j=1 µij, σ2
+A = �M1
+i=1
+�M2
+j=1 σ2
+ij. Here,
+µij and σ2
+ij are the mean and variance of the random variable
+δij, which follows the Nakagami-m distribution. Hence, µij =
+Γ(m1+ 1
+2 )
+Γ(m1)
+�� Ωm1
+m1
+�
+and σ2
+ij = Ωm1
+�
+1 −
+1
+m1
+�
+Γ(m1+ 1
+2 )
+Γ(m1)
+�2�
+,
+for all i = {1, . . . , M1} and j = {1, . . ., M2}.
+Likewise the cumulative distribution function (CDF) of A
+can be derived from its PDF as
+FA(y)=
+� y
+−∞
+fA(y)dy =
+�
+1−Q
+�
+y−µA
+σ2
+A
+�
+,
+if y > 0,
+0,
+if y = 0.
+(7)
+B. Outage Probability
+The normalized instantaneous rate, denoted by Rin, for the
+DRAT scenario can be formulated from (4) and expressed as
+Rin = log2 (1 + γmax) = log2
+�
+1 + A2¯γ
+�
+.
+(8)
+Now, the end-to-end outage from V to BS via RIS, denoted by
+Pout, can be defined in terms of a rate threshold, Rth, as
+Pout = Pr [Rin < Rth] = Pr
+�
+log2
+�
+1 + A2¯γ
+�
+< Rth
+�
+= Pr
+
+A <
+�
+2Rth − 1
+¯γ
+
+ = Pr [A < Υth] ,
+(9)
+where Υth =
+�
+2Rth −1
+¯γ
+. Thus, the closed-form expression of
+the outage probability DRAT can be evaluated as
+Pout =
+� Υth
+0
+fA(y)dy,
+=FA (Υth) = 1 − Q
+�Υth − µA
+σ2
+A
+�
+.
+(10)
+C. Spectral Efficiency
+SE for the DRAT scenario can be defined from (8) as
+SE =E [Rin] = E
+�
+log2
+�
+1 + A2 B ¯γ
+��
+,
+=
+� ∞
+0
+log2
+�
+1 + y2 B ¯γ
+�
+fA(y)dy.
+(11)
+The exact derivation of the integral in (11) is mathematically
+intractable, and thus a closed-form expression may not be
+derived. Hence, we resort to approximate SE with tight upper
+and lower bounds by invoking Jensen’s inequality.
+1) Upper Bound: Applying Jensen’s inequality, we define
+the upper bound for SE as SEu, where SE ≤ SEu. Now,
+SEu can be evaluated from (11) as
+SEu = log2
+�
+1 + ¯γ B E
+�
+A2��
+,
+(12)
+and expressed as
+SEu = log2 [1 + ¯γ B M1M2 Ωm1
+×
+�
+1 + (M1 M2 − 1)
+m1
+�Γ(m1 + 1
+2)
+Γ(m1)
+�2��
+.
+(13)
+Evaluation of Upper Bound: In (12), E
+�
+A2�
+can be evaluated
+utilizing the relation Var [X] = E
+�
+X2�
+− (E [X])2 as
+E
+�
+A2�
+=Var [A] + (E [A])2 = σ2
+A + µ2
+A.
+(14)
+After substituting the values of µ2
+A and σ2
+A in (12), the upper
+bound for DRAT-based SE can be evaluated.
+2) Lower Bound: Likewise, we define the lower bound for
+SE as SEl, where SE ≥ SEl. Now, SEl can again be be
+defined from (11) as
+SEl = log2
+�
+1 +
+¯γ B
+E
+� 1
+A2
+�
+�
+,
+(15)
+and expressed as given in (16), on the top of next page.
+Evaluation of Lower Bound: In (15), the expectation
+E
+�
+1/A2�
+can be solved utilizing the Taylor series expansion
+and approximated as [15]
+E
+� 1
+A2
+�
+≈
+1
+E [A2] + Var
+�
+A2�
+[E [A2]]3 .
+(17)
+Since the statistical characteristics of A is known to be Gaussian
+distributed (as discussed earlier in subsection A), A2 will
+follow a non-central chi-square distribution. Thus, the mean
+and variance of A2 can be expressed as
+Var
+�
+A2�
+= 2 σ2
+A
+�
+σ2
+A + 2 µ2
+A
+�
+,
+(18)
+E
+�
+A2�
+= σ2
+A + µ2
+A,
+(19)
+respectively. Thus, utilizing these expressions and substituting
+the values of µ2
+A and σ2
+A, the lower bound for SE of the DRAT
+scenario can be evaluated.
+3) Approximation for Large M: We define SE as approx-
+imate SE (ASE) for large M1 and M2. Now, with the upper
+and lower bounds of SE of the DRAT scenario, exact SE lies
+in-between and can be expressed as
+SEl ≤ SE ≤ SEu.
+(20)
+However, for larger M1 and M2, i.e., M1, M2 ≫ 1, both SEl
+and SEu converge to SE. Thus, ASE can be given as
+SE = log2
+�
+1 + ¯γ B M 2
+1 M 2
+2 Ωm1
+m1
+�Γ(m1 + 1
+2)
+Γ(m1)
+�2�
+.
+(21)
+It can be noted from (21) that, through utilizing dual RIS,
+the fourth order channel gain can be realized, i.e., M 2
+1 M 2
+2 ,
+whereas, for single RIS, the maximum channel gain is of the
+second order, i.e., N 2.
+
+SEl = log2
+
+1 + ¯γ B
+M1M2 Ωm1
+�
+1 + (M1 M2−1)
+m1
+� Γ(m1+ 1
+2 )
+Γ(m1)
+�2�3
+2
+�
+1 + (2M1 M2−1)
+m1
+� Γ(m1+ 1
+2 )
+Γ(m1)
+�2� �
+1 −
+1
+m1
+� Γ(m1+ 1
+2 )
+Γ(m1)
+�2�
++
+�
+1 + (M1 M2−1)
+m1
+� Γ(m1+ 1
+2 )
+Γ(m1)
+�2�2
+
+
+(16)
+TABLE I
+SIMULATION PARAMETERS
+Parameter
+Simulation Values
+Circuit Power
+PBS=10 dBm, PU=10 dBm [5]
+Fading Parameter for DRAT
+m1= 10
+Fading Parameter for Direct Links
+m3= 1
+RIS Power Consumption
+PRE = 10 dBm [5]
+HPA Power Consumption Factor
+α = 1.2
+Noise Floor
+σ2 = -120 dBm
+D. Energy Efficiency
+Now, EE of the dual RIS-aided system is defined as the
+ratio of SE over the total power consumed and can be ex-
+pressed as EE =
+SE
+Ptot , where Ptot denotes the total power
+consumed, which consists of the transmit power, the circuit
+power consumption at BS and V, and the power consumed at
+RIS. Considering all the power consumed, the EE in can be
+expressed
+EE =
+SE
+(1 + ξ)Pt + P c
+V + (M1 + M2)P c
+RIS + P c
+BS
+,
+(22)
+where P c
+RIS denotes the power utilized by each RU, ξ =
+1
+ω
+and ω is the drain efficiency of HPA. Likewise, P c
+V , i.e., the
+power consumed in other circuit components excluding HPA at
+V and P c
+BS is the circuit power consumption at BS.
+This completes the analytical derivation of the outage, SE,
+and EE for DRAT of the uplink of V2I communication.
+IV. SIMULATION RESULTS
+This section discusses and presents the simulation results for
+the performance of the dual RIS-assisted V2I communication.
+Further, the results for the SRAT and DCT scenarios are
+presented for the sake of comparison. The distances between
+V-to-RIS1, RIS1-to-RIS2 and RIS2-to-BS are assumed to be
+5, 100 and 5 meters, respectively. Similarly for the simulation
+purpose, M = M1 = M2 and N is taken as to be N = 2 M,
+in order to maintain the fairness in the comparison. The rest of
+the simulation parameters are summarized in Table I.
+Fig. 2 shows the SE performance for the DRAT scenario,
+where the solid lines without marker points show the exact
+(simulation) performance of DRAT, whereas the markers show
+the analytically derived upper and lower bounds on SE. Ad-
+ditionally, ASE for large M is also plotted. The simulation
+verifies that the derived upper and lower bounds are quite
+tight as the analytically derived bounds are remarkably close
+to the actual performance. Further, it can also be noted that
+the difference between exact SE and ASE (as shown in (21))
+diminishes as M increases. For instance, at M = 10 and
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+SNR (dB)
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+SE (bps/Hz)
+Sim
+UB
+LB
+Approx
+M = 10, 20, 50, 100
+Fig. 2. SE with respect to ¯γ for different M of the proposed DRAT scenario.
+0
+500
+1000
+1500
+2000
+2500
+M 
+0
+5
+10
+15
+20
+25
+SE (bps/Hz)
+DRAT
+SRAT
+DCT
+Fig. 3. SE with respect to M for the proposed DRAT scenario.
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+SNR (dB)
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+Outage
+Rth=5
+Rth=7.5
+Rth=10
+Fig. 4. Outage with respect to Pt for different rate thresholds for DRAT.
+¯γ = 30 dB, SE is 1.5037 bps/Hz whereas SE is 1.5034
+bps/Hz; however, at M = 50 and ¯γ = 15 dB, SE is 5.2201
+whereas SE is 5.2200 bps/Hz. Thus, it shows that the bounds
+are quite accurate and near to the exact simulation value.
+Fig. 3 shows the SE results for the DRAT scenario, and
+compares them with the SRAT and DCT scenarios. Specifically,
+it shows SE for a varying number of RUs. The following
+observations can be easily inferred from this plot: 1) Apart
+from smaller M, SE of DRAT is always better than SE of the
+SRAT scheme due to the fourth order gain provided by dual
+RIS. This can also be inferred from the analytical evaluation in
+(31). 2) Due to the multiplicative pathloss, for less number of
+RUs, i.e., smaller M, the DCT scenario may provide better SE
+performance than the RIS-reflected link for both DRAT and
+
+0
+500
+1000
+1500
+2000
+2500
+M 
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+EE (bits/Hz/Joule)
+DRAT
+SRAT
+DCT
+(a) EE with respect to M, here Pt = 10 dBm.
+0
+5
+10
+15
+20
+25
+30
+SNR (dB)
+0
+0.5
+1
+1.5
+EE (bits/Hz/Joule)
+DRAT
+SRAT
+DCT
+(b) EE with respect to SNR, here M = 1000.
+Fig. 5. EE comparison for DRAT with respect to the SRAT and DCT scenarios.
+SRAT scenarios. However, as the number of RUs increases,
+the RIS-based scenarios outperform DCT. 3) Similar to single
+RIS, dual RIS-based DRAT also suffers from the multiplicative
+effect of pathloss. Thus, for smaller RUs, the SRAT scenario
+shows better SE than the DRAT one. 4) As the number of RUs
+increases sufficiently, DRAT outperforms SRAT significantly.
+Fig. 4 shows the outage probability of the DRAT scenario for
+three different rate thresholds, i.e., Rth = {5, 7.5, 10} bps/Hz.
+As evident from the result, the outage can be improved either
+through increasing the transmit power or the number of RUs.
+Since, the transmit power at BS is usually constrained, RIS
+provides an alternate to improve the outage through increasing
+RUs, instead of increasing the transmit power. Thus, to circum-
+vent the power constraint, the number of RUs at RIS can be
+scaled accordingly.
+Fig. 5 shows the EE results of the DRAT scenario, the EE
+plots of the SRAT and DCT scenarios are also plotted here
+for comparison. Specifically, in Fig. 5(a), the performance is
+with respect to M, while in Fig. 5(b), the EE curve is plotted
+against SNR. It can be observed that, for large M, the DRAT
+scenario is the most energy-efficient. Although, for smaller M,
+single RIS provides better EE; this is due to the fact that the
+received signal of the dual RIS-reflected link suffers from the
+multiplicative pathloss that can be mitigated by by large M.
+From the above results on SE and EE, it can be easily inferred
+that the proposed DRAT scheme outperforms SRAT in terms of
+both SE as well as EE. Similarly, the above results also show
+that, for a fixed rate requirement, DRAT requires lower transmit
+power and hence is more energy efficient.
+V. CONCLUSION
+V2X has opened up a slew of novel possibilities in the
+wireless vehicular communication arena, but its potential for
+enabling true ITS has yet to be explored completely, despite
+its significant importance in the safety of autonomous driving.
+In this work, we have envisioned the integration of RIS into
+vehicular networks to realize the true potential in enhancing the
+performance of the V2I communication. Specifically, we have
+evaluated the performance of a dual-RIS assisted V2I uplink
+communication scenario in terms of the outage probability, SE
+and EE. Novel closed-form expressions are derived and verified
+through the extensive numerical simulations. The results show
+a significant gain in the performance can be achieved through
+the proposed RIS scenario.
+VI. ACKNOWLEDGEMENT
+This work was supported by the Nazarbayev University CRP
+Grant no. 11022021CRP1513.
+REFERENCES
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+communications: Enabling technologies, challenges, and opportunities,”
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+
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+page_content='IT]  10 Jan 2023  On the Performance of Dual RIS-assisted V2I  Communication under Nakagami-m Fading  Mohd Hamza Naim Shaikh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Khaled Rabie◦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Xingwang Li#,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Theodoros Tsiftsis†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' and Galymzhan Nauryzbayev  School of Engineering and Digital Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Nazarbayev University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Nur-Sultan City,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
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+page_content=' Kazakhstan  Department of Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Manchester Metropolitan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Manchester,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
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+page_content=' Henan Polytechnic University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Jiaozuo 454000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' China  †Department of Informatics & Telecommunications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' University of Thessaly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Greece;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  †School of Intelligent Systems Science and Engineering, Jinan University, China  Email: {hamza.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='shaikh, galymzhan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='nauryzbayev}@nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='kz, ◦k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='rabie@mmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='uk,  #lixingwang@hpu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='cn, †tsiftsis@ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='org  Abstract—Vehicle-to-everything (V2X) connectivity in 5G-and-  beyond communication networks supports the futuristic intelligent  transportation system (ITS) by allowing vehicles to intelligently  connect with everything.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' The advent of reconfigurable intelligent  surfaces (RISs) has led to realizing the true potential of V2X  communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' In this work, we propose a dual RIS-based  vehicle-to-infrastructure (V2I) communication scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Following  that, the performance of the proposed communication scheme  is evaluated in terms of deriving the closed-form expressions  for outage probability, spectral efficiency and energy efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Finally, the analytical findings are corroborated with simulations  which illustrate the superiority of the RIS-assisted vehicular  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Keywords— Reconfigurable intelligent surface (RIS), dual RIS,  energy efficiency, spectral efficiency, vehicular communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' INTRODUCTION  As a key enabler for intelligent transportation systems (ITSs),  vehicle-to-everything (V2X) communication has sparked a re-  newed interest in the research community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' V2X encompasses  a wide range of wireless technologies such as vehicle-to-  pedestrian (V2P), vehicle-to-infrastructure (V2I), and vehicle-  to-vehicle (V2V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Additionally, it also includes the vehicu-  lar communications with vulnerable road users (VRUs), grid  (V2G), network (V2N) and cloud (V2C) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' The V2X com-  munications will be a critical component of the futuristic  connected and self-driving cars, envisioned and enabled by  the sixth-generation (6G) wireless technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Furthermore,  the V2X communications will also enhance and transform  the quality-of-service (QoS) in terms of unparalleled user  experience, ultra-high road safety and air quality improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  In addition, a slew of advanced applications will also be  supported like platooning, trajectory alignments, exchanging  sensor data and high precision maps, and so on [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Thanks to  the enhanced capabilities of 6G, vehicles will receive accurate  safety information, intelligent traffic management support, and  innovative infotainment features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Thus, the 6G services will be  used to create a fully automated, autonomous, and ubiquitously  connected vehicular network [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Recently, reconfigurable intelligent surfaces (RISs) have  emerged as a breakthrough technology that offers a great deal  in terms of wireless communication [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Inherently, RIS is a  software-defined artificial structure made up of a large number  of scattering passive elements, termed as reflecting units (RUs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  These RUs are capable to adjust the electromagnetic (EM)  properties of a reflected wave that is incident on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Thus,  RIS can use not only this ability to boost the received signal’s  power, but also the capability to create an additional reflective  link to mitigate the impact of blockages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' With the large number  of RUs, RISs are particularly known to have large spectral and  energy efficiency [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' As a result, RIS may be used to improve  the quality of vehicular communication through establishing  a low-cost, highly energy efficient indirect line-of-sight (LoS)  communications [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  In [7], the authors investigated the outage performance for  RIS-assisted vehicular communication networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Likewise, the  secrecy outage performance of RIS-aided vehicular communi-  cations has been studied in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' RISs were also investigated for  detecting VRUs such as cyclists, pedestrians and wheelchair  users [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Specifically, the authors utilized RISs for enhancing  the radar visibility for VRUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Further, in [10], the authors  provided a optimization framework for resource allocation  in the RIS-aided vehicular communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Specifically, they  jointly optimized the power allocation, RIS reflection coeffi-  cients and spectrum allocation for different QoS requirements  of the V2V and V2I communication links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Likewise, in [11],  the authors discussed a system model where RSU leverages RIS  to connect the dark zones, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=', areas blocked due by obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Moreover, a comprehensive overview on the recent advances  in 6G vehicular networks was provided in [12, 13], where the  authors also described various open challenges and possible  research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Motivated by the above, in this work, we investigate the  performance of a dual RIS-assisted V2I communication net-  work scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Specifically, the proposed scenario considers the  uplink transmission where the vehicle is communicating with  the base station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' To enhance the communication capabilities, the  vehicle is supported through two RISs which create a virtual  line-of-sight (LoS) link, which, otherwise, was inherently non-  LoS (NLoS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' The major contributions can be summarized as  Explicitly, we invoked the central limit theorem (CLT) to  characterize the received signal-to-noise ratio (SNR) for  Vehicle-to-Vehicle (V2V)  Vehicle-to-Infrastructure (V2I)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Schematic for the considered dual RIS-aided V2I communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  the proposed dual RIS case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Further, based on this, we  derived the closed-form expression for outage probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Further, we derived the closed-form expressions for the  upper and lower bounds of SE and EE of the proposed  dual RIS-assisted V2I communication scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Finally, as a performance benchmark, the proposed sce-  nario is compared with the single RIS-assisted V2I com-  munication and with RIS V2I communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' The results  show the superiority of the proposed scenario of dual RIS-  assisted V2I over the single RIS-assisted V2I communi-  cation case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' SYSTEM MODEL  As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' 1, in this work, we consider a V2I  communication model, wherein the vehicular user (V) tries to  communicate with a nearby base station (BS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Apart from the  direct cellular link, a reflected path through RISs is considered  to support this uplink transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' In particular, we consider  a dual RIS-assisted uplink V2I transmission with two RISs,  one each placed near V and BS both, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' For the two  RISs, the number of RUs is assumed to be M1 and M2 for  RIS-1 and RIS-2, respectively, while keeping the total number  of RUs unchanged, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=', M1+M2 = N, where N is the number  of RUs in large RIS for the single RIS scenario, which is the  benchmark for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Thus, based on RIS, the following  scenarios are considered in this work  Dual RIS-assisted Transmission (DRAT): In DRAT, the  transmission takes place only through the two RISs and  the reflected link, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Single RIS-assisted Transmission (SRAT): In SRAT, the  transmission takes place through single large RIS which  is placed near to BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Direct Cellular Transmission (DCT): In DCT, V commu-  nicates with BS directly without utilizing RISs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Thus, the  transmission is inherently NLoS and experiences a higher  pathloss exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' This would also serve as the baseline  scheme for the performance comparison of the above two  cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Channel Model  The channels between V-to-RIS-1 and RIS-2-to-BS can  be modeled as deterministic LOS channels as the distances  are small and the probability of having LoS is very high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  However, the distance between RIS-1 and RIS-2 is large and  thus the small scale fading for the channel between the ith  element of RIS-1 and the jth element of RIS-2, denoted by  hRR  ij , is modeled through Nakagami-m fading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Hence, for  i = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=', M1} and j = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=', M2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Further, the  distances related to the V-to-RIS-1, RIS-1-to-RIS-2 and RIS-2-  to-BS links are represented by d1, dRR and d2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Received Signal Model  The received base-band signal at BS, denoted by r, for the  dual RIS-aided transmission case can be expressed as  r =  �  B Pt  ��M1  i=1  �M2  j=1 ejφ(1)  i hRR  ij ejφ(2)  j  �  s + No,  (1)  where Pt is the transmit power constraint at V, B is the distance-  dependent pathloss, s ∼ CN (0, 1) is the transmitted symbol,  and No ∼ CN  �  0, σ2�  is the additive white Gaussian noise  (AWGN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Further, φ1 and φ2 are the phase of the V-to-RIS1  and RIS2-to-BS channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Further, for a link distance d, B at  the carrier frequency of 3 GHz can be given by [14]  B(d) [dB] =  �  −37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='5 − 22 log10(d/1 m)  if LOS,  −35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='1 − 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='7 log10(d/1 m)  if NLOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  (2)  Likewise, instantaneous SNR at BS can be formulated as  γ =  ����  �M1  i=1  �M2  j=1 δije  j  �  φ(1)  i  +φ(2)  j  −ϕij  �����  2  B Pt  σ2  ,  (3)  where δij and ϕij denote the amplitude and phase of hRR  ij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  1) RIS Reflection Parameters: Now, SNR at BS can be  maximized through adjusting the phase at RISs to cancel  the resultant phase, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=', φ(1)  i  + φ(2)  j  − ϕij = 0, for i =  {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=', M1} and j = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=', M2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Thus, by substituting  ϕij = φ(1)  i  + φ(2)  j , ∀i, j, the received signal power at BS can  be maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Consequently, maximized SNR corresponding  to the optimal phase can be given as  γmax =  ����M1  i=1  �M2  j=1 δij  ���  2  B Pt  σ2  = A2B Pt  σ2  = A2 B ¯γ,  (4)  where A2 =  ���  �M1  i=1  �M2  j=1 δij  ���  2  is the cascaded channel gain  provided by RISs, and ¯γ = Pt/σ2 is transmit SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Likewise, proceeding in the similar way, for the SRAT  scenario, maximized SNR at BS can be given as1  ˆγmax =  ��N  i=1 βi  �2  ¯γ = B2¯γ,  (5)  where βi is the amplitude of a channel between RIS and  V, denoted by hRU  i  , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=', hRU  i  = βie−jϕi, and B2 is the  corresponding channel gain provided by single RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  1For the SRAT scenario, the analysis is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Thus, the detailed description  is omitted for the sake of brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' In particular, for SRAT, large RIS with N  RUs is present near BS, where N = M1 + M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Likewise, the RIS-to-BS link  is also modeled as Nakagami-m fading with the rest of the parameters being  the same, as in DRAT, like transmit power constraint at V, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' PERFORMANCE ANALYSIS  This section initially evaluates SNR for the dual RIS-aided  V2I scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Utilizing the SNR expressions formulated earlier,  the outage probability, SE and EE are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Statistical Characterization of the Dual RIS Channel Gain  Now utilizing CLT for M  ≫ 1, A = �M1  i=1  �M2  j=1 δij  can be approximated through a Gaussian distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=',  A ∼ N(µy, σ2  y) [15], with a probability density function (PDF)  given by  fA(y) =  \uf8f1  \uf8f2  \uf8f3  1  √  2πσ2  A exp  �  −(y−µA)2  2σ2  A  �  ,  if y > 0,  0,  if y = 0,  (6)  where µA = �M1  i=1  �M2  j=1 µij, σ2  A = �M1  i=1  �M2  j=1 σ2  ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Here,  µij and σ2  ij are the mean and variance of the random variable  δij, which follows the Nakagami-m distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Hence, µij =  Γ(m1+ 1  2 )  Γ(m1)  �� Ωm1  m1  �  and σ2  ij = Ωm1  �  1 −  1  m1  �  Γ(m1+ 1  2 )  Γ(m1)  �2�  ,  for all i = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' , M1} and j = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=', M2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Likewise the cumulative distribution function (CDF) of A  can be derived from its PDF as  FA(y)=  � y  −∞  fA(y)dy =  �  1−Q  �  y−µA  σ2  A  �  ,  if y > 0,  0,  if y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  (7)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Outage Probability  The normalized instantaneous rate, denoted by Rin, for the  DRAT scenario can be formulated from (4) and expressed as  Rin = log2 (1 + γmax) = log2  �  1 + A2¯γ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  (8)  Now, the end-to-end outage from V to BS via RIS, denoted by  Pout, can be defined in terms of a rate threshold, Rth, as  Pout = Pr [Rin < Rth] = Pr  �  log2  �  1 + A2¯γ  �  < Rth  �  = Pr  \uf8ee  \uf8f0A <  �  2Rth − 1  ¯γ  \uf8f9  \uf8fb = Pr [A < Υth] ,  (9)  where Υth =  �  2Rth −1  ¯γ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Thus, the closed-form expression of  the outage probability DRAT can be evaluated as  Pout =  � Υth  0  fA(y)dy,  =FA (Υth) = 1 − Q  �Υth − µA  σ2  A  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  (10)  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Spectral Efficiency  SE for the DRAT scenario can be defined from (8) as  SE =E [Rin] = E  �  log2  �  1 + A2 B ¯γ  ��  ,  =  � ∞  0  log2  �  1 + y2 B ¯γ  �  fA(y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  (11)  The exact derivation of the integral in (11) is mathematically  intractable, and thus a closed-form expression may not be  derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Hence, we resort to approximate SE with tight upper  and lower bounds by invoking Jensen’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  1) Upper Bound: Applying Jensen’s inequality, we define  the upper bound for SE as SEu, where SE ≤ SEu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Now,  SEu can be evaluated from (11) as  SEu = log2  �  1 + ¯γ B E  �  A2��  ,  (12)  and expressed as  SEu = log2 [1 + ¯γ B M1M2 Ωm1  ×  �  1 + (M1 M2 − 1)  m1  �Γ(m1 + 1  2)  Γ(m1)  �2��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  (13)  Evaluation of Upper Bound: In (12), E  �  A2�  can be evaluated  utilizing the relation Var [X] = E  �  X2�  − (E [X])2 as  E  �  A2�  =Var [A] + (E [A])2 = σ2  A + µ2  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  (14)  After substituting the values of µ2  A and σ2  A in (12), the upper  bound for DRAT-based SE can be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  2) Lower Bound: Likewise, we define the lower bound for  SE as SEl, where SE ≥ SEl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Now, SEl can again be be  defined from (11) as  SEl = log2  �  1 +  ¯γ B  E  � 1  A2  �  �  ,  (15)  and expressed as given in (16), on the top of next page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Evaluation of Lower Bound: In (15), the expectation  E  �  1/A2�  can be solved utilizing the Taylor series expansion  and approximated as [15]  E  � 1  A2  �  ≈  1  E [A2] + Var  �  A2�  [E [A2]]3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  (17)  Since the statistical characteristics of A is known to be Gaussian  distributed (as discussed earlier in subsection A), A2 will  follow a non-central chi-square distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Thus, the mean  and variance of A2 can be expressed as  Var  �  A2�  = 2 σ2  A  �  σ2  A + 2 µ2  A  �  ,  (18)  E  �  A2�  = σ2  A + µ2  A,  (19)  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Thus, utilizing these expressions and substituting  the values of µ2  A and σ2  A, the lower bound for SE of the DRAT  scenario can be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  3) Approximation for Large M: We define SE as approx-  imate SE (ASE) for large M1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Now, with the upper  and lower bounds of SE of the DRAT scenario, exact SE lies  in-between and can be expressed as  SEl ≤ SE ≤ SEu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  (20)  However, for larger M1 and M2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=', M1, M2 ≫ 1, both SEl  and SEu converge to SE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Thus, ASE can be given as  SE = log2  �  1 + ¯γ B M 2  1 M 2  2 Ωm1  m1  �Γ(m1 + 1  2)  Γ(m1)  �2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  (21)  It can be noted from (21) that, through utilizing dual RIS,  the fourth order channel gain can be realized, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=', M 2  1 M 2  2 ,  whereas, for single RIS, the maximum channel gain is of the  second order, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=', N 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  SEl = log2  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f01 + ¯γ B  M1M2 Ωm1  �  1 + (M1 M2−1)  m1  � Γ(m1+ 1  2 )  Γ(m1)  �2�3  2  �  1 + (2M1 M2−1)  m1  � Γ(m1+ 1  2 )  Γ(m1)  �2� �  1 −  1  m1  � Γ(m1+ 1  2 )  Γ(m1)  �2�  +  �  1 + (M1 M2−1)  m1  � Γ(m1+ 1  2 )  Γ(m1)  �2�2  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb  (16)  TABLE I  SIMULATION PARAMETERS  Parameter  Simulation Values  Circuit Power  PBS=10 dBm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' PU=10 dBm [5]  Fading Parameter for DRAT  m1= 10  Fading Parameter for Direct Links  m3= 1  RIS Power Consumption  PRE = 10 dBm [5]  HPA Power Consumption Factor  α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='2  Noise Floor  σ2 = -120 dBm  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Energy Efficiency  Now, EE of the dual RIS-aided system is defined as the  ratio of SE over the total power consumed and can be ex-  pressed as EE =  SE  Ptot , where Ptot denotes the total power  consumed, which consists of the transmit power, the circuit  power consumption at BS and V, and the power consumed at  RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Considering all the power consumed, the EE in can be  expressed  EE =  SE  (1 + ξ)Pt + P c  V + (M1 + M2)P c  RIS + P c  BS  ,  (22)  where P c  RIS denotes the power utilized by each RU, ξ =  1  ω  and ω is the drain efficiency of HPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Likewise, P c  V , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=', the  power consumed in other circuit components excluding HPA at  V and P c  BS is the circuit power consumption at BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  This completes the analytical derivation of the outage, SE,  and EE for DRAT of the uplink of V2I communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' SIMULATION RESULTS  This section discusses and presents the simulation results for  the performance of the dual RIS-assisted V2I communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Further, the results for the SRAT and DCT scenarios are  presented for the sake of comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' The distances between  V-to-RIS1, RIS1-to-RIS2 and RIS2-to-BS are assumed to be  5, 100 and 5 meters, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Similarly for the simulation  purpose, M = M1 = M2 and N is taken as to be N = 2 M,  in order to maintain the fairness in the comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' The rest of  the simulation parameters are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' 2 shows the SE performance for the DRAT scenario,  where the solid lines without marker points show the exact  (simulation) performance of DRAT, whereas the markers show  the analytically derived upper and lower bounds on SE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Ad-  ditionally, ASE for large M is also plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' The simulation  verifies that the derived upper and lower bounds are quite  tight as the analytically derived bounds are remarkably close  to the actual performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Further, it can also be noted that  the difference between exact SE and ASE (as shown in (21))  diminishes as M increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' For instance, at M = 10 and  0  5  10  15  20  25  30  35  40  45  50  SNR (dB)  0  2  4  6  8  10  12  14  16  18  20  SE (bps/Hz)  Sim  UB  LB  Approx  M = 10, 20, 50, 100  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' SE with respect to ¯γ for different M of the proposed DRAT scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  0  500  1000  1500  2000  2500  M  0  5  10  15  20  25  SE (bps/Hz)  DRAT  SRAT  DCT  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' SE with respect to M for the proposed DRAT scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  0  2  4  6  8  10  12  14  16  18  20  SNR (dB)  10-6  10-5  10-4  10-3  10-2  10-1  100  Outage  Rth=5  Rth=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='5  Rth=10  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Outage with respect to Pt for different rate thresholds for DRAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  ¯γ = 30 dB, SE is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='5037 bps/Hz whereas SE is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='5034  bps/Hz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' however, at M = 50 and ¯γ = 15 dB, SE is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='2201  whereas SE is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='2200 bps/Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Thus, it shows that the bounds  are quite accurate and near to the exact simulation value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' 3 shows the SE results for the DRAT scenario, and  compares them with the SRAT and DCT scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Specifically,  it shows SE for a varying number of RUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' The following  observations can be easily inferred from this plot: 1) Apart  from smaller M, SE of DRAT is always better than SE of the  SRAT scheme due to the fourth order gain provided by dual  RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' This can also be inferred from the analytical evaluation in  (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' 2) Due to the multiplicative pathloss, for less number of  RUs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=', smaller M, the DCT scenario may provide better SE  performance than the RIS-reflected link for both DRAT and  0  500  1000  1500  2000  2500  M  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='7  EE (bits/Hz/Joule)  DRAT  SRAT  DCT  (a) EE with respect to M, here Pt = 10 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  0  5  10  15  20  25  30  SNR (dB)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='5  EE (bits/Hz/Joule)  DRAT  SRAT  DCT  (b) EE with respect to SNR, here M = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' EE comparison for DRAT with respect to the SRAT and DCT scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  SRAT scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' However, as the number of RUs increases,  the RIS-based scenarios outperform DCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' 3) Similar to single  RIS, dual RIS-based DRAT also suffers from the multiplicative  effect of pathloss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Thus, for smaller RUs, the SRAT scenario  shows better SE than the DRAT one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' 4) As the number of RUs  increases sufficiently, DRAT outperforms SRAT significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' 4 shows the outage probability of the DRAT scenario for  three different rate thresholds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=', Rth = {5, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='5, 10} bps/Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  As evident from the result, the outage can be improved either  through increasing the transmit power or the number of RUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Since, the transmit power at BS is usually constrained, RIS  provides an alternate to improve the outage through increasing  RUs, instead of increasing the transmit power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Thus, to circum-  vent the power constraint, the number of RUs at RIS can be  scaled accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' 5 shows the EE results of the DRAT scenario, the EE  plots of the SRAT and DCT scenarios are also plotted here  for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Specifically, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' 5(a), the performance is  with respect to M, while in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' 5(b), the EE curve is plotted  against SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' It can be observed that, for large M, the DRAT  scenario is the most energy-efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Although, for smaller M,  single RIS provides better EE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' this is due to the fact that the  received signal of the dual RIS-reflected link suffers from the  multiplicative pathloss that can be mitigated by by large M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  From the above results on SE and EE, it can be easily inferred  that the proposed DRAT scheme outperforms SRAT in terms of  both SE as well as EE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Similarly, the above results also show  that, for a fixed rate requirement, DRAT requires lower transmit  power and hence is more energy efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' CONCLUSION  V2X has opened up a slew of novel possibilities in the  wireless vehicular communication arena, but its potential for  enabling true ITS has yet to be explored completely, despite  its significant importance in the safety of autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  In this work, we have envisioned the integration of RIS into  vehicular networks to realize the true potential in enhancing the  performance of the V2I communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Specifically, we have  evaluated the performance of a dual-RIS assisted V2I uplink  communication scenario in terms of the outage probability, SE  and EE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Novel closed-form expressions are derived and verified  through the extensive numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' The results show  a significant gain in the performance can be achieved through  the proposed RIS scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' ACKNOWLEDGEMENT  This work was supported by the Nazarbayev University CRP  Grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
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+AmbieGen: A Search-based Framework for Autonomous
+Systems Testing
+Dmytro Humeniuk, Foutse Khomh, Giuliano Antoniol
+Polytechnique Montr´eal, 2500 Chemin de Polytechnique, QC H3T 1J4, Montr´eal,
+Canada
+Abstract
+Thorough testing of safety-critical autonomous systems, such as self-driving
+cars, autonomous robots, and drones, is essential for detecting potential fail-
+ures before deployment. One crucial testing stage is model-in-the-loop test-
+ing, where the system model is evaluated by executing various scenarios in
+a simulator. However, the search space of possible parameters defining these
+test scenarios is vast, and simulating all combinations is computationally in-
+feasible. To address this challenge, we introduce AmbieGen, a search-based
+test case generation framework for autonomous systems.
+AmbieGen uses
+evolutionary search to identify the most critical scenarios for a given system,
+and has a modular architecture that allows for the addition of new systems
+under test, algorithms, and search operators. Currently, AmbieGen supports
+test case generation for autonomous robots and autonomous car lane keep-
+ing assist systems. In this paper, we provide a high-level overview of the
+framework’s architecture and demonstrate its practical use cases.
+Keywords:
+evolutionary search, autonomous systems, self driving cars,
+autonomous robots, neural network testing
+Metadata
+The project metadata is presented in Table 1.
+1. Motivation and significance
+Autonomous systems, including autonomous vehicles, robots, or drones
+can provide a number of benefits such as driving assistance, high-risk zone
+Preprint submitted to Science of Computer Programming
+January 4, 2023
+arXiv:2301.01234v1  [cs.RO]  1 Jan 2023
+
+Nr.
+Code metadata description
+Please fill in this column
+C1
+Current code version
+v0.1.0
+C2
+Permanent link to code/repository
+used for this code version
+For
+example:
+https://github.
+com/swat-lab-optimization/
+AmbieGen-tool
+C3
+Permanent
+link
+to
+Reproducible
+Capsule
+https://codeocean.com/
+capsule/1741442/tree
+C4
+Legal Code License
+MIT license (MIT)
+C5
+Code versioning system used
+git
+C6
+Software code languages, tools, and
+services used
+python
+C7
+Compilation requirements, operat-
+ing environments and dependencies
+indicated in requirements.txt
+C8
+If available, link to developer docu-
+mentation/manual
+https://github.com/
+swat-lab-optimization/
+AmbieGen-tool/blob/master/
+README.md
+C9
+Support email for questions
+dmytro.humeniuk@polymtl.ca
+Table 1: Code metadata (mandatory)
+exploration, and aid in rescue operations. At the same time, these are safety-
+critical systems and it is very important to ensure they are robust to unseen
+environments and conditions. This can be done by thorough testing prior
+to their deployment. Typically, at the initial development stages model-in-
+the-loop testing is performed [1], where the system is tested in a simulation
+environment. Given the complexity of autonomous systems, the number of
+potential test scenarios is vast and exhaustive execution is not feasible. For
+example, an autonomous vehicle scenario could involve a variety of param-
+eters such as road topology, the movement and behavior of other vehicles
+and pedestrians, traffic signs, weather conditions, etc. We surmise that in
+order to identify the most critical scenarios for a given system, application
+of search algorithms is necessary.
+In this work, we propose AmbieGen, a search based framework for gen-
+erating adversarial test scenarios for autonomous systems.
+By leveraging
+evolutionary search AmbieGen allows to find challenging and diverse test
+scenarios.
+2
+
+The problem of identifying critical scenarios for a system has been ad-
+dressed in several previous works on falsifying temporal logic requirements
+of cyber-physical systems, such as S-Taliro [2], Breach [3], and ARIsTEO [4].
+These works typically consider falsifying a model of the system that takes a
+set of input signals and produces a set of output signals.
+In our work, we focus on testing autonomous systems for which the input
+signals are complex and may include data from various sensors and cameras.
+Generating a valid combination of falsifying input signals (such as lidar read-
+ings and RGB camera readings) directly would be challenging. Therefore,
+we propose a method for generating test cases that specify a virtual environ-
+ment for the autonomous system, rather than the input signals. The input
+signals are generated in the virtual environment during simulation based on
+the actions of the autonomous agent.
+Several approaches have been proposed for generating virtual environ-
+ments for testing autonomous driving and robotics systems, including As-
+Fault [5], Frenetic [6], DeepJanus [7], DeepHyperion [8] and others presented
+at the SBST 2021 [9] and SBST 2022 [10] tool competitions.
+The tool we present in this paper, AmbieGen, is the winner of SBST 2022
+tool competition. It could produce the biggest number of diverse fault reveal-
+ing scenarios for an autonomous vehicle lane keeping assist system (LKAS)
+given a limited time budget. More details about the search algorithm im-
+plementation can be found in our research paper [11]. In our work we have
+shown that the simplified model of the system can be effective in guiding the
+search for producing the test scenarios for the full, simulator based, model
+of the system.
+Our framework can be used for further research in the search algorithms,
+search operator and fitness function design for autonomous systems adver-
+sarial testing. We built the framework to be modular, and thus easily cus-
+tomizable. By referring to project documentation as well as the example
+implementations we provide, researchers can specify their own test scenario
+generation problems, fitness functions, crossover and mutation operators.
+This tool is developed in Python and can be easily run as a python package.
+More instructions and examples are provided in the AmbieGen repository.
+2. Software description
+In this work, we present AmbieGen, an open-source Python framework
+that utilizes evolutionary search for the generation of test scenarios for au-
+3
+
+tonomous systems. Currently, AmbieGen supports the creation of test sce-
+narios for lane keeping assist systems (LKAS) in autonomous vehicles and
+for autonomous robots navigating a closed room with obstacles.
+The test scenarios for LKAS in vehicles are designed to challenge the
+system with various road topologies, while the scenarios for autonomous
+robots involve navigating a closed room with obstacles.
+Examples of the
+generated scenarios can be seen in Figure 1.
+Figure 1: An example of the test case for LKAS system (a) and an autonomous robot (b).
+The x-axis represents the map length in meters, and the y-axis represents the map width
+in meters.
+2.1. Software architecture
+This subsection provides a detailed description of the software imple-
+mentation of AmbieGen. The key components of AmbieGen are illustrated
+in Figure 2, which are common components for implementing evolutionary
+search. We use the Pymoo framework [12] to implement the search algo-
+rithms. The most important modules and classes are outlined below:
+• Solution - this is one of the most important classes, which contains all
+the necessary attributes and functions needed to represent the candi-
+date solution of the algorithm. It should contain a scenario attribute
+with the list of test case parameters, function for fitness evaluation,
+novelty calculation, as well as, optionally, image generation.
+4
+
+200
+a
+40
+b
+口
+Ci
+■
+35-
+■
+■
+■
+30
+■
+25
+Robotpath
+20-
+Walls
+V
+国
+15 -
+■
+■
+口
+■
+10
+5
+■
+■
+fo
+0
+0
+5
+10
+15
+20
+25
+200
+30
+35
+40Figure 2: AmbieGen architecture
+• Sampling - this is the class for initial population generation. At the
+output it provides N instances of the Solution class, with the initial-
+ized scenario attribute, defining the test scenario. Typically the test
+scenario is represented by a two dimensional array, randomly initial-
+ized based on the minimum and maximum values of the test case pa-
+rameters, defined in the configuration file. Each column of the array
+corresponds to some part of the environment. More information about
+the representation of the test scenarios that we used can be found in
+the repository page as well as in our research article.
+• Problem - in this class, we define the logic for evaluating the fitness
+of each solution. For single-objective search (using GA), we specify
+the fitness function for evaluating the scenario fault revealing power.
+For two-objective search (using NSGA-II), we define two objectives:
+fault revealing power and novelty calculation. The novelty objective
+is calculated as the average novelty of a given test scenario relative to
+the 5 solutions with the highest fault revealing power fitness. If the
+problem has any constraints, such as a minimum required fitness value,
+they should also be specified in this class.
+• TC to environment - this is a function to transform the test case (TC)
+encoded as a 2D array of parameters, to the input format suitable for
+the system model. For example, for the LKAS problem, the model
+input is a list of the 2D coordinates of points, defining the road topol-
+ogy. The test case itself is represented as a sequence of transformations
+5
+
+Pymoo
+post_processing()
+Sampling
+Solution object 1
+TC to environment ()
++gen_randomscenario()
+Solution object 2
+Problem
+fitness evaluation ()
+Solution object 3
+Solution
+Crossover
++map_size
+Solution object 4
+configuration file
++scenario
+Mutation
++fitness eval()
+Population size
+Solution object N
+Numberofgenerations
++novelty eval()
+Crossover/mutationrate
++build image()
+TCparameterranges
+Folderto save resultsneeded to perform to obtain the points. For the autonomous robot the
+test scenario is represented as a sequence of parameters describing the
+2D map with obstacles. The TC to environment module is used to
+create a 2D bitmap from the given parameters. The bitmap is given
+as the input to the autonomous robot model, which runs a planning
+algorithm to find the shortest path between the start and goal location.
+• fitness evaluation - a function to calculate the fitness i.e fault revealing
+power of the scenario. It takes the output of the TC to environment
+function as the input and execute the system model. It collects the
+data about the model behaviour during execution and computes the
+fitness score. For the LKAS system, the fitness is defined by the biggest
+deviation from the lane center and for the autonomous robot - by the
+length of the path to reach the goal.
+• Crossover - in this class the crossover operator is defined. Currently
+we are using a one point crossover, which can be applied to fixed and
+variable length solutions.
+• Mutation - in this class the mutation operator is implemented. We
+have 2 types of mutations: exchange and change of variable. In ex-
+change mutation, two randomly selected columns of the test case are
+exchanged. In the case of the road topology, it would correspond to
+exchanging the positions of two random road segments. In change of
+variable mutation, a randomly selected parameter value in the test case
+matrix is changed. In the road topology example it could correspond
+to the change of the length of one of the straight road segments.
+• post processing - The post-processing module of our framework includes
+several functions for handling the test suite and its metadata.
+The
+function get test suite() retrieves the test suite, get stats() retrieves
+metadata such as fitness and novelty scores, and save tcs images()
+saves the images of the test cases. The size of the test suite, denoted
+as T, can be specified in the configuration file. In our experiments, T
+was typically set to 30, representing the best solutions found by the
+algorithm.
+Metadata for the test suite includes the fitness of the top T solutions,
+their novelty (calculated as the average novelty between all pairs of
+scenarios in the test suite), and the convergence (best solution fitness
+6
+
+found at each epoch).
+The post-processing module also includes a
+compare.py script for comparing the results of different algorithms,
+using the collected metadata to generate convergence plots and fitness
+and diversity boxplots.
+• configuration file - finally we have a configuration file, where the pa-
+rameters of the algorithm, such as: the population size, the number
+of generations, crossover/mutation rate, and the test suite size are de-
+fined. Users should also specify the allowable ranges for the test case
+parameters and the paths for saving the resulting test suite and its
+metadata.
+Currently, when adding a new problem, one should provide the implemen-
+tation of each of the modules as well as the TC to environment and fitness
+evaluation functions. We are working on reducing the number of additional
+implementations needed. Our framework includes the implementation of all
+the modules for the LKAS and autonomous robot test case generation prob-
+lems.
+2.2. Software functionalities
+AmbieGen public version 0.1.0 as presented in this paper offers the fol-
+lowing major functionalities:
+• Autonomous vehicle LKAS system testing: generating scenarios, rep-
+resented as a list of 2D coordinates defining the road topology.
+• Autonomous robot testing: generating scenarios, represented as the 2D
+bitmap, defining obstacle locations in a fixed sized map.
+• Search-based generation: our framework provides options for search-
+based test suite generation, including random search, single-objective
+genetic algorithm (GA), and two-objective genetic algorithm (NSGA-
+II). The search algorithms are implemented using the Pymoo frame-
+work [12], and can be easily extended to support additional algorithms
+supported by Pymoo with minor modifications.
+The single-objective GA optimizes the test suite for scenario fault re-
+vealing power, while the two-objective NSGA-II optimizes for both
+fault revealing power and diversity.
+As demonstrated in our previ-
+ous work [11], the two-objective algorithm allows to produce a more
+diverse set of test scenarios compared to the single-objective search.
+7
+
+• Experiment data tracking: AmbieGen tracks the results of each ex-
+periment and saves them in a user-defined location. The saved data
+includes the T (as determined by the user) best test scenarios identified
+based on their fitness or crowding distance, as well as their associated
+metadata such as fitness, average diversity, and visualizations. This
+allows for easy analysis and comparison of the results of different ex-
+periments.
+2.3. Use cases of the software
+In this subsection we provide an illustrative example of how to use Am-
+bieGen to generate test cases for an autonomous robot planning algorithm
+testing. Suppose we want to perform 30 runs of the NSGA-II algorithm with
+150 individuals and 200 generations to evaluate this configuration. We want
+to save the generated test cases, their illustrations as well as their metadata,
+such as fitness and diversity. Below you can see the configuration file entries
+with the parameters we chose for the genetic algorithm and well as the path
+to save the experiment results:
+ga = {" pop_size ": 150, "n_gen ": 200, "mut_rate ": 0.4, "cross_rate ": 0.9,
+" test_suite_size ": 30 }
+files = {" stats_path ": "stats", "tcs_path ": "tcs", "images_path ": images "}
+Now we are ready to start the test case generation. We can launch Am-
+bieGen with the following command and parameters:
+python
+optimize.py --problem =" robot" --algo =" nsga2" --runs =30 \\
+--save_results=True
+The search will start and you could see some printouts, such as in Fig. 3 with
+the current number of generation (n gen), number of evaluations (n eval),
+constraint violation (cv min), number of non-dominant solution for NSGA-
+II algorithm (n nds) and the best solution found (f opt) for GA algorithm.
+More details about the printed information can be found on the Pymoo page
+(https://pymoo.org/interface/display.html).
+After a successful run, you will see the confirmation about the run exe-
+cution time, saved test cases, their metadata and the images, as in Fig. 4
+In Fig. 5 you can see examples of the metadata saved, such as the algo-
+rithm convergence 5a (the best fitness value at each generation in the format
+”evaluation number”: best fitness found), the fitness of the test cases in the
+test suite as well as their average diversity i.e., novelty 5b. Novelty is cal-
+culated as the average diversity of all of the pairs of the test cases in the
+8
+
+Figure 3: Printouts during the search
+Figure 4: Successful run confirmation
+test suite. In Fig. 6 we show an example of the test case images saved for a
+particular run.
+(a) Scenario fitness convergence
+(b) Final test suite fitness and diversity
+Figure 5: Metadata for the generated scenarios
+Finally, let us suppose we also want to run a random search with the same
+evaluation budget to be able to compare the performance of our configuration
+of NSGA-II algorithm to some baseline. We can run the random search by
+9
+
+01:0602.320 INFO
+started test generation,writing logs to file: logs.txt
+-12-9101:0602,320INFO
+Running the optimization
+2-12-9801:0602,321INFO
+Problem: robot,Algorithm:nsga2,Runs number:3e,Saving the results:True
+2-12-9101:06.02,343INFO
+Executing run o:
+2-12-9101:06:02344INFO
+Using random seed:1753925990
+n_gen
+n_eval
+innds
+cvmin
+cv_avg
+eps
+indicator
+1
+150
+1
+5.474517E+01
+8.330072E+91
+2
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+7.742613E+01
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+5
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+1
+7.8751083190
+6.161694E+0103:21:13,072INFO
+Execution time,6909.677314 sec
+,088INFO
+Test suite of 3o test scenarios generated
+$,103INFO
+Thehighest fitnessfound:224.994949
+3:211L3,103INFO
+Average diversity:0.720751
+03:21:25,148INFO
+Stats savedas stats nsga231-12-2022-stats.json
+03:21:25,157INFO
+Stats saved asstats nsga231-12-2022-conv.json
+3:21:25,361INFO
+Test cases saved as tcs nsga2l31-12-2022-tcs.json
+03:21:53,871INFO
+Images saved in tc images nsga2
+21:53,871INFO
+Images saved in tcimagesnsga2rung":f
+"158"：97.81219330881972
+200"：99.59797974644661
+"250"：99.59797974644661,
+"300"：186.18376618487352,
+"350"：186.18376618407352,
+"480"：106.18376618407352,
+"450：107.25483399593897,
+"500"：107.25483399593897,
+"550":130.56854249492375,
+"600"：130.56854249492375,
+"650"：130.56854249492375,
+"700"：130.56854249492375,
+"888"130.56854249492375
+"850":
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+"900":rune":f
+"fitness":[
+198.7106781186548
+171.8538238691624,
+192.02438661763966,
+194.46803743153552,
+190.36753236814718,
+211.88225099390866,
+209.88225099390866,
+194.85281374238588,
+168.71067811865476,
+191.8822509939086,
+181.15432893255073,
+183.39696961967007
+novelty":0.23096571372433472
+runi"Figure 6: Images of the generated scenarios
+executing the following command:
+python
+optimize.py --problem =" robot" --algo =" random" --runs =30 \\
+--save_results=True
+The random search will be run and the metadata saved, as in the previous
+case. Now we can compare the results produced by the two different search
+algorithms via executing the following command:
+python
+compare.py --stats_path =" stats_nsga2" " stats_random" \\
+--stats_names "NSGA -II" "Random"
+In the stats path argument we specify the paths of the metadata for the
+runs we wish to compare and in the stats names the names we assign for the
+runs.
+In Fig.7 and Fig. 8 we can see examples of the outputs produced by the
+compare.py script. Fig. 7a shows the fitness and Fig. 7b the diversity of the
+scenarios in the test suites produced over the specified number of runs. Fig.
+8 shows the best values found by the compared search algorithms over the
+generations.
+3. Illustrative examples
+In this section, we present the summarized results of several test genera-
+tion case studies using the AmbieGen tool. The full results can be found in
+our research paper [11] and the SBST 2022 competition report [10].
+We conducted a case study on an autonomous robot with an obstacle
+avoidance algorithm based on nearness diagrams [13]. The robot model was
+a Pioneer 3-AT equipped with a SICK LMS200 laser with a sensing range
+of 10 meters. The simulations were run in the Player/Stage simulator [14].
+10
+
+2022-10-15-images_fin_rob
+Vruno
+o.png
+1.png
+2.png
+Test case fitenss 198.7106781186548
+3.png
+Robotpathm
+质4.png
+Walls
+35
+5.png
+6.png
+30
+7.png
+25 
+8.png
+9.png
+U
+20
+10.png
+U
+15 
+11.png
+U
+10
+12.png
+U
+13.png
+U
+5
+14.png
+U
+15.png
+U
+0
+5
+10
+15
+20
+30
+35
+40(a) Scenario fitness
+(b) Scenario diversity
+Figure 7: Evaluating the NSGA-II algorithm for autonomous robot test case generation
+Figure 8: Comparing the convergence of NSGA-II and random search for autonomous
+robot case study
+You can see an illustration of the simulation environment in Fig. 9a. We
+used AmbieGen to generate diverse maps with obstacles to test the robot’s
+performance. We identified several scenarios in which the robot became stuck
+and failed to reach its goal location. An example of such a scenario can be
+found in the following video: Video.
+To evaluate the effectiveness of our tool, we allocated a two-hour budget
+for AmbieGen to generate test scenarios. The generated scenarios were then
+passed to the simulator and executed. We repeated the experiment 30 times,
+using both the NSGA-II and random search configurations of AmbieGen.
+The average number of failures detected is shown in Fig. 9b. On average,
+11
+
+220
+NSGA-II
+Random
+200
+180
+Fitness
+160
+140
+120
+100
+80
+0
+20
+40
+60
+80
+100
+120
+140
+Numberofgenerations250
+200
+itness
+.150
+左 100
+50
+0
+Random
+NSGA-II
+Algorithm0.8
+0.6
+Novelty
+0.4
+0.2
+0.0
+Random
+NSGA-II
+AlgorithmAmbieGen detected 9 failures in two hours, compared to 2 failures for random
+search
+(a) Executing autonomous robot scenario in the Play-
+er/Stage simulator
+(b) The number of failures revealed by AmbieGen for
+the robot case study
+Figure 9: Using AmbieGen for testing autonomous robot navigation algorithm
+In the second case study, we evaluated the performance of our test gener-
+ation tool on an autonomous vehicle lane keeping assist system (LKAS) using
+the BeamNg simulator [15]. We used the AmbieGen tool to generate diverse,
+fault-revealing road topologies, which were then simulated in the BeamNg
+environment (shown in Fig. 10a). During the simulations, we identified a
+number of scenarios in which the vehicle left its lane (an example of which
+can be seen in the video at Video).
+We ran our tool for a time budget of 2 hours, using the SBST22 compe-
+tition code pipeline. The failure criterion for the LKAS system was defined
+as more than 85% of the car’s area leaving the lane. The driving agent had
+a maximum speed of 70 Km/h. We compared the results of AmbieGen’s
+NSGA-II configuration, Random Search configuration, and the Frenetic tool
+[6], which was also given a 2-hour time budget for test generation.
+As shown in Fig. 10b, out of 30 runs, AmbieGen and Frenetic on average
+produced almost the same number of failures (14), while Random Search
+produced an average of 9 failures.
+The obtained results suggest that AmbieGen could effectively identify
+failures in the autonomous systems under test.
+4. Impact
+Autonomous systems testing is an important area of research, and finding
+test scenarios that reveal a diverse range of system failures within a limited
+12
+
+25
+faults
+20
+15
+Revealed
+10
+5
+0
+AmbieGen
+RandomSearch
+Generationmethod(a) Executing the LKAS scenario in the BeamNg sim-
+ulator
+(b) The number of failures revealed by AmbieGen for
+the LKAS case study
+Figure 10: Using AmbieGen to test autonomous vehicle LKAS model
+time and evaluation budget is a significant challenge [16]. One of the common
+solutions is to use evolutionary search to guide the sampling towards more
+challenging scenarios [5, 7]. These search based techniques allow to identify
+potential failures and improve the overall reliability of the system.
+AmbieGen is a test generation tool that uses evolutionary search to gen-
+erate test scenarios for autonomous systems. Its modular design allows for
+customization of the initial population generation function, fitness evaluation
+function, search operators (such as crossover and mutation), and the search
+algorithm itself. Out of the box, AmbieGen supports testing of autonomous
+robots and vehicle LKAS systems, and additional systems can be added using
+the provided implementations as examples.
+AmbieGen is a valuable resource for research on search-based test case
+generation for autonomous systems. Its built-in modules enable easy com-
+parison of different search algorithms and their modifications, based on the
+quality and diversity of the generated solutions, as well as the convergence
+of the algorithm over time.
+AmbieGen can help answer research questions that are not frequently
+discussed in the literature, such as:
+• To what extent the diversity preservation technique A helps improve
+the diversity of the test suite? The importance of the diversity in test
+case generation is extensively discussed in the work of Klikovits et al.
+[17].
+• To what extent does the search operator A helps improve the conver-
+gence over the operator B? To what extent the algorithm A outperforms
+13
+
+30
+25
+ults
+20
+led
+15
+veal
+Rev
+10
+5
+0
+AmbieGen
+Frenetic
+RandomSearch
+Generationmethodthe algorithm B for the test case generation? Improvements to the base-
+line genetic algorithms implementations can lead to better results, as
+discussed by Abdessalem et al. [18], where multi-objective population-
+based search algorithms and decision tree classification were combined.
+• What fitness criteria are more relevant for guiding the system towards
+fault revealing scenarios? This question includes the comparison of the
+single, multi-objective based search as well surrogate model assisted
+search.
+AmbieGen can also be useful in the pursuit of actively studied research ques-
+tions, where the fault revealing test case generation is required, such as:
+transferability of failures from simulation to the real world [19], autonomous
+system failure prediction [20], test case prioritization [21] and others.
+AmbieGen has proven its effectiveness in fault revealing by winning this
+year’s edition of the SBST 2022 cyber-physical testing tool competition. Our
+submission is described in the following article [22] and is available at the fol-
+lowing link https://github.com/dgumenyuk/tool-competition-av.
+We
+have always kept our tool open sourced and we expect more people to start
+using it. We welcome all the contributions for expanding our framework.
+5. Conclusions
+In this paper, we present the AmbieGen framework for search based test
+case generation for autonomous systems, in its public version 0.1.0.
+We
+briefly outline the motivation for developing this framework, its workflow and
+main functionalities. We also provide illustrative examples for using the tool
+for autonomous vehicle lane keeping assist system testing and autonomous
+robot obstacle avoiding algorithm testing.
+The main features of our tool
+include:
+• modular architecture, which allows researchers to easily modify the
+existing modules, such as initial population generation, crossover, mu-
+tation, fitness function as well as introduce new problems and run ex-
+periments;
+• we provide implementations of test case generation for two systems
+under test: autonomous vehicle LKAS system and autonomous robot;
+this implementation includes three search algorithms: random search,
+14
+
+single objective genetic algorithm and a two-objective NSGA-II genetic
+algorithm;
+• our framework is built to be compatible with Pymoo framework [12],
+allowing to fully benefit from the Pymoo framework features, such as
+high number of implemented algorithms in Pymoo.
+6. Future Plans
+Our framework currently includes the implementation of two test case
+generation problems, as well as three algorithms (random search, GA, NSGA-
+II) for generating test cases. The fitness function is calculated based on a
+simplified model of the system, and test scenarios are represented as 2D
+arrays, with each column describing a discrete aspect of the scenario. In the
+future, we plan to expand the capabilities of our framework to include:
+• new algorithms, especially the ones based on the quality-diversity search
+[23]
+• new test case generation problems, for instance more complex test sce-
+narios that include moving pedestrians, other vehicles and traffic signs;
+• new fitness functions e.g based on surrogate models of the system under
+test, as in the work of Ramakrishna et al. [24], functions based on
+neuron coverage [25] and surprise adequacy [26] dedicated to testing
+systems containing neural networks;
+• add new problem representations, supporting popular scenario specifi-
+cation languages such as SCENIC [27];
+• add an integration with popular simulators, for instance CARLA [28]
+or LGSVL [29]. This will allow to directly evaluate the system model
+with the generated scenarios. Also the feedback from the simulators
+could be incorporated in fitness functions for guiding the test scenario
+sampling.
+Acknowledgements
+This work is partly funded by the by the Fonds de Recherche du Qu´ebec
+(FRQ), the Natural Sciences and Engineering Research Council of Canada
+(NSERC), and the Canadian Institute for Advanced Research (CIFAR).
+15
+
+References
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+19
+
diff --git a/9dAzT4oBgHgl3EQfSfu4/content/tmp_files/load_file.txt b/9dAzT4oBgHgl3EQfSfu4/content/tmp_files/load_file.txt
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@@ -0,0 +1,519 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf,len=518
+page_content='AmbieGen: A Search-based Framework for Autonomous  Systems Testing  Dmytro Humeniuk, Foutse Khomh, Giuliano Antoniol  Polytechnique Montr´eal, 2500 Chemin de Polytechnique, QC H3T 1J4, Montr´eal,  Canada  Abstract  Thorough testing of safety-critical autonomous systems, such as self-driving  cars, autonomous robots, and drones, is essential for detecting potential fail-  ures before deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' One crucial testing stage is model-in-the-loop test-  ing, where the system model is evaluated by executing various scenarios in  a simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' However, the search space of possible parameters defining these  test scenarios is vast, and simulating all combinations is computationally in-  feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' To address this challenge, we introduce AmbieGen, a search-based  test case generation framework for autonomous systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  AmbieGen uses  evolutionary search to identify the most critical scenarios for a given system,  and has a modular architecture that allows for the addition of new systems  under test, algorithms, and search operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' Currently, AmbieGen supports  test case generation for autonomous robots and autonomous car lane keep-  ing assist systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' In this paper, we provide a high-level overview of the  framework’s architecture and demonstrate its practical use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  Keywords:  evolutionary search, autonomous systems, self driving cars,  autonomous robots, neural network testing  Metadata  The project metadata is presented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' Motivation and significance  Autonomous systems, including autonomous vehicles, robots, or drones  can provide a number of benefits such as driving assistance, high-risk zone  Preprint submitted to Science of Computer Programming  January 4, 2023  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='01234v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='RO]  1 Jan 2023  Nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  Code metadata description  Please fill in this column  C1  Current code version  v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='0  C2  Permanent link to code/repository  used for this code version  For  example:  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  com/swat-lab-optimization/  AmbieGen-tool  C3  Permanent  link  to  Reproducible  Capsule  https://codeocean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='com/  capsule/1741442/tree  C4  Legal Code License  MIT license (MIT)  C5  Code versioning system used  git  C6  Software code languages, tools, and  services used  python  C7  Compilation requirements, operat-  ing environments and dependencies  indicated in requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='txt  C8  If available, link to developer docu-  mentation/manual  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='com/  swat-lab-optimization/  AmbieGen-tool/blob/master/  README.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='md  C9  Support email for questions  dmytro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='humeniuk@polymtl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='ca  Table 1: Code metadata (mandatory)  exploration, and aid in rescue operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' At the same time, these are safety-  critical systems and it is very important to ensure they are robust to unseen  environments and conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' This can be done by thorough testing prior  to their deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' Typically, at the initial development stages model-in-  the-loop testing is performed [1], where the system is tested in a simulation  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' Given the complexity of autonomous systems, the number of  potential test scenarios is vast and exhaustive execution is not feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' For  example, an autonomous vehicle scenario could involve a variety of param-  eters such as road topology, the movement and behavior of other vehicles  and pedestrians, traffic signs, weather conditions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' We surmise that in  order to identify the most critical scenarios for a given system, application  of search algorithms is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  In this work, we propose AmbieGen, a search based framework for gen-  erating adversarial test scenarios for autonomous systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  By leveraging  evolutionary search AmbieGen allows to find challenging and diverse test  scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  2  The problem of identifying critical scenarios for a system has been ad-  dressed in several previous works on falsifying temporal logic requirements  of cyber-physical systems, such as S-Taliro [2], Breach [3], and ARIsTEO [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  These works typically consider falsifying a model of the system that takes a  set of input signals and produces a set of output signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  In our work, we focus on testing autonomous systems for which the input  signals are complex and may include data from various sensors and cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  Generating a valid combination of falsifying input signals (such as lidar read-  ings and RGB camera readings) directly would be challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' Therefore,  we propose a method for generating test cases that specify a virtual environ-  ment for the autonomous system, rather than the input signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' The input  signals are generated in the virtual environment during simulation based on  the actions of the autonomous agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  Several approaches have been proposed for generating virtual environ-  ments for testing autonomous driving and robotics systems, including As-  Fault [5], Frenetic [6], DeepJanus [7], DeepHyperion [8] and others presented  at the SBST 2021 [9] and SBST 2022 [10] tool competitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  The tool we present in this paper, AmbieGen, is the winner of SBST 2022  tool competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' It could produce the biggest number of diverse fault reveal-  ing scenarios for an autonomous vehicle lane keeping assist system (LKAS)  given a limited time budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' More details about the search algorithm im-  plementation can be found in our research paper [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' In our work we have  shown that the simplified model of the system can be effective in guiding the  search for producing the test scenarios for the full, simulator based, model  of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  Our framework can be used for further research in the search algorithms,  search operator and fitness function design for autonomous systems adver-  sarial testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' We built the framework to be modular, and thus easily cus-  tomizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' By referring to project documentation as well as the example  implementations we provide, researchers can specify their own test scenario  generation problems, fitness functions, crossover and mutation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  This tool is developed in Python and can be easily run as a python package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  More instructions and examples are provided in the AmbieGen repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' Software description  In this work, we present AmbieGen, an open-source Python framework  that utilizes evolutionary search for the generation of test scenarios for au-  3  tonomous systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' Currently, AmbieGen supports the creation of test sce-  narios for lane keeping assist systems (LKAS) in autonomous vehicles and  for autonomous robots navigating a closed room with obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  The test scenarios for LKAS in vehicles are designed to challenge the  system with various road topologies, while the scenarios for autonomous  robots involve navigating a closed room with obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  Examples of the  generated scenarios can be seen in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  Figure 1: An example of the test case for LKAS system (a) and an autonomous robot (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  The x-axis represents the map length in meters, and the y-axis represents the map width  in meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' Software architecture  This subsection provides a detailed description of the software imple-  mentation of AmbieGen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' The key components of AmbieGen are illustrated  in Figure 2, which are common components for implementing evolutionary  search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' We use the Pymoo framework [12] to implement the search algo-  rithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' The most important modules and classes are outlined below:  Solution - this is one of the most important classes, which contains all  the necessary attributes and functions needed to represent the candi-  date solution of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' It should contain a scenario attribute  with the list of test case parameters, function for fitness evaluation,  novelty calculation, as well as, optionally, image generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  4  200  a  40  b  口  Ci  ■  35-  ■  ■  ■  30  ■  25  Robotpath  20-  Walls  V  国  15 -  ■  ■  口  ■  10  5  ■  ■  fo  0  0  5  10  15  20  25  200  30  35  40Figure 2: AmbieGen architecture  Sampling - this is the class for initial population generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' At the  output it provides N instances of the Solution class, with the initial-  ized scenario attribute, defining the test scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' Typically the test  scenario is represented by a two dimensional array, randomly initial-  ized based on the minimum and maximum values of the test case pa-  rameters, defined in the configuration file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' Each column of the array  corresponds to some part of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' More information about  the representation of the test scenarios that we used can be found in  the repository page as well as in our research article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  Problem - in this class, we define the logic for evaluating the fitness  of each solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' For single-objective search (using GA), we specify  the fitness function for evaluating the scenario fault revealing power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  For two-objective search (using NSGA-II), we define two objectives:  fault revealing power and novelty calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' The novelty objective  is calculated as the average novelty of a given test scenario relative to  the 5 solutions with the highest fault revealing power fitness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' If the  problem has any constraints, such as a minimum required fitness value,  they should also be specified in this class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  TC to environment - this is a function to transform the test case (TC)  encoded as a 2D array of parameters, to the input format suitable for  the system model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' For example, for the LKAS problem, the model  input is a list of the 2D coordinates of points, defining the road topol-  ogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' The test case itself is represented as a sequence of transformations  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='Pymoo  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='post_processing()  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='Sampling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='Solution object 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='TC to environment ()  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='+gen_randomscenario()  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='Solution object 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='Problem  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='fitness evaluation ()  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='Solution object 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='Solution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='Crossover  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='+map_size  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='Solution object 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='configuration file  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='+scenario  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='Mutation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='+fitness eval()  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='Population size  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='Solution object N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='Numberofgenerations  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='+novelty eval()  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='Crossover/mutationrate  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='+build image()  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='TCparameterranges  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='Folderto save resultsneeded to perform to obtain the points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' For the autonomous robot the  test scenario is represented as a sequence of parameters describing the  2D map with obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' The TC to environment module is used to  create a 2D bitmap from the given parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' The bitmap is given  as the input to the autonomous robot model, which runs a planning  algorithm to find the shortest path between the start and goal location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  fitness evaluation - a function to calculate the fitness i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='e fault revealing  power of the scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' It takes the output of the TC to environment  function as the input and execute the system model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' It collects the  data about the model behaviour during execution and computes the  fitness score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' For the LKAS system, the fitness is defined by the biggest  deviation from the lane center and for the autonomous robot - by the  length of the path to reach the goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  Crossover - in this class the crossover operator is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' Currently  we are using a one point crossover, which can be applied to fixed and  variable length solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  Mutation - in this class the mutation operator is implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' We  have 2 types of mutations: exchange and change of variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' In ex-  change mutation, two randomly selected columns of the test case are  exchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' In the case of the road topology, it would correspond to  exchanging the positions of two random road segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' In change of  variable mutation, a randomly selected parameter value in the test case  matrix is changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' In the road topology example it could correspond  to the change of the length of one of the straight road segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  post processing - The post-processing module of our framework includes  several functions for handling the test suite and its metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  The  function get test suite() retrieves the test suite, get stats() retrieves  metadata such as fitness and novelty scores, and save tcs images()  saves the images of the test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' The size of the test suite, denoted  as T, can be specified in the configuration file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' In our experiments, T  was typically set to 30, representing the best solutions found by the  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  Metadata for the test suite includes the fitness of the top T solutions,  their novelty (calculated as the average novelty between all pairs of  scenarios in the test suite), and the convergence (best solution fitness  6  found at each epoch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  The post-processing module also includes a  compare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='py script for comparing the results of different algorithms,  using the collected metadata to generate convergence plots and fitness  and diversity boxplots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  configuration file - finally we have a configuration file, where the pa-  rameters of the algorithm, such as: the population size, the number  of generations, crossover/mutation rate, and the test suite size are de-  fined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' Users should also specify the allowable ranges for the test case  parameters and the paths for saving the resulting test suite and its  metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  Currently, when adding a new problem, one should provide the implemen-  tation of each of the modules as well as the TC to environment and fitness  evaluation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' We are working on reducing the number of additional  implementations needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' Our framework includes the implementation of all  the modules for the LKAS and autonomous robot test case generation prob-  lems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' Software functionalities  AmbieGen public version 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='0 as presented in this paper offers the fol-  lowing major functionalities:  Autonomous vehicle LKAS system testing: generating scenarios, rep-  resented as a list of 2D coordinates defining the road topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  Autonomous robot testing: generating scenarios, represented as the 2D  bitmap, defining obstacle locations in a fixed sized map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  Search-based generation: our framework provides options for search-  based test suite generation, including random search, single-objective  genetic algorithm (GA), and two-objective genetic algorithm (NSGA-  II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' The search algorithms are implemented using the Pymoo frame-  work [12], and can be easily extended to support additional algorithms  supported by Pymoo with minor modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  The single-objective GA optimizes the test suite for scenario fault re-  vealing power, while the two-objective NSGA-II optimizes for both  fault revealing power and diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  As demonstrated in our previ-  ous work [11], the two-objective algorithm allows to produce a more  diverse set of test scenarios compared to the single-objective search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  7  Experiment data tracking: AmbieGen tracks the results of each ex-  periment and saves them in a user-defined location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' The saved data  includes the T (as determined by the user) best test scenarios identified  based on their fitness or crowding distance, as well as their associated  metadata such as fitness, average diversity, and visualizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' This  allows for easy analysis and comparison of the results of different ex-  periments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' Use cases of the software  In this subsection we provide an illustrative example of how to use Am-  bieGen to generate test cases for an autonomous robot planning algorithm  testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' Suppose we want to perform 30 runs of the NSGA-II algorithm with  150 individuals and 200 generations to evaluate this configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' We want  to save the generated test cases, their illustrations as well as their metadata,  such as fitness and diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' Below you can see the configuration file entries  with the parameters we chose for the genetic algorithm and well as the path  to save the experiment results:  ga = {" pop_size ": 150, "n_gen ": 200, "mut_rate ": 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='4, "cross_rate ": 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='9,  " test_suite_size ": 30 }  files = {" stats_path ": "stats", "tcs_path ": "tcs", "images_path ": images "}  Now we are ready to start the test case generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' We can launch Am-  bieGen with the following command and parameters:  python  optimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='py --problem =" robot" --algo =" nsga2" --runs =30 \\\\  --save_results=True  The search will start and you could see some printouts, such as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' 3 with  the current number of generation (n gen), number of evaluations (n eval),  constraint violation (cv min), number of non-dominant solution for NSGA-  II algorithm (n nds) and the best solution found (f opt) for GA algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  More details about the printed information can be found on the Pymoo page  (https://pymoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='org/interface/display.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='html).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  After a successful run, you will see the confirmation about the run exe-  cution time, saved test cases, their metadata and the images, as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' 4  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' 5 you can see examples of the metadata saved, such as the algo-  rithm convergence 5a (the best fitness value at each generation in the format  ”evaluation number”: best fitness found), the fitness of the test cases in the  test suite as well as their average diversity i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=', novelty 5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' Novelty is cal-  culated as the average diversity of all of the pairs of the test cases in the  8  Figure 3: Printouts during the search  Figure 4: Successful run confirmation  test suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' 6 we show an example of the test case images saved for a  particular run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  (a) Scenario fitness convergence  (b) Final test suite fitness and diversity  Figure 5: Metadata for the generated scenarios  Finally, let us suppose we also want to run a random search with the same  evaluation budget to be able to compare the performance of our configuration  of NSGA-II algorithm to some baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' We can run the random search by  9  01:0602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='320 INFO  started test generation,writing logs to file: logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='txt  12-9101:0602,320INFO  Running the optimization  2-12-9801:0602,321INFO  Problem: robot,Algorithm:nsga2,Runs number:3e,Saving the results:True  2-12-9101:06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='02,343INFO  Executing run o:  2-12-9101:06:02344INFO  Using random seed:1753925990  n_gen  n_eval  innds  cvmin  cv_avg  eps  indicator  1  150  1  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
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+page_content='8751083190  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='161694E+0103:21:13,072INFO  Execution time,6909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='677314 sec  ,088INFO  Test suite of 3o test scenarios generated  $,103INFO  Thehighest fitnessfound:224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='994949  3:211L3,103INFO  Average diversity:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='720751  03:21:25,148INFO  Stats savedas stats nsga231-12-2022-stats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='json  03:21:25,157INFO  Stats saved asstats nsga231-12-2022-conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='json  3:21:25,361INFO  Test cases saved as tcs nsga2l31-12-2022-tcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='json  03:21:53,871INFO  Images saved in tc images nsga2  21:53,871INFO  Images saved in tcimagesnsga2rung":f  "158"：97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
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+page_content='39696961967007  novelty":0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='23096571372433472  runi"Figure 6: Images of the generated scenarios  executing the following command:  python  optimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='py --problem =" robot" --algo =" random" --runs =30 \\\\  --save_results=True  The random search will be run and the metadata saved, as in the previous  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' Now we can compare the results produced by the two different search  algorithms via executing the following command:  python  compare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='py --stats_path =" stats_nsga2" " stats_random" \\\\  --stats_names "NSGA -II" "Random"  In the stats path argument we specify the paths of the metadata for the  runs we wish to compare and in the stats names the names we assign for the  runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='7 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' 8 we can see examples of the outputs produced by the  compare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='py script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' 7a shows the fitness and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' 7b the diversity of the  scenarios in the test suites produced over the specified number of runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  8 shows the best values found by the compared search algorithms over the  generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' Illustrative examples  In this section, we present the summarized results of several test genera-  tion case studies using the AmbieGen tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' The full results can be found in  our research paper [11] and the SBST 2022 competition report [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  We conducted a case study on an autonomous robot with an obstacle  avoidance algorithm based on nearness diagrams [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' The robot model was  a Pioneer 3-AT equipped with a SICK LMS200 laser with a sensing range  of 10 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' The simulations were run in the Player/Stage simulator [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  10  2022-10-15-images_fin_rob  Vruno  o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='png  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='png  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='png  Test case fitenss 198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='7106781186548  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='png  Robotpathm  质4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='png  Walls  35  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='png  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='png  30  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='png  25  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='png  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='png  U  20  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='png  U  15  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='png  U  10  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='png  U  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='png  U  5  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='png  U  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='png  U  0  5  10  15  20  30  35  40(a) Scenario fitness  (b) Scenario diversity  Figure 7: Evaluating the NSGA-II algorithm for autonomous robot test case generation  Figure 8: Comparing the convergence of NSGA-II and random search for autonomous  robot case study  You can see an illustration of the simulation environment in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' 9a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' We  used AmbieGen to generate diverse maps with obstacles to test the robot’s  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' We identified several scenarios in which the robot became stuck  and failed to reach its goal location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' An example of such a scenario can be  found in the following video: Video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  To evaluate the effectiveness of our tool, we allocated a two-hour budget  for AmbieGen to generate test scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' The generated scenarios were then  passed to the simulator and executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' We repeated the experiment 30 times,  using both the NSGA-II and random search configurations of AmbieGen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  The average number of failures detected is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' 9b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' On average,  11  220  NSGA-II  Random  200  180  Fitness  160  140  120  100  80  0  20  40  60  80  100  120  140  Numberofgenerations250  200  itness  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='150  左 100  50  0  Random  NSGA-II  Algorithm0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='6  Novelty  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='0  Random  NSGA-II  AlgorithmAmbieGen detected 9 failures in two hours,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' compared to 2 failures for random  search  (a) Executing autonomous robot scenario in the Play-  er/Stage simulator  (b) The number of failures revealed by AmbieGen for  the robot case study  Figure 9: Using AmbieGen for testing autonomous robot navigation algorithm  In the second case study,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' we evaluated the performance of our test gener-  ation tool on an autonomous vehicle lane keeping assist system (LKAS) using  the BeamNg simulator [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' We used the AmbieGen tool to generate diverse,  fault-revealing road topologies, which were then simulated in the BeamNg  environment (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' 10a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' During the simulations, we identified a  number of scenarios in which the vehicle left its lane (an example of which  can be seen in the video at Video).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  We ran our tool for a time budget of 2 hours, using the SBST22 compe-  tition code pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' The failure criterion for the LKAS system was defined  as more than 85% of the car’s area leaving the lane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' The driving agent had  a maximum speed of 70 Km/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' We compared the results of AmbieGen’s  NSGA-II configuration, Random Search configuration, and the Frenetic tool  [6], which was also given a 2-hour time budget for test generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' 10b, out of 30 runs, AmbieGen and Frenetic on average  produced almost the same number of failures (14), while Random Search  produced an average of 9 failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  The obtained results suggest that AmbieGen could effectively identify  failures in the autonomous systems under test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' Impact  Autonomous systems testing is an important area of research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' and finding  test scenarios that reveal a diverse range of system failures within a limited  12  25  faults  20  15  Revealed  10  5  0  AmbieGen  RandomSearch  Generationmethod(a) Executing the LKAS scenario in the BeamNg sim-  ulator  (b) The number of failures revealed by AmbieGen for  the LKAS case study  Figure 10: Using AmbieGen to test autonomous vehicle LKAS model  time and evaluation budget is a significant challenge [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' One of the common  solutions is to use evolutionary search to guide the sampling towards more  challenging scenarios [5, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' These search based techniques allow to identify  potential failures and improve the overall reliability of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  AmbieGen is a test generation tool that uses evolutionary search to gen-  erate test scenarios for autonomous systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' Its modular design allows for  customization of the initial population generation function, fitness evaluation  function, search operators (such as crossover and mutation), and the search  algorithm itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' Out of the box, AmbieGen supports testing of autonomous  robots and vehicle LKAS systems, and additional systems can be added using  the provided implementations as examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  AmbieGen is a valuable resource for research on search-based test case  generation for autonomous systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' Its built-in modules enable easy com-  parison of different search algorithms and their modifications, based on the  quality and diversity of the generated solutions, as well as the convergence  of the algorithm over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  AmbieGen can help answer research questions that are not frequently  discussed in the literature, such as:  To what extent the diversity preservation technique A helps improve  the diversity of the test suite?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' The importance of the diversity in test  case generation is extensively discussed in the work of Klikovits et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  To what extent does the search operator A helps improve the conver-  gence over the operator B?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' To what extent the algorithm A outperforms  13  30  25  ults  20  led  15  veal  Rev  10  5  0  AmbieGen  Frenetic  RandomSearch  Generationmethodthe algorithm B for the test case generation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' Improvements to the base-  line genetic algorithms implementations can lead to better results, as  discussed by Abdessalem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' [18], where multi-objective population-  based search algorithms and decision tree classification were combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  What fitness criteria are more relevant for guiding the system towards  fault revealing scenarios?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' This question includes the comparison of the  single, multi-objective based search as well surrogate model assisted  search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  AmbieGen can also be useful in the pursuit of actively studied research ques-  tions, where the fault revealing test case generation is required, such as:  transferability of failures from simulation to the real world [19], autonomous  system failure prediction [20], test case prioritization [21] and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  AmbieGen has proven its effectiveness in fault revealing by winning this  year’s edition of the SBST 2022 cyber-physical testing tool competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' Our  submission is described in the following article [22] and is available at the fol-  lowing link https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='com/dgumenyuk/tool-competition-av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  We  have always kept our tool open sourced and we expect more people to start  using it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' We welcome all the contributions for expanding our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' Conclusions  In this paper, we present the AmbieGen framework for search based test  case generation for autonomous systems, in its public version 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  We  briefly outline the motivation for developing this framework, its workflow and  main functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' We also provide illustrative examples for using the tool  for autonomous vehicle lane keeping assist system testing and autonomous  robot obstacle avoiding algorithm testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  The main features of our tool  include:  modular architecture, which allows researchers to easily modify the  existing modules, such as initial population generation, crossover, mu-  tation, fitness function as well as introduce new problems and run ex-  periments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  we provide implementations of test case generation for two systems  under test: autonomous vehicle LKAS system and autonomous robot;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  this implementation includes three search algorithms: random search,  14  single objective genetic algorithm and a two-objective NSGA-II genetic  algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  our framework is built to be compatible with Pymoo framework [12],  allowing to fully benefit from the Pymoo framework features, such as  high number of implemented algorithms in Pymoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' Future Plans  Our framework currently includes the implementation of two test case  generation problems, as well as three algorithms (random search, GA, NSGA-  II) for generating test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' The fitness function is calculated based on a  simplified model of the system, and test scenarios are represented as 2D  arrays, with each column describing a discrete aspect of the scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' In the  future, we plan to expand the capabilities of our framework to include:  new algorithms, especially the ones based on the quality-diversity search  [23]  new test case generation problems, for instance more complex test sce-  narios that include moving pedestrians, other vehicles and traffic signs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  new fitness functions e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='g based on surrogate models of the system under  test, as in the work of Ramakrishna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' [24], functions based on  neuron coverage [25] and surprise adequacy [26] dedicated to testing  systems containing neural networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  add new problem representations, supporting popular scenario specifi-  cation languages such as SCENIC [27];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content='  add an integration with popular simulators, for instance CARLA [28]  or LGSVL [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
+page_content=' This will allow to directly evaluate the system model  with the generated scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfSfu4/content/2301.01234v1.pdf'}
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diff --git a/9tE5T4oBgHgl3EQfRQ5h/content/tmp_files/2301.05519v1.pdf.txt b/9tE5T4oBgHgl3EQfRQ5h/content/tmp_files/2301.05519v1.pdf.txt
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+Second sound resonators and tweezers as vorticity or velocity
+probes : fabrication, model and method
+Eric Woillez∗, Jérôme Valentin†and Philippe-E. Roche‡
+Univ. Grenoble Alpes, CNRS, Institut NEEL, F-38042 Grenoble, France
+Distributed under a Creative Commons Attribution.
+CC-BY | 4.0 International licence
+Abstract
+An analytical model of second-sound resonators with open-cavity is presented and validated against
+simulations and experiments in superfluid helium using a new design of resonators reaching unprecedented
+resolution. The model accounts for diffraction, geometrical misalignments and flow through the cavity. It is
+validated against simulations and experiments using cavities of aspect ratio of the order of unity operated
+up to their 20th resonance in superfluid helium. An important result is that resonators can be optimized
+to selectively sense the quantum vortex density carried by the throughflow -as customarily done in the
+literature- or alternatively to sense the mean velocity of this throughflow. Two velocity probing methods are
+proposed, one taking advantage of geometrical misalignements between the tweezers plates, and another one
+by driving the resonator non-linearly, beyond a threshold entailing the self-sustainment of a vortex tangle
+within the cavity.
+After reviewing several methods, a new mathematical treatment of the resonant signal is proposed, to
+properly separate the quantum vorticity from the parasitic signals arising for instance from temperature and
+pressure drift. This so-called elliptic method consists in a geometrical projection of the resonance in the
+inverse complex plane. Its strength is illustrated over a broad range of operating conditions.
+The resonator model and the elliptic method are applied to characterize a new design of second-sound
+resonator of high resolution thanks to miniaturization and design optimization. When immersed in a su-
+perfluid flow, these so-called
+second-sound tweezers provide time-space resolved information like classical
+local probes in turbulence, here down to sub-millimeter and sub-millisecond scales. The principle, design
+and micro-fabrication of second sound tweezers are detailed, as well as their potential for the exploration of
+quantum turbulence
+Contents
+1
+Introduction to second sound resonators
+2
+1.1
+Quantum fluids and second sound
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+2
+1.2
+Generation and detection of second sound waves
+. . . . . . . . . . . . . . . . . . . . . . . . . . .
+2
+1.3
+From macroscopic second sound sensors to microscopic tweezers . . . . . . . . . . . . . . . . . . .
+3
+1.4
+Overview of the manuscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+2
+Design, fabrication and mode of operation of second sound tweezers
+4
+2.1
+Mechanical design
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+2.2
+Second sound detection and generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+2.2.1
+Thermometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+2.2.2
+Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+2.2.3
+Digression on the operation in the non-linear heating regime
+. . . . . . . . . . . . . . . .
+9
+2.3
+Microfabrication and assembling
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+2.4
+Electric circuit
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+13
+∗present affiliation: CEA-Liten, Grenoble
+†present affiliation: Observatoire de Paris - PSL, CNRS, LERMA, F-75014, Paris, France
+‡Corresponding author
+1
+arXiv:2301.05519v1  [cond-mat.other]  13 Jan 2023
+
+3
+Models of second sound resonators
+14
+3.1
+Resonant spectrum of second sound resonator: phenomenological aspects
+. . . . . . . . . . . . .
+15
+3.2
+Analytical approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+15
+3.3
+Numeric algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+19
+3.3.1
+For a backgroud medium at rest
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+19
+3.3.2
+In the presence of a turbulent flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+21
+3.4
+Quantitative predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+22
+3.4.1
+Spectral response of second sound resonators . . . . . . . . . . . . . . . . . . . . . . . . .
+23
+3.4.2
+Response with a flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+23
+3.4.3
+Effect of lateral shift of the emitter and receiver plates . . . . . . . . . . . . . . . . . . . .
+26
+3.4.4
+Limits of the model
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+27
+3.5
+Quantum vortex or velocity measurements ?
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+28
+4
+Measurements with second sound tweezers
+29
+4.1
+The vortex line density from the attenuation coefficient
+. . . . . . . . . . . . . . . . . . . . . . .
+30
+4.2
+Analytical method in an idealized case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+32
+4.3
+The elliptic method
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+33
+4.4
+Applications of the elliptic method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+35
+4.4.1
+Suppression of temperature and pressure drifts . . . . . . . . . . . . . . . . . . . . . . . .
+35
+4.4.2
+Measure of vortex line density fluctuations
+. . . . . . . . . . . . . . . . . . . . . . . . . .
+36
+4.4.3
+Filtering the vibration of the plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+38
+4.5
+Velocity measurements
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+38
+5
+Summary and Perspectives
+40
+1
+Introduction to second sound resonators
+1.1
+Quantum fluids and second sound
+Below the so-called lambda transition, liquid 4He enters a quantum state named He-II. This liquid transition
+occurs around Tλ ≃ 2.18 K at saturated vapor conditions. According to Tisza and Landau two-fluid model,
+the hydrodynamics of He-II can be described as the hydrodynamics of two interpenetrating fluids, called the
+superfluid component and the normal fluid component [Bal07, Gri09]. The superfluid density ρs is vanishingly
+small right below the transition, and it increases as temperature decreases. The opposite dependence occurs
+for the normal fluid density ρn, which becomes vanishingly small in the zero temperature limit. The properties
+of both fluids strikingly differ. The superfluid has zero viscosity and zero entropy. Besides, the circulation of
+its velocity field is quantized in units of κ = h/m ≃ 0.997.10−7m2/s, where h is Planck constant and m the
+atomic mass of 4He. This quantization constrain results in the existence of filamentary vortices of Ångströmic
+diameter, later referred to as the superfluid or quantum vortices[Don91]. Contrariwise, the normal fluid follows
+a classical viscous dynamics and carries all the entropy of the He-II[Put74, Kha00].
+The existence of distinct velocity fields vs and vn for the superfluid and normal fluid results in the existence
+of two independent sound modes in He-II, as can be shown by linearizing the equations of motion [Put74,
+Kha00, Don09]. The so-called “first sound” corresponds to a standard acoustic wave : both fluids are oscillating
+in phase (vs = vn), producing oscillations of the local pressure and density ρ = ρs + ρn. The “second sound”
+corresponds to both fluids oscillating in antiphase without net mass flow (ρsvs = −ρnvn). Hence, the relative
+densities of superfluid and normal fluid locally oscillates, as well as entropy and temperature.
+1.2
+Generation and detection of second sound waves
+Experimentally, two techniques are mostly used to generate and detect second sound: one mechanical and
+the other thermal. Alternative techniques not discussed here have been occasionally used, including optical
+scattering (e.g. [PGB70]) and acoustic detection above the liquid-vapor interface[LFF47] or in the flow itself
+(e.g. [HR76]).
+The mechanical technique consists in exciting and sensing a single component of He-II, either its superfluid
+one or the normal fluid component. Indeed, a single fluid motion can be viewed as a superposition of a second
+sound and a first sound (i.e. an acoustic wave), with an exact compensation of motion for one of the two fluids
+at the location of the transducer. Since the first sound velocity is typically one decade larger than the second
+second velocity[DB98], most second sound resonances don’t coincide with acoustic resonances.
+In practice,
+a selective displacement of the superfluid component is achieved in Peshkov transducers. They are made of
+a standard acoustic transducer side-by-side with a fixed porous membrane-filter which tiny pores are viscous
+dampers for the normal fluid but are transparent (“superleaks”) for the superfluid [Pes48, HL88]. Alternatively,
+2
+
+Emitter
+Receiver
+Emitter
+Receiver
+Flow
+Flow
+Figure 1:
+Left: A macroscopic resonator for second sound, embedded in the sidewall, is used to sense
+the averaged flow properties in the shaded region.
+Right: Second sound tweezers. In contrast with the
+macroscopic design of the left schematics, this miniaturized resonator, positioned within the flow, allows space
+and time resolution of the flow variations.
+a selective displacement of the normal fluid component is achieved in oscillating superleak transducers. They
+are based on a vibrating porous membrane that is coupled to the motion of normal fluid by viscous forces, and
+uncoupled to the inviscid superfluid. They can be manufactured by replacing the membrane of a loudspeaker
+or microphone by a millipore or a nucleopore sheet [WBF+69, SE70, DLL80].
+The thermal technique to generate and detect second sound consists in forcing second sound by Joule
+effect, and detecting it with a thermometer. Depending on the operating range of temperature and practical
+considerations, several types of thermometers can be suitable to detect second sound waves. The literature
+being vast, we only list a few thermistor materials and bibliographic entry points. Materials with a negative
+temperature coefficients1 include carbon in various forms (aquadag paint, fiber, pencil graphite,..) [HVS01],
+doped Ge[Sny62], RuOx [YI18], ZrNx/Cernox [YYK97, FS04] and Ge-on-GaAs [MMP+07]. Transition edge
+superconductor thermometers are often preferred when large sensitivity or low resistivity is important, for
+instance Au2Bi [Not64], PbSn [CR83, RR01], granular Al [CA68, MSS76] and AuSn [Not64, Lag76, BSS83].
+More information on AuSn is provided in section 2.2.1. The second sound tweezers presented in the present
+study resorts to this thermal technique, both for generation and detection of second sound2.
+1.3
+From macroscopic second sound sensors to microscopic tweezers
+In the presence of superfluid vortices, the superfluid and normal fluid experience a viscous mutual coupling
+[Don91], which entails a damping of the second sound waves. This attenuation of second sound by vortices
+have been extensively used as a tool to explore the properties of He-II flows over the last 60 years[Vin57], in
+particular to explore the properties of quantum turbulence (e.g. see [VJSS19]), a field of applications which
+as motivated the development of second sound tweezers. For example, mechanical second sound transducers
+were successfully used to study the turbulence of He-II in the wake of a grid by groups in Eugene, Prague and
+Tallahassee (e.g. see [SNVD02, BVS+14, MG18]). Examples of thermal second sound transducers successfully
+used to study turbulence of He-II flows are described in studies by groups from Paris, Tallahassee, Grenoble
+and Gainesville (e.g. see [WPHE81, HVS92, RDD+07, YI18]).
+A specific type of probe allows very sensitive probing of the density of quantum vortices in He-II flow :
+standing-wave second sound resonators. Such a resonator consists in two parallel plates facing each other, one
+functionalized with a second sound emitter and the other with a receiver. The emitter excites the cavity at
+resonance to benefit from the amplification of the cavity. The characteristics of the standing wave between the
+plates provides information on the properties of the fluid and flow between the plates, in particular the density
+of vortex lines, which impacts the amplitude of the standing wave. Aside from vortex density measurements, the
+second sound can also provide information on the fluid temperature -since the second sound velocity depends
+on it- and on the velocity of the background He-II flow when it induces a phase shift or Doppler effect on the
+second sound (e.g. see [DL77, WPHE81, WVR21]).
+The characteristics listed above for the standard (macroscopic) second sound resonators remain relevant for
+their miniaturized version : second sound tweezers. Beside miniaturization, a key specificity of tweezers is their
+1Phosphor bronze wire, a positive temperature coefficient thermometer has also been used in the early days [Pes46].
+2In principle, mechanical and thermal techniques could be combined to generate and detect, although we are not aware of any
+composite configuration reported in the literature.
+3
+
+low footprint on the streamlines when positioned in the core of a flow. This key differences between standard and
+tweezers resonators are sketched in figure 1. Consequently, standard resonators provide information on averaged
+properties of the flow, while tweezers give access to space and time resolved information. Thus, tweezers are local
+probes in the same ways as the hot-wire anemometers or cold-wire thermometers used in turbulence studies.
+In the present design for tweezers, the thermal actuation technique is preferred to the mechanical actuation
+because it makes it easier to respect the constrains of miniaturization and reduced flow blockage.
+1.4
+Overview of the manuscript
+The following sections cover independent topics,
+Section 3 presents a comprehensive modelling of second-sound resonators accounting for plate misalignment,
+advection, finite size and near field diffraction. Diffraction, which has been neglected in previous quantitative
+models, turns out to be a dominant source of degradation of the quality factor in our case studies. Applications
+to the measurement of vortex concentration or velocity are considered.
+Section 4 presents existing methods to process the signal from second sound resonators, and their limits. To
+circumvent them, we introduce a new general approach, named the elliptic method, based on a mathematical
+properties of resonance. This method allows to dynamically separate the amplitude variations of the standing
+wave due to variations of vortex density or variations of velocity, from the phase variations (more precisely, from
+the acoustical path variations), for instance due to variations of the second sound velocity themselves resulting
+from a temperature drift.
+Section 2 reports the design, clean-room fabrication and operation of miniaturized second-sound resonators,
+named second-sound tweezers. These tweezers allow to probe the throughflow of helium with an unprecedented
+spatial and time resolution.
+For better clarity, we first present the second-sound tweezers, which allows to illustrate the topics on mod-
+elling and method with a challenging practical case. Though we emphasize that the modelling and methods
+introduced in this article are general and relevant to second-sound resonator regardless of their size, including
+the macroscopic sensors embedded in parallel walls encountered in the literature.
+2
+Design, fabrication and mode of operation of second sound tweezers
+The core part of second sound tweezers is a stack composed of a heating cantilever and a thermometer cantilever,
+separated by a spacer (see Figs. 2 and 3). Heaters and thermometers are cantilevers composed of a baseplate,
+an elongated arm and a tip. The baseplate is the thickest part while the tip is the thinnest one. The active
+areas, the emitter and receiver plates, are located on the tips. For a given device, heater and thermometer have
+strictly the same mechanical structure, the only difference between them being the chemical elements used in
+the serpentine electrical path deposited on the tip. Three cantilever types were fabricated in order to allow
+assembling of resonators with three different tip sizes (see Fig.4). The tip widths are 1000µm, 500µm and
+250µm.
+Next subsection 2.1 presents considerations which prevailed in the mechanical design of the second sound
+tweezers. The following one (section 2.2) discusses the detection (thermometry) and generation (heating) of
+second sound by the tweezers. The last two subsections present the microfabrication techniques (section 2.3)
+and the electrical circuitry used to operate the probes (section 2.4).
+2.1
+Mechanical design
+Resolution. The tweezers space-resolution Lres is set by the largest dimension of its cavity, which can be
+either the inter-plates distance D, also called the “gap”, or the side length L of the plates here assumed to be
+squared shaped. The present study mostly focuses on cavities with an aspect ratio of order 1, to benefit from
+optimal space averaging of the signal at given space-resolution. The tweezers time-resolution τres is set by the
+decay-time of a wave bouncing between the plates. In section 3.2, we introduce and validate a simple model
+accounting for dissipation in the cavity due to diffraction loss and residual inclination of the plates. An upper
+bound for τres is obtained from the diffraction loss term: τres ≃ L2f/bc2
+2, where b ≈ 0.38, c2 is the second sound
+velocity and f is the wave frequency which can be approximated as nc2/2D for the nth mode of resonance (see
+eq.6). Thus, the tweezers time-resolution due to diffraction loss can be estimated as
+τres. ≃ L2f
+bc2
+2
+≃ n L2
+Dc2
+The ratio Lres/τres of the space resolution and time resolution defines a characteristic velocity for which
+the probe optimally averages the space-time fluctuations. For instance, in cavities of aspect ratio one (D = L)
+operated on its nth resonance, the nominal velocity is estimated as c2/n. These estimates show that cavities
+4
+
+Figure 2: Second sound tweezers, pictured from two angular perspectives. (Left) Probe as seen in the direction
+of the mean flow (or nearly so). The second sound cavity is localized at the upper tip of the probe, in the
+encircled area. The bend copper wire through the probe is a temporary joystick used for a fine-alignment of
+cavity plates under the microscope. The two coaxial cables for heating and thermometry are visible in the
+lower left corner of the picture. (Right) The picture insert shows the same probe after some rotation, and after
+removal of the joystick, making visible the through-hole across the silicon stack. The two staggered notches
+used for thermal confinement of the standing wave in the cavity are clearly apparent.
+Tip
+Arm
+Baseplate
+Cantilever with heater
+ Kapton / Tracks
+spacer
+Cantilever with thermometer
+Tracks / Kapton
+Second sound standing wave
+Figure 3: Schematic side view of the constitutive stack of second sound tweezers (the pieces are shown sepa-
+rated for illustration, their thicknesses are exaggerated). The active areas, the emitter and receiver plates, are
+constituents of the tip.
+250 µm
+500 µm
+1000 µm
+Figure 4: Top view of the 3 cantilever types. The tips widths are respectively 1000µm, 500µm and 250µm.
+Left: Mechanical structures, all parts are silicon made, different thicknesses are represented by different colors.
+The baseplate width is 2.5mm for all types.
+Right: Electrical path on each tip type.
+Yellow areas are
+a deposition of TiPt for heaters and AuSn for thermometers. Orange areas are a thick AuPt deposition for
+current leads.
+5
+
+2.5mm
+Baseplate
+Arm
+Tip
+10mm
+12.5mm
+2.5mmwith aspect ratio of order unity are fittingly sized for second-sound-subsonic flow, that is flows with a mean
+velocity of few m/s.
+Blocking effect. The above considerations on space resolution are relevant if the measured flow is not
+altered by the probe support. The present design conforms to the empirical ×10 rule stating that components
+of the support that obstruct the flow on a length scale X should be positioned at least 10X away from the
+measurement zone. Accordingly, the cavity is at the end of elongated arms and the cantilevers are 25 mm in
+length of decreasing thicknesses and widths, as illustrated by the picture of Fig.2 and by Figure 3. The thickness
+successive values are around 520µm, 170µm and 20µm while the width decreases from 2.5mm to 145 µm in the
+narrowest zone (resp. 275 µm and 500 µm) for cavities with L = 250 µm (resp. L = 500 µm and L = 1000 µm).
+Wave confinement. The spatial resolution of tweezers would be degraded if the second sound standing
+wave was spreading out of the L × L × D cavity, by reflection between the supporting arms. A design trick was
+implemented to confine the standing wave in the cavity region by breaking the mirror symmetry between the
+two cantilevers. Thus, as shown in Fig.2, anti-symmetric notches in the tips prevent the second sound to escape
+by bouncing away from the cavity, at least in the geometric-optic approximation where diffraction is neglected.
+Mechanical resonances. Besides the ×10 rule consideration, these dimensions are chosen such that the
+mechanical vibrations of the arm are pushed up to ∼ 1 kHz or above. The fundamental resonance frequency of
+the trapezoidal-shaped arm in vacuum was estimated from the analytical formula in [Lob07] (section 1.3.1.1).
+f0 = 8.367
+2π
+e
+t2
+�
+ESi(3w2 + w1)
+ρSi(49w2 + 215w1)
+We find f0 = 2195 Hz (resp. 1889 Hz and 1569 Hz) using the material properties ESi = 140 GPa, ρSi =
+2330 kg/m3 and the dimensions of the intermediate section of the arm having thickness e = 172 µm, length
+t = 12.5 mm, and width decreasing from w2 = 1.5 mm to w1 = 250 µm (resp. to 500 µm and 1000 µm). An
+experimental validation was done at room temperature in air with an arm with w1 = 1000 µm. Its mechanical
+vibration frequency spectrum was measured from a photoreceptor detecting a laser beam reflecting of the
+arm. The mechanical excitation was provided either by hitting the table supporting the set-up with a small
+hammer or by pointing a jet of compressed air toward the arm. In both cases, the fundamental mechanical
+resonance frequency was found to be 1215 Hz, in reasonable agreement with the 1569 Hz prediction given the
+uncertainty on the Young modulus and deviations from the trapezoidal shape. As discussed in 4.4.3, indirect
+measurements of the resonance frequency were done in 1.2 m/s superfluid flow and gave f ≈ 825 Hz, f ≈ 1050
+Hz and an amplitude of vibration smaller than 1 µm. The decrease in frequency compared to room-temperature
+measurement is interpreted as mostly due to a fluidic added mass effect[Sad98].
+Deflection of the tips’ ends. The thicknesses of the tweezers parts are such that the mechanical deflection
+at the tip endpoint remains significantly lower than the inter-plate distance under typical operating conditions.
+The deflection at the tip endpoint can be estimated by considering separately the arm deflection (with
+thickness 172µm) and the tip deflection (with thickness 20µm). As a first approximation, both arm and tip
+are considered as cantilever beams of uniform width submitted to a uniformly distributed load, and having one
+embedded end and one free end. This geometrical approximation overestimates the deflection of the arm, as its
+endpoint is narrower than its base, and it underestimates the deflection of the tip, as the notch is ignored. Still,
+this is enough to get order-of-magnitude estimates. The load is estimated as the dynamic pressure of a liquid
+helium flow impinging the tweezers in transverse direction at a velocity U=0.1 m/s, that is 10% of the typical
+longitudinal flow velocity of 1 m/s. The dynamic pressure P is taken as:
+P = 1
+2ρU 2
+where the liquid helium density is ρ ≃ 140 kg/m3. According to Euler-Bernouilli beam theory, the free end
+deflection δmax of the cantilever is:
+δmax = 3
+2
+Pt4
+ESi.e3
+The total deflection (arm and tip) is upper-bounded by considering the sum of the deflection of a 2.5mm
+long tip and a 15mm (not 12.5mm) long arm. This way, the small angle generated on the tip by the arm
+deflection is taken into account. Using the values t = 15mm (length), e = 172 µm (thickness) and E = 140GPa
+(Young modulus), the deflection of the arm endpoint is found to be 75nm. The deflection of the tip endpoint
+with t = 2.5mm and e = 20 µm gives a 37nm deflection Thus, the total mechanical deflection of the tweezers
+tip due to a steady lateral flow of 0.1 m/s is a fraction of a micron, that is decades smaller than the inter-plate
+distance.
+The mechanical resonance of the tweezers arm and tip, discussed above, could lead to deflections larger than
+the one due to a steady forcing. The amplitude of those mechanical oscillations was measured in a turbulent
+He-II flow up to velocities exceeding 1 m/s, taking advantage of the dependence of the second sound resonance
+6
+
+with respect to the cavity gap. The measured signal will be presented to illustrate the efficiency of the elliptic
+projection method in separating the fluctuations of the acoustical path of the cavity and fluctuations of the
+bulk attenuation of second sound between plates. The mechanical oscillations of the cavity gap are found to be
+typically 0.5 µm (around 1 kHz). As expected, such a deflection is decades lower than the interplate distance
+(1.3 mm in this case) and than the second sound wavelength.
+Boundary layer. In the presence of a mean flow through the cavity, a velocity boundary layer will develop
+along each tweezers plate. In principle, this boundary layer could contribute to the measured signal and therefore
+alter the measurement of the incoming flow, for instance by increasing the density of superfluid vortices in the
+cavity and therefore second sound attenuation. As illustrated later, the second sound standing wave that settles
+between the plates have nodes of velocity near the plates (e.g. see fig. 10 and fig. 16) while the sensitivity
+of second sound to vortices arises in antinodal regions of velocity. As long as the boundary layer thickness is
+thin enough, say within a fraction of λ/4 (λ = c2/f is the second sound wavelength), it is not expected to alter
+significantly the measured signal.
+A first requirement for this condition is that the mean flow direction is parallel to the plates so that the flow
+penetrates through the cavity with minimal deflection. A consequence is that plates should be widely separated
+when operated in flows with undefined or zero mean velocity, such as the core of a mixing layer.
+A second requirement is that the plate thickness is much thinner than λ/4. Present plates are 20 microns
+thin, to be compared with λ/4 ≃ D/2n for the nth mode of resonance. For instance, with D = 500 µm and
+n = 3, the condition 20 ≪ λ/4 ≃ 83 µm is indeed well satisfied.
+A third condition regards the downstream development of the boundary layer thickness, which should also
+stay well within λ/4. The physics of boundary layers in He-II is ill-understood[SPB17] but existing experiments
+(e.g.
+see [SHVS99]) suggests that classical hydrodynamics phenomenology could remain valid in the high
+temperature limit.
+In classical hydrodynamics, the so-called displacement thickness of a laminar “Blasius”
+boundary layer at distance L from its origin is given by
+δbl = 1.73
+�
+Lν
+U
+where U is the mean velocity far from the boundary layer and ν is the kinematic viscosity of the fluid. In
+He-II, several diffusive coefficients could arguably play the role of ν, in particular the quantum of circulation
+around a quantum vortex and the kinematic viscosity associated with the dynamics viscosity of the normal fluid
+normalized either by the normal fluid density or by the total density. In the temperature range of interest, all
+these diffusive coefficients are within one order of magnitude, typ. 10−8 − 10−7m2/s. Taking ν = 3.10−8 m2/s,
+L = 1000 µm and U = 0.5 m/s, one finds δbl = 13 µm, and a boundary layer Reynolds number δblU/ν = 217
+consistent with the laminar picture. This thickness estimate, similar in magnitude with the plate thickness,
+satisfies the third requirement δbl ≪ λ/4.
+2.2
+Second sound detection and generation
+2.2.1
+Thermometry
+The temperature-sensitive material used in the present study is AuSn, which fulfills two requirements: (1) it
+is compatible with the microfabrication process and (2) it can be tuned to become temperature-sensitive over
+a range of special interest to quantum turbulence studies[WVR21], from 1.5 K up to the superfluid transition
+temperature Tλ ≃ 2.18K in saturated vapor conditions. Surely, other materials would be more appropriate in
+other conditions, e.g. Al has been used previously for the tweezers operated around 1.5 K in [RDD+07].
+The gold-tin AuSn thermometer is a metal-superconductor composite material, with superconducting Sn
+islands electrically connected by a gold layer. This granular structure is imaged by electronic microscopy in
+Fig.5 (right). The temperature dependence phenomenology can be interpreted in a simple way. Indeed, by
+proximity effect, the gold in contact with tin behaves as a superconductor over a spatial extent which depends
+on temperature : by adjusting the characteristic length scales and thicknesses of the granular pattern, the
+temperature response of the material can be tuned.
+As a preliminary study, the temperature of the resistance of a 100 squares long AuSn track was tested for
+three different tin thicknesses, as illustrated on the left plot of Fig.6. A description of the conduction mechanism
+in AuSn is presented in [BSS83].
+In the present study, AuSn layers are deposited by evaporation with successively a small erosion of the
+substrate by argon ion bombardment during 20s then the deposition of a 25nm gold layer and a 100nm tin
+layer (hypothetic thickness for a planar - not granular - layer). The thermometer is shaped into a meander
+deposited by lithographic technique on the tip of tweezers arm, as pictured in Fig.5 (right plot). The total
+resistance at superfluid temperatures doesn’t exceed a few hundreds of ohm, a value chosen to be much larger
+than the resistance of the leads but small enough to prevent parasitic effect from the leads’ capacitance (typ.
+few hundreds of picoFarads) up to the highest frequencies of operation.
+7
+
+Figure 5:
+Left: Tip of tweezers arm as seen from the facing arm.
+The thin meander on the top is the
+active heater (Pt) or thermometer (AuSn).
+The overlap with gold tracks is apparent on the lower part of
+the picture Right: Close up picture by electronic microscopy of the AuSn thermometer showing its granular
+aspect. Depending on tweezers models (see fig. 8), the width of the AuSn track is 4, 11 or 24 µm
+In order to obtain resistance values in this range, the meander length was fixed close to 700 squares for all
+tip sizes. According to the tip size, the track width was adapted so as the serpentine shape occupy the whole
+available area on the tip. At ambient temperature, the AuSn layer resistance was found to drift from low values
+to their final values during a few days (less than one week) after deposition. After this period, the resistance
+was found to be stable at least over 6 months.
+Figure 6 (right plot) shows a typical AuSn thermometer resistance R(T0) versus temperature T0 for different
+direct current I. Regarding the temperature dependence of resistance, the current density is a more determining
+parameter than the total current. Thus, the comparison between left and right plots of figure 6 should be done
+at constant values of the ratio of current over track width. At low current density (I � 10µA), the sensitivity
+exceeds 1 Ω.mK−1.
+At larger current densities, the current-induced magnetic field significantly shifts the
+superconducting-metal transition to lower temperature and broaden it, allowing to extend the measurement
+range down to 1.6 K and below.
+In the range of currents explored in Fig.
+6, the reduction of sensitivity
+in Ω.K−1 at larger current I is more than compensated by the larger sensitivity in V.K−1 units across the
+thermistor. Most measurements presented hereafter are done with a polarization of I ≃ 27 µA.
+2.2.2
+Heating
+The heater consists in a meander of metal deposited by lithographic technique on the tip of a cantilever, alike
+the thermometer (see fig. 5 (left)). The same lithographic mask was used, and therefore the meander length is
+also close to 700 squares for all the tip sizes.
+As can be seen on figures 4 and 5, a buffer zone was designed between the gold tracks and the meander. Into
+this zone, the electrical path is wide but the material is the same as in the meander (platinum for heater). The
+buffer zone length is approximately 20 squares. This design aims at providing some thermal isolation between
+the meander and the gold track.
+Numerous resistive materials are suitable, e.g. chrome was used for the tweezers in [RDD+07] and platinum
+in [WVR21]. Present data have been obtained with platinum to benefit from the temperature-independence of
+its resistivity at superfluid temperatures [PK82], and nevertheless to allow re-use of these miniature heaters as
+miniature thermometers or hot-film anemometers in experiments conducted at higher temperatures where Pt
+regains temperature dependence [Kem91]. A 5nm titanium layer was deposited prior to platinum as an adhesion
+layer.
+The thickness of the Pt layer, around 80 nm, was chosen such that the electrical resistance of the heater at
+superfluid temperature is around a few hundreds of ohms, like for the thermometer maximum resistance and
+for the same reasons.
+The heater is driven with a sinusoidal current at frequency f/2. The resulting Joule effect can be decomposed
+into a constant mean heating and the sinusoidal heat flux at the frequency f that drives the second sound
+resonance. A benefit of this f/2 excitation is that the signal monitored by the thermometer - centered around f
+- is not spoiled by spurious sub-harmonic electromagnetic coupling at f/2 from the excitation circuitry. Thus,
+no special care is needed to minimize the electromagnetic cross-talk between the electrical tracks of the heater
+and the electrical tracks of the thermometer, despite their proximity.
+The non-zero mean heating results is a steady thermal flux in He-II, the corresponding entropy being carried
+8
+
+10μm     
+2   
+     
+2.2   
+     
+2.4   
+     
+2.6   
+     
+0   
+     
+0.2   
+     
+0.4   
+     
+0.6   
+     
+0.8   
+     
+1   
+     
+Au 25nm + Sn 90nm   
+     
+Au 25nm + Sn 100nm   
+     
+Au 25nm + Sn 110nm   
+     
+T     
+λ     
+ transition under saturated vapor   
+R0  /  max(R0)
+T0  (K)
+     
+1.7   
+     
+1.8   
+     
+1.9   
+     
+2   
+     
+2.1   
+     
+2.2   
+     
+0   
+     
+100   
+     
+200   
+     
+300   
+     
+1      
+μ     
+A     
+d     
+c     
+ + 1      
+μ     
+A     
+a     
+c   
+     
+3      
+μ     
+A     
+d     
+c     
+ + 1      
+μ     
+A     
+a     
+c   
+     
+10      
+μ     
+A     
+d     
+c     
+ + 1      
+μ     
+A     
+a     
+c   
+     
+20      
+μ     
+A     
+d     
+c     
+ + 1      
+μ     
+A     
+a     
+c   
+     
+30      
+μ     
+A     
+d     
+c     
+ + 1      
+μ     
+A     
+a     
+c   
+R0  (Ω)
+T0  (K)
+Figure 6:
+Left: Example of temperature and current dependence of AuSn layers with different Sn thicknesses
+deposited on a track of width 50µm . The current polarization spans from 1µA up to 1mA.
+Right: Tempera-
+ture response of the AuSn track of small tweezers for different electrical currents. The thickness of the tin layer
+is 100 nm of Tin, and the width of the track is 4µm. The small difference between both plots -when compared
+at similar current densities- is compatible with the uncertainty on the layer thicknesses. This good agreement
+indicates that AuSn properties are robust to the full fabrication process of the tweezers. No significant aging of
+AuSn properties has been noticed over a few years period.
+away from the heater in the form of steady normal fluid flow. This outgoing normal flow is balanced by an
+opposite steady mass flow of superfluid towards the heater. Such cross-flows are referred to as counterflows in
+the quantum fluid literature[Tou82, NF95]. This steady counterflow adds up to a pure second sound generated
+by the heater, but -contrary to it- its effects are not amplified by resonance in the cavity.
+Quasi-linear vs non-linear regimes. The second sound resonators are operated with standing waves
+of low amplitude, say ∼ 100 µK.
+In this limit, the amplitude T of the temperature standing wave nearly
+responds linearly to the heating power P. For larger heating power, the ratio T/P decreases with P, which is
+interpreted as the result of a turbulent transition within the tweezers which fuels a dense tangle of quantum
+vortices dissipating the second sound wave3. The crossover from the quasi-linear to non-linear response of T(P)
+is illustrated by figure 7 (left plot) for tweezers at 1.6 K in the absence of external flow. In these conditions and
+for these tweezers, the transition occurs around P ≃ 1 W/cm2, where P is the total Joule power normalized
+by the heating surface. In other conditions, this transition was observed at smaller power densities, but no
+systematic study was carried on the threshold value.
+2.2.3
+Digression on the operation in the non-linear heating regime
+The present study focuses on the linear regime of heating, but a short digression on higher powers allows
+uncovering a noteworthy property of non-linear operation and incidentally backing-up the above interpretations
+of the nature of the non-linear regime. Figure 7-right displays the amplitude of the (normalized) temperature
+standing wave T/P versus P in flows of different mean velocities U and turbulence intensity of few percents. In
+the linear regime (P � 1W/cm2 in the conditions of Fig. 7), the plateaus of T/P decreases when U increases.
+Following the classical interpretation (see later), the standing wave T is damped by the vortices present in the
+external flow, which concentration increases with U. Interestingly, in the non-linear regime (say for P � 2W/cm2
+in Fig. 7), the dependence of T/P versus U is opposite. The interpretation is that the extra damping of the
+3In one dataset, a close look at the quasi-linear region in quiescent superfluid around 1.5 K evidenced a small discontinuity of
+T/P versus P dependence around P ⋆ ≃ 10−2W/cm2 (not shown), suggesting another flow transition, but this small effect was
+not detectable in other datasets. A Reynolds number of this possible transition can be defined with the transverse characteristic
+length scale L ≃ 1 mm, the quantum of circulation κ ≃ 0.997.10−7m2/s and the counterflow superfluid velocity towards the
+heater vs ≃ 0.3 mm/s (amplification of velocity by the quality factor of the cavity has not been taken into account). One finds
+Res = vsL/κ ≃ 3. Critical Reynolds numbers Res of a few units have already been reported to characterize the threshold of
+appearance of a few superfluid vortices across the section of pipes that are closed at one of their ends with a heating plug (see
+fig. 3 in [BLR17]), a transition referred as the T1-transition in the counterflow literature [Tou82, NF95]. By analogy, this could
+suggest that the discontinuity at P ⋆ might be associated with the appearance of a sparse tangle of the quantum vortices near
+the heater, which density is expected to increase for larger P. Such vortices would damp the standing wave, but no such effect
+has been detected. Indeed, first the observed damping of the standing wave in quiescent He-II can be accounted for by the sole
+effect of diffraction (as shown later), which indicates that all the other sources of loss are comparatively small. Second, loss due to
+such “counterflow” vortices would make the T(P) dependence sub-linear rather than linear, which is not clearly observed. Since no
+quantitative evidence of these vortices could be clearly identified, this effect was not further explored.
+9
+
+10-1
+100
+P (W/cm2)
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+2.2
+T/P (a.u)
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+P (W/cm2)
+-1
+0
+1
+2
+3
+X (mK)
+-2
+-1
+0
+1
+2
+3
+Y (mK)
+10-1
+100
+P (W/cm2)
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+2.2
+T/P (a.u)
+Overheating
+0
+0.2
+0.4
+0.6
+0.8
+1
+U (m/s)
+Figure 7:
+Left: Normalized amplitude T of the temperature standing wave versus heating power P for
+second sound tweezers in a quiescent He-II around 1.6 K. The transition around P = 1 W/cm2 is interpreted
+by the development of a self-sustained vortex tangle within the probe. The insert shows the amplitude of the
+temperature standing waves in the complex plane for a subset of frequencies belonging to the same second sound
+resonance. In this representation, the extra-dissipation associated with the self-sustained vortex tangle results
+in a curvature of the iso-frequency radial “lines” revealing the broadening of the resonance.
+Right: Same
+quantity for second sound tweezers swept by turbulent flows of different mean velocities but similar turbulence
+intensity. In the linear regime P � 1 W/cm2, the velocity dependence is opposite to the one in the non-linear
+regime, P � 2 W/cm2, demonstrating respectively vortex and velocity sensing by the probe.
+standing wave T now mostly results from the vortices generated within the tweezers by the heating itself. This
+vortex density decreases at larger U because vortices are more efficiently swept out of the tweezers. In the
+non-linear regime, the second-sound tweezers thus behave as a local anemometer. In the linear regime, we will
+show that second-sound tweezers can not only behave as vortex probes -as illustrated by Fig.7-right but also as
+anemometers through a mechanism discussed later.
+2.3
+Microfabrication and assembling
+Mechanical and electrical assembly.
+The distance between the plates is set by the spacer, composed of one or several micro-machined silicon
+elements. Additionally, two Kapton films with golden copper tracks are inserted in contact with the gold tracks
+of the heater and thermometer. The cantilevers, Kapton films and spacer elements are stacked on each other
+as shown in Fig.3. The resulting assembly is clamped with a standard picture clip, downsized by electro-wire
+erosion, and soldered to the head of a mounting screw. An improvement compared to the clamping technique
+introduced in [RDD+07] is the possibility to insert a temporary “joystick” through the whole assembly to allow
+precise alignment (or offsetting) of the cavity plates under microscope (see Fig.2-a).
+All the mechanical structures of cantilevers and spacers are made of silicon. The cantilevers are fabricated
+by processing SOI (Silicon On Insulator) substrates by microelectronic techniques. The substrates diameter
+is 100mm, the silicon substrate, oxide and device thicknesses are respectively 500µm, 1µm and 20µm. The
+substrates are double side polished. The wafers are oxidized so as to form a 100nm thick SiO2 layer on both
+sides. Distinct wafers are used to fabricate heaters and thermometers but the process differs only in the metals
+used for tips electrical paths. By using a circular symmetry design, 46 cantilevers are made per wafer.
+The cantilevers’ fabrication process is presented in table 1. The serpentine electrical path (red color) is
+deposited first on the SOI wafer frontside. The deposition is done using standard photolithography, evaporation
+and lift-off sequence. The photoresist used is AZ 5214E from Microchemicals GmbH, processed as a negative
+photoresist. Depending on the cantilever type, heater or thermometer, two different evaporation sequences are
+used: Ti 5nm + Pt 80nm or Au 25nm + Sn 100nm. Evaporation is preceded by in situ wafer surface cleaning by
+an argon ions bombardment during 20s. Lift-off is initiated in an acetone bath during 5min and then completed
+by ultrasounds during a few tens of seconds. The current leads (orange color) are deposited during a second
+photolithography, evaporation and lift-off sequence. The evaporation sequence is Ti 5nm + Au 200nm + Ti
+5nm + Pt 50nm. The usage of a platinum layer was found to facilitate lift-off and may also be useful for brazing
+purpose. A thin protective resist layer is deposited on frontside in order to protect it during all subsequent
+operations on the backside.
+10
+
+Figure 8: Overview of mask design (the disk diameter is 100mm).
+Side
+Step
+Surface material
+Design
+Frontside
+1. Electrical circuitry fabrication.
+TiAuTiPt (orange)
+AuSn/TiPt (red)
+Backside
+2. Aluminum mask fabrication.
+Aluminum
+Backside
+3. Resist mask fabrication.
+4. Etching of surface oxide and
+silicon.
+5. Resist removal.
+Photoresist
+Backside
+6. Etching of surface oxide and
+silicon until buried oxide is reached.
+7. Etching of buried oxide.
+8. Aluminum removal.
+Aluminum
+Frontside
+9. Resist mask fabrication.
+10. Etching of surface oxide and
+silicon until opening.
+Photoresist
+Table 1: Cantilevers fabrication process.
+11
+
+Phase
+Deposition
+Etch
+Sub-phase
+Main
+Boost
+Main
+Gas
+C4F8: 250 sccm
+SF6: 250 sccm
+O2: 45 sccm
+SF6: 450 sccm
+O2: 45 sccm
+Duration
+2.2 s
+2.0 s
+5.5 s
+Pressure
+14 mTor
+20 mTor
+75 mTor
+Coil power
+1200 W
+1780 W
+1780 W
+Platen power
+20 W
+110 W
+50 W
+Electromagnet current
+0 A
+0 A
+2 A
+Platen frequency
+RF 13.56 MHz
+He backside pressure
+10 Tor
+Table 2: Bosch process recipe used during deep silicon etching on backside.
+The cantilevers 3D structuration starts with a backside deep etching of silicon. Two superimposed etch
+masks are fabricated first, one aluminum mask and one photoresist mask deposited over the aluminum one.
+The aluminum mask is made in the same way as the frontside electrical paths. A backside alignment is necessary
+during photolithography. The aluminum thickness is 120nm. The protective resist layer on frontside had to be
+deposited again after lift-off. The resist mask is made by photolithography on the positive AZ4562 photoresist
+spin-coated at 4000rpm. The aluminum mask is identical to the resist mask except within the arm area, as
+shown in table 1. This area is covered by resist but not by aluminum. Thus, the resist fully covers aluminum.
+After the fabrication of both masks, the deep etching is started, with the resist mask protecting both silicon
+oxide and aluminum surfaces. The surface oxide is etched first then the solid silicon. The thin oxide layer is
+etched by reactive ion etching (RIE) based on SF6 gas. The silicon was etched in a STS HRM deep reactive
+ion etching (DRIE) equipment using a recipe based on Bosch process. The recipe is presented on table 2, the
+etch duration is 120 cycles. As shown on table 1, a 200µm wide trench is dug from the backside around the
+baseplate and arm areas. The tip area is fully exposed to etching as this part was extracted from the device
+layer of SOI only. The longer this phase, the thicker the cantilever arm at the end of process.
+Following this first etch phase, the resist is removed by an oxygen plasma and the backside surface oxide is
+etched into the freshly uncovered arm area. Then, another silicon etch sequence is applied with the remaining
+aluminum mask. Its objective was to thin the arm and to reach the buried oxide of SOI in the trench, at the
+same time. The same recipe is used, 200 cycles are applied. The Bosch process is interrupted 3 times, every 50
+cycles, in order to apply a 1min oxygen plasma followed by 20s of an isotropic silicon etching recipe (plasma
+pressure 75mTor, SF6 flow 450sccm, O2 flow 45sccm, coil power 1780W, platen power 50W). The objective is
+to reduce the parasitic effect generated by the passivation layer deposited on arm sidewalls during the first deep
+silicon etching sequence. As the arm area is masked during the first silicon etch sequence and unmasked during
+the second one, the passivation layer located along the arm edges is released and could generate locally some
+irregular micromasking effect. The oxygen plasma is intended to help remove the floating passivation films. The
+isotropic silicon etching is intended to cut the silicon pillars generated by micromasking. The final 50 cycles
+of the Bosch process recipe end up reaching the SOI buried oxide layer, at the trench bottom, all around the
+cantilever (however the oxide should not be reached into the arm area). It is necessary at this step to check that
+the buried oxide is fully uncovered by silicon everywhere in the trench bottom and in the tip area. However,
+etching cycles should not be applied in excess so as to avoid some mechanical weakening of the wafer. At this
+step, the arm thickness has its final value. The buried oxide is then removed by plasma etching. This oxide
+is fully removed at the trench bottom and in the tip area. If this layer is not removed, some bending may
+occur on the tip at the end of process, due to mechanical stress of oxide. The aluminum mask is removed in
+an aluminum etchant solution at 50°C during a few minutes. The frontside protective resist layer is removed
+during an acetone cleaning bath.
+The 3D cantilever structuration is ended by a third silicon etching made from the frontside. The etch mask
+is formed by photolithography on AZ1512HS resist, deposited at 4000rpm. As shown on table 1, the mask
+design includes two bridges on the baseplate sides in order to maintain the cantilever after having opened the
+trench that surrounds it. The design also includes the tip contour. The frontside thin surface oxide is etched
+first, then the silicon of the device layer. The silicon etching is done with specific conditions. Due to the wafer
+mechanical weakness at this step, caused by the multiple deep trenches made on the backside, the processed
+wafer is layed on and attached to a blank silicon wafer by Kapton tape. This ensemble is loaded into the etching
+chamber. As no thermal bridge is present between the two wafers, the recipe is adapted: low RF powers are
+used and a 22s idle time is added after each etching cycle (see table 3). The objective is to avoid overheating
+during etching. The silicon etch duration is 50 cycles.
+After this sequence, the trench around cantilever is fully opened. The resist is removed by a low power oxygen
+plasma. Some spacers are fabricated together with the cantilevers but most of them are made separately from
+12
+
+Phase
+Deposition
+Etch
+Sub-phase
+Main
+Delay
+Boost
+Main
+Gas
+C4F8: 250 sccm
+SF6: 250 sccm
+O2: 10 sccm
+Duration
+3 s
+2.0 s
+5.5 s
+22.5 s
+Pressure
+14 mTor
+20 mTor
+40 mTor
+40 mTor
+Coil power
+300 W
+300 W
+300 W
+1 W
+Platen power
+20 W
+100 W
+50 W
+1 W
+Electromagnet current
+0 A
+0 A
+2 A
+0 A
+Platen frequency
+RF 13.56 MHz
+He backside pressure
+10 Tor
+Table 3: Bosch process recipe used during deep silicon etching on frontside.
+two silicon wafers with thicknesses 300µm and 525µm. These wafers are covered by thin dielectric layers on
+both sides.
+2.4
+Electric circuit
+Figure 9 shows a circuit used for time-resolved measurements with second-sound tweezers, with example values
+of resistances and gain. The time-resolved data presented in this paper have been obtained with such a circuit
+and using the following equipment. The front-end of the preamplifier is the EPC1-B model by Celian or an SA-
+400F3 model by NF when frequencies above 100 kHz are explored. The lock-in amplifier is a LI-5640 by NF or
+Model 7280 by SignalRecovery above 100 kHz. In most conditions, its built-in internal generator provides both
+the drive of the tweezers heater (at frequency f/2) and the reference frequency to detect the temperature signal
+(at frequency f). The acquisition system is built around PXI-4462 analog input cards by National Instrument,
+and it records both the in-phase (X) and quadrature (Y ) signal from the analog outputs of the lock-in amplifier.
+In occasional conditions, the temperature signal at the lock-in input is buried in a much larger electro-
+magnetic parasitic signal at f/2, and it cannot be properly resolved by the limited voltage dynamic range of
+the lock-in amplifier. This situation can occur when the tweezers are operated far from a resonance, where
+the second sound signal is small, or when the tweezers are operated at very high frequency (say > 100 kHz),
+as electromagnetic coupling increases with frequency. The magnitude of this parasitic coupling depends on
+the geometrical and electrical specificities of the tweezers and cables. In the range of parameters explored in
+the present study, the order of magnitude of the parasitic voltage induced across the thermometer resistor,
+normalized by the voltage applied across the heating resistor, is
+0.5% ×
+f/2
+100kHz
+Such situations are handle thanks to the differential input of the lock-in amplifier, by removing a signal
+mimicking the parasitic one. In such cases, a two-channel waveform generator is used: one channel driving the
+heater (at frequency f/2), another channel mimicking the parasitic signal (at frequency f/2, with manually
+tuned amplitude and phase shift) and the "sync" output of the generator synchronizing the lock-in demodulation
+(at frequency f). The 33612A generator by Agilent is used for this purpose. Alternatively, the compensation
+signal can be generated directly from the lock-in internal generator, completed with a simple RC phase shifter
+and eventually ratio transformer.
+In principle, any positive temperature coefficient thermistor -like AuSn thermometer- that is not well ther-
+malized with the fluid can become unstable when driven by a current source. Indeed, an infinitesimal thermistor
+fluctuation from T0 to T0 + δT0 entails a resistance variation of δR = ∂R/∂T0.δT0 > 0, leading to an excess of
+Joule dissipation δR.I2 for a constant current drive I. Calling Rth is the thermal resistance of the thermistor-
+fluid interface, this extra Joule dissipation results in an overheating Rth × δR.I2 which could lead to a thermal
+instability. The stability condition is difficult to predict for spatially distributed thermistor deposited on a
+Si crystal and immersed in superfluid. Thus, first tests have been done with a voltage polarization, before
+validating empirically the stability of our current polarization.
+The frequency bandwidth of the measurements is arbitrarily set by the integration time constant of the
+lock-in amplifier. In practice, the performance of the circuit are limited by the 0.65 nV/
+√
+Hz input voltage
+noise of the EPC1-B pre-amplifier, all other sources of noise being smaller. For a 27 µA current polarization, a
+thermometer sensitivity of 0.5 Ω.mK−1, and a 10 Hz or 1000 Hz bandwidth of demodulation, the temperature
+resolution Trms is :
+Trms =
+√
+10.0.65
+0.5 × 27 µK ≃ 150nK for a 10 Hz measurement bandwidth
+13
+
+300 kΩ
+acquisition
+system
+10 kΩ
+400 Ω
+Tweezers heater
+Reference
+clock
+9V batteries
+quadrature
+phase
+Optional noise compensation
+Low noise AC preamp
+(40 - 50 dB)
+Lock-in amplifier
+(at freq. f )
++
+-
+Freq. f/2, phase φ
+r1
+r2
+Freq. f/2
+Tweezers thermometer
+(0 - 300 Ω)
+R(T0)
+Current source (typ 30 µADC) 
+Figure 9: Example of circuitry for measurements with a high dynamical reserve.
+or
+Trms =
+√
+1000.0.65
+0.5 × 27
+µK ≃ 1.5µK for a 1 kHz measurement bandwidth
+These resolutions are sufficient in standard conditions. Indeed, they are respectively 3 and 2 decades smaller
+than the typical amplitude of a second sound at resonance, and reaching the same temperature resolution at
+significantly larger bandwidth would be useless given the space-time resolution of the probe itself. If needed,
+better resolution could nevertheless be achieved with a larger polarization across the thermometer or using a
+cryogenic amplifier (e.g. see [DLF+14] and http://cryohemt.com) before being limited by the thermal noise
+floor of the thermistor (typ. 0.15 nV/
+√
+Hz for 200 Ω at 2 K).
+3
+Models of second sound resonators
+The second sound equations within the linear approximation can be written in terms of the temperature fluc-
+tuations T and the velocity of the normal component vn as
+∂tvn + σρs
+ρn
+∇T = 0,
+(1)
+∂tT + σT0
+cp
+∇.vn = 0.
+with σ the entropy per unit of mass, cp the heat capacity, and ρs , ρn are the densities of the superfluid and
+normal components respectively. All along the present section, T0 is the notation for the bath mean temperature
+whereas T denotes the temperature fluctuations, with ⟨T⟩ = 0.
+We introduce the second sound velocity c2 defined by the relation
+c2
+2 = ρs
+ρn
+σ2T0
+cp
+.
+(2)
+It can be deduced from Eqs. (1) that both the temperature T and the normal velocity vn follow the wave
+equation
+∂2
+t T − c2
+2∆T = 0.
+(3)
+We explain in the present section how Eqs. (1-2-3) can be used to build quantitative models of second sound
+resonators. We first focus on phenomenological aspects in sec. 3.1. Then, we give analytical approximations in
+sec. 3.2 and an accurate numerical model in sec. 3.3. Finally, we discuss the model quantitative predictions in
+secs. 3.4 and 3.5.
+14
+
+Heater
+Thermometer
+z=0
+z=D
+Figure 10: Schematic representation of the second resonant mode of a second sound cavity. The red curve
+displays the temperature field at time t = 0, and the blue curve displays the normal fluid velocity vn at time
+t =
+1
+4f , where f is the excitation frequency.
+3.1
+Resonant spectrum of second sound resonator: phenomenological aspects
+The basic idea of second sound resonators is to create a second sound resonance between two parallel plates
+facing each other. A second sound wave is excited with a first plate, while the magnitude and phase of the
+temperature oscillation is recorded with the second plate used as a thermometer (see Fig. 10). For simplicity,
+we assume from now on that the second sound wave is excited by a heating, but the whole discussion can be
+straightforward extended to nucleopore mechanized resonators. The temperature oscillations within the cavity
+are coupled to normal fluid velocity oscillations according to the second sound equations (1). Both components
+oscillate in quadrature, which means that they reach their maximal amplitude with a
+1
+4f time shift (Fig. 10).
+We note jQ = j0e2iπft the periodic component of the heat flux emitted from the heater.
+We assume
+throughout the present article perfectly insulating plates, which means that the boundary conditions for the
+second sound wave are
+vn.n =
+�
+0
+for z = D
+jQ
+ρσT0
+for z = 0 ,
+(4)
+where n is the unit vector directed inward the cavity and normal to the plates. The second equation in (4)
+reflects the fact that the normal component carries all the entropy in the fluid. As illustrated in Fig. 10,
+the thermometer plate is a node of the normal velocity oscillation, whereas the normal velocity amplitude only
+vanishes on the heater plate in quadrature (t =
+1
+4f + n
+2f ). According to the first relation in Eq. (1), the boundary
+conditions (4) for the normal velocity translate into the following boundary conditions for the temperature field
+∇T.n =
+�
+0
+for z = D
+− ρn∂tjQ
+ρρsσ2T0
+for z = 0 .
+(5)
+In particular, the thermometer plate is an antinode for the temperature.
+We display in Fig. 11 a typical experimental spectrum of second sound tweezers, that is, the temperature
+magnitude averaged over the thermometer plate, as a function of the heating frequency f. The spectrum is
+reminiscent of a Fabry–Perot resonator (described in sec. 3.2): it displays very clear resonant peaks almost
+equally separated, and a stable non-zero minimum at the non-resonant frequencies. However, the spectrum of
+Fig. 11 displays three major characteristics that can be observed for every tweezers spectrum. First, the location
+of the resonant frequencies are slightly shifted compared to the standard values fn given by 2πfnD
+c2
+= nπ, (n ∈ N)
+expected for an ideal Fabry–Perot resonator. Only for large mode numbers do the resonant peaks again coincide
+with the expected values. Second, the temperature magnitude vanishes in the zero frequency limit, and the
+first modes of the spectrum grow linearly with f. In between, the resonant amplitudes saturate and then slowly
+decrease at high frequency.
+These latter peculiarities of the frequency response were not described in previous references about second
+sound resonators. This prompted us to study different models for second sound resonators, including the finite
+size effects and near field diffraction phenomena.
+We first describe analytical approximations in sec.
+3.2,
+then we develop in sec. 3.3 a numerical algorithm based on the exact solution of the wave equation (3). The
+numerical scheme can be adapted for various types of planar second sound resonators. We then give quantitative
+predictions specifically for the response of second sound tweezers without and in the presence of a flow in sec.
+3.4 and a summary of the main results in sec. 3.5.
+3.2
+Analytical approximations
+The starting point to build our model of second sound tweezers is to assume that all zeroth order physical
+effects observed with the tweezers are geometrical effects of diffraction.
+This means in particular that we
+assume perfectly reflecting resonator plates, and we also neglect bulk attenuation of second sound waves when
+15
+
+0
+10
+20
+30
+40
+50
+60
+70
+f (kHz)
+0
+0.2
+0.4
+0.6
+0.8
+1
+T (K)
+10-4
+Linear growth of
+the first modes
+Stable baseline
+Decrease at high
+frequency
+Figure 11: Experimental spectrum of second sound tweezers of lateral size L = 1 mm and gap D ≈ 1.435 mm.
+f is the heating frequency, and T is the thermal wave magnitude. The figure displays the main characteristics
+of tweezers typical spectrum: first, the resonant frequencies are not located at the values fn given by 2πfnD
+c2
+=
+nπ, (n ∈ N), displayed by the gray vertical lines. The amplitudes of the resonant modes first increases linearly
+with f until they saturate and eventually decrease at high frequency. The baseline level only weakly depends
+on f.
+the fluid is at rest [CR83, RG84]. These assumptions turn to be self-consistent, because the predictions of
+the model developed in sec. 3.3 reproduce the main features observed in experiments. We thus start with
+the simplest model of a resonant cavity for planar waves, namely the Fabry–Perot model. We then propose
+variations of the Fabry–Perot model, taking progressively into account the particular resonator geometry. We
+discuss the predictions of these models and their relevance for our second sound tweezers.
+The standard Fabry–Perot model
+The Fabry–Perot model corresponds to a one-dimensional resonator
+composed of two infinite parallel plates separated by a gap D. In that case, the wave Eq. (3) together with the
+boundary conditions Eqs. (5) can be solved exactly, for a periodic heating jQ = j0e2iπft . However, as there is
+no energy loss between two perfectly insulating and infinite plates, a bulk dissipation has to be introduced by
+hand in the model, to balance energy injection from the heater. This can be done with an additional dissipation
+coefficient ξ (in m−1). The temperature at the thermometer plate is T(t) = Re
+�
+Te2iπft�
+(where Re is the real
+part operator) with the complex wave amplitude T given by
+T =
+A
+sinh
+�
+i 2πfD
+c2
++ ξD
+�,
+(6)
+with A = −
+j0
+ρcpc2 . An illustration of a Fabry–Perot spectrum is displayed in grey in Fig. 14, with ξD = 0.15
+and A = 1. We introduce the wave number k = 2πf
+c2 . It can be proved from Eq. (6) that the spectrum maxima
+are reached for the values knD = nπ, (n ∈ N), and correspond to constructive interferences in the cavity. The
+baseline level is set by the minima reached for the values knD = nπ + π
+2 , (n ∈ N), and correspond to destructive
+interferences in the cavity. For the simple Fabry–Perot model of Eq. (6), all the resonant peaks have equal
+height and are uniformly separated. Therefore, some main features of experimental spectra are missing, an
+indication that important other physical effects have to be included in the model.
+Second sound resonators embedded in infinite walls
+A possible modification of the Fabry–Perot res-
+onator is to consider finite-size heater and thermometer of size L embedded in two parallel and infinite walls
+facing each other. This geometry is most commonly encountered in the literature. With such a configuration,
+16
+
+the thermal wave is not a plane wave any more because it is emitted by a finite size heater. The model thus
+contains diffraction effects, that the simplest Fabry–Perot resonator do not display. An illustration of the model
+setup is displayed in Fig. 12. An exact solution of the wave equation Eq. (3) can be found using the technique
+of image source points. Let Σ1 be the heating plate and Σ2 be the thermometer plate, and we assume that the
+thermometer is sensitive to the average temperature over Σ2. Then the response of the tweezers is given by
+T(t) = Re
+�
+Te2iπft�
+with
+T =
+ikj0
+2πρcpc2
+1
+L2
+�
+Σ2
+d2r2
+�
+Σ1
+d2r1G (r2 − r1) ,
+with the Green function G(r) defined for every vector r in the (x, y) plane
+G(r) = 2
++∞
+�
+n=0
+1
+|(2n + 1)Dez + r|e−ik|(2n+1)Dez+r|.
+Such a model correctly predicts that the tweezers spectrum vanishes when the heating frequency f goes to zero.
+Yet, it does not reproduce the linear increase of the resonant magnitude of the first modes, neither the decrease
+of the resonant peaks at large frequency observed in experiments with second sound tweezers. This means that
+other effects have to be taken into account to model a fully-immersed open resonant cavity such as non-perfect
+plates alignment and energy loss by diffraction outside the cavity when the latter is not embedded in infinite
+walls.
+Empirically modified Fabry–Perot model
+Contrary to a Fabry–Perot resonator composed of infinite
+plates, second-sound resonators are built with plates of finite size L, approximately of the same order as the
+gap D between them. Those finite size effects are important as they introduce a frequency-dependent energy
+diffracted outside the cavity. This mechanism is sketched in Fig. 12. According to standard diffraction theory,
+a finite wave initially of size L with a wavelength λ = c2
+f spreads with a typical opening angle given by λ
+L. By
+this geometrical effect, a part of the wave energy is lost as the wave reaches the other side of the cavity. The
+energy loss is roughly proportional to the surface of the wave cross-section that “misses” the reflector (see the
+right panel of Fig. 12). Therefore, the fraction of energy lost at the wave reflection is controlled by the ratio
+�
+L + 2λD
+L
+�2 − L2
+�
+L + 2λD
+L
+�2
+≈ 4λD
+L2 ,
+(7)
+≈
+4
+NF
+,
+where we have introduced the Fresnel number NF = L2
+λD.
+The tweezers plates are mounted at the top of arms of a few millimeters. The perfect parallelism of the plates
+is usually not reached for our tweezers, but a small inclination γ of the order of a few degrees can be observed
+instead. A relative inclination γ -even small- of both plates creates an additional energy loss mechanism (see
+Fig. 13). Intuitively, this second mechanism is controlled by the non-dimensional number
+Ni = λ
+γL.
+(8)
+We assume that the Fabry–Perot model (6) can be corrected using the two non-dimensional numbers NF =
+L2f
+c2D in Eq.
+(7) and Ni =
+c2
+γfL in Eq.
+(8).
+More precisely, based on empirical observations, we find that
+second-sound tweezers spectra can be accurately represented by the formula
+T =
+A
+sinh
+�
+i
+�
+2πfD
+c2
+− a c2D
+L2f
+�
++ b c2D
+L2f + c
+�
+γfL
+c2
+�2�,
+(9)
+where a, b and c are fitting coefficients. As there is no other small parameter in the problem, a reasonable
+assumption is to look for coefficients of order one. We find that the values a ≈ 0.95, b ≈ 0.38 and c ≈ 1.3 give
+accurate spectra predictions. An illustration of a modified Fabry–Perot spectrum with Eq. (9) is given in Fig.
+14. The linear amplitude growth of the first resonant peaks can be interpreted as a progressive focalization of
+the wave, and is thus controlled by the Fresnel diffraction number NF in Eq. (12). The shift proportional to
+1
+f in peaks frequency positions, observed in the experimental spectra, is also controlled by NF . The decrease
+in resonant magnitude for large mode numbers can be interpreted as a wave deflection outside the cavity, after
+back and forth propagation between the plates. This latter effect is controlled by the second non-dimensional
+number Ni in Eq. (8).
+17
+
+L
+L
+D
+L
+Thermal wave
+Figure 12: Representation of the wave dispersion. The energy loss is controlled by the non-dimensional number
+λ
+L, according to standard diffraction theory. In this section, we discuss both the case of tweezers embedded in
+walls, and the case of free tweezers in open space.
+Figure 13: Effect of the plates’ inclination γ. Inclination creates an additional energy loss mechanism controlled
+by the second non-dimensional number
+λ
+γL.
+18
+
+20
+2
+4
+6
+8
+10
+12
+14
+kD
+0
+1
+2
+3
+4
+5
+6
+7
+T
+inclination
+effect
+destructive interferences
+focalization
+of the wave
+Figure 14: Analytical models of second-sound tweezers. The grey curve represents the standard Fabry-Perot
+spectrum. The blue curve represents the empirical correction of the Fabry-Perot formula using the two non-
+dimensional numbers
+λ
+L and
+λ
+γL.
+The modified spectrum displays the characteristic features of a tweezers
+experimental spectrum, as displayed in Fig. 11.
+The major interest of the Fabry–Perot model is to offer an analytical expression to fit locally a resonant peak
+of second sound resonator spectrum. The local fit of a peak is of particular interest to interpret the experimental
+data, as will be explained in sec. 4. Based on Eq. (9), given a measured resonant frequency f0, we will look for
+a fitting expression
+T =
+A
+sinh
+�
+i 2π(f−f0)D
+c2
++ ξ0D
+�,
+(10)
+valid for second sound frequencies f close to f0. In that expression, ξ0 encapsulates the different geometrical
+mechanisms responsible for energy loss when the fluid is at rest. A and ξ0 are thus fitting parameters that can
+be found easily with the experimental data obtained by varying f in the vicinity of f0.
+3.3
+Numeric algorithm
+Sec. 3.2 presents a class of models of increasing complexity, still with an analytical solution. Those models
+show how all zeroth-order effects observed with second sound resonators can be recovered from geometrical
+diffraction effects. However, they do not allow for quantitative predictions of the resonator spectra, nor do
+they allow including the effects of a flow. We develop in the present section a numerical algorithm, based on
+the exact resolution of the wave equations with the particular tweezers geometry, with and without flow. The
+algorithm could be extended to any second sound resonator with a planar geometry. As will become clear in
+the following, this numerical model allows going far beyond the approximate models of sec. 3.2.
+3.3.1
+For a backgroud medium at rest
+The aim of the present section is to build a numerical algorithm to solve the wave equation (3) for a periodic
+heating jQ = j0e2iπft. We look for a solution with the ansatz T(r, t) = Re
+�
+T(r)e2iπft�
+. Then, the wave
+equation for T is
+∆T + k2T = 0,
+where we have introduced the wave number k = 2πf
+c2 . The boundary conditions are
+�
+∇T(r).n1 = − ikj0
+ρcpc2
+for r ∈ Σ1
+∇T(r).n2 = 0
+for r ∈ Σ2
+,
+(11)
+where Σ1 is the heater plate and Σ2 the thermometer plate. The notations are given in Fig. 15. We propose
+the method described below, based on the Huyggens–Fresnel principle. The principle states that every point
+19
+
+of the wave emitter can be considered as a point source. The linearity of the wave equation can then be used
+to reconstruct the entire wave by summation of all point source contributions. The Huyggens–Fresnel principle
+has been widely used in the context of electromagnetism, for example to compute diffraction patterns produced
+by small apertures, or interference patterns... The major difficulty in the context of second sound tweezers is
+that none of the standard approximations of electromagnetism can be done, neither the far-field approximation
+nor the small wavelength approximation. This explains why numerical resolution is very useful in this context.
+We neglect the tweezers arms, which means that both plates are considered as freestanding, infinitely thin
+and perfectly insulating plates. We allow a relative inclination γ around the x-axis and a possible relative lateral
+shift Xsh of one plate with respect to the other along the x-axis. We assume that the thermometer is sensitive
+to the temperature averaged over Σ2.
+Let us introduce the Green function
+G(r) = 1
+|r|e−ik|r|,
+(12)
+which is the fundamental solution of the wave equation
+∆G + k2G = 4πδ(r).
+(13)
+Let Σ be one of our two square plates, and U(r′) be a smooth function defined over Σ. We introduce the wave
+defined by
+T(r) = −1
+2π
+�
+Σ
+G(r − r′)U(r′) d2r′.
+(14)
+By linearity, T is a solution of Eq. (3), for all r /∈ Σ, because G is a solution. A asymptotic calculation in the
+vicinity of Σ then shows that T satisfies the boundary condition
+∇T(r).n
+−→
+r→r0∈Σ U(r0),
+(15)
+where n is the unit vector normal to Σ and directed inward the cavity (see Fig. 15). We are going to use Eqs.
+(14) and (15) as the two fundamental relations to build our algorithm. We will compute the solution of the
+wave equation as an infinite summation of all the emitted and reflected waves in the cavity.
+The first wave T 1 is emitted by the heating plate Σ1 and satisfies the first relation in Eq. (11)
+∇T 1(r).n1 = − ikj0
+ρcpc2
+for r ∈ Σ1.
+Given Eq. (14) and (15), it is clear that the first wave is given by
+T 1(r) =
+ikj0
+2πρcpc2
+�
+Σ1
+G(r − r′) d2r1.
+(16)
+Then each time a wave denoted T n hits a plate Σ (Σ1 or Σ2), it produces a reflected wave T n+1 to satisfy the
+boundary condition
+∇
+�
+T n(r) + T n+1(r)
+�
+.n = 0.
+(17)
+The situation is sketched in the left panel of Fig. (15). If we choose for T n+1 the expression
+T n+1(r) = 1
+2π
+�
+Σ
+G(r − r′)
+�
+∇T n(r′).n
+�
+d2r′,
+(18)
+then Eq. (15) shows that Tn+1 satisfies the boundary condition
+∇T n+1(r).n
+−→
+r→r0∈Σ −∇T n(r0).n,
+which is exactly Eq. (17). Eqs. (16) and (18) define our recursive algorithm. Eq. (18) shows that the reflected
+wave is generated by the gradient of the incident wave. Practically, the recursive computation of all forth and
+back reflected waves thus requires at each step n the computation of ∇T n only on the plates, rather than T n.
+For a reflection at (say) Σ1, we have
+∇T n+1(r).n2 = −1
+2π
+�
+Σ1
+G(r − r1)
+�
+1
+|r − r1| + ik
+�
+n2. r − r1
+|r − r1|
+�
+∇T n(r1).n1
+�
+d2r1,
+(19)
+20
+
+D
+L
+Thermometer
+L
+n2
+Heater
+heat flux
+n1
+x
+y
+z
+Xsh/2
+Xsh/2
+n
+Figure 15: Left: Geometrical setup of the numerical algorithm and notations. Right: Representation of an
+incoming and outcoming wave at the nth reflection.
+The solution of the wave equation is finally given by the superposition of all waves T n, that is
+T(r) =
++∞
+�
+n=1
+T n(r),
+= 1
+2π
+�
+Σ1
+G(r − r1)
++∞
+�
+n=0
+�
+∇T 2n+1(r1).n1
+�
+d2r1
++ 1
+2π
+�
+Σ2
+G(r − r2)
++∞
+�
+n=1
+�
+∇T 2n(r2).n2
+�
+d2r2.
+and the thermometer response is given by
+�
+T
+�
+Σ2 = 1
+L2
+�
+Σ2
+T(r) d2r.
+A simulation of the temperature field at t = 0 of the 5th resonant mode of second sound tweezers with aspect
+ratio L
+D = 0.4, without lateral shift nor inclination of the plates, is displayed in Fig. 16. It can be clearly seen
+in particular that the amplitude of the temperature field decreases along the z-axis contrary to a Fabry–Perot
+resonator. This symmetry breaking is due to the diffraction effects associated with the finite size of the plates.
+A bulk dissipation can be included in the algorithm, for example, to account for quantum vortex lines inside
+the cavity. In that case, let ξ be the second-sound attenuation coefficient (in m−1), the wave number k = 2πf
+c2
+of the Green function (12) should be replaced by
+k = 2πf
+c2
+− iξ.
+(20)
+3.3.2
+In the presence of a turbulent flow
+One of the aims of second-sound resonator modelling is to understand their response in the present of a flow
+U sweeping the cavity. One effect of the flow is to advect the second sound wave. In the present section, we
+explain how the algorithm of sec. 3.3.1 should be modified to account for this effect. We assume in the following
+that the inequality |U| < c2 is strictly satisfied, which means that the flow is not supersonic for second sound
+waves.
+In the presence of a non-zero flow U, the Green function (12) becomes
+G(r, t) =
+e−2iπft∗
+|r − Ut∗|
+�
+1 + U
+c2 . r−Ut∗
+|r−Ut∗|
+�,
+(21)
+21
+
+2mTn+1Figure 16:
+Left: Temperature field fluctuations of the 5th resonant mode of second sound tweezers with aspect
+ratio L
+D = 0.4, and heating power jQ = 0.185 W/cm2, at the bath temperature T0 = 2K. Right: Temperature
+field fluctuations for the same conditions with an additional flow of velocity U
+c2 = 0.18 directed upward . The
+nodes of the temperature standing wave correspond to antinodes of sound sound velocity, and vice-versa.
+where t∗ is the time shift corresponding to the signal propagation from the source
+|r − Ut∗| = c2t∗.
+(22)
+In practice, the flow velocity range reached in quantum turbulence experiments is most often much lower than
+the second sound velocity, with |U| hardly reaching a few m/s. Most experiments are done in the temperature
+range where 10 < c2 < 20 m/s. We thus introduce the small parameter β = |U|
+c2 ≪ 1. Similarly to the standard
+approximations of electromagnetism, we assume that the effect of β is mostly concentrated in the phase shift
+e−2iπft∗ of Eq. (21). We use the approximation |r − Ut∗|
+�
+1 + U
+c2 . r−Ut∗
+|r−Ut∗|
+�
+≈ |r|, and we solve Eq. (22) to
+obtain t∗ to leading order in β. The Green function then becomes
+G(r, t) = e−ik|r|Γ(r,U)
+|r|
+,
+(23)
+where as previously k = 2πf
+c2
+and
+Γ (r, U) = 1 − U
+c2
+. r
+|r|.
+The algorithm detailed in sec. 3.3.1 can be applied straightforward with the Green function Eq. (23). In
+particular, Eq. (19) becomes
+∇T n+1(r).n2 = −1
+2π
+�
+Σ1
+G (r − r1, U)
+��
+1
+|r − r1| + ik
+� r − r1
+|r − r1|.n2 − ik U
+c2
+.n2
+� �
+∇T n.n1
+�
+(r1) d2r1.
+(24)
+A simulation of the temperature field at t = 0 of the 5th resonant mode of second sound tweezers with aspect
+ratio
+L
+D = 0.4, without lateral shift nor inclination, and with a flow of velocity
+U
+c2 = 0.18, is displayed in the
+right panel of Fig. 16. The effect of the flow can be clearly seen with the upward distortion of the antinodes of
+the wave, compared to the reference temperature profile without flow displayed in the left panel.
+3.4
+Quantitative predictions
+We present in this section the quantitative results obtained with the algorithm of sec. 3.3. The algorithm is
+specifically run in the configuration of second sound tweezers, but most predictions are relevant for other types
+22
+
+Thermometer
+Thermometer
+Heater
+Heater
+-0.15-0.1
+-0.05
+-0.15 -0.1 -0.05
+0.05
+0.15
+0.05
+0.1
+0.15
+0
+T (mK)
+T (mK)of second sound resonators. We first show that the algorithm can quantitatively account for the experimental
+spectra. We then use it to predict the response in the presence of a flow and a bulk dissipation in the cavity.
+The predictions are systematically compared to experimental results for second sound tweezers. We eventually
+display some experimental observations that illustrate the limits of our model.
+3.4.1
+Spectral response of second sound resonators
+Given a resonator lateral size L, the model of sec. 3.3 has three geometrical parameters : the gap D, the
+inclination γ and the lateral shift Xsh (see notations in Fig. 15). We first sketch qualitatively the importance
+of those three parameters.
+The gap D is the main parameter: it sets the location of the resonant frequencies, and the quality factor of
+the resonances at low mode numbers. For second sound tweezers, the value of D can be usually obtained within
+a precision of a few micrometers (D is of the order of 1 millimeter). The relative inclination of the plates γ is
+responsible for the saturation of the resonant magnitude and its decrease at large mode numbers. It is typically
+smaller than a few degrees. Contrary to the gap, only the order of magnitude of γ, not its precise value, can
+be determined from the tweezers spectrum. The lateral shift Xsh has very little impact on the spectrum if the
+value Xsh
+L
+remains small enough (we can typically reach Xsh
+L
+< 0.1 in the tweezers fabrication). However, the
+effect of this parameter is of paramount importance to understand open cavity resonators response in a flow
+(such as second sound tweezers), and will be investigated in sec. 3.4.2. We consider the case Xsh = 0 in the
+present section. The tweezers size L is known from the probe fabrication process.
+The method goes as follows: we first find a gap rough estimation �D, for example from the average spacing
+between the experimental resonant peaks. Then we can run a simulation for parallel plates (γ = 0), unit gap
+D = 1, and aspect ratio L
+�
+D, in the range 0 < k∗ < nπ (where n is the number of modes to be fitted, and k∗ = kD
+is the non-dimensional wave number). This gives a function fL/ �
+D(k∗). The experimental spectrum can then
+be fitted with the function T(f) = AfL/ �
+D( 2πfD
+c2
+), where A and D are the two free parameters to be fitted,
+provided the experimental value of c2 is known. The high sensitivity of the location of the resonant frequencies
+makes this method very accurate to obtain the gap D.
+Once D has been found, new simulations have to be run to find the order of magnitude of γ.
+As was
+previously said, γ controls the saturation and the decrease of the resonant magnitudes for large mode numbers.
+Its value can thus be approximated from a fit of the resonant modes with the largest magnitude. A fit of an
+experimental tweezers spectrum is displayed in Fig. 17. The values of the fitting parameters for this spectrum
+are D = 1.435 ± 0.003 mm and γ = 4.2 ± 0.5 deg. Given the simplicity of the model assumptions, in particular
+the assumptions of perfectly insulating and infinitely thin plates without support arms, the agreement with
+experimental results is very good.
+Interestingly, the resonators can also be used in some conditions as thermometers. Once the gap D is known
+with high enough precision, the spectrum can be fitted using c2 as a fitting parameter instead of D. Away from
+the second sound plateau of the curve c2(T0) located around 1.65 K, the value of c2 obtained from the spectrum
+gives access to the average temperature with a typical accuracy of one mK, simply by inverting the function
+c2(T0).
+3.4.2
+Response with a flow
+Once the characteristics of the resonator have been determined with a background medium at rest, their response
+in a flow can be studied using the modified algorithm presented in sec. 3.3.2. We experimentally observe that
+the tweezers response is attenuated in the presence of a superfluid helium flow. This attenuation is related to
+two physical mechanisms, illustrated in Fig. 18: first, the thermal wave crossing the cavity is damped by the
+quantum vortices carried by the flow. This type of damping is usually considered as being proportional to the
+density of quantum vortex lines between the plates. Second, the flow mean velocity is responsible for a ballistic
+advection of the thermal wave outside the cavity. The thermal wave emitted by the heater partly “misses” the
+thermometer plate, and, even if the wave is not attenuated, a decrease of the tweezers’ response will be observed.
+Both mechanisms described above exist in experimental superfluid flows, and cannot be observed independently:
+once there is a superfluid flow, quantum vortices are created. One key objective is to be able to separate the
+attenuation of the experimental signal due to bulk attenuation inside the cavity, from the attenuation due to
+ballistic advection of the wave outside the cavity. We will introduce a mathematical procedure to perform such
+a separation on a fluctuating signal.
+What cannot be experimentally achieved can be simulated with the tweezers model developed in sec. 3.3.
+The bulk dissipation can be implemented in the algorithm with a wave number complex part ξ (see Eq. (20)),
+and the flow ballistic deflection can be implemented with a non-zero velocity U (see Eqs. (23-24)). Both effects
+can be independently studied, setting alternatively ξ or U to zero. We first detail below the respective effects
+23
+
+0
+10
+20
+30
+40
+50
+60
+70
+f (kHz)
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+T (K)
+10-5
+Experimental data
+Numerical simulation
+Figure 17: Prediction of the second sound tweezers experimental spectrum of Fig.
+11 using the numerical
+algorithm. The fitting parameters are the gap D, the inclination γ and the total heating power.
+Heater
+Thermometer
+Bulk
+dissipation
+Heater
+Thermometer
+Flow
+Figure 18: Schematic representation of the two attenuation mechanisms.
+The top panel illustrates a bulk
+dissipation of the wave, due for example to the presence of quantum vortices. The bottom panel illustrates the
+ballistic deflection of the wave by a flow directed parallel to the plates.
+24
+
+2
+2.2
+2.4
+2.6
+kD
+1
+1.5
+2
+2.5
+3
+T
+0
+1
+2
+X
+-1.5
+-1
+-0.5
+0
+0.5
+1
+1.5
+Y
+0
+0.05
+0.1
+0.15
+0.2
+Figure 19: Numerical simulation: collapse of a resonance due to increasing values of the bulk attenuation ξ.
+The left panel display the magnitude of the thermal wave as a function of the wavevector k, and the right panel
+display the same resonance in the phase-quadrature plane. It can be seen that the bulk attenuation results in
+an homothetic collapse of the resonance, that means, without global phase shift. For a given value of k, the
+model predicts that attenuation is directed toward the center of the resonant Kennelly circle (red curves of right
+panel).
+of ξ and U for perfectly aligned plates (Xsh = 0).
+Fig. 19 display the result of a numerical simulation for second sound tweezers of aspect ratio L/D = 1, γ = 0
+and increasing values of bulk dissipation in the range 0 < ξD < 0.2. The left panel display the magnitude of the
+second resonant mode as a function of the wave number, and the right panel display the same resonant mode in
+the phase-quadrature plane. More precisely, if we call T(k) the thermal wave magnitude recorded by the ther-
+mometer, and ϕ(k) its phase, the right panel display the curve Y (k) = T(k) sin(ϕ(k)), X(k) = T(k) cos(ϕ(k)).
+The resonant curve (Y (k), X(k)) is called in the following the resonant “Kennelly circle”, because the curve is
+very close to a circle crossing the origin. It can even be shown that the resonant curve becomes closer to a
+perfect circle for increasing resonant quality factors. The major characteristic to be observed in Fig. 19 is that
+the collapse of the resonant Kennelly circle due to bulk attenuation is homothetic. It means that the different
+curves have no relative phase shift between each other, when the bulk attenuation increases. The red curves
+in the right panel display the displacement in the phase-quadrature plane for a fixed value of the wavevector.
+The model predicts that the displacement is directed toward the Kennelly circle center, which implies that
+the path at fixed wavevector approximately follows a straight line. By comparison, the left panel of Fig. 21
+display an experimental resonance in the phase-quadrature plane, for second sound tweezers of size L = 1 mm in
+superfluid Helium at 1.65 K. The global orientation of the resonant Kennelly circles is simply due to a uniform
+phase shift introduced by the measurement devices, and should be overlooked. It can be seen that the resonance
+collapse with increasing values of the flow velocity follows the predictions of Fig. 19: it is homothetic. The
+red paths correspond to the tweezers signal at fixed heating frequency. Those paths follow approximately a
+straight line directed to the Kennelly circle center. The slight deviation in the path orientation compared to
+the predictions of Fig.19 can be explained by a second sound velocity reduction and will be discussed in sec. 3.4.4.
+Fig. 20 display the result of a numerical simulation for second sound tweezers of aspect ratio L/D = 1,
+γ = 0, with ξ = 0 and a flow mean velocity 0 <
+U
+c2 < 0.2. As there is no tweezers lateral shift Xsh = 0,
+negative velocities would lead to the same result from symmetry considerations. The figure illustrates the effect
+of pure ballistic advection on a resonance in the phase-quadrature plane. First, it can be seen that the collapse
+of the resonant Kennelly circle is accompanied by a relative anti-clockwise phase shift of the curves when the
+velocity increases. Also, the displacement of the tweezers signal at fixed wavenumber follow the red straight
+paths directed anti-clockwise. This type of signal strongly contrasts with the one of Fig. 19 obtained for a
+pure bulk attenuation. The prediction of the left panel in Fig. 20 cannot be directly compared to experiments
+because, as stated before, a superfluid flow always carries quantum vortices that overwhelm the tweezers signal
+for tweezers satisfying Xsh ≈ 0.
+25
+
+0
+1
+2
+X
+-1.5
+-1
+-0.5
+0
+0.5
+1
+1.5
+Y
+0
+0.05
+0.1
+0.15
+0.2
+-0.5
+0
+0.5
+1
+1.5
+X
+-1
+-0.5
+0
+0.5
+1
+Y
+-0.2
+-0.15
+-0.1
+-0.05
+0
+0.05
+0.1
+0.15
+0.2
+U/c2
+Increasing U
+-0.2
+0
+0.2
+U/c2
+-0.5
+0
+0.5
+s
+s
+Figure 20: Numerical simulation: collapse of a resonance due to ballistic deflection of the thermal wave
+in the presence of a flow of velocity U, without bulk attenuation (ξ=0). The left panel display the result for
+tweezers without lateral shift (Xsh = 0). Contrary to the results of Fig. 19, it can be seen in the present case
+that the collapse is associated with a global anti-clockwise phase shift of the resonant Kennelly circle. Each
+red curve represents the attenuation at a given value of the wavevector k. The right panel display the result
+for tweezers with a strong lateral shift Xsh = 0.5 × L (where L is the tweezers size). Such tweezers are very
+sensitive to the velocity U, with both an attenuation of the resonance and a strong clockwise angular shift of
+the Kennelly circle. We note s the curvilinear abscissa of the curve obtained at a given value of k (red curve).
+The inset displays the function s(U). This shows that, once calibrated, second-sound tweezers can be used as
+anemometers.
+3.4.3
+Effect of lateral shift of the emitter and receiver plates
+We discuss in this section the consequences of a lateral shift, that is Xsh ̸= 0 with the notations of Fig. 15.
+Contrary to the previous sections, the present discussion is restricted to second sound tweezers, for which a
+lateral shift has major quantitative effects. A lateral shift would not be as important, for example in the case
+of wall embedded resonators.
+The lateral shift has a marginal effect on the tweezers spectrum when the background fluid is at rest. An
+effect only appears in the presence of a nonzero velocity specifically oriented in the shifting direction U = Uex,
+because of the mechanism of ballistic advection of the thermal wave by the flow (see the representation of the
+mechanism in Fig. 18). The importance of this effect depends on the tweezers aspect ratio, on the reduced
+velocity β = U
+c2 , and on the lateral shift Xsh. The lateral shift in the plates’ positioning magnifies the signal
+component related to ballistic advection. This property opens the opportunity to build second sound tweezers
+for which ballistic advection of the wave completely overwhelms bulk attenuation from the quantum vortices,
+which means that the tweezers signal is in fact a measure of the velocity component in the shifting direction.
+We illustrate this mechanism in Fig. 20.
+The right panel displays a numerical simulation of a tweezers resonant mode in the phase-quadrature plane,
+for the parameters L
+D = 1, γ = 0 and Xsh = 0.5, for positive and negative values of the flow velocity in the range
+−0.2 < U
+c2 < 0.2. As can be seen at the first sight, the deformation of the resonant curve - that we equivalently
+call the “Kennelly circle” - is very different from a deformation due to a bulk attenuation (see Fig. 19). First,
+we observe that the deformation can result in an increase of the magnitude of the thermometer signal, when
+the velocity is negative. This can be explained in this configuration, because the thermal wave emitted by the
+heating plate is redirected toward the thermometer plate: less energy is scattered outside the cavity when the
+wave is first emitted by the heater, and the signal magnitude increases. On contrary, the signal magnitude
+decreases when the velocity is positive because the flow advects the emitted thermal wave further away from the
+thermometer plate and more energy is scattered outside the cavity. Second, the deformation of the Kennelly
+circle is associated to a global clockwise rotation, a phenomenon that is not observed for bulk attenuation in Fig.
+19. Coming back to Fig. 20, the red curve displays the displacement in the phase-quadrature plane for a fixed
+wave frequency value. The displacement follows a very characteristic curved path always directed clockwise.
+Let s(U) be the curvilinear abscissa of the red path. Once calibrated, the value of s can be used as a measure
+of the flow velocity component in the ex direction.
+The right panel of Fig. 21 displays the experimental signal observed with second sound tweezers of size
+26
+
+0
+0.05
+0.1
+0.15
+X (mK)
+-0.05
+0
+0.05
+0.1
+0.15
+Y (mK)
+0
+0.2
+0.4
+0.6
+0.8
+1
+U (m/s)
+-0.4
+-0.2
+0
+0.2
+X (mK)
+-0.1
+0
+0.1
+0.2
+0.3
+0.4
+Y (mK)
+0
+0.2
+0.4
+0.6
+0.8
+1
+U (m/s)
+Figure 21: Experiment: Collapse of a second sound resonance for increasing values of the flow mean velocity
+U, in superfluid Helium at T0 ≈ 1.65 K. The right panel display the result for tweezers of size L = 1 mm and
+minor lateral shift Xsh < 0.1×L. The figure shows a homothetic collapse of the resonant Kennelly circle without
+global phase shift, as predicted by the model of Fig. 19. The red curves display the displacement in the phase-
+quadrature plane at a fixed value of the second sound frequency f. The right panel displays the experimental
+data obtained with shifted second sound tweezers with parameters L = 250 µm and Xsh = 0.5 × L. The
+figure qualitatively confirms the clockwise angular shift with increasing values of U, predicted by the numerical
+simulations of Fig. 20.
+L = 250 µm, D = 431 µm and Xsh ≈ 125 µm, for a positive velocity range 0 < U < 1 m/s. The main
+characteristics of a ballistic advection signal can be observed: the Kennelly circle are attenuated with a clear
+clockwise rotation, and the signal at fixed frequency follows a curved path in the clockwise direction. This is a
+strong indication that those type of tweezers can be used as anemometers. The signal fluctuations of those type
+of tweezers were recently characterized in a turbulent flow of superfluid helium [WVR21]. It has been shown in
+particular that both the signal spectra and its probability distributions indeed display all the characteristics of
+that of turbulent velocity fluctuations.
+3.4.4
+Limits of the model
+Although the model of sec.
+3.3 gives excellent experimental predictions, we still observe some unexpected
+phenomena with real second sound tweezers. We discuss two of them in this section.
+We have seen in secs. 3.4.2 that the thermal wave complex amplitude T(f) can be represented in the phase-
+quadrature plane by a curve (X(f), Y (f)) very close to a circle crossing the origin. This osculating circle will
+be called in the following the resonant “Kennelly circle”. The wave is damped in the presence of a superfluid
+flow, which can be seen in the phase-quadrature plane as a homothetic shrink of the Kennelly circle toward the
+origin. Fig. 22 displays an experimental resonance in the phase-quadrature plane, for U = 0 m/s and U = 0.7
+m/s, together with the fitted Kennelly circles. As can be seen in the figure, the resonant curve at U = 0 has
+periodic oscillations in and out the Kennelly circle. We call this phenomenon the “daisy effect”. The daisy
+effect progressively disappears for increasing values of U, and cannot be seen any more on the resonant curve at
+U = 0.7 m/s. We interpret the daisy effect as a secondary resonance in the experimental setup with a typical
+acoustic path of a few centimeters. We assume that the flow kicks out the thermal wave from this secondary
+resonant path when U is increased. The daisy effect alters the attenuation measurements close to U = 0, and
+should be considered with care before assessing the vortex line densities for low mean velocities.
+It has been shown in sec. 3.4.2 that the displacement of the tweezers signal in the phase quadrature plane
+for a fixed wave frequency, follows a straight line.
+We call “attenuation axis” the direction of this straight
+path. The model predicts that the attenuation axis should always be directed toward the center of the resonant
+Kennelly circle. Fig. 23 displays a zoom on a part of the Kennelly circle at U = 0, together with the signal
+displacement at fixed frequency and for increasing flow velocity. It can be seen that the displacement is indeed
+a straight line, but not exactly directed toward the Kennelly circle center. An angle between 20° and 30° is
+systematically observed between the attenuation axis and the circle center direction (see Fig. 23). Moreover,
+the angle is always positive (with the figure convention) and cannot be interpreted as a ballistic advection,
+27
+
+-8
+-7
+-6
+-5
+-4
+-3
+-2
+-1
+0
+1
+X
+10-7
+-3
+-2
+-1
+0
+1
+2
+3
+4
+Y
+10-7
+U=0 m/s
+U=0.7 m/s
+Figure 22: Experimental resonance obtained with a second sound tweezers at 1.98 K, for two values of the He
+flow mean velocity U. The blue curve displays a periodic perturbation of the resonance that we refer to as the
+“daisy effect”. The circle is a fit of the Kennelly osculating circle for this resonance. This effect is not predicted
+by our model, and we interpret it as a secondary resonance in the experimental setup. The daisy effect perturbs
+the measurements at low values of U, but it can be seen on the red curve that the effect disappears for higher
+values of U.
+that would give a negative angle instead. This effect is thus very likely been attributed to a decrease of the
+second sound velocity in the presence of the quantum vortices. Whereas a second sound velocity reduction has
+previously been observed in the presence of quantum vortices [LV74, Meh74, MLM78], the exact value of this
+reduction turns to be difficult to assess in particular experimental conditions. We therefore keep the second
+sound velocity reduction as a qualitative explanation, and we do not try to assess quantitative result from the
+attenuation axis angle.
+3.5
+Quantum vortex or velocity measurements ?
+Let us summarize the discussion of sec 3.4. We have shown that second sound resonators are sensitive to two
+physical mechanisms. The first one is the thermal wave bulk attenuation inside the tweezers cavity, due to
+the quantum vortices. The second one is thermal wave ballistic advection perpendicular to the plates4. Both
+mechanisms exist for all the second sound resonators, but depending on their geometry, they can preferentially
+be sensitive to the one or the other mechanism. We call selectivity the fraction of the signal due to quantum
+vortices or to ballistic advection. Let T (ξ, U) be the probe signal as a function of the bulk attenuation coefficient
+ξ (m−1) and flow velocity U(m/s), we define the vortex selectivity as
+Rξ =
+��T (ξ, 0) − T (0, 0)
+��
+��T (ξ, U) − T (0, 0)
+��.
+(25)
+and by symmetry we define the velocity selectivity as
+RU =
+��T (0, U) − T (0, 0)
+��
+��T (ξ, U) − T (0, 0)
+��.
+(26)
+Further investigations in second sound tweezers experiments have shown that the velocity/vortex selectivity
+process only weakly depends on the aspect ratio
+L
+D.
+Indeed, for a given resonator lateral size L, ballistic
+advection of the wave outside the cavity increases when the gap D increases, but the number of quantum vortex
+lines inside the cavity also increases linearly with D. Altogether, both the ballistic advection and the bulk
+attenuation due to the quantum vortices have similar dependence with D, that’s why changing the gap has
+no significant effect on selectivity. For second sound tweezers, we observe that the selectivity neither depends
+strongly on the mean temperature (that controls the superfluid fraction and the second sound velocity).
+4Advection of second sound by velocity is illustrated e.g. in [DL77].
+28
+
+-3.5
+-3
+-2.5
+-2
+-1.5
+X
+10-4
+-2.5
+-2
+-1.5
+Y
+10-4
+U=0 m/s
+0<U<1.25 m/s
+Attenuation axis
+at fixed frequency
+Kenelly
+circle center
+Figure 23: Attenuation of a resonance in the presence of a flow mean velocity U, obtained with second sound
+tweezers in superfluid Helium at 1.65 K. The blue curve is the resonance at U = 0 in the phase-quadrature
+plane. Contrary to the prediction of the model of section 3.4 (see also Fig. 19), the attenuation signal at fixed
+heating frequency is not directed exactly toward the centre of the Kennelly osculating circle. We observe a
+systematic clockwise angular shift 20◦ < θ < 30◦. We still have no definite explanation for this observation.
+As explained in sec. 3.4.2, the velocity selectivity is more important for open cavity resonators such as second
+sound tweezers. For a given tweezers size, we find that the selectivity depends mainly on the shift Xsh and on
+the wave mode number. Perfectly aligned tweezers excited with low mode numbers are preferentially sensitive
+to quantum vortices. Increasing Xsh or choosing larger mode numbers leads to a larger velocity sensitivity and
+changes the signal balance from vortex selectivity to velocity selectivity. The tweezers selectivity also strongly
+depends on the size L: smaller tweezers can encompass less quantum vortices in the cavity, and are thus less
+sensitive to quantum vortices, while the velocity sensitivity remains unchanged. The velocity selectivity is thus
+larger when the tweezers are smaller. Fig. 24 displays the selectivity of two second sound tweezers of size
+L = 250 µm and L = 1 mm respectively, depending on the shift Xsh and the mode number n (where k = nπ).
+The simulation was run with a quantum vortex line density L = 2 × 1010 m−2 and U = 1 m/s, in accordance
+with the typical values observed in the experiments of [WVR21]. It can be seen that large tweezers (L = 1 mm)
+can reach a vortex selectivity Rξ > 90% for a small shift and low mode number, which means that they can be
+used for direct quantum vortex measurements. On contrary, small tweezers (L = 250 µm) can reach a velocity
+selectivity RU > 90% for large shift or high mode number, and can thus be used as anemometers, as confirmed
+by the experiments reported in [WVR21].
+4
+Measurements with second sound tweezers
+Second sound tweezers are singular sensors in the sense that they can measure two degrees of freedom at the
+same time, whereas most of hydrodynamics sensors only measure one (e.g. Pitot tubes, Cantilevers, Hot wires).
+The tweezers record the magnitude and phase of the thermal wave averaged over the thermometer plate. Both
+quantities contain physical information about the system. To summarize it shortly, magnitude variations give
+information about quantum vortices in the cavity, whereas phase variations give information about the local
+mean temperature and pressure. The local mean velocity has an impact on both magnitude and phase, and
+will be specifically treated in sec. 4.5. The aim of the following sections is to explain how properly separate
+quantum vortices signal from other signal components.
+In the following, we call L⊥ the density of projected quantum vortex lines density (projected VLD)
+L⊥ = 1
+V
+�
+V
+sin2 θ(l)dl,
+(27)
+where V is the tweezers cavity volume, l is the curvilinear abscissa along the vortex lines inside the cavity ,
+θ(l) is the angle between the quantum vortex line and the direction perpendicular to the plates (vector ez).
+29
+
+Figure 24: Selectivity to quantum vortices or velocity advection for two second sound tweezers, obtained with
+numerical simulations. The color code indicates the fraction of the signal due to bulk attenuation by quantum
+vortices (see Fig. 18). The selectivity of the tweezers mainly depends on the lateral shift of both plates one
+from another, and the resonant mode number excited in the cavity. The left panel shows that large tweezers
+(L = 1mm) are mainly sensitive to quantum vortices. Almost pure quantum vortex signal can be achieved
+with carefully aligned tweezers excited at low mode numbers (Rξ > 90%). On the reverse, small tweezers
+(L = 250µm) are mainly sensitive to the velocity. Almost pure velocity signal can be achieved by shifting the
+heater and the thermometer plates and by exciting the cavity at large mode numbers (RU > 90%). The present
+simulation was run with a vortex line density L = 2 × 1010 m−2 and U = 1 m/s, in accordance with the typical
+values observed in our experiments.
+Assuming isotropy of the vortex tangle, the total quantum vortex lines density (VLD) is
+L = 3
+2L⊥.
+(28)
+A second sound wave is damped in the presence of a tangle of quantum vortices. Let ξV LD (in m−1) be
+the bulk attenuation coefficient of second sound waves, it has been found[HV56a, HV56b, Tsa62, SP66, MPS84]
+that ξV LD is proportional to L⊥ according to the relation
+ξV LD = BκL⊥
+4c2
+,
+(29)
+where B is the first Vinen coefficient and κ ≈ 9.98 × 10−8 m2/s (for 4He) is the quantum of circulation around
+one vortex.
+Therefore, Eq. (29) shows that a measure of the bulk attenuation coefficient gives access to the projected
+VLD defined by Eq. (27). We recall in sec. 4.1 the standard methods to measure the bulk attenuation coefficient
+from a second sound resonance, and we propose in sec. 4.3 a new method called “the elliptic method”. We give
+in sec. 4.4 some examples to apply the elliptic method to the experimental data.
+4.1
+The vortex line density from the attenuation coefficient
+We assume that a single second sound resonance can be accurately represented by the following expression (see
+Eq. (10))
+T(f) = T 0
+sinh (ξ0D)
+sinh
+�
+i 2π(f−f0)D
+c2
++ (ξ0 + ξV LD)D
+�,
+(30)
+where f0 is the second sound frequency of the local amplitude maximum, D is the resonator gap, c2 is the
+second sound velocity, ξ0 is the attenuation coefficient without flow and ξV LD is the additional bulk attenuation
+in the presence of quantum vortices given by Eq. (29). ξV LD = 0 without flow.
+A standard method to measure ξV LD goes as follows: we fix the second sound frequency at the resonant
+value f0, and we measure the thermal wave amplitude with, and without flow. Eq. (30) then shows that ξV LD
+is given by
+ξV LD = 1
+Dasinh
+� T 0
+T(f0) sinh (ξ0D)
+�
+− ξ0.
+(31)
+30
+
+26p-
+0.5
+0.45
+0.4
+0.35
+0.3
+shift
+0.25ity / vortex sensitivity -→0.2
+0.15
+Vortex se
+0.1
+0.05
+0
+1234567
+modeK velocity sensitiv
+ectivity > 90%
+89101112131415
+number0.5
+0.45
+Velocity se
+0.4
+0.35
+0.3
+shift
+0.25ty / vortex sensitivity →
+electivity > 90%0.2
+0.15
+0.1
+0.05
+0
+2345678
+modeK velocity sensitiv
+910111213141516
+numberFigure 25: Spectral response of tweezers in the SHREK facility, right below the superfluid transition temperature
+Tλ, where second sound velocity c2 is very sensitive to temperature (here D ≃ 500 µm and c2 drifts around a
+mean value of 5.4 m/s). The frequency axis is adimensionalize using the second sound velocity c2 calculated
+from the temperature recorded near the sidewall of the flow. While the temperature of the bath is regulated,
+frequency sweeps are repeated half a dozen of times for different turbulent flow conditions flagged by colors.
+A systematic drift of resonance frequencies versus flow conditions is observed ; it is interpreted as an under-
+estimation of temperature, and therefore over-estimate of c2, due to turbulent dissipation in the core of the flow.
+At a given mean flow, some scatter of the resonance frequencies is apparent ; it is interpreted as noise from
+the temperature regulation. The elliptic method introduced in section 4.3 allows separating such temperature
+artifacts from the attenuation due to second sound attenuation by quantum vortices.
+Eq. (31) shows that beside the value of D , that can be determined from a fit of the tweezers spectrum (sec.
+3.4), the value of ξ0 has to be accurately measured. This is usually done by the measurement of the resonant
+half width. With some algebra manipulations, it can be found from Eq. (30) that the resonant magnitude
+satisfies
+��T
+��2 =
+��T 0
+��2
+sinh2 (ξ0D)
+sinh2 (ξ0D) + sin2 �
+2π(f−f0)D
+c2
+�.
+(32)
+Let ∆f be the frequency half-width defined by the relation
+���T(f0 ± ∆f
+2 )
+���
+2
+= 1
+2
+��T 0
+��2, it can be shown from Eq.
+(32) that ξ0 and ∆f are related by
+sin
+�π∆fD
+c2
+�
+= sinh (ξ0D) .
+(33)
+We note in particular that the relation (33) can be used to find ξ0 as long as the resonance quality factor is
+high enough, that means, for sinh (ξ0D) < 1. The linear approximations of Eqs. (31) and (33) are usually used
+when ξ0D ≪ 1, and they give the well-know approximation:
+L⊥ ≃ 4π∆f
+Bκ
+� T 0
+T(f0) − 1
+�
+,
+(34)
+For low quality factor resonances, another method should be used instead of the resonant half width. The
+elliptic method presented in sec. 4.3 allows determining ξ0 for resonances of any quality factors.
+The main problem of the method presented above is that it implicitly assumes that there is no variation
+of the acoustic path value 2πf0D
+c2
+during the measurement. In particular, as c2 depends on temperature and
+pressure, it means that the experiment should have an excellent temperature and pressure regulation. This can
+become increasingly difficult when the second sound derivatives become steep, close to the superfluid transition.
+Moreover, measurements in the presence of a flow are necessary done out of equilibrium as the flow dissipates
+energy.
+As an example, measurements in such conditions are illustrated by figure 25, which shows second
+sound resonances measured close to the superfluid transition in the turbulent Von Karman experiment SHREK
+[RBD+14]. Furthermore, we observe that a measurement with a non-zero value of ξV LD can be associated with
+an acoustic path shift (i.e. a variation of the factor 2πfD
+c2
+). The situation is illustrated in the left panel of Fig.
+26, where the acoustic path shift leads to an overestimation of the attenuation and an important error on ξV LD.
+31
+
+0.08
+0.5
+0.4
+0.07
+0.3
+0.06
+0.2
+0.05
+0.1
+/w)
+U
+-0.1
+0.03
+-0.2
+0.02
+-0.3
+0.01
+-0.4
+0
+-0.5
+2 元
+4元
+6元
+8元
+10元
+12元
+14元
+2元fD/C 21.6
+1.8
+2
+2.2
+2.4
+kD
+0
+0.2
+0.4
+0.6
+0.8
+1
+T
+First measurement
+VLD=0
+Second measurement
+VLD=1e-4
+Acoustic
+path shift
+Attenuation
+error
+Without
+phase shift
+With
+phase shift
+-0.2
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+X
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+Y
+R
+r
+Figure 26: An illustration of an attenuation measurement. Left: the resonant mode without (blue curve),
+and with quantum vortex attenuation (red curve). The figure shows that an acoustic path shift can lead to
+an important error in the attenuation measurement. Right: the resonant mode represented in the phase-
+quadrature plane together with the fitted Kennelly circle. The figure shows that the acoustic path shift creates
+a phase shift θ. Using the phase measurement θ, the maximal magnitude can be recovered using Eq. (37).
+Passive and active approaches have been reported in the literature to handle the most common cause of
+acoustical path shift during second sound measurement: the temperature drift and its resulting shift of the
+resonance frequency.
+A passive approach consists in performing a sweep of the second sound frequency, across the resonance curve.
+Afterward, with proper modelling of the resonance, the attenuation and the phase shift can be fitted separately,
+e.g. as done in [MSS76]. A limit of this approach is its time resolution, that is restricted by the duration of
+frequency scan. Another passive approach consists in performing systematic calibration of the full frequency
+responses of the resonator in various conditions, and subsequently interpolating measurements obtained at a
+fixed working frequency onto this mapping [VBL+17].
+A standard example of active approaches consist in controlling the helium bath temperature. An alternative
+or complementary approach consists in controlling the second-sound frequency so that it always matches the
+resonance peak, despite possible drift of the temperature. The resonator itself can provide the feedback signal
+of these control loops, for example by monitoring the thermometer or locking the phase of the second sound
+signal. An even more direct approach has been recently proposed: the resonator is driven by a self-oscillating
+circuit, which frequency adapts dynamically to the drift of the second sound velocity [YIE17].
+Below, we introduce two analytical methods to separate phase shift from attenuation. The first method is
+relevant for simple cases (sec. 4.2), while the second one has a broader range of validity (sec. 4.3).
+4.2
+Analytical method in an idealized case
+The acoustic path shift can be corrected using the resonant representation in the phase-quadrature plane. Let
+X(f) and Y (f) be respectively the real and imaginary parts of T(f), the curve (X(f), Y (f)) in the phase-
+quadrature plane is very close to a circle crossing the origin. It can even be proved (see sec. 4.3) that the
+resonant curve converges to a circle when the quality factor increases, or equivalently when ξ0 decreases5. All
+along the present article, this limit circle is called the “Kennelly circle”. An illustration of two resonant curves
+with their osculating Kennelly circles is displayed in the right panel of Fig. 26. The acoustic path shift translates
+in a phase shift θ in the phase-quadrature plane, such that the amplitude of the second measurement (with
+ξV LD > 0) does not correspond to the maximal amplitude R of the attenuated resonant peak (see Fig. 26 for
+5In this case, the complex amplitude of the nth temperature resonance is often approximated by the Lorentzian formula [VS71,
+DLL80]
+T(f) ≃
+Tn
+1 + iQn f−fn
+fn
+(35)
+where Tn, Qn and fn are the amplitude at resonance, the quality factor and resonance frequency of the mode of interest. To
+highlight that this Lorentzian approximation describes a circle in the complex plane X-Y , it can be written as:
+T(f)
+T n/2
+= 1 + eiφ(f)
+(36)
+where tan φ = 2F/
+�
+F 2 − 1
+�
+and F = Qn (f − fn) /fn.
+32
+
+Figure 27: Sequence of five resonances of second sound tweezers at 1.6K. The flow is weakly turbulent: the
+velocity standard deviation is a few percents of the mean velocity displayed on the colorbar. The right side plot
+illustrates that each resonance can be approximated by a circle in the complex plane. The global phase shift of
+the last resonance (lower circle) compared to the others is attributed to a cut-off of the measurement electronics
+at high frequency.
+the notations). Using the geometric properties of the Kennelly circle, R can be approximately recovered from
+the measured amplitude r with
+R =
+r
+cos θ.
+(37)
+Thus, a modified version of Eq. (31) can be written to find the VLD attenuation coefficient in the presence
+of a phase shift
+ξV LD = 1
+Dasinh
+� T 0 cos θ
+T(f0)e−iθ sinh (ξ0D)
+�
+− ξ0.
+(38)
+4.3
+The elliptic method
+We present in this section an original method to obtain the values of the acoustic path shift and the attenuation
+coefficient, from experimental data. The method, that we call the “elliptic method”, is much simpler to imple-
+ment, and much more reliable, than the fit of the Kennelly resonant circle and the use of Eq. (38). Besides, the
+method can be used for resonances with very low quality factors.
+The method comes from the observation that a pair of two ideal consecutive resonances is transformed into
+an ellipse with the complex inversion z → 1
+z in the phase-quadrature plane. The inversion is represented in Fig.
+28. To prove this assertion, consider the inversion of the classical Fabry–Perot expression Eq. (6)
+1
+T =
+sinh
+�
+i 2πfD
+c2
++ ξD
+�
+A
+.
+(39)
+Expanding the sinh in the previous expression gives
+1
+T = cos
+�2πfD
+c2
+� sinh (ξD)
+A
++ i sin
+�2πfD
+c2
+� cosh (ξD)
+A
+.
+(40)
+Finally, let Xl = Re
+�
+1
+T
+�
+and Yl = Im
+�
+1
+T
+�
+be respectively the real and imaginary parts of Eq. (40), the
+coordinates (Xl, Yl) satisfy the equation
+�
+Xl
+sinh (ξD) /A
+�2
++
+�
+Yl
+cosh (ξD) /A
+�2
+= 1
+(41)
+which is exactly the cartesian equation of an ellipse with semi-major axis a = cosh(ξD)
+A
+, and semi-minor axis
+b = sinh(ξD)
+A
+. In particular, we note that the attenuation coefficient ξ can be recovered from the ratio of the
+semi-major and semi-minor elliptic axes using the formula
+ξ = 1
+Datanh
+� b
+a
+�
+.
+33
+
+0.15
+(yu)
+0.1
+0.05
+20
+40
+60
+80
+100
+f (kHz)0.1
+2
+0.05
+1
+(mK)
+(s/w)
+0
+0
+> -0.05
+U
+-1
+-0.1
+-2
+-0.15
+-0.1
+0
+0.1
+X (mK)-1
+-0.5
+0
+0.5
+1
+X
+-0.5
+0
+0.5
+Y
+-2
+0
+2
+X
+-4
+-2
+0
+2
+4
+Y
+1/z
+Figure 28: Transformation of a pair of consecutive resonances to an ellipse using the inversion of the complex
+plane z → 1/z.
+0.4
+0.6
+0.8
+1
+1.2
+X
+-0.2
+0
+0.2
+Y
+1/z
+acoustic path
+axis
+acoustic path
+axis
+attenuation
+axis
+u
+v
+attenuation
+axis
+ul
+vl
+Figure 29: A resonant mode in the phase-quadrature plane and its elliptic transform, for an ideal Fabry–perot
+resonance with ξ0 = 0.2 and 1.95π < kD < 2.05π.
+When the quality factor increases (equivalently when ξ decreases) the ellipse is flattened. The limit of infinite
+quality factor (ξ → 0) corresponds to two parallel straight lines in the complex plane.
+Second sound tweezers resonances are not ideal Fabry–Perot resonances. Yet, we have argued in sec. 3.2
+that a single second sound resonance can be locally fitted by the following Fabry–Perot equation (see also Eq.
+(10))
+T =
+A
+sinh
+�
+i 2π(f−f0)D
+c2
++ (ξ0 + ξV LD)D
+�.
+(42)
+This in particular means that the resonant curve in the vicinity of its maximal amplitude is transformed into a
+part of an ellipse with the complex inversion z → 1
+z . The curve (Xl(f), Yl(f)) is very close to a straight line, for
+frequencies f close to the resonant frequency f0. The situation is illustrated in Fig. 29. The figure shows a part
+of ideal Fabry–Perot resonances given by Eq. (42), in the range 1.95π < kD < 2.05π (where k = 2πf
+c2 ), and for
+increasing values of the VLD attenuation coefficient ξV LD. The left panel displays the different resonant curves
+close to their maximal amplitudes, in the phase-quadrature plane. Using the complex inversion, those curves
+become almost parallel straight lines, as can be seen in the right panel. The transformation of the resonant
+Kennelly circles into parallel straight lines has very nice applications that we explain in the following.
+In the phase-quadrature plane, a variation of the acoustic path value 2πfD
+c2
+corresponds to a displacement
+along the Kennelly circle, whereas a variation of the bulk attenuation ξ corresponds to a displacement orthogonal
+to the Kennelly circle. The acoustic path direction and the attenuation direction thus form a local orthogonal
+basis. Such a basis is displayed by the red arrows in the left panel of Fig. 29. When the reference point where
+the basis is defined moves along the Kennelly circle, the basis rotates and the acoustic path and attenuation
+axes have to be redefined. This can become a tiresome task while analyzing the experimental data. Fortunately,
+the complex inversion is a conformal mapping, which means that it preserves locally the angles: the local basis
+composed of the acoustic path and attenuation axes is transformed into an orthogonal basis (see the red arrows
+in the right panel of Fig. 29). More precisely, let z0 be the complex position in the phase-quadrature plane,
+34
+
+and (u, v) be the two complex (unit) vectors defining the local basis at z0, then the local basis (ul, vl) at the
+point
+1
+z0 is given by
+�
+�
+�
+ul
+= −u |z0|2
+z2
+0
+vl
+= −v |z0|2
+z2
+0
+.
+(43)
+The major advantage of defining the local basis (ul, vl) with the elliptic transform, is that it becomes a global
+basis: when the reference point
+1
+z0 moves because of a change in the acoustic path value or the attenuation
+value, the basis is simply translated in the plane but the vectors (ul, vl) do not change. One can find the global
+basis (ul, vl) for a given resonance and use it to find the local basis (u, v) at every point in the phase-quadrature
+plane. We will see in sec. 4.4.1 how the global basis (ul, vl) can be easily used to suppress temperature and
+pressure drifts during second sound attenuation measurements.
+We finally explain how the elliptic method can be used to measure the bulk attenuation coefficient ξ0. We
+have seen in sec. 4.1 that the standard methods to find ξ0 are based on the measure of the half-width ∆f
+and on Eq. (33). As was said previously, the classical method can only be applied to resonances satisfying
+(ξ0D) < 1. It is a global method, in the sense that one has to sweep the frequency to measure a large part of
+the resonant curve. The method is only accurate provided the resonance does not deviate too much from an
+ideal Fabry–Perot resonance, which is often not satisfied for the first modes of second sound tweezers (see for
+example the first mode of Fig. 17). The alternative method consists in expanding (Xl, Yl) given by Eq. (41) to
+leading order in f − f0
+�
+Xl
+∼ sinh(ξ0D)
+A
+,
+Yl
+∼ 2π(f−f0)D
+c2
+cosh(ξ0D)
+A
+,
+(44)
+where f0 is the resonant frequency. Using Eq. (44), we get
+Yl(f)
+Xl(f) ∼
+2πD
+tanh (ξ0D) c2
+(f − f0).
+(45)
+Yl(f)
+Xl(f) is proportional to f in the vicinity of f0, with the proportionality factor
+2πD
+tanh(ξ0D)c2 . The attenuation
+coefficient ξ0 can be found by a linear fit of the function Yl
+Xl provided D and c2 are known.
+4.4
+Applications of the elliptic method
+The motivation to develop and use the elliptic method has come from experimental constrains: in a large super-
+fluid experiment, it can be very difficult to control the values of mean temperature and pressure. This is even
+more the case if the superfluid experiment dissipates energy. One then expects a drift of the thermodynamics
+conditions during the measurement. Regarding second sound resonators, the critical parameter is the second
+sound velocity c2, because variations lead to uncontrolled acoustic path shifts. The elliptic method has been
+designed to easily filter those variations from experimental data. This includes filtering temperature and pres-
+sure drifts (sec. 4.4.1) and the vibration of the tweezers arms (sec. 4.4.3), and properly extract the quantum
+vortex lines fluctuations (sec. 4.4.2).
+4.4.1
+Suppression of temperature and pressure drifts
+This section presents an example of the elliptic method implementation for second sound tweezers, to find the
+relation between ⟨ξV LD⟩ and the mean velocity U in the presence of a superfluid flow.
+As before, we note (X, Y ) the temperature signal obtained from the second sound tweezers in the phase-
+quadrature plane, and (Xl, Yl) the cartesian coordinates obtained by the complex inversion given by
+�
+�
+�
+Xl
+= Re
+�
+1
+X+iY
+�
+,
+Yl
+= Im
+�
+1
+X+iY
+�
+.
+(46)
+The coordinates (Xl, Yl) will be called “elliptic coordinates” for convenience. Fig. 30 displays experimental data
+obtained from second sound tweezers of size L = 1 mm, in a saturated bath at mean temperature T0 ≈ 2.14 K,
+and for different mean flow velocities 0 < U < 1.2 m/s. At such a temperature close to the superfluid transition,
+it was difficult to regulate the mean temperature and therefore the second sound velocity. Uncontrolled acoustic
+path variations can be observed, for example in the red points of Fig. 30.
+The first step consists in sweeping the second sound frequency f in the vicinity of the resonant frequency f0.
+The data (X, Y ) obtained, displayed by the black curve of the left panel of Fig. 30, form a part of the Kennelly
+circle. As explained in sec. 4.3, the elliptic coordinates (Xl, Yl) given by Eq. (46) form a straight line (see the
+35
+
+0
+0.1
+0.2
+0.3
+X (mK)
+-0.05
+0
+0.05
+0.1
+0.15
+0.2
+Y (mK)
+2
+4
+6
+8
+-4
+-3
+-2
+-1
+0
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+U (m/s)
+Attenuation
+axis
+ul
+vl
+Acoustic path
+axis
+Figure 30: Experiment: measurement of a resonance with second sound tweezers at T0 ≈ 2.14 K, for different
+values of the mean flow velocity U. The left panel presents the data in the phase-quadrature plane, and the
+right panel presents the same data after the complex inversion. The red points are obtained for a fixed value
+f = 6.35 kHz, and sweeping the flow mean velocity between 0 and 1.2 m/s.
+right panel of Fig. 30). Using a linear fit, it is then straightforward to obtain the unit vector vl parallel to the
+line defining the acoustic path axis, and the orthogonal vector ul defining the attenuation axis. We then call
+Zl = (Xl, Yl) the vector of the elliptic coordinates. The attenuation coefficient ξ0 can be found by the relation
+(see Eq. (45))
+Zl(f).vl
+Zl(f).ul
+∼
+2πD
+tanh (ξ0D) c2
+(f − f0).
+Fig. 30 displays experimental resonant curves obtained for non-zero mean velocities U > 0 , to illustrate the
+robustness of the elliptic method. However, we emphasize that only the resonant curve with U = 0 is necessary
+to find the global basis (ul, vl) in the plane of elliptic coordinates.
+The second step consists in fixing the second sound frequency to f0 and vary the mean velocity U to look
+at the resonance attenuation. The experimental data are displayed by the red points in Fig. 30. In can be
+seen in the figure that the bulk attenuation is accompanied by a systematic acoustic path deviation as the
+mean velocity increases. For mean velocities U ≈ 1 m/s, energy dissipation in the experiment leads to a data
+dispersion along the acoustic path direction. To properly recover the mean VLD attenuation coefficient ⟨ξV LD⟩
+, we use the elliptic coordinates Zl = (Xl, Yl) , and we project it on the attenuation axis ul. We get from Eq.
+(44)
+Zl(U).ul
+Zl(0).ul
+= sinh ((ξ0 + ⟨ξV LD⟩) D)
+sinh (ξ0D)
+.
+And ⟨ξV LD⟩ is then given by
+⟨ξV LD⟩ = 1
+Dasinh
+�Zl(U).ul
+Zl(0).ul
+sinh (ξ0D)
+�
+− ξ0.
+(47)
+We note that the previous expression remains accurate even if the second sound frequency chosen for the
+measurement is close but not exactly equal to the resonant frequency f0. The average VLD attenuation can
+then be converted to the average projected vortex line density ⟨L⊥⟩ using Eq. (29).
+4.4.2
+Measure of vortex line density fluctuations
+Second sound tweezers are designed to directly probe the small scale vortex line density fluctuations, not only
+its average value. The method to probe fluctuations slightly differs from the method used to probe the average
+value explained in sec. 4.4.1. The average VLD value can be directly computed using the complex inversion of
+experimental data, but this is no longer possible for its fluctuations. Indeed, the tweezers signal have different
+sources of noise, like e.g. thermal white noise, interfering frequencies, electromagnetic bursts, etc... Those
+signals can usually be considered as independent additive noises in the signal data, and easily filtered out or
+attenuated by an appropriate post-processing. On the contrary, the complex inversion is a non-linear transfor-
+mation. Using the latter on noisy data can lead to an overestimation of the signal fluctuations closest to zero,
+36
+
+1/z-2
+-1.5
+-1
+-0.5
+X (V)
+x 10-6
+-16
+-12
+-8
+-4
+Y (V)
+x 10-7
+0 m/s
+1.20 m/s
+Elliptic transformation
+acoustic path axes
+attenuation axes
+Figure 31:
+perturb the additivity of noise sources and make them much more difficult to filter out. We thus choose to
+compute the VLD fluctuations only using linear transformations.
+The first step is similar to that of sec. 4.4.1. We sweep the second sound frequency f close to the resonant
+frequency f0, in order to measure a part of the Kennelly circle. We then transform this Kennelly circle into a
+straight line using the complex inversion, and we find the global basis (ul, vl) in the plane of elliptic coordinates.
+A fit of the Kennelly circle, and its transformation into a straight line can be seen in Fig. 31. This experimental
+step has to be done just before the fluctuations measurement.
+We then record the signal fluctuations (X(t), Y (t)), for different values of the flow mean velocity U. Fig.
+31 displays the fluctuating signal for U = 0 and U = 1.2 m/s in the form of clouds of data points. It can be in
+particular observed that the U = 1.2 m/s data are shifted compared to the U = 0 m/s data because of both
+an average attenuation and an acoustic path shift (see sec. 4.4.1). Let us define ⟨Z(t)⟩ = ⟨X(t)⟩ + i ⟨Y (t)⟩ the
+average complex position in the phase-quadrature plane, for a given value of U. Following Eq. (43), the local
+basis (u, v) of the attenuation and acoustic path axes can be computed from the global elliptic basis (ul, vl) by
+�
+�
+�
+u
+= −ul
+⟨Z⟩2
+|⟨Z⟩|2
+v
+= −vl
+⟨Z⟩2
+|⟨Z⟩|2
+.
+(48)
+Fig.
+31 shows that the local basis (u, v) depends on ⟨Z⟩ and thus on the value of U: for different mean
+velocities, the bases are rotated from one another. If there is a significant drift of the mean signal value during
+the measurement (as can e.g. be observed in Fig. 22), the local basis will also depend on time, with a typical
+timescale that should be much larger than the fluctuation timescale.
+The acoustic path fluctuations and the attenuation fluctuations can then be recovered using a projection on
+the (u, v) basis. More precisely, let x be the average acoustic path value and δξ = ξ − ⟨ξ⟩ be a small fluctuation
+of the attenuation coefficient, a leading order expansion of expression Eq. (42) shows that
+T ≈
+A
+sinh (ix + ⟨ξ⟩ D) − δξ
+AD
+sinh (ix + ⟨ξ⟩ D) tanh (ix + ⟨ξ⟩ D).
+(49)
+We then do the approximation ⟨Z⟩ ≈ A/ sinh (ix + (ξ0 + ⟨ξV LD⟩) D) which is equivalent to neglecting non-linear
+corrections in Eq. (49). We finally get
+δξ(t) = 1
+Du. (Z(t) − ⟨Z⟩) ×
+����
+tanh (ix + (ξ0 + ⟨ξV LD⟩) D)
+⟨Z⟩
+���� ,
+(50)
+where the value of ξ0 can be found with Eq. (45) and ⟨ξV LD⟩ with Eq. (47).
+The use of the elliptic method is illustrated by Figure 32, which reports the probability density function of
+the fluctuations of the quantum vortex density in a nearly isotropic superfluid turbulent flow The data of this
+37
+
+0.5
+1
+1.5
+2
+p (arb. shift)
+s /<s>
+vortex line density fluct.
+velocity fluctuations
+Figure 32: Example of the probability density function of vortex line density (dashed line) and velocity (contin-
+uous line) measured by second sound tweezers in a superfluid turbulent flow [WVR21]. In abscissa, each signal
+s is normalized by its mean value < s >. The nearly Gaussian velocity statistics and skewed vorticity statistics
+are reminiscent of those in classical turbulence.
+plot were reported together with spectra of vorticity fluctuations. For details about the setup and analysis of
+these results, see [WVR21].
+4.4.3
+Filtering the vibration of the plates
+One possible source of noise for second sound resonator measurement is the vibration of the plates arms whenever
+U ̸= 0. The signature of those vibrations can be very clearly identified in the form of two thin peaks in the
+fluctuations power spectrum. Those two peaks are located at the two arms resonant frequencies: their exact
+values can vary for different tweezers, but we always observe them around f ≈ 1 kHz (see sec. 4.4).
+Fortunately, the tweezers arms vibrations correspond to a variation of the gap D, and thus to acoustic path
+fluctuations. Fig. 33 display a part of the tweezers fluctuations power spectrum. The fluctuations are projected
+along the attenuation axis (blue curve) and along the acoustic path axis (red curve), following the method
+presented in sec. 4.4.2. The two peaks located at f ≈ 825 Hz and f ≈ 1050 Hz are identified on the acoustic
+path axis fluctuations power spectrum, whereas the same peaks are damped by many orders of magnitude on
+the attenuation axis fluctuations. Using the power spectrum of Fig. 33, we can estimate the order of magnitude
+of the gap standard deviation. We find
+��
+(δD)2�
+≈ 0.5 µm, and
+�
+⟨(δD)2⟩
+D
+≈ 4 × 10−4. This confirms that the
+arms vibrations have a negligible impact on the measurement.
+4.5
+Velocity measurements
+As shown in Sec. 3.5, the second sound tweezers geometry can be optimized to sense specifically velocity rather
+than vortex density. For this purpose, one trick consists in shifting one plate with respect to the other in the flow
+direction. Figure 34 shows three second sound tweezers and one anemometer that is based on the same principle
+as Pitot tubes. All sensors are positioned across a nearly isotropic superfluid flow bounded by a cylindrical pipe
+-not shown here- (for details on this set-up, see [RCSR17a]). The figure insert is a close view of the tip of the
+left-side tweezers, which is dedicated to velocity measurements. This shift of one plate versus the other in the
+downstream direction is clearly visible.
+The projection of the anemometer-tweezers signal in the complex plane is not performed along orthogonal
+axes. These axes are determined with an in-situ calibration, ramping the mean velocity. The complementary
+“Pitot tube” signal is used to calibrate the axis in units of m/s. The duration of velocity ramp is chosen to be
+much larger than the time resolution of the Pitot tube.
+As an illustration, Figure 32 presents the probability density function of the velocity fluctuations measured
+by the anemometer tweezers, together with vortex line density fluctuations recorded by the other tweezers during
+the same experimental run (see Fig. 34). As expected in nearly homogeneous and isotropic quantum turbulence
+[SCLR12], the velocity statistics are close to a Gaussian when probed at scales significantly larger than the
+intervortex distance, which is the case here. Velocity spectra derived from the same dataset are reported in
+[WVR21].
+38
+
+700
+800
+900
+1000
+1100
+f (Hz)
+10-18
+10-17
+10-16
+P(f)
+attenuation axis fluctuations
+acoustic path axis fluctuations
+Figure 33: Experiment: a part of second sound tweezers fluctuation power spectrum, with T0 = 1.65 K,
+U = 1.2 m/s and for a tweezers gap D = 1.320 mm. The tweezers arms’ resonances can be clearly identified in
+the acoustic path fluctuations.
+Figure 34: Example of an arrangement of three second sound tweezers dedicated to velocity (x1, left side) and
+vorticity (x2, right side) time series acquisitions, together with a miniature total head pressure tube (loosely
+labelled "Pitot tube" on the bottom left side of the picture) used for velocity calibration.
+All probes are
+mounted on a ring connecting two 76mm-inner-diameter coaxial pipes (see Fig.1 of ref. [WVR21]). The insert
+is a close-up view of the shifted plates of the anemometer tweezers.
+39
+
+Thermometer
+Flow
+Heater
+VLD probing
+second sound
+tweezers
+Second sound
+tweezers
+anemometer
+Miniature Pitot
+tube5
+Summary and Perspectives
+This study has covered three independent topics : the comprehensive analytical modelling of second-sound
+resonators with a cavity allowing a throughflow of superfluid (section 3), new mathematical methods to process
+the signal provided by such resonators (section 4) and the miniaturization of immersed second-sound resonators
+allowing time and/or space resolved flow sensing (section 2). These so-called second-sound tweezers have been
+used throughout the manuscript to demonstrate the strength and limits of the modelling and analysis methods.
+Two observations remains unexplained: the origin of some noise on the acoustic path length and a second
+order oscillations of the resonator spectral response in quiescent 4He, named the “daisy effect” (section 3.4.4).
+Fortunately, both effects don’t impair the measurement of the flow vorticity or velocity.
+Some results that were not anticipated when this study was initiated, are worth recalling and discussing:
+• the possibility to operate tweezers by over-driving the second-sound standing wave beyond an intrinsic
+turbulent transition. In this non-linear mode, the probe becomes sensitive to velocity, which is interpreted
+as a signature of the sweeping of the local vortex tangle by the outer flow (section 2.2.3). This operating
+mode is somehow analogous of hot film and hot wire anemometry in classical fluid, where sensitivity
+to velocity is due to the more-or-less pronounced sweeping of the thermal boundary layer around an
+overheated thermometer.
+• in the linear regime (standing wave of small amplitude), the probe can be sized and operated to be either
+mostly sensitive to quantum vortices or mostly sensitive to the velocity of the throughflow (section 3.5).
+This prediction is verified experimentally by comparing statistics in turbulent flows (section 4.5). The
+spatial and time resolution of tweezers operated as anemometers -both in non-linear and linear mode-
+is close or better than the alternative miniaturized anemometers working in He-II, that is hot-wires
+[DBM+15, DRS+21], cantilevers[SMR12, RCSR17b], Pitot tubes [SMR12, RCSR17b] and total head-
+pressure probes[MT98, WVR21].
+• in the absence of throughflow, the full spectral response of the resonators, and in particular its quality
+factors, can be accurately determined simply taking into account the loss by diffraction and misalignment
+of the reflecting plates of the cavity.
+In other words, the other sources of dissipation have negligible
+contributions in the range of conditions explored here.
+This is no longer the case in a presence of a
+throughflow carrying quantum vortices since the latter can significantly contribute to the total dissipation.
+We have not explored experimentally the production and detection of second-sound by mechanical means,
+which implies cavity with rough surfaces (e.g. millipore or nucleopore membranes). In this case, the effect
+of vortices pinned on the surface may no longer be negligible ; for a discussion see [DLL80].
+• the possibility to sense the variations of the vortex line density or velocity without knowledge on variations
+of the second sound velocity (or acoustical path). More generally, the elliptic method allows a mathematical
+decoupling of both effects by a projection method in the (inverse) complex plane. This results is of major
+practical interest in flows where the second-sound velocity is not accurately controlled due to residual
+temperature variations or thermal gradients. This situation can occur for instance in flows sequentially
+driven at various levels of forcing (e.g. to explore a Reynolds number dependence), in inhomogeneous
+dissipative flows and in flows close to the lambda superfluid transitions where the second-sound velocities
+strongly depends of temperature.
+Two applications of second-sound tweezers have been illustrated. Measurement within a turbulent boundary
+layer (Fig. 25) are possible thanks to the small size of probe, and measurements of time series in the bulk
+of quantum turbulent flow (Fig. 32) are possible thanks to both the time and space resolutions of the probe.
+Among other applications the probe can map the velocity or the vorticity field of an inhomogeneous flow. A
+mapping of vorticity in a counterflow jet has been recently done and will be reported elsewhere.
+Another application of tweezers would be to probe simultaneously the temperature fluctuations and those of
+either velocity or vorticity, for instance to explore their correlations in turbulent counterflows or even co-flows.
+Indeed, the tweezers thermometer provides a direct measurement of temperature in a bandwidth spanning from
+zero frequency up to a fraction of the frequency of the second sound standing wave, and this signal could be
+acquired without impairing the measurement of the second sound standing wave. Alternating measurements in
+the linear and non-linear modes is also interesting to explore locally both velocity and vorticity in a given flow.
+A last example of application is to operate a double-tweezers made of a heating plate between two thermometer
+plates, or vice-versa. Such a stack can be used to probe joint statistics of vorticity on one side, and velocity on
+the other side, or this arrangement can be used alternatively to probe transverse gradient of either vorticity or
+velocity.
+40
+
+Acknowledgements
+We warmly thank our colleague Benoît Chabaud for support during cryogenic tests and operation of the TOUPIE
+wind-tunnel. We acknowledge and thank the staff of the PTA and Nanofab clean-rooms in Grenoble, where
+microfabrication was done. Data of figure 25 have been acquired in the SHREK facility. We thank all the
+members of the SHREK collaboration, in particular Michel Bon Mardion and Bernard Rousset for the specific
+operation very near the superfluid transition. We acknowledge the indirect but key contribution of A. Elbakyan
+regarding the comprehensiveness of the bibliography.
+The probe design, fabrication, cryogenic operation and theoretical analysis took place over 7 years, and was
+possible thanks to following research grants: ANR Ecouturb grant (ANR-16-CE30-0016), ANR QUTE-HPC
+grant (18-CE46-0013- 03) and EU Horizon 2020 Research and Innovation Program "the European Microkelvin
+Platform (EMP)" (824109).
+This research was funded in part by the Agence nationale de la recherche (ANR). A CC-BY public copyright
+license has been applied by the authors to the present document and will be applied to all subsequent versions
+up to the Author Accepted Manuscript arising from this submission, in accordance with the grant’s open access
+conditions.
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diff --git a/9tE5T4oBgHgl3EQfRQ5h/content/tmp_files/load_file.txt b/9tE5T4oBgHgl3EQfRQ5h/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf,len=3124
+page_content='Second sound resonators and tweezers as vorticity or velocity  probes : fabrication, model and method  Eric Woillez∗, Jérôme Valentin†and Philippe-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Roche‡  Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Grenoble Alpes, CNRS, Institut NEEL, F-38042 Grenoble, France  Distributed under a Creative Commons Attribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  CC-BY | 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='0 International licence  Abstract  An analytical model of second-sound resonators with open-cavity is presented and validated against  simulations and experiments in superfluid helium using a new design of resonators reaching unprecedented  resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The model accounts for diffraction, geometrical misalignments and flow through the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' It is  validated against simulations and experiments using cavities of aspect ratio of the order of unity operated  up to their 20th resonance in superfluid helium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' An important result is that resonators can be optimized  to selectively sense the quantum vortex density carried by the throughflow -as customarily done in the  literature- or alternatively to sense the mean velocity of this throughflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Two velocity probing methods are  proposed, one taking advantage of geometrical misalignements between the tweezers plates, and another one  by driving the resonator non-linearly, beyond a threshold entailing the self-sustainment of a vortex tangle  within the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  After reviewing several methods, a new mathematical treatment of the resonant signal is proposed, to  properly separate the quantum vorticity from the parasitic signals arising for instance from temperature and  pressure drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This so-called elliptic method consists in a geometrical projection of the resonance in the  inverse complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Its strength is illustrated over a broad range of operating conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The resonator model and the elliptic method are applied to characterize a new design of second-sound  resonator of high resolution thanks to miniaturization and design optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' When immersed in a su-  perfluid flow, these so-called  second-sound tweezers provide time-space resolved information like classical  local probes in turbulence, here down to sub-millimeter and sub-millisecond scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The principle, design  and micro-fabrication of second sound tweezers are detailed, as well as their potential for the exploration of  quantum turbulence  Contents  1  Introduction to second sound resonators  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1  Quantum fluids and second sound  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
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+page_content='  23  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3  Effect of lateral shift of the emitter and receiver plates .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
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+page_content='4  Limits of the model  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
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+page_content='5  Quantum vortex or velocity measurements ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
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+page_content='  35  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
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+page_content='2  Measure of vortex line density fluctuations  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
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+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3  Filtering the vibration of the plates .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
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+page_content='  38  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  Velocity measurements  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
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+page_content='  38  5  Summary and Perspectives  40  1  Introduction to second sound resonators  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1  Quantum fluids and second sound  Below the so-called lambda transition, liquid 4He enters a quantum state named He-II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This liquid transition  occurs around Tλ ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='18 K at saturated vapor conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' According to Tisza and Landau two-fluid model,  the hydrodynamics of He-II can be described as the hydrodynamics of two interpenetrating fluids, called the  superfluid component and the normal fluid component [Bal07, Gri09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The superfluid density ρs is vanishingly  small right below the transition, and it increases as temperature decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The opposite dependence occurs  for the normal fluid density ρn, which becomes vanishingly small in the zero temperature limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The properties  of both fluids strikingly differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The superfluid has zero viscosity and zero entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Besides, the circulation of  its velocity field is quantized in units of κ = h/m ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='10−7m2/s, where h is Planck constant and m the  atomic mass of 4He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This quantization constrain results in the existence of filamentary vortices of Ångströmic  diameter, later referred to as the superfluid or quantum vortices[Don91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Contrariwise, the normal fluid follows  a classical viscous dynamics and carries all the entropy of the He-II[Put74, Kha00].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The existence of distinct velocity fields vs and vn for the superfluid and normal fluid results in the existence  of two independent sound modes in He-II, as can be shown by linearizing the equations of motion [Put74,  Kha00, Don09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The so-called “first sound” corresponds to a standard acoustic wave : both fluids are oscillating  in phase (vs = vn), producing oscillations of the local pressure and density ρ = ρs + ρn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The “second sound”  corresponds to both fluids oscillating in antiphase without net mass flow (ρsvs = −ρnvn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Hence, the relative  densities of superfluid and normal fluid locally oscillates, as well as entropy and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  Generation and detection of second sound waves  Experimentally, two techniques are mostly used to generate and detect second sound: one mechanical and  the other thermal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Alternative techniques not discussed here have been occasionally used, including optical  scattering (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' [PGB70]) and acoustic detection above the liquid-vapor interface[LFF47] or in the flow itself  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' [HR76]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The mechanical technique consists in exciting and sensing a single component of He-II, either its superfluid  one or the normal fluid component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Indeed, a single fluid motion can be viewed as a superposition of a second  sound and a first sound (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' an acoustic wave), with an exact compensation of motion for one of the two fluids  at the location of the transducer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Since the first sound velocity is typically one decade larger than the second  second velocity[DB98], most second sound resonances don’t coincide with acoustic resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  In practice,  a selective displacement of the superfluid component is achieved in Peshkov transducers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' They are made of  a standard acoustic transducer side-by-side with a fixed porous membrane-filter which tiny pores are viscous  dampers for the normal fluid but are transparent (“superleaks”) for the superfluid [Pes48, HL88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Alternatively,  2  Emitter  Receiver  Emitter  Receiver  Flow  Flow  Figure 1:  Left: A macroscopic resonator for second sound, embedded in the sidewall, is used to sense  the averaged flow properties in the shaded region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Right: Second sound tweezers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' In contrast with the  macroscopic design of the left schematics, this miniaturized resonator, positioned within the flow, allows space  and time resolution of the flow variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  a selective displacement of the normal fluid component is achieved in oscillating superleak transducers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' They  are based on a vibrating porous membrane that is coupled to the motion of normal fluid by viscous forces, and  uncoupled to the inviscid superfluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' They can be manufactured by replacing the membrane of a loudspeaker  or microphone by a millipore or a nucleopore sheet [WBF+69, SE70, DLL80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The thermal technique to generate and detect second sound consists in forcing second sound by Joule  effect, and detecting it with a thermometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Depending on the operating range of temperature and practical  considerations, several types of thermometers can be suitable to detect second sound waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The literature  being vast, we only list a few thermistor materials and bibliographic entry points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Materials with a negative  temperature coefficients1 include carbon in various forms (aquadag paint, fiber, pencil graphite,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='.) [HVS01],  doped Ge[Sny62], RuOx [YI18], ZrNx/Cernox [YYK97, FS04] and Ge-on-GaAs [MMP+07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Transition edge  superconductor thermometers are often preferred when large sensitivity or low resistivity is important, for  instance Au2Bi [Not64], PbSn [CR83, RR01], granular Al [CA68, MSS76] and AuSn [Not64, Lag76, BSS83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  More information on AuSn is provided in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The second sound tweezers presented in the present  study resorts to this thermal technique, both for generation and detection of second sound2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3  From macroscopic second sound sensors to microscopic tweezers  In the presence of superfluid vortices, the superfluid and normal fluid experience a viscous mutual coupling  [Don91], which entails a damping of the second sound waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This attenuation of second sound by vortices  have been extensively used as a tool to explore the properties of He-II flows over the last 60 years[Vin57], in  particular to explore the properties of quantum turbulence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' see [VJSS19]), a field of applications which  as motivated the development of second sound tweezers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' For example, mechanical second sound transducers  were successfully used to study the turbulence of He-II in the wake of a grid by groups in Eugene, Prague and  Tallahassee (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' see [SNVD02, BVS+14, MG18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Examples of thermal second sound transducers successfully  used to study turbulence of He-II flows are described in studies by groups from Paris, Tallahassee, Grenoble  and Gainesville (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' see [WPHE81, HVS92, RDD+07, YI18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  A specific type of probe allows very sensitive probing of the density of quantum vortices in He-II flow :  standing-wave second sound resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Such a resonator consists in two parallel plates facing each other, one  functionalized with a second sound emitter and the other with a receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The emitter excites the cavity at  resonance to benefit from the amplification of the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The characteristics of the standing wave between the  plates provides information on the properties of the fluid and flow between the plates, in particular the density  of vortex lines, which impacts the amplitude of the standing wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Aside from vortex density measurements, the  second sound can also provide information on the fluid temperature -since the second sound velocity depends  on it- and on the velocity of the background He-II flow when it induces a phase shift or Doppler effect on the  second sound (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' see [DL77, WPHE81, WVR21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The characteristics listed above for the standard (macroscopic) second sound resonators remain relevant for  their miniaturized version : second sound tweezers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Beside miniaturization, a key specificity of tweezers is their  1Phosphor bronze wire, a positive temperature coefficient thermometer has also been used in the early days [Pes46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  2In principle, mechanical and thermal techniques could be combined to generate and detect, although we are not aware of any  composite configuration reported in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  3  low footprint on the streamlines when positioned in the core of a flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This key differences between standard and  tweezers resonators are sketched in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Consequently, standard resonators provide information on averaged  properties of the flow, while tweezers give access to space and time resolved information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Thus, tweezers are local  probes in the same ways as the hot-wire anemometers or cold-wire thermometers used in turbulence studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  In the present design for tweezers, the thermal actuation technique is preferred to the mechanical actuation  because it makes it easier to respect the constrains of miniaturization and reduced flow blockage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4  Overview of the manuscript  The following sections cover independent topics,  Section 3 presents a comprehensive modelling of second-sound resonators accounting for plate misalignment,  advection, finite size and near field diffraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Diffraction, which has been neglected in previous quantitative  models, turns out to be a dominant source of degradation of the quality factor in our case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Applications  to the measurement of vortex concentration or velocity are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Section 4 presents existing methods to process the signal from second sound resonators, and their limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' To  circumvent them, we introduce a new general approach, named the elliptic method, based on a mathematical  properties of resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This method allows to dynamically separate the amplitude variations of the standing  wave due to variations of vortex density or variations of velocity, from the phase variations (more precisely, from  the acoustical path variations), for instance due to variations of the second sound velocity themselves resulting  from a temperature drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Section 2 reports the design, clean-room fabrication and operation of miniaturized second-sound resonators,  named second-sound tweezers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' These tweezers allow to probe the throughflow of helium with an unprecedented  spatial and time resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  For better clarity, we first present the second-sound tweezers, which allows to illustrate the topics on mod-  elling and method with a challenging practical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Though we emphasize that the modelling and methods  introduced in this article are general and relevant to second-sound resonator regardless of their size, including  the macroscopic sensors embedded in parallel walls encountered in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  2  Design, fabrication and mode of operation of second sound tweezers  The core part of second sound tweezers is a stack composed of a heating cantilever and a thermometer cantilever,  separated by a spacer (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 2 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Heaters and thermometers are cantilevers composed of a baseplate,  an elongated arm and a tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The baseplate is the thickest part while the tip is the thinnest one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The active  areas, the emitter and receiver plates, are located on the tips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' For a given device, heater and thermometer have  strictly the same mechanical structure, the only difference between them being the chemical elements used in  the serpentine electrical path deposited on the tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Three cantilever types were fabricated in order to allow  assembling of resonators with three different tip sizes (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The tip widths are 1000µm, 500µm and  250µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Next subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1 presents considerations which prevailed in the mechanical design of the second sound  tweezers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The following one (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2) discusses the detection (thermometry) and generation (heating) of  second sound by the tweezers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The last two subsections present the microfabrication techniques (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3)  and the electrical circuitry used to operate the probes (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1  Mechanical design  Resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The tweezers space-resolution Lres is set by the largest dimension of its cavity, which can be  either the inter-plates distance D, also called the “gap”, or the side length L of the plates here assumed to be  squared shaped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The present study mostly focuses on cavities with an aspect ratio of order 1, to benefit from  optimal space averaging of the signal at given space-resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The tweezers time-resolution τres is set by the  decay-time of a wave bouncing between the plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' In section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2, we introduce and validate a simple model  accounting for dissipation in the cavity due to diffraction loss and residual inclination of the plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' An upper  bound for τres is obtained from the diffraction loss term: τres ≃ L2f/bc2  2, where b ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='38, c2 is the second sound  velocity and f is the wave frequency which can be approximated as nc2/2D for the nth mode of resonance (see  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Thus, the tweezers time-resolution due to diffraction loss can be estimated as  τres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' ≃ L2f  bc2  2  ≃ n L2  Dc2  The ratio Lres/τres of the space resolution and time resolution defines a characteristic velocity for which  the probe optimally averages the space-time fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' For instance, in cavities of aspect ratio one (D = L)  operated on its nth resonance, the nominal velocity is estimated as c2/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' These estimates show that cavities  4  Figure 2: Second sound tweezers, pictured from two angular perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (Left) Probe as seen in the direction  of the mean flow (or nearly so).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The second sound cavity is localized at the upper tip of the probe, in the  encircled area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The bend copper wire through the probe is a temporary joystick used for a fine-alignment of  cavity plates under the microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The two coaxial cables for heating and thermometry are visible in the  lower left corner of the picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (Right) The picture insert shows the same probe after some rotation, and after  removal of the joystick, making visible the through-hole across the silicon stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The two staggered notches  used for thermal confinement of the standing wave in the cavity are clearly apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Tip  Arm  Baseplate  Cantilever with heater  Kapton / Tracks  spacer  Cantilever with thermometer  Tracks / Kapton  Second sound standing wave  Figure 3: Schematic side view of the constitutive stack of second sound tweezers (the pieces are shown sepa-  rated for illustration, their thicknesses are exaggerated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The active areas, the emitter and receiver plates, are  constituents of the tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  250 µm  500 µm  1000 µm  Figure 4: Top view of the 3 cantilever types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The tips widths are respectively 1000µm, 500µm and 250µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Left: Mechanical structures, all parts are silicon made, different thicknesses are represented by different colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The baseplate width is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5mm for all types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Right: Electrical path on each tip type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Yellow areas are  a deposition of TiPt for heaters and AuSn for thermometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Orange areas are a thick AuPt deposition for  current leads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5mm  Baseplate  Arm  Tip  10mm  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5mm  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5mmwith aspect ratio of order unity are fittingly sized for second-sound-subsonic flow, that is flows with a mean  velocity of few m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Blocking effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The above considerations on space resolution are relevant if the measured flow is not  altered by the probe support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The present design conforms to the empirical ×10 rule stating that components  of the support that obstruct the flow on a length scale X should be positioned at least 10X away from the  measurement zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Accordingly, the cavity is at the end of elongated arms and the cantilevers are 25 mm in  length of decreasing thicknesses and widths, as illustrated by the picture of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2 and by Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The thickness  successive values are around 520µm, 170µm and 20µm while the width decreases from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5mm to 145 µm in the  narrowest zone (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 275 µm and 500 µm) for cavities with L = 250 µm (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' L = 500 µm and L = 1000 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Wave confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The spatial resolution of tweezers would be degraded if the second sound standing  wave was spreading out of the L × L × D cavity, by reflection between the supporting arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' A design trick was  implemented to confine the standing wave in the cavity region by breaking the mirror symmetry between the  two cantilevers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Thus, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2, anti-symmetric notches in the tips prevent the second sound to escape  by bouncing away from the cavity, at least in the geometric-optic approximation where diffraction is neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Mechanical resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Besides the ×10 rule consideration, these dimensions are chosen such that the  mechanical vibrations of the arm are pushed up to ∼ 1 kHz or above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The fundamental resonance frequency of  the trapezoidal-shaped arm in vacuum was estimated from the analytical formula in [Lob07] (section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  f0 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='367  2π  e  t2  �  ESi(3w2 + w1)  ρSi(49w2 + 215w1)  We find f0 = 2195 Hz (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 1889 Hz and 1569 Hz) using the material properties ESi = 140 GPa, ρSi =  2330 kg/m3 and the dimensions of the intermediate section of the arm having thickness e = 172 µm, length  t = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5 mm, and width decreasing from w2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5 mm to w1 = 250 µm (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' to 500 µm and 1000 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' An  experimental validation was done at room temperature in air with an arm with w1 = 1000 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Its mechanical  vibration frequency spectrum was measured from a photoreceptor detecting a laser beam reflecting of the  arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The mechanical excitation was provided either by hitting the table supporting the set-up with a small  hammer or by pointing a jet of compressed air toward the arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' In both cases, the fundamental mechanical  resonance frequency was found to be 1215 Hz, in reasonable agreement with the 1569 Hz prediction given the  uncertainty on the Young modulus and deviations from the trapezoidal shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' As discussed in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3, indirect  measurements of the resonance frequency were done in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2 m/s superfluid flow and gave f ≈ 825 Hz, f ≈ 1050  Hz and an amplitude of vibration smaller than 1 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The decrease in frequency compared to room-temperature  measurement is interpreted as mostly due to a fluidic added mass effect[Sad98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Deflection of the tips’ ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The thicknesses of the tweezers parts are such that the mechanical deflection  at the tip endpoint remains significantly lower than the inter-plate distance under typical operating conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The deflection at the tip endpoint can be estimated by considering separately the arm deflection (with  thickness 172µm) and the tip deflection (with thickness 20µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' As a first approximation, both arm and tip  are considered as cantilever beams of uniform width submitted to a uniformly distributed load, and having one  embedded end and one free end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This geometrical approximation overestimates the deflection of the arm, as its  endpoint is narrower than its base, and it underestimates the deflection of the tip, as the notch is ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Still,  this is enough to get order-of-magnitude estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The load is estimated as the dynamic pressure of a liquid  helium flow impinging the tweezers in transverse direction at a velocity U=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1 m/s, that is 10% of the typical  longitudinal flow velocity of 1 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The dynamic pressure P is taken as:  P = 1  2ρU 2  where the liquid helium density is ρ ≃ 140 kg/m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' According to Euler-Bernouilli beam theory, the free end  deflection δmax of the cantilever is:  δmax = 3  2  Pt4  ESi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='e3  The total deflection (arm and tip) is upper-bounded by considering the sum of the deflection of a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5mm  long tip and a 15mm (not 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5mm) long arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This way, the small angle generated on the tip by the arm  deflection is taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Using the values t = 15mm (length), e = 172 µm (thickness) and E = 140GPa  (Young modulus), the deflection of the arm endpoint is found to be 75nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The deflection of the tip endpoint  with t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5mm and e = 20 µm gives a 37nm deflection Thus, the total mechanical deflection of the tweezers  tip due to a steady lateral flow of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1 m/s is a fraction of a micron, that is decades smaller than the inter-plate  distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The mechanical resonance of the tweezers arm and tip, discussed above, could lead to deflections larger than  the one due to a steady forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The amplitude of those mechanical oscillations was measured in a turbulent  He-II flow up to velocities exceeding 1 m/s, taking advantage of the dependence of the second sound resonance  6  with respect to the cavity gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The measured signal will be presented to illustrate the efficiency of the elliptic  projection method in separating the fluctuations of the acoustical path of the cavity and fluctuations of the  bulk attenuation of second sound between plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The mechanical oscillations of the cavity gap are found to be  typically 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5 µm (around 1 kHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' As expected, such a deflection is decades lower than the interplate distance  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3 mm in this case) and than the second sound wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Boundary layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' In the presence of a mean flow through the cavity, a velocity boundary layer will develop  along each tweezers plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' In principle, this boundary layer could contribute to the measured signal and therefore  alter the measurement of the incoming flow, for instance by increasing the density of superfluid vortices in the  cavity and therefore second sound attenuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' As illustrated later, the second sound standing wave that settles  between the plates have nodes of velocity near the plates (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 10 and fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 16) while the sensitivity  of second sound to vortices arises in antinodal regions of velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' As long as the boundary layer thickness is  thin enough, say within a fraction of λ/4 (λ = c2/f is the second sound wavelength), it is not expected to alter  significantly the measured signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  A first requirement for this condition is that the mean flow direction is parallel to the plates so that the flow  penetrates through the cavity with minimal deflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' A consequence is that plates should be widely separated  when operated in flows with undefined or zero mean velocity, such as the core of a mixing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  A second requirement is that the plate thickness is much thinner than λ/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Present plates are 20 microns  thin, to be compared with λ/4 ≃ D/2n for the nth mode of resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' For instance, with D = 500 µm and  n = 3, the condition 20 ≪ λ/4 ≃ 83 µm is indeed well satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  A third condition regards the downstream development of the boundary layer thickness, which should also  stay well within λ/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The physics of boundary layers in He-II is ill-understood[SPB17] but existing experiments  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  see [SHVS99]) suggests that classical hydrodynamics phenomenology could remain valid in the high  temperature limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  In classical hydrodynamics, the so-called displacement thickness of a laminar “Blasius”  boundary layer at distance L from its origin is given by  δbl = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='73  �  Lν  U  where U is the mean velocity far from the boundary layer and ν is the kinematic viscosity of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' In  He-II, several diffusive coefficients could arguably play the role of ν, in particular the quantum of circulation  around a quantum vortex and the kinematic viscosity associated with the dynamics viscosity of the normal fluid  normalized either by the normal fluid density or by the total density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' In the temperature range of interest, all  these diffusive coefficients are within one order of magnitude, typ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 10−8 − 10−7m2/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Taking ν = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='10−8 m2/s,  L = 1000 µm and U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5 m/s, one finds δbl = 13 µm, and a boundary layer Reynolds number δblU/ν = 217  consistent with the laminar picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This thickness estimate, similar in magnitude with the plate thickness,  satisfies the third requirement δbl ≪ λ/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  Second sound detection and generation  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1  Thermometry  The temperature-sensitive material used in the present study is AuSn, which fulfills two requirements: (1) it  is compatible with the microfabrication process and (2) it can be tuned to become temperature-sensitive over  a range of special interest to quantum turbulence studies[WVR21], from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5 K up to the superfluid transition  temperature Tλ ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='18K in saturated vapor conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Surely, other materials would be more appropriate in  other conditions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Al has been used previously for the tweezers operated around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5 K in [RDD+07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The gold-tin AuSn thermometer is a metal-superconductor composite material, with superconducting Sn  islands electrically connected by a gold layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This granular structure is imaged by electronic microscopy in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The temperature dependence phenomenology can be interpreted in a simple way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Indeed, by  proximity effect, the gold in contact with tin behaves as a superconductor over a spatial extent which depends  on temperature : by adjusting the characteristic length scales and thicknesses of the granular pattern, the  temperature response of the material can be tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  As a preliminary study, the temperature of the resistance of a 100 squares long AuSn track was tested for  three different tin thicknesses, as illustrated on the left plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' A description of the conduction mechanism  in AuSn is presented in [BSS83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  In the present study, AuSn layers are deposited by evaporation with successively a small erosion of the  substrate by argon ion bombardment during 20s then the deposition of a 25nm gold layer and a 100nm tin  layer (hypothetic thickness for a planar - not granular - layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The thermometer is shaped into a meander  deposited by lithographic technique on the tip of tweezers arm, as pictured in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5 (right plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The total  resistance at superfluid temperatures doesn’t exceed a few hundreds of ohm, a value chosen to be much larger  than the resistance of the leads but small enough to prevent parasitic effect from the leads’ capacitance (typ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  few hundreds of picoFarads) up to the highest frequencies of operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  7  Figure 5:  Left: Tip of tweezers arm as seen from the facing arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The thin meander on the top is the  active heater (Pt) or thermometer (AuSn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The overlap with gold tracks is apparent on the lower part of  the picture Right: Close up picture by electronic microscopy of the AuSn thermometer showing its granular  aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Depending on tweezers models (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 8), the width of the AuSn track is 4, 11 or 24 µm  In order to obtain resistance values in this range, the meander length was fixed close to 700 squares for all  tip sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' According to the tip size, the track width was adapted so as the serpentine shape occupy the whole  available area on the tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' At ambient temperature, the AuSn layer resistance was found to drift from low values  to their final values during a few days (less than one week) after deposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' After this period, the resistance  was found to be stable at least over 6 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Figure 6 (right plot) shows a typical AuSn thermometer resistance R(T0) versus temperature T0 for different  direct current I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Regarding the temperature dependence of resistance, the current density is a more determining  parameter than the total current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Thus, the comparison between left and right plots of figure 6 should be done  at constant values of the ratio of current over track width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' At low current density (I � 10µA), the sensitivity  exceeds 1 Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='mK−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  At larger current densities, the current-induced magnetic field significantly shifts the  superconducting-metal transition to lower temperature and broaden it, allowing to extend the measurement  range down to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='6 K and below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  In the range of currents explored in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  6, the reduction of sensitivity  in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='K−1 at larger current I is more than compensated by the larger sensitivity in V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='K−1 units across the  thermistor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Most measurements presented hereafter are done with a polarization of I ≃ 27 µA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  Heating  The heater consists in a meander of metal deposited by lithographic technique on the tip of a cantilever, alike  the thermometer (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 5 (left)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The same lithographic mask was used, and therefore the meander length is  also close to 700 squares for all the tip sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  As can be seen on figures 4 and 5, a buffer zone was designed between the gold tracks and the meander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Into  this zone, the electrical path is wide but the material is the same as in the meander (platinum for heater).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The  buffer zone length is approximately 20 squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This design aims at providing some thermal isolation between  the meander and the gold track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Numerous resistive materials are suitable, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' chrome was used for the tweezers in [RDD+07] and platinum  in [WVR21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Present data have been obtained with platinum to benefit from the temperature-independence of  its resistivity at superfluid temperatures [PK82], and nevertheless to allow re-use of these miniature heaters as  miniature thermometers or hot-film anemometers in experiments conducted at higher temperatures where Pt  regains temperature dependence [Kem91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' A 5nm titanium layer was deposited prior to platinum as an adhesion  layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The thickness of the Pt layer, around 80 nm, was chosen such that the electrical resistance of the heater at  superfluid temperature is around a few hundreds of ohms, like for the thermometer maximum resistance and  for the same reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The heater is driven with a sinusoidal current at frequency f/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The resulting Joule effect can be decomposed  into a constant mean heating and the sinusoidal heat flux at the frequency f that drives the second sound  resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' A benefit of this f/2 excitation is that the signal monitored by the thermometer - centered around f  is not spoiled by spurious sub-harmonic electromagnetic coupling at f/2 from the excitation circuitry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Thus,  no special care is needed to minimize the electromagnetic cross-talk between the electrical tracks of the heater  and the electrical tracks of the thermometer, despite their proximity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The non-zero mean heating results is a steady thermal flux in He-II, the corresponding entropy being carried  8  10μm  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='6  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='8  1  Au 25nm + Sn 90nm  Au 25nm + Sn 100nm  Au 25nm + Sn 110nm  T  λ  transition under saturated vapor  R0  /  max(R0)  T0  (K)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='9  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  0  100  200  300  1  μ  A  d  c  + 1  μ  A  a  c  3  μ  A  d  c  + 1  μ  A  a  c  10  μ  A  d  c  + 1  μ  A  a  c  20  μ  A  d  c  + 1  μ  A  a  c  30  μ  A  d  c  + 1  μ  A  a  c  R0  (Ω)  T0  (K)  Figure 6:  Left: Example of temperature and current dependence of AuSn layers with different Sn thicknesses  deposited on a track of width 50µm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The current polarization spans from 1µA up to 1mA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Right: Tempera-  ture response of the AuSn track of small tweezers for different electrical currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The thickness of the tin layer  is 100 nm of Tin, and the width of the track is 4µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The small difference between both plots -when compared  at similar current densities- is compatible with the uncertainty on the layer thicknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This good agreement  indicates that AuSn properties are robust to the full fabrication process of the tweezers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' No significant aging of  AuSn properties has been noticed over a few years period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  away from the heater in the form of steady normal fluid flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This outgoing normal flow is balanced by an  opposite steady mass flow of superfluid towards the heater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Such cross-flows are referred to as counterflows in  the quantum fluid literature[Tou82, NF95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This steady counterflow adds up to a pure second sound generated  by the heater, but -contrary to it- its effects are not amplified by resonance in the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Quasi-linear vs non-linear regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The second sound resonators are operated with standing waves  of low amplitude, say ∼ 100 µK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  In this limit, the amplitude T of the temperature standing wave nearly  responds linearly to the heating power P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' For larger heating power, the ratio T/P decreases with P, which is  interpreted as the result of a turbulent transition within the tweezers which fuels a dense tangle of quantum  vortices dissipating the second sound wave3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The crossover from the quasi-linear to non-linear response of T(P)  is illustrated by figure 7 (left plot) for tweezers at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='6 K in the absence of external flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' In these conditions and  for these tweezers, the transition occurs around P ≃ 1 W/cm2, where P is the total Joule power normalized  by the heating surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' In other conditions, this transition was observed at smaller power densities, but no  systematic study was carried on the threshold value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3  Digression on the operation in the non-linear heating regime  The present study focuses on the linear regime of heating, but a short digression on higher powers allows  uncovering a noteworthy property of non-linear operation and incidentally backing-up the above interpretations  of the nature of the non-linear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Figure 7-right displays the amplitude of the (normalized) temperature  standing wave T/P versus P in flows of different mean velocities U and turbulence intensity of few percents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' In  the linear regime (P � 1W/cm2 in the conditions of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 7), the plateaus of T/P decreases when U increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Following the classical interpretation (see later), the standing wave T is damped by the vortices present in the  external flow, which concentration increases with U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Interestingly, in the non-linear regime (say for P � 2W/cm2  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 7), the dependence of T/P versus U is opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The interpretation is that the extra damping of the  3In one dataset, a close look at the quasi-linear region in quiescent superfluid around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5 K evidenced a small discontinuity of  T/P versus P dependence around P ⋆ ≃ 10−2W/cm2 (not shown), suggesting another flow transition, but this small effect was  not detectable in other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' A Reynolds number of this possible transition can be defined with the transverse characteristic  length scale L ≃ 1 mm, the quantum of circulation κ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='10−7m2/s and the counterflow superfluid velocity towards the  heater vs ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3 mm/s (amplification of velocity by the quality factor of the cavity has not been taken into account).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' One finds  Res = vsL/κ ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Critical Reynolds numbers Res of a few units have already been reported to characterize the threshold of  appearance of a few superfluid vortices across the section of pipes that are closed at one of their ends with a heating plug (see  fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 3 in [BLR17]), a transition referred as the T1-transition in the counterflow literature [Tou82, NF95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' By analogy, this could  suggest that the discontinuity at P ⋆ might be associated with the appearance of a sparse tangle of the quantum vortices near  the heater, which density is expected to increase for larger P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Such vortices would damp the standing wave, but no such effect  has been detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Indeed, first the observed damping of the standing wave in quiescent He-II can be accounted for by the sole  effect of diffraction (as shown later), which indicates that all the other sources of loss are comparatively small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Second, loss due to  such “counterflow” vortices would make the T(P) dependence sub-linear rather than linear, which is not clearly observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Since no  quantitative evidence of these vortices could be clearly identified, this effect was not further explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  9  10-1  100  P (W/cm2)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='8  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  T/P (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='u)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  P (W/cm2)  1  0  1  2  3  X (mK)  2  1  0  1  2  3  Y (mK)  10-1  100  P (W/cm2)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='8  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  T/P (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='u)  Overheating  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='8  1  U (m/s)  Figure 7:  Left: Normalized amplitude T of the temperature standing wave versus heating power P for  second sound tweezers in a quiescent He-II around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='6 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The transition around P = 1 W/cm2 is interpreted  by the development of a self-sustained vortex tangle within the probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The insert shows the amplitude of the  temperature standing waves in the complex plane for a subset of frequencies belonging to the same second sound  resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' In this representation, the extra-dissipation associated with the self-sustained vortex tangle results  in a curvature of the iso-frequency radial “lines” revealing the broadening of the resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Right: Same  quantity for second sound tweezers swept by turbulent flows of different mean velocities but similar turbulence  intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' In the linear regime P � 1 W/cm2, the velocity dependence is opposite to the one in the non-linear  regime, P � 2 W/cm2, demonstrating respectively vortex and velocity sensing by the probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  standing wave T now mostly results from the vortices generated within the tweezers by the heating itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This  vortex density decreases at larger U because vortices are more efficiently swept out of the tweezers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' In the  non-linear regime, the second-sound tweezers thus behave as a local anemometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' In the linear regime, we will  show that second-sound tweezers can not only behave as vortex probes -as illustrated by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='7-right but also as  anemometers through a mechanism discussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3  Microfabrication and assembling  Mechanical and electrical assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The distance between the plates is set by the spacer, composed of one or several micro-machined silicon  elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Additionally, two Kapton films with golden copper tracks are inserted in contact with the gold tracks  of the heater and thermometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The cantilevers, Kapton films and spacer elements are stacked on each other  as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The resulting assembly is clamped with a standard picture clip, downsized by electro-wire  erosion, and soldered to the head of a mounting screw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' An improvement compared to the clamping technique  introduced in [RDD+07] is the possibility to insert a temporary “joystick” through the whole assembly to allow  precise alignment (or offsetting) of the cavity plates under microscope (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2-a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  All the mechanical structures of cantilevers and spacers are made of silicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The cantilevers are fabricated  by processing SOI (Silicon On Insulator) substrates by microelectronic techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The substrates diameter  is 100mm, the silicon substrate, oxide and device thicknesses are respectively 500µm, 1µm and 20µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The  substrates are double side polished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The wafers are oxidized so as to form a 100nm thick SiO2 layer on both  sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Distinct wafers are used to fabricate heaters and thermometers but the process differs only in the metals  used for tips electrical paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' By using a circular symmetry design, 46 cantilevers are made per wafer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The cantilevers’ fabrication process is presented in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The serpentine electrical path (red color) is  deposited first on the SOI wafer frontside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The deposition is done using standard photolithography, evaporation  and lift-off sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The photoresist used is AZ 5214E from Microchemicals GmbH, processed as a negative  photoresist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Depending on the cantilever type, heater or thermometer, two different evaporation sequences are  used: Ti 5nm + Pt 80nm or Au 25nm + Sn 100nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Evaporation is preceded by in situ wafer surface cleaning by  an argon ions bombardment during 20s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Lift-off is initiated in an acetone bath during 5min and then completed  by ultrasounds during a few tens of seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The current leads (orange color) are deposited during a second  photolithography, evaporation and lift-off sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The evaporation sequence is Ti 5nm + Au 200nm + Ti  5nm + Pt 50nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The usage of a platinum layer was found to facilitate lift-off and may also be useful for brazing  purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' A thin protective resist layer is deposited on frontside in order to protect it during all subsequent  operations on the backside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  10  Figure 8: Overview of mask design (the disk diameter is 100mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Side  Step  Surface material  Design  Frontside  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Electrical circuitry fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  TiAuTiPt (orange)  AuSn/TiPt (red)  Backside  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Aluminum mask fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Aluminum  Backside  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Resist mask fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Etching of surface oxide and  silicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Resist removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Photoresist  Backside  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Etching of surface oxide and  silicon until buried oxide is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Etching of buried oxide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Aluminum removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Aluminum  Frontside  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Resist mask fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Etching of surface oxide and  silicon until opening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Photoresist  Table 1: Cantilevers fabrication process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  11  Phase  Deposition  Etch  Sub-phase  Main  Boost  Main  Gas  C4F8: 250 sccm  SF6: 250 sccm  O2: 45 sccm  SF6: 450 sccm  O2: 45 sccm  Duration  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2 s  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='0 s  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5 s  Pressure  14 mTor  20 mTor  75 mTor  Coil power  1200 W  1780 W  1780 W  Platen power  20 W  110 W  50 W  Electromagnet current  0 A  0 A  2 A  Platen frequency  RF 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='56 MHz  He backside pressure  10 Tor  Table 2: Bosch process recipe used during deep silicon etching on backside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The cantilevers 3D structuration starts with a backside deep etching of silicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Two superimposed etch  masks are fabricated first, one aluminum mask and one photoresist mask deposited over the aluminum one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The aluminum mask is made in the same way as the frontside electrical paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' A backside alignment is necessary  during photolithography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The aluminum thickness is 120nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The protective resist layer on frontside had to be  deposited again after lift-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The resist mask is made by photolithography on the positive AZ4562 photoresist  spin-coated at 4000rpm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The aluminum mask is identical to the resist mask except within the arm area, as  shown in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This area is covered by resist but not by aluminum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Thus, the resist fully covers aluminum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  After the fabrication of both masks, the deep etching is started, with the resist mask protecting both silicon  oxide and aluminum surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The surface oxide is etched first then the solid silicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The thin oxide layer is  etched by reactive ion etching (RIE) based on SF6 gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The silicon was etched in a STS HRM deep reactive  ion etching (DRIE) equipment using a recipe based on Bosch process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The recipe is presented on table 2, the  etch duration is 120 cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' As shown on table 1, a 200µm wide trench is dug from the backside around the  baseplate and arm areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The tip area is fully exposed to etching as this part was extracted from the device  layer of SOI only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The longer this phase, the thicker the cantilever arm at the end of process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Following this first etch phase, the resist is removed by an oxygen plasma and the backside surface oxide is  etched into the freshly uncovered arm area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Then, another silicon etch sequence is applied with the remaining  aluminum mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Its objective was to thin the arm and to reach the buried oxide of SOI in the trench, at the  same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The same recipe is used, 200 cycles are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The Bosch process is interrupted 3 times, every 50  cycles, in order to apply a 1min oxygen plasma followed by 20s of an isotropic silicon etching recipe (plasma  pressure 75mTor, SF6 flow 450sccm, O2 flow 45sccm, coil power 1780W, platen power 50W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The objective is  to reduce the parasitic effect generated by the passivation layer deposited on arm sidewalls during the first deep  silicon etching sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' As the arm area is masked during the first silicon etch sequence and unmasked during  the second one, the passivation layer located along the arm edges is released and could generate locally some  irregular micromasking effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The oxygen plasma is intended to help remove the floating passivation films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The  isotropic silicon etching is intended to cut the silicon pillars generated by micromasking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The final 50 cycles  of the Bosch process recipe end up reaching the SOI buried oxide layer, at the trench bottom, all around the  cantilever (however the oxide should not be reached into the arm area).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' It is necessary at this step to check that  the buried oxide is fully uncovered by silicon everywhere in the trench bottom and in the tip area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' However,  etching cycles should not be applied in excess so as to avoid some mechanical weakening of the wafer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' At this  step, the arm thickness has its final value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The buried oxide is then removed by plasma etching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This oxide  is fully removed at the trench bottom and in the tip area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' If this layer is not removed, some bending may  occur on the tip at the end of process, due to mechanical stress of oxide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The aluminum mask is removed in  an aluminum etchant solution at 50°C during a few minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The frontside protective resist layer is removed  during an acetone cleaning bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The 3D cantilever structuration is ended by a third silicon etching made from the frontside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The etch mask  is formed by photolithography on AZ1512HS resist, deposited at 4000rpm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' As shown on table 1, the mask  design includes two bridges on the baseplate sides in order to maintain the cantilever after having opened the  trench that surrounds it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The design also includes the tip contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The frontside thin surface oxide is etched  first, then the silicon of the device layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The silicon etching is done with specific conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Due to the wafer  mechanical weakness at this step, caused by the multiple deep trenches made on the backside, the processed  wafer is layed on and attached to a blank silicon wafer by Kapton tape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This ensemble is loaded into the etching  chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' As no thermal bridge is present between the two wafers, the recipe is adapted: low RF powers are  used and a 22s idle time is added after each etching cycle (see table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The objective is to avoid overheating  during etching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The silicon etch duration is 50 cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  After this sequence, the trench around cantilever is fully opened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The resist is removed by a low power oxygen  plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Some spacers are fabricated together with the cantilevers but most of them are made separately from  12  Phase  Deposition  Etch  Sub-phase  Main  Delay  Boost  Main  Gas  C4F8: 250 sccm  SF6: 250 sccm  O2: 10 sccm  Duration  3 s  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='0 s  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5 s  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5 s  Pressure  14 mTor  20 mTor  40 mTor  40 mTor  Coil power  300 W  300 W  300 W  1 W  Platen power  20 W  100 W  50 W  1 W  Electromagnet current  0 A  0 A  2 A  0 A  Platen frequency  RF 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='56 MHz  He backside pressure  10 Tor  Table 3: Bosch process recipe used during deep silicon etching on frontside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  two silicon wafers with thicknesses 300µm and 525µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' These wafers are covered by thin dielectric layers on  both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4  Electric circuit  Figure 9 shows a circuit used for time-resolved measurements with second-sound tweezers, with example values  of resistances and gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The time-resolved data presented in this paper have been obtained with such a circuit  and using the following equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The front-end of the preamplifier is the EPC1-B model by Celian or an SA-  400F3 model by NF when frequencies above 100 kHz are explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The lock-in amplifier is a LI-5640 by NF or  Model 7280 by SignalRecovery above 100 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' In most conditions, its built-in internal generator provides both  the drive of the tweezers heater (at frequency f/2) and the reference frequency to detect the temperature signal  (at frequency f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The acquisition system is built around PXI-4462 analog input cards by National Instrument,  and it records both the in-phase (X) and quadrature (Y ) signal from the analog outputs of the lock-in amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  In occasional conditions, the temperature signal at the lock-in input is buried in a much larger electro-  magnetic parasitic signal at f/2, and it cannot be properly resolved by the limited voltage dynamic range of  the lock-in amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This situation can occur when the tweezers are operated far from a resonance, where  the second sound signal is small, or when the tweezers are operated at very high frequency (say > 100 kHz),  as electromagnetic coupling increases with frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The magnitude of this parasitic coupling depends on  the geometrical and electrical specificities of the tweezers and cables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' In the range of parameters explored in  the present study, the order of magnitude of the parasitic voltage induced across the thermometer resistor,  normalized by the voltage applied across the heating resistor, is  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5% ×  f/2  100kHz  Such situations are handle thanks to the differential input of the lock-in amplifier, by removing a signal  mimicking the parasitic one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' In such cases, a two-channel waveform generator is used: one channel driving the  heater (at frequency f/2), another channel mimicking the parasitic signal (at frequency f/2, with manually  tuned amplitude and phase shift) and the "sync" output of the generator synchronizing the lock-in demodulation  (at frequency f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The 33612A generator by Agilent is used for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Alternatively, the compensation  signal can be generated directly from the lock-in internal generator, completed with a simple RC phase shifter  and eventually ratio transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  In principle, any positive temperature coefficient thermistor -like AuSn thermometer- that is not well ther-  malized with the fluid can become unstable when driven by a current source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Indeed, an infinitesimal thermistor  fluctuation from T0 to T0 + δT0 entails a resistance variation of δR = ∂R/∂T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='δT0 > 0, leading to an excess of  Joule dissipation δR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='I2 for a constant current drive I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Calling Rth is the thermal resistance of the thermistor-  fluid interface, this extra Joule dissipation results in an overheating Rth × δR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='I2 which could lead to a thermal  instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The stability condition is difficult to predict for spatially distributed thermistor deposited on a  Si crystal and immersed in superfluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Thus, first tests have been done with a voltage polarization, before  validating empirically the stability of our current polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The frequency bandwidth of the measurements is arbitrarily set by the integration time constant of the  lock-in amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' In practice, the performance of the circuit are limited by the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='65 nV/  √  Hz input voltage  noise of the EPC1-B pre-amplifier, all other sources of noise being smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' For a 27 µA current polarization, a  thermometer sensitivity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5 Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='mK−1, and a 10 Hz or 1000 Hz bandwidth of demodulation, the temperature  resolution Trms is :  Trms =  √  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5 × 27 µK ≃ 150nK for a 10 Hz measurement bandwidth  13  300 kΩ  acquisition  system  10 kΩ  400 Ω  Tweezers heater  Reference  clock  9V batteries  quadrature  phase  Optional noise compensation  Low noise AC preamp  (40 - 50 dB)  Lock-in amplifier  (at freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' f )  + Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' f/2, phase φ  r1  r2  Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' f/2  Tweezers thermometer  (0 - 300 Ω)  R(T0)  Current source (typ 30 µADC)  Figure 9: Example of circuitry for measurements with a high dynamical reserve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  or  Trms =  √  1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5 × 27  µK ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5µK for a 1 kHz measurement bandwidth  These resolutions are sufficient in standard conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Indeed, they are respectively 3 and 2 decades smaller  than the typical amplitude of a second sound at resonance, and reaching the same temperature resolution at  significantly larger bandwidth would be useless given the space-time resolution of the probe itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' If needed,  better resolution could nevertheless be achieved with a larger polarization across the thermometer or using a  cryogenic amplifier (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' see [DLF+14] and http://cryohemt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='com) before being limited by the thermal noise  floor of the thermistor (typ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='15 nV/  √  Hz for 200 Ω at 2 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  3  Models of second sound resonators  The second sound equations within the linear approximation can be written in terms of the temperature fluc-  tuations T and the velocity of the normal component vn as  ∂tvn + σρs  ρn  ∇T = 0,  (1)  ∂tT + σT0  cp  ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='vn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  with σ the entropy per unit of mass, cp the heat capacity, and ρs , ρn are the densities of the superfluid and  normal components respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' All along the present section, T0 is the notation for the bath mean temperature  whereas T denotes the temperature fluctuations, with ⟨T⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  We introduce the second sound velocity c2 defined by the relation  c2  2 = ρs  ρn  σ2T0  cp  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (2)  It can be deduced from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (1) that both the temperature T and the normal velocity vn follow the wave  equation  ∂2  t T − c2  2∆T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (3)  We explain in the present section how Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (1-2-3) can be used to build quantitative models of second sound  resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We first focus on phenomenological aspects in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Then, we give analytical approximations in  sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2 and an accurate numerical model in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Finally, we discuss the model quantitative predictions in  secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  14  Heater  Thermometer  z=0  z=D  Figure 10: Schematic representation of the second resonant mode of a second sound cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The red curve  displays the temperature field at time t = 0, and the blue curve displays the normal fluid velocity vn at time  t =  1  4f , where f is the excitation frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1  Resonant spectrum of second sound resonator: phenomenological aspects  The basic idea of second sound resonators is to create a second sound resonance between two parallel plates  facing each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' A second sound wave is excited with a first plate, while the magnitude and phase of the  temperature oscillation is recorded with the second plate used as a thermometer (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' For simplicity,  we assume from now on that the second sound wave is excited by a heating, but the whole discussion can be  straightforward extended to nucleopore mechanized resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The temperature oscillations within the cavity  are coupled to normal fluid velocity oscillations according to the second sound equations (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Both components  oscillate in quadrature, which means that they reach their maximal amplitude with a  1  4f time shift (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  We note jQ = j0e2iπft the periodic component of the heat flux emitted from the heater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  We assume  throughout the present article perfectly insulating plates, which means that the boundary conditions for the  second sound wave are  vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='n =  �  0  for z = D  jQ  ρσT0  for z = 0 ,  (4)  where n is the unit vector directed inward the cavity and normal to the plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The second equation in (4)  reflects the fact that the normal component carries all the entropy in the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 10,  the thermometer plate is a node of the normal velocity oscillation, whereas the normal velocity amplitude only  vanishes on the heater plate in quadrature (t =  1  4f + n  2f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' According to the first relation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (1), the boundary  conditions (4) for the normal velocity translate into the following boundary conditions for the temperature field  ∇T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='n =  �  0  for z = D  − ρn∂tjQ  ρρsσ2T0  for z = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (5)  In particular, the thermometer plate is an antinode for the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  We display in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 11 a typical experimental spectrum of second sound tweezers, that is, the temperature  magnitude averaged over the thermometer plate, as a function of the heating frequency f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The spectrum is  reminiscent of a Fabry–Perot resonator (described in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2): it displays very clear resonant peaks almost  equally separated, and a stable non-zero minimum at the non-resonant frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' However, the spectrum of  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 11 displays three major characteristics that can be observed for every tweezers spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' First, the location  of the resonant frequencies are slightly shifted compared to the standard values fn given by 2πfnD  c2  = nπ, (n ∈ N)  expected for an ideal Fabry–Perot resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Only for large mode numbers do the resonant peaks again coincide  with the expected values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Second, the temperature magnitude vanishes in the zero frequency limit, and the  first modes of the spectrum grow linearly with f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' In between, the resonant amplitudes saturate and then slowly  decrease at high frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  These latter peculiarities of the frequency response were not described in previous references about second  sound resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This prompted us to study different models for second sound resonators, including the finite  size effects and near field diffraction phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  We first describe analytical approximations in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2,  then we develop in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3 a numerical algorithm based on the exact solution of the wave equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The  numerical scheme can be adapted for various types of planar second sound resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We then give quantitative  predictions specifically for the response of second sound tweezers without and in the presence of a flow in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4 and a summary of the main results in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  Analytical approximations  The starting point to build our model of second sound tweezers is to assume that all zeroth order physical  effects observed with the tweezers are geometrical effects of diffraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  This means in particular that we  assume perfectly reflecting resonator plates, and we also neglect bulk attenuation of second sound waves when  15  0  10  20  30  40  50  60  70  f (kHz)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='8  1  T (K)  10-4  Linear growth of  the first modes  Stable baseline  Decrease at high  frequency  Figure 11: Experimental spectrum of second sound tweezers of lateral size L = 1 mm and gap D ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='435 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  f is the heating frequency, and T is the thermal wave magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The figure displays the main characteristics  of tweezers typical spectrum: first, the resonant frequencies are not located at the values fn given by 2πfnD  c2  =  nπ, (n ∈ N), displayed by the gray vertical lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The amplitudes of the resonant modes first increases linearly  with f until they saturate and eventually decrease at high frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The baseline level only weakly depends  on f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  the fluid is at rest [CR83, RG84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' These assumptions turn to be self-consistent, because the predictions of  the model developed in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3 reproduce the main features observed in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We thus start with  the simplest model of a resonant cavity for planar waves, namely the Fabry–Perot model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We then propose  variations of the Fabry–Perot model, taking progressively into account the particular resonator geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We  discuss the predictions of these models and their relevance for our second sound tweezers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The standard Fabry–Perot model  The Fabry–Perot model corresponds to a one-dimensional resonator  composed of two infinite parallel plates separated by a gap D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' In that case, the wave Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (3) together with the  boundary conditions Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (5) can be solved exactly, for a periodic heating jQ = j0e2iπft .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' However, as there is  no energy loss between two perfectly insulating and infinite plates, a bulk dissipation has to be introduced by  hand in the model, to balance energy injection from the heater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This can be done with an additional dissipation  coefficient ξ (in m−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The temperature at the thermometer plate is T(t) = Re  �  Te2iπft�  (where Re is the real  part operator) with the complex wave amplitude T given by  T =  A  sinh  �  i 2πfD  c2  + ξD  �,  (6)  with A = −  j0  ρcpc2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' An illustration of a Fabry–Perot spectrum is displayed in grey in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 14, with ξD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='15  and A = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We introduce the wave number k = 2πf  c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' It can be proved from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (6) that the spectrum maxima  are reached for the values knD = nπ, (n ∈ N), and correspond to constructive interferences in the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The  baseline level is set by the minima reached for the values knD = nπ + π  2 , (n ∈ N), and correspond to destructive  interferences in the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' For the simple Fabry–Perot model of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (6), all the resonant peaks have equal  height and are uniformly separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Therefore, some main features of experimental spectra are missing, an  indication that important other physical effects have to be included in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Second sound resonators embedded in infinite walls  A possible modification of the Fabry–Perot res-  onator is to consider finite-size heater and thermometer of size L embedded in two parallel and infinite walls  facing each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This geometry is most commonly encountered in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' With such a configuration,  16  the thermal wave is not a plane wave any more because it is emitted by a finite size heater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The model thus  contains diffraction effects, that the simplest Fabry–Perot resonator do not display.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' An illustration of the model  setup is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' An exact solution of the wave equation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (3) can be found using the technique  of image source points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Let Σ1 be the heating plate and Σ2 be the thermometer plate, and we assume that the  thermometer is sensitive to the average temperature over Σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Then the response of the tweezers is given by  T(t) = Re  �  Te2iπft�  with  T =  ikj0  2πρcpc2  1  L2  �  Σ2  d2r2  �  Σ1  d2r1G (r2 − r1) ,  with the Green function G(r) defined for every vector r in the (x, y) plane  G(r) = 2  +∞  �  n=0  1  |(2n + 1)Dez + r|e−ik|(2n+1)Dez+r|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Such a model correctly predicts that the tweezers spectrum vanishes when the heating frequency f goes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Yet, it does not reproduce the linear increase of the resonant magnitude of the first modes, neither the decrease  of the resonant peaks at large frequency observed in experiments with second sound tweezers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This means that  other effects have to be taken into account to model a fully-immersed open resonant cavity such as non-perfect  plates alignment and energy loss by diffraction outside the cavity when the latter is not embedded in infinite  walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Empirically modified Fabry–Perot model  Contrary to a Fabry–Perot resonator composed of infinite  plates, second-sound resonators are built with plates of finite size L, approximately of the same order as the  gap D between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Those finite size effects are important as they introduce a frequency-dependent energy  diffracted outside the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This mechanism is sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' According to standard diffraction theory,  a finite wave initially of size L with a wavelength λ = c2  f spreads with a typical opening angle given by λ  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' By  this geometrical effect, a part of the wave energy is lost as the wave reaches the other side of the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The  energy loss is roughly proportional to the surface of the wave cross-section that “misses” the reflector (see the  right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Therefore, the fraction of energy lost at the wave reflection is controlled by the ratio  �  L + 2λD  L  �2 − L2  �  L + 2λD  L  �2  ≈ 4λD  L2 ,  (7)  ≈  4  NF  ,  where we have introduced the Fresnel number NF = L2  λD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The tweezers plates are mounted at the top of arms of a few millimeters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The perfect parallelism of the plates  is usually not reached for our tweezers, but a small inclination γ of the order of a few degrees can be observed  instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' A relative inclination γ -even small- of both plates creates an additional energy loss mechanism (see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Intuitively, this second mechanism is controlled by the non-dimensional number  Ni = λ  γL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (8)  We assume that the Fabry–Perot model (6) can be corrected using the two non-dimensional numbers NF =  L2f  c2D in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (7) and Ni =  c2  γfL in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  More precisely, based on empirical observations, we find that  second-sound tweezers spectra can be accurately represented by the formula  T =  A  sinh  �  i  �  2πfD  c2  − a c2D  L2f  �  + b c2D  L2f + c  �  γfL  c2  �2�,  (9)  where a, b and c are fitting coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' As there is no other small parameter in the problem, a reasonable  assumption is to look for coefficients of order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We find that the values a ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='95, b ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='38 and c ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3 give  accurate spectra predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' An illustration of a modified Fabry–Perot spectrum with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (9) is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The linear amplitude growth of the first resonant peaks can be interpreted as a progressive focalization of  the wave, and is thus controlled by the Fresnel diffraction number NF in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The shift proportional to  1  f in peaks frequency positions, observed in the experimental spectra, is also controlled by NF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The decrease  in resonant magnitude for large mode numbers can be interpreted as a wave deflection outside the cavity, after  back and forth propagation between the plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This latter effect is controlled by the second non-dimensional  number Ni in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  17  L  L  D  L  Thermal wave  Figure 12: Representation of the wave dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The energy loss is controlled by the non-dimensional number  λ  L, according to standard diffraction theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' In this section, we discuss both the case of tweezers embedded in  walls, and the case of free tweezers in open space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Figure 13: Effect of the plates’ inclination γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Inclination creates an additional energy loss mechanism controlled  by the second non-dimensional number  λ  γL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  18  20  2  4  6  8  10  12  14  kD  0  1  2  3  4  5  6  7  T  inclination  effect  destructive interferences  focalization  of the wave  Figure 14: Analytical models of second-sound tweezers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The grey curve represents the standard Fabry-Perot  spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The blue curve represents the empirical correction of the Fabry-Perot formula using the two non-  dimensional numbers  λ  L and  λ  γL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The modified spectrum displays the characteristic features of a tweezers  experimental spectrum, as displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The major interest of the Fabry–Perot model is to offer an analytical expression to fit locally a resonant peak  of second sound resonator spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The local fit of a peak is of particular interest to interpret the experimental  data, as will be explained in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (9), given a measured resonant frequency f0, we will look for  a fitting expression  T =  A  sinh  �  i 2π(f−f0)D  c2  + ξ0D  �,  (10)  valid for second sound frequencies f close to f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' In that expression, ξ0 encapsulates the different geometrical  mechanisms responsible for energy loss when the fluid is at rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' A and ξ0 are thus fitting parameters that can  be found easily with the experimental data obtained by varying f in the vicinity of f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3  Numeric algorithm  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2 presents a class of models of increasing complexity, still with an analytical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Those models  show how all zeroth-order effects observed with second sound resonators can be recovered from geometrical  diffraction effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' However, they do not allow for quantitative predictions of the resonator spectra, nor do  they allow including the effects of a flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We develop in the present section a numerical algorithm, based on  the exact resolution of the wave equations with the particular tweezers geometry, with and without flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The  algorithm could be extended to any second sound resonator with a planar geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' As will become clear in  the following, this numerical model allows going far beyond the approximate models of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1  For a backgroud medium at rest  The aim of the present section is to build a numerical algorithm to solve the wave equation (3) for a periodic  heating jQ = j0e2iπft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We look for a solution with the ansatz T(r, t) = Re  �  T(r)e2iπft�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Then, the wave  equation for T is  ∆T + k2T = 0,  where we have introduced the wave number k = 2πf  c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The boundary conditions are  �  ∇T(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='n1 = − ikj0  ρcpc2  for r ∈ Σ1  ∇T(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='n2 = 0  for r ∈ Σ2  ,  (11)  where Σ1 is the heater plate and Σ2 the thermometer plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The notations are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We propose  the method described below, based on the Huyggens–Fresnel principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The principle states that every point  19  of the wave emitter can be considered as a point source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The linearity of the wave equation can then be used  to reconstruct the entire wave by summation of all point source contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The Huyggens–Fresnel principle  has been widely used in the context of electromagnetism, for example to compute diffraction patterns produced  by small apertures, or interference patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The major difficulty in the context of second sound tweezers is  that none of the standard approximations of electromagnetism can be done, neither the far-field approximation  nor the small wavelength approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This explains why numerical resolution is very useful in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  We neglect the tweezers arms, which means that both plates are considered as freestanding, infinitely thin  and perfectly insulating plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We allow a relative inclination γ around the x-axis and a possible relative lateral  shift Xsh of one plate with respect to the other along the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We assume that the thermometer is sensitive  to the temperature averaged over Σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Let us introduce the Green function  G(r) = 1  |r|e−ik|r|,  (12)  which is the fundamental solution of the wave equation  ∆G + k2G = 4πδ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (13)  Let Σ be one of our two square plates, and U(r′) be a smooth function defined over Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We introduce the wave  defined by  T(r) = −1  2π  �  Σ  G(r − r′)U(r′) d2r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (14)  By linearity, T is a solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (3), for all r /∈ Σ, because G is a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' A asymptotic calculation in the  vicinity of Σ then shows that T satisfies the boundary condition  ∇T(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='n  −→  r→r0∈Σ U(r0),  (15)  where n is the unit vector normal to Σ and directed inward the cavity (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We are going to use Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (14) and (15) as the two fundamental relations to build our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We will compute the solution of the  wave equation as an infinite summation of all the emitted and reflected waves in the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The first wave T 1 is emitted by the heating plate Σ1 and satisfies the first relation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (11)  ∇T 1(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='n1 = − ikj0  ρcpc2  for r ∈ Σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Given Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (14) and (15), it is clear that the first wave is given by  T 1(r) =  ikj0  2πρcpc2  �  Σ1  G(r − r′) d2r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (16)  Then each time a wave denoted T n hits a plate Σ (Σ1 or Σ2), it produces a reflected wave T n+1 to satisfy the  boundary condition  ∇  �  T n(r) + T n+1(r)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (17)  The situation is sketched in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' If we choose for T n+1 the expression  T n+1(r) = 1  2π  �  Σ  G(r − r′)  �  ∇T n(r′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='n  �  d2r′,  (18)  then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (15) shows that Tn+1 satisfies the boundary condition  ∇T n+1(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='n  −→  r→r0∈Σ −∇T n(r0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='n,  which is exactly Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (16) and (18) define our recursive algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (18) shows that the reflected  wave is generated by the gradient of the incident wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Practically, the recursive computation of all forth and  back reflected waves thus requires at each step n the computation of ∇T n only on the plates, rather than T n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  For a reflection at (say) Σ1, we have  ∇T n+1(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='n2 = −1  2π  �  Σ1  G(r − r1)  �  1  |r − r1| + ik  �  n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' r − r1  |r − r1|  �  ∇T n(r1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='n1  �  d2r1,  (19)  20  D  L  Thermometer  L  n2  Heater  heat flux  n1  x  y  z  Xsh/2  Xsh/2  n  Figure 15: Left: Geometrical setup of the numerical algorithm and notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Right: Representation of an  incoming and outcoming wave at the nth reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The solution of the wave equation is finally given by the superposition of all waves T n, that is  T(r) =  +∞  �  n=1  T n(r),  = 1  2π  �  Σ1  G(r − r1)  +∞  �  n=0  �  ∇T 2n+1(r1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='n1  �  d2r1  + 1  2π  �  Σ2  G(r − r2)  +∞  �  n=1  �  ∇T 2n(r2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='n2  �  d2r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  and the thermometer response is given by  �  T  �  Σ2 = 1  L2  �  Σ2  T(r) d2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  A simulation of the temperature field at t = 0 of the 5th resonant mode of second sound tweezers with aspect  ratio L  D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4, without lateral shift nor inclination of the plates, is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' It can be clearly seen  in particular that the amplitude of the temperature field decreases along the z-axis contrary to a Fabry–Perot  resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This symmetry breaking is due to the diffraction effects associated with the finite size of the plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  A bulk dissipation can be included in the algorithm, for example, to account for quantum vortex lines inside  the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' In that case, let ξ be the second-sound attenuation coefficient (in m−1), the wave number k = 2πf  c2  of the Green function (12) should be replaced by  k = 2πf  c2  − iξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (20)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  In the presence of a turbulent flow  One of the aims of second-sound resonator modelling is to understand their response in the present of a flow  U sweeping the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' One effect of the flow is to advect the second sound wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' In the present section, we  explain how the algorithm of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1 should be modified to account for this effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We assume in the following  that the inequality |U| < c2 is strictly satisfied, which means that the flow is not supersonic for second sound  waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  In the presence of a non-zero flow U, the Green function (12) becomes  G(r, t) =  e−2iπft∗  |r − Ut∗|  �  1 + U  c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' r−Ut∗  |r−Ut∗|  �,  (21)  21  2mTn+1Figure 16:  Left: Temperature field fluctuations of the 5th resonant mode of second sound tweezers with aspect  ratio L  D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4, and heating power jQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='185 W/cm2, at the bath temperature T0 = 2K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Right: Temperature  field fluctuations for the same conditions with an additional flow of velocity U  c2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='18 directed upward .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The  nodes of the temperature standing wave correspond to antinodes of sound sound velocity, and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  where t∗ is the time shift corresponding to the signal propagation from the source  |r − Ut∗| = c2t∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (22)  In practice, the flow velocity range reached in quantum turbulence experiments is most often much lower than  the second sound velocity, with |U| hardly reaching a few m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Most experiments are done in the temperature  range where 10 < c2 < 20 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We thus introduce the small parameter β = |U|  c2 ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Similarly to the standard  approximations of electromagnetism, we assume that the effect of β is mostly concentrated in the phase shift  e−2iπft∗ of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We use the approximation |r − Ut∗|  �  1 + U  c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' r−Ut∗  |r−Ut∗|  �  ≈ |r|, and we solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (22) to  obtain t∗ to leading order in β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The Green function then becomes  G(r, t) = e−ik|r|Γ(r,U)  |r|  ,  (23)  where as previously k = 2πf  c2  and  Γ (r, U) = 1 − U  c2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' r  |r|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The algorithm detailed in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1 can be applied straightforward with the Green function Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' In  particular, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (19) becomes  ∇T n+1(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='n2 = −1  2π  �  Σ1  G (r − r1, U)  ��  1  |r − r1| + ik  � r − r1  |r − r1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='n2 − ik U  c2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='n2  � �  ∇T n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='n1  �  (r1) d2r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (24)  A simulation of the temperature field at t = 0 of the 5th resonant mode of second sound tweezers with aspect  ratio  L  D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4, without lateral shift nor inclination, and with a flow of velocity  U  c2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='18, is displayed in the  right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The effect of the flow can be clearly seen with the upward distortion of the antinodes of  the wave, compared to the reference temperature profile without flow displayed in the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4  Quantitative predictions  We present in this section the quantitative results obtained with the algorithm of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The algorithm is  specifically run in the configuration of second sound tweezers, but most predictions are relevant for other types  22  Thermometer  Thermometer  Heater  Heater  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='15-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='15 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='15  0  T (mK)  T (mK)of second sound resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We first show that the algorithm can quantitatively account for the experimental  spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We then use it to predict the response in the presence of a flow and a bulk dissipation in the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The predictions are systematically compared to experimental results for second sound tweezers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We eventually  display some experimental observations that illustrate the limits of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1  Spectral response of second sound resonators  Given a resonator lateral size L, the model of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3 has three geometrical parameters : the gap D, the  inclination γ and the lateral shift Xsh (see notations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We first sketch qualitatively the importance  of those three parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The gap D is the main parameter: it sets the location of the resonant frequencies, and the quality factor of  the resonances at low mode numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' For second sound tweezers, the value of D can be usually obtained within  a precision of a few micrometers (D is of the order of 1 millimeter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The relative inclination of the plates γ is  responsible for the saturation of the resonant magnitude and its decrease at large mode numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' It is typically  smaller than a few degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Contrary to the gap, only the order of magnitude of γ, not its precise value, can  be determined from the tweezers spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The lateral shift Xsh has very little impact on the spectrum if the  value Xsh  L  remains small enough (we can typically reach Xsh  L  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1 in the tweezers fabrication).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' However, the  effect of this parameter is of paramount importance to understand open cavity resonators response in a flow  (such as second sound tweezers), and will be investigated in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We consider the case Xsh = 0 in the  present section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The tweezers size L is known from the probe fabrication process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The method goes as follows: we first find a gap rough estimation �D, for example from the average spacing  between the experimental resonant peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Then we can run a simulation for parallel plates (γ = 0), unit gap  D = 1, and aspect ratio L  �  D, in the range 0 < k∗ < nπ (where n is the number of modes to be fitted, and k∗ = kD  is the non-dimensional wave number).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This gives a function fL/ �  D(k∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The experimental spectrum can then  be fitted with the function T(f) = AfL/ �  D( 2πfD  c2  ), where A and D are the two free parameters to be fitted,  provided the experimental value of c2 is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The high sensitivity of the location of the resonant frequencies  makes this method very accurate to obtain the gap D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Once D has been found, new simulations have to be run to find the order of magnitude of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  As was  previously said, γ controls the saturation and the decrease of the resonant magnitudes for large mode numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Its value can thus be approximated from a fit of the resonant modes with the largest magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' A fit of an  experimental tweezers spectrum is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The values of the fitting parameters for this spectrum  are D = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='435 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='003 mm and γ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Given the simplicity of the model assumptions, in particular  the assumptions of perfectly insulating and infinitely thin plates without support arms, the agreement with  experimental results is very good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Interestingly, the resonators can also be used in some conditions as thermometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Once the gap D is known  with high enough precision, the spectrum can be fitted using c2 as a fitting parameter instead of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Away from  the second sound plateau of the curve c2(T0) located around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='65 K, the value of c2 obtained from the spectrum  gives access to the average temperature with a typical accuracy of one mK, simply by inverting the function  c2(T0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  Response with a flow  Once the characteristics of the resonator have been determined with a background medium at rest, their response  in a flow can be studied using the modified algorithm presented in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We experimentally observe that  the tweezers response is attenuated in the presence of a superfluid helium flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This attenuation is related to  two physical mechanisms, illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 18: first, the thermal wave crossing the cavity is damped by the  quantum vortices carried by the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This type of damping is usually considered as being proportional to the  density of quantum vortex lines between the plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Second, the flow mean velocity is responsible for a ballistic  advection of the thermal wave outside the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The thermal wave emitted by the heater partly “misses” the  thermometer plate, and, even if the wave is not attenuated, a decrease of the tweezers’ response will be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Both mechanisms described above exist in experimental superfluid flows, and cannot be observed independently:  once there is a superfluid flow, quantum vortices are created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' One key objective is to be able to separate the  attenuation of the experimental signal due to bulk attenuation inside the cavity, from the attenuation due to  ballistic advection of the wave outside the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We will introduce a mathematical procedure to perform such  a separation on a fluctuating signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  What cannot be experimentally achieved can be simulated with the tweezers model developed in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The bulk dissipation can be implemented in the algorithm with a wave number complex part ξ (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (20)),  and the flow ballistic deflection can be implemented with a non-zero velocity U (see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (23-24)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Both effects  can be independently studied, setting alternatively ξ or U to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We first detail below the respective effects  23  0  10  20  30  40  50  60  70  f (kHz)  0  1  2  3  4  5  6  7  8  9  T (K)  10-5  Experimental data  Numerical simulation  Figure 17: Prediction of the second sound tweezers experimental spectrum of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  11 using the numerical  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The fitting parameters are the gap D, the inclination γ and the total heating power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Heater  Thermometer  Bulk  dissipation  Heater  Thermometer  Flow  Figure 18: Schematic representation of the two attenuation mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The top panel illustrates a bulk  dissipation of the wave, due for example to the presence of quantum vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The bottom panel illustrates the  ballistic deflection of the wave by a flow directed parallel to the plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  24  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='6  kD  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  3  T  0  1  2  X  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  Y  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  Figure 19: Numerical simulation: collapse of a resonance due to increasing values of the bulk attenuation ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The left panel display the magnitude of the thermal wave as a function of the wavevector k, and the right panel  display the same resonance in the phase-quadrature plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' It can be seen that the bulk attenuation results in  an homothetic collapse of the resonance, that means, without global phase shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' For a given value of k, the  model predicts that attenuation is directed toward the center of the resonant Kennelly circle (red curves of right  panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  of ξ and U for perfectly aligned plates (Xsh = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 19 display the result of a numerical simulation for second sound tweezers of aspect ratio L/D = 1, γ = 0  and increasing values of bulk dissipation in the range 0 < ξD < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The left panel display the magnitude of the  second resonant mode as a function of the wave number, and the right panel display the same resonant mode in  the phase-quadrature plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' More precisely, if we call T(k) the thermal wave magnitude recorded by the ther-  mometer, and ϕ(k) its phase, the right panel display the curve Y (k) = T(k) sin(ϕ(k)), X(k) = T(k) cos(ϕ(k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The resonant curve (Y (k), X(k)) is called in the following the resonant “Kennelly circle”, because the curve is  very close to a circle crossing the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' It can even be shown that the resonant curve becomes closer to a  perfect circle for increasing resonant quality factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The major characteristic to be observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 19 is that  the collapse of the resonant Kennelly circle due to bulk attenuation is homothetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' It means that the different  curves have no relative phase shift between each other, when the bulk attenuation increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The red curves  in the right panel display the displacement in the phase-quadrature plane for a fixed value of the wavevector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The model predicts that the displacement is directed toward the Kennelly circle center, which implies that  the path at fixed wavevector approximately follows a straight line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' By comparison, the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 21  display an experimental resonance in the phase-quadrature plane, for second sound tweezers of size L = 1 mm in  superfluid Helium at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='65 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The global orientation of the resonant Kennelly circles is simply due to a uniform  phase shift introduced by the measurement devices, and should be overlooked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' It can be seen that the resonance  collapse with increasing values of the flow velocity follows the predictions of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 19: it is homothetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The  red paths correspond to the tweezers signal at fixed heating frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Those paths follow approximately a  straight line directed to the Kennelly circle center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The slight deviation in the path orientation compared to  the predictions of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='19 can be explained by a second sound velocity reduction and will be discussed in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 20 display the result of a numerical simulation for second sound tweezers of aspect ratio L/D = 1,  γ = 0, with ξ = 0 and a flow mean velocity 0 <  U  c2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' As there is no tweezers lateral shift Xsh = 0,  negative velocities would lead to the same result from symmetry considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The figure illustrates the effect  of pure ballistic advection on a resonance in the phase-quadrature plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' First, it can be seen that the collapse  of the resonant Kennelly circle is accompanied by a relative anti-clockwise phase shift of the curves when the  velocity increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Also, the displacement of the tweezers signal at fixed wavenumber follow the red straight  paths directed anti-clockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This type of signal strongly contrasts with the one of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 19 obtained for a  pure bulk attenuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The prediction of the left panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 20 cannot be directly compared to experiments  because, as stated before, a superfluid flow always carries quantum vortices that overwhelm the tweezers signal  for tweezers satisfying Xsh ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  25  0  1  2  X  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  Y  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  X  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  1  Y  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='05  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  U/c2  Increasing U  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  U/c2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  s  s  Figure 20: Numerical simulation: collapse of a resonance due to ballistic deflection of the thermal wave  in the presence of a flow of velocity U, without bulk attenuation (ξ=0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The left panel display the result for  tweezers without lateral shift (Xsh = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Contrary to the results of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 19, it can be seen in the present case  that the collapse is associated with a global anti-clockwise phase shift of the resonant Kennelly circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Each  red curve represents the attenuation at a given value of the wavevector k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The right panel display the result  for tweezers with a strong lateral shift Xsh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5 × L (where L is the tweezers size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Such tweezers are very  sensitive to the velocity U, with both an attenuation of the resonance and a strong clockwise angular shift of  the Kennelly circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We note s the curvilinear abscissa of the curve obtained at a given value of k (red curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The inset displays the function s(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This shows that, once calibrated, second-sound tweezers can be used as  anemometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3  Effect of lateral shift of the emitter and receiver plates  We discuss in this section the consequences of a lateral shift, that is Xsh ̸= 0 with the notations of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Contrary to the previous sections, the present discussion is restricted to second sound tweezers, for which a  lateral shift has major quantitative effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' A lateral shift would not be as important, for example in the case  of wall embedded resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The lateral shift has a marginal effect on the tweezers spectrum when the background fluid is at rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' An  effect only appears in the presence of a nonzero velocity specifically oriented in the shifting direction U = Uex,  because of the mechanism of ballistic advection of the thermal wave by the flow (see the representation of the  mechanism in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The importance of this effect depends on the tweezers aspect ratio, on the reduced  velocity β = U  c2 , and on the lateral shift Xsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The lateral shift in the plates’ positioning magnifies the signal  component related to ballistic advection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This property opens the opportunity to build second sound tweezers  for which ballistic advection of the wave completely overwhelms bulk attenuation from the quantum vortices,  which means that the tweezers signal is in fact a measure of the velocity component in the shifting direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  We illustrate this mechanism in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The right panel displays a numerical simulation of a tweezers resonant mode in the phase-quadrature plane,  for the parameters L  D = 1, γ = 0 and Xsh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5, for positive and negative values of the flow velocity in the range  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2 < U  c2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' As can be seen at the first sight, the deformation of the resonant curve - that we equivalently  call the “Kennelly circle” - is very different from a deformation due to a bulk attenuation (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' First,  we observe that the deformation can result in an increase of the magnitude of the thermometer signal, when  the velocity is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This can be explained in this configuration, because the thermal wave emitted by the  heating plate is redirected toward the thermometer plate: less energy is scattered outside the cavity when the  wave is first emitted by the heater, and the signal magnitude increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' On contrary, the signal magnitude  decreases when the velocity is positive because the flow advects the emitted thermal wave further away from the  thermometer plate and more energy is scattered outside the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Second, the deformation of the Kennelly  circle is associated to a global clockwise rotation, a phenomenon that is not observed for bulk attenuation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Coming back to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 20, the red curve displays the displacement in the phase-quadrature plane for a fixed  wave frequency value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The displacement follows a very characteristic curved path always directed clockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Let s(U) be the curvilinear abscissa of the red path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Once calibrated, the value of s can be used as a measure  of the flow velocity component in the ex direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 21 displays the experimental signal observed with second sound tweezers of size  26  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='15  X (mK)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='05  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='15  Y (mK)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='8  1  U (m/s)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  X (mK)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4  Y (mK)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='8  1  U (m/s)  Figure 21: Experiment: Collapse of a second sound resonance for increasing values of the flow mean velocity  U, in superfluid Helium at T0 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='65 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The right panel display the result for tweezers of size L = 1 mm and  minor lateral shift Xsh < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1×L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The figure shows a homothetic collapse of the resonant Kennelly circle without  global phase shift, as predicted by the model of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The red curves display the displacement in the phase-  quadrature plane at a fixed value of the second sound frequency f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The right panel displays the experimental  data obtained with shifted second sound tweezers with parameters L = 250 µm and Xsh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5 × L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The  figure qualitatively confirms the clockwise angular shift with increasing values of U, predicted by the numerical  simulations of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  L = 250 µm, D = 431 µm and Xsh ≈ 125 µm, for a positive velocity range 0 < U < 1 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The main  characteristics of a ballistic advection signal can be observed: the Kennelly circle are attenuated with a clear  clockwise rotation, and the signal at fixed frequency follows a curved path in the clockwise direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This is a  strong indication that those type of tweezers can be used as anemometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The signal fluctuations of those type  of tweezers were recently characterized in a turbulent flow of superfluid helium [WVR21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' It has been shown in  particular that both the signal spectra and its probability distributions indeed display all the characteristics of  that of turbulent velocity fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4  Limits of the model  Although the model of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3 gives excellent experimental predictions, we still observe some unexpected  phenomena with real second sound tweezers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We discuss two of them in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  We have seen in secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2 that the thermal wave complex amplitude T(f) can be represented in the phase-  quadrature plane by a curve (X(f), Y (f)) very close to a circle crossing the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This osculating circle will  be called in the following the resonant “Kennelly circle”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The wave is damped in the presence of a superfluid  flow, which can be seen in the phase-quadrature plane as a homothetic shrink of the Kennelly circle toward the  origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 22 displays an experimental resonance in the phase-quadrature plane, for U = 0 m/s and U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='7  m/s, together with the fitted Kennelly circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' As can be seen in the figure, the resonant curve at U = 0 has  periodic oscillations in and out the Kennelly circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We call this phenomenon the “daisy effect”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The daisy  effect progressively disappears for increasing values of U, and cannot be seen any more on the resonant curve at  U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='7 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We interpret the daisy effect as a secondary resonance in the experimental setup with a typical  acoustic path of a few centimeters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We assume that the flow kicks out the thermal wave from this secondary  resonant path when U is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The daisy effect alters the attenuation measurements close to U = 0, and  should be considered with care before assessing the vortex line densities for low mean velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  It has been shown in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2 that the displacement of the tweezers signal in the phase quadrature plane  for a fixed wave frequency, follows a straight line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  We call “attenuation axis” the direction of this straight  path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The model predicts that the attenuation axis should always be directed toward the center of the resonant  Kennelly circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 23 displays a zoom on a part of the Kennelly circle at U = 0, together with the signal  displacement at fixed frequency and for increasing flow velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' It can be seen that the displacement is indeed  a straight line, but not exactly directed toward the Kennelly circle center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' An angle between 20° and 30° is  systematically observed between the attenuation axis and the circle center direction (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Moreover,  the angle is always positive (with the figure convention) and cannot be interpreted as a ballistic advection,  27  8  7  6  5  4  3  2  1  0  1  X  10-7  3  2  1  0  1  2  3  4  Y  10-7  U=0 m/s  U=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='7 m/s  Figure 22: Experimental resonance obtained with a second sound tweezers at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='98 K, for two values of the He  flow mean velocity U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The blue curve displays a periodic perturbation of the resonance that we refer to as the  “daisy effect”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The circle is a fit of the Kennelly osculating circle for this resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This effect is not predicted  by our model, and we interpret it as a secondary resonance in the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The daisy effect perturbs  the measurements at low values of U, but it can be seen on the red curve that the effect disappears for higher  values of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  that would give a negative angle instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This effect is thus very likely been attributed to a decrease of the  second sound velocity in the presence of the quantum vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Whereas a second sound velocity reduction has  previously been observed in the presence of quantum vortices [LV74, Meh74, MLM78], the exact value of this  reduction turns to be difficult to assess in particular experimental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We therefore keep the second  sound velocity reduction as a qualitative explanation, and we do not try to assess quantitative result from the  attenuation axis angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  Quantum vortex or velocity measurements ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Let us summarize the discussion of sec 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We have shown that second sound resonators are sensitive to two  physical mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The first one is the thermal wave bulk attenuation inside the tweezers cavity, due to  the quantum vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The second one is thermal wave ballistic advection perpendicular to the plates4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Both  mechanisms exist for all the second sound resonators, but depending on their geometry, they can preferentially  be sensitive to the one or the other mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We call selectivity the fraction of the signal due to quantum  vortices or to ballistic advection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Let T (ξ, U) be the probe signal as a function of the bulk attenuation coefficient  ξ (m−1) and flow velocity U(m/s), we define the vortex selectivity as  Rξ =  ��T (ξ, 0) − T (0, 0)  ��  ��T (ξ, U) − T (0, 0)  ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (25)  and by symmetry we define the velocity selectivity as  RU =  ��T (0, U) − T (0, 0)  ��  ��T (ξ, U) − T (0, 0)  ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (26)  Further investigations in second sound tweezers experiments have shown that the velocity/vortex selectivity  process only weakly depends on the aspect ratio  L  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Indeed, for a given resonator lateral size L, ballistic  advection of the wave outside the cavity increases when the gap D increases, but the number of quantum vortex  lines inside the cavity also increases linearly with D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Altogether, both the ballistic advection and the bulk  attenuation due to the quantum vortices have similar dependence with D, that’s why changing the gap has  no significant effect on selectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' For second sound tweezers, we observe that the selectivity neither depends  strongly on the mean temperature (that controls the superfluid fraction and the second sound velocity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  4Advection of second sound by velocity is illustrated e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' in [DL77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  28  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  X  10-4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  Y  10-4  U=0 m/s  0<U<1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='25 m/s  Attenuation axis  at fixed frequency  Kenelly  circle center  Figure 23: Attenuation of a resonance in the presence of a flow mean velocity U, obtained with second sound  tweezers in superfluid Helium at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='65 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The blue curve is the resonance at U = 0 in the phase-quadrature  plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Contrary to the prediction of the model of section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4 (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 19), the attenuation signal at fixed  heating frequency is not directed exactly toward the centre of the Kennelly osculating circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We observe a  systematic clockwise angular shift 20◦ < θ < 30◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We still have no definite explanation for this observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  As explained in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2, the velocity selectivity is more important for open cavity resonators such as second  sound tweezers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' For a given tweezers size, we find that the selectivity depends mainly on the shift Xsh and on  the wave mode number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Perfectly aligned tweezers excited with low mode numbers are preferentially sensitive  to quantum vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Increasing Xsh or choosing larger mode numbers leads to a larger velocity sensitivity and  changes the signal balance from vortex selectivity to velocity selectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The tweezers selectivity also strongly  depends on the size L: smaller tweezers can encompass less quantum vortices in the cavity, and are thus less  sensitive to quantum vortices, while the velocity sensitivity remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The velocity selectivity is thus  larger when the tweezers are smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 24 displays the selectivity of two second sound tweezers of size  L = 250 µm and L = 1 mm respectively, depending on the shift Xsh and the mode number n (where k = nπ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The simulation was run with a quantum vortex line density L = 2 × 1010 m−2 and U = 1 m/s, in accordance  with the typical values observed in the experiments of [WVR21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' It can be seen that large tweezers (L = 1 mm)  can reach a vortex selectivity Rξ > 90% for a small shift and low mode number, which means that they can be  used for direct quantum vortex measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' On contrary, small tweezers (L = 250 µm) can reach a velocity  selectivity RU > 90% for large shift or high mode number, and can thus be used as anemometers, as confirmed  by the experiments reported in [WVR21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  4  Measurements with second sound tweezers  Second sound tweezers are singular sensors in the sense that they can measure two degrees of freedom at the  same time, whereas most of hydrodynamics sensors only measure one (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Pitot tubes, Cantilevers, Hot wires).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The tweezers record the magnitude and phase of the thermal wave averaged over the thermometer plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Both  quantities contain physical information about the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' To summarize it shortly, magnitude variations give  information about quantum vortices in the cavity, whereas phase variations give information about the local  mean temperature and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The local mean velocity has an impact on both magnitude and phase, and  will be specifically treated in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The aim of the following sections is to explain how properly separate  quantum vortices signal from other signal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  In the following, we call L⊥ the density of projected quantum vortex lines density (projected VLD)  L⊥ = 1  V  �  V  sin2 θ(l)dl,  (27)  where V is the tweezers cavity volume, l is the curvilinear abscissa along the vortex lines inside the cavity ,  θ(l) is the angle between the quantum vortex line and the direction perpendicular to the plates (vector ez).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  29  Figure 24: Selectivity to quantum vortices or velocity advection for two second sound tweezers, obtained with  numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The color code indicates the fraction of the signal due to bulk attenuation by quantum  vortices (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The selectivity of the tweezers mainly depends on the lateral shift of both plates one  from another, and the resonant mode number excited in the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The left panel shows that large tweezers  (L = 1mm) are mainly sensitive to quantum vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Almost pure quantum vortex signal can be achieved  with carefully aligned tweezers excited at low mode numbers (Rξ > 90%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' On the reverse, small tweezers  (L = 250µm) are mainly sensitive to the velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Almost pure velocity signal can be achieved by shifting the  heater and the thermometer plates and by exciting the cavity at large mode numbers (RU > 90%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The present  simulation was run with a vortex line density L = 2 × 1010 m−2 and U = 1 m/s, in accordance with the typical  values observed in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Assuming isotropy of the vortex tangle, the total quantum vortex lines density (VLD) is  L = 3  2L⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (28)  A second sound wave is damped in the presence of a tangle of quantum vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Let ξV LD (in m−1) be  the bulk attenuation coefficient of second sound waves, it has been found[HV56a, HV56b, Tsa62, SP66, MPS84]  that ξV LD is proportional to L⊥ according to the relation  ξV LD = BκL⊥  4c2  ,  (29)  where B is the first Vinen coefficient and κ ≈ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='98 × 10−8 m2/s (for 4He) is the quantum of circulation around  one vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Therefore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (29) shows that a measure of the bulk attenuation coefficient gives access to the projected  VLD defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We recall in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1 the standard methods to measure the bulk attenuation coefficient  from a second sound resonance, and we propose in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3 a new method called “the elliptic method”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We give  in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4 some examples to apply the elliptic method to the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1  The vortex line density from the attenuation coefficient  We assume that a single second sound resonance can be accurately represented by the following expression (see  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (10))  T(f) = T 0  sinh (ξ0D)  sinh  �  i 2π(f−f0)D  c2  + (ξ0 + ξV LD)D  �,  (30)  where f0 is the second sound frequency of the local amplitude maximum, D is the resonator gap, c2 is the  second sound velocity, ξ0 is the attenuation coefficient without flow and ξV LD is the additional bulk attenuation  in the presence of quantum vortices given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' ξV LD = 0 without flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  A standard method to measure ξV LD goes as follows: we fix the second sound frequency at the resonant  value f0, and we measure the thermal wave amplitude with, and without flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (30) then shows that ξV LD  is given by  ξV LD = 1  Dasinh  � T 0  T(f0) sinh (ξ0D)  �  − ξ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (31)  30  26p-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3  shift  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='25ity / vortex sensitivity -→0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='15  Vortex se  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='05  0  1234567  modeK velocity sensitiv  ectivity > 90%  89101112131415  number0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='45  Velocity se  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3  shift  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='25ty / vortex sensitivity →  electivity > 90%0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='05  0  2345678  modeK velocity sensitiv  910111213141516  numberFigure 25: Spectral response of tweezers in the SHREK facility, right below the superfluid transition temperature  Tλ, where second sound velocity c2 is very sensitive to temperature (here D ≃ 500 µm and c2 drifts around a  mean value of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4 m/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The frequency axis is adimensionalize using the second sound velocity c2 calculated  from the temperature recorded near the sidewall of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' While the temperature of the bath is regulated,  frequency sweeps are repeated half a dozen of times for different turbulent flow conditions flagged by colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  A systematic drift of resonance frequencies versus flow conditions is observed ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' it is interpreted as an under-  estimation of temperature, and therefore over-estimate of c2, due to turbulent dissipation in the core of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  At a given mean flow, some scatter of the resonance frequencies is apparent ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' it is interpreted as noise from  the temperature regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The elliptic method introduced in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3 allows separating such temperature  artifacts from the attenuation due to second sound attenuation by quantum vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (31) shows that beside the value of D , that can be determined from a fit of the tweezers spectrum (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4), the value of ξ0 has to be accurately measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This is usually done by the measurement of the resonant  half width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' With some algebra manipulations, it can be found from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (30) that the resonant magnitude  satisfies  ��T  ��2 =  ��T 0  ��2  sinh2 (ξ0D)  sinh2 (ξ0D) + sin2 �  2π(f−f0)D  c2  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (32)  Let ∆f be the frequency half-width defined by the relation  ���T(f0 ± ∆f  2 )  ���  2  = 1  2  ��T 0  ��2, it can be shown from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (32) that ξ0 and ∆f are related by  sin  �π∆fD  c2  �  = sinh (ξ0D) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (33)  We note in particular that the relation (33) can be used to find ξ0 as long as the resonance quality factor is  high enough, that means, for sinh (ξ0D) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The linear approximations of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (31) and (33) are usually used  when ξ0D ≪ 1, and they give the well-know approximation:  L⊥ ≃ 4π∆f  Bκ  � T 0  T(f0) − 1  �  ,  (34)  For low quality factor resonances, another method should be used instead of the resonant half width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The  elliptic method presented in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3 allows determining ξ0 for resonances of any quality factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The main problem of the method presented above is that it implicitly assumes that there is no variation  of the acoustic path value 2πf0D  c2  during the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' In particular, as c2 depends on temperature and  pressure, it means that the experiment should have an excellent temperature and pressure regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This can  become increasingly difficult when the second sound derivatives become steep, close to the superfluid transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Moreover, measurements in the presence of a flow are necessary done out of equilibrium as the flow dissipates  energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  As an example, measurements in such conditions are illustrated by figure 25, which shows second  sound resonances measured close to the superfluid transition in the turbulent Von Karman experiment SHREK  [RBD+14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Furthermore, we observe that a measurement with a non-zero value of ξV LD can be associated with  an acoustic path shift (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' a variation of the factor 2πfD  c2  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The situation is illustrated in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  26, where the acoustic path shift leads to an overestimation of the attenuation and an important error on ξV LD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  31  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1  /w)  U  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  2 元  4元  6元  8元  10元  12元  14元  2元fD/C 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='8  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4  kD  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='8  1  T  First measurement  VLD=0  Second measurement  VLD=1e-4  Acoustic  path shift  Attenuation  error  Without  phase shift  With  phase shift  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  X  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='6  Y  R  r  Figure 26: An illustration of an attenuation measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Left: the resonant mode without (blue curve),  and with quantum vortex attenuation (red curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The figure shows that an acoustic path shift can lead to  an important error in the attenuation measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Right: the resonant mode represented in the phase-  quadrature plane together with the fitted Kennelly circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The figure shows that the acoustic path shift creates  a phase shift θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Using the phase measurement θ, the maximal magnitude can be recovered using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Passive and active approaches have been reported in the literature to handle the most common cause of  acoustical path shift during second sound measurement: the temperature drift and its resulting shift of the  resonance frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  A passive approach consists in performing a sweep of the second sound frequency, across the resonance curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Afterward, with proper modelling of the resonance, the attenuation and the phase shift can be fitted separately,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' as done in [MSS76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' A limit of this approach is its time resolution, that is restricted by the duration of  frequency scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Another passive approach consists in performing systematic calibration of the full frequency  responses of the resonator in various conditions, and subsequently interpolating measurements obtained at a  fixed working frequency onto this mapping [VBL+17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  A standard example of active approaches consist in controlling the helium bath temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' An alternative  or complementary approach consists in controlling the second-sound frequency so that it always matches the  resonance peak, despite possible drift of the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The resonator itself can provide the feedback signal  of these control loops, for example by monitoring the thermometer or locking the phase of the second sound  signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' An even more direct approach has been recently proposed: the resonator is driven by a self-oscillating  circuit, which frequency adapts dynamically to the drift of the second sound velocity [YIE17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Below, we introduce two analytical methods to separate phase shift from attenuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The first method is  relevant for simple cases (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2), while the second one has a broader range of validity (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  Analytical method in an idealized case  The acoustic path shift can be corrected using the resonant representation in the phase-quadrature plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Let  X(f) and Y (f) be respectively the real and imaginary parts of T(f), the curve (X(f), Y (f)) in the phase-  quadrature plane is very close to a circle crossing the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' It can even be proved (see sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3) that the  resonant curve converges to a circle when the quality factor increases, or equivalently when ξ0 decreases5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' All  along the present article, this limit circle is called the “Kennelly circle”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' An illustration of two resonant curves  with their osculating Kennelly circles is displayed in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The acoustic path shift translates  in a phase shift θ in the phase-quadrature plane, such that the amplitude of the second measurement (with  ξV LD > 0) does not correspond to the maximal amplitude R of the attenuated resonant peak (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 26 for  5In this case, the complex amplitude of the nth temperature resonance is often approximated by the Lorentzian formula [VS71,  DLL80]  T(f) ≃  Tn  1 + iQn f−fn  fn  (35)  where Tn, Qn and fn are the amplitude at resonance, the quality factor and resonance frequency of the mode of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' To  highlight that this Lorentzian approximation describes a circle in the complex plane X-Y , it can be written as:  T(f)  T n/2  = 1 + eiφ(f)  (36)  where tan φ = 2F/  �  F 2 − 1  �  and F = Qn (f − fn) /fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  32  Figure 27: Sequence of five resonances of second sound tweezers at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='6K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The flow is weakly turbulent: the  velocity standard deviation is a few percents of the mean velocity displayed on the colorbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The right side plot  illustrates that each resonance can be approximated by a circle in the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The global phase shift of  the last resonance (lower circle) compared to the others is attributed to a cut-off of the measurement electronics  at high frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  the notations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Using the geometric properties of the Kennelly circle, R can be approximately recovered from  the measured amplitude r with  R =  r  cos θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (37)  Thus, a modified version of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (31) can be written to find the VLD attenuation coefficient in the presence  of a phase shift  ξV LD = 1  Dasinh  � T 0 cos θ  T(f0)e−iθ sinh (ξ0D)  �  − ξ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (38)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3  The elliptic method  We present in this section an original method to obtain the values of the acoustic path shift and the attenuation  coefficient, from experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The method, that we call the “elliptic method”, is much simpler to imple-  ment, and much more reliable, than the fit of the Kennelly resonant circle and the use of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Besides, the  method can be used for resonances with very low quality factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The method comes from the observation that a pair of two ideal consecutive resonances is transformed into  an ellipse with the complex inversion z → 1  z in the phase-quadrature plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The inversion is represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' To prove this assertion, consider the inversion of the classical Fabry–Perot expression Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (6)  1  T =  sinh  �  i 2πfD  c2  + ξD  �  A  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (39)  Expanding the sinh in the previous expression gives  1  T = cos  �2πfD  c2  � sinh (ξD)  A  + i sin  �2πfD  c2  � cosh (ξD)  A  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (40)  Finally, let Xl = Re  �  1  T  �  and Yl = Im  �  1  T  �  be respectively the real and imaginary parts of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (40), the  coordinates (Xl, Yl) satisfy the equation  �  Xl  sinh (ξD) /A  �2  +  �  Yl  cosh (ξD) /A  �2  = 1  (41)  which is exactly the cartesian equation of an ellipse with semi-major axis a = cosh(ξD)  A  , and semi-minor axis  b = sinh(ξD)  A  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' In particular, we note that the attenuation coefficient ξ can be recovered from the ratio of the  semi-major and semi-minor elliptic axes using the formula  ξ = 1  Datanh  � b  a  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  33  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='15  (yu)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='05  20  40  60  80  100  f (kHz)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='05  1  (mK)  (s/w)  0  0  > -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='05  U  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1  X (mK)-1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  1  X  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  Y  2  0  2  X  4  2  0  2  4  Y  1/z  Figure 28: Transformation of a pair of consecutive resonances to an ellipse using the inversion of the complex  plane z → 1/z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  X  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  Y  1/z  acoustic path  axis  acoustic path  axis  attenuation  axis  u  v  attenuation  axis  ul  vl  Figure 29: A resonant mode in the phase-quadrature plane and its elliptic transform, for an ideal Fabry–perot  resonance with ξ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='95π < kD < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='05π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  When the quality factor increases (equivalently when ξ decreases) the ellipse is flattened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The limit of infinite  quality factor (ξ → 0) corresponds to two parallel straight lines in the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Second sound tweezers resonances are not ideal Fabry–Perot resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Yet, we have argued in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  that a single second sound resonance can be locally fitted by the following Fabry–Perot equation (see also Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (10))  T =  A  sinh  �  i 2π(f−f0)D  c2  + (ξ0 + ξV LD)D  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (42)  This in particular means that the resonant curve in the vicinity of its maximal amplitude is transformed into a  part of an ellipse with the complex inversion z → 1  z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The curve (Xl(f), Yl(f)) is very close to a straight line, for  frequencies f close to the resonant frequency f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The situation is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The figure shows a part  of ideal Fabry–Perot resonances given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (42), in the range 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='95π < kD < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='05π (where k = 2πf  c2 ), and for  increasing values of the VLD attenuation coefficient ξV LD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The left panel displays the different resonant curves  close to their maximal amplitudes, in the phase-quadrature plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Using the complex inversion, those curves  become almost parallel straight lines, as can be seen in the right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The transformation of the resonant  Kennelly circles into parallel straight lines has very nice applications that we explain in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  In the phase-quadrature plane, a variation of the acoustic path value 2πfD  c2  corresponds to a displacement  along the Kennelly circle, whereas a variation of the bulk attenuation ξ corresponds to a displacement orthogonal  to the Kennelly circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The acoustic path direction and the attenuation direction thus form a local orthogonal  basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Such a basis is displayed by the red arrows in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' When the reference point where  the basis is defined moves along the Kennelly circle, the basis rotates and the acoustic path and attenuation  axes have to be redefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This can become a tiresome task while analyzing the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Fortunately,  the complex inversion is a conformal mapping, which means that it preserves locally the angles: the local basis  composed of the acoustic path and attenuation axes is transformed into an orthogonal basis (see the red arrows  in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' More precisely, let z0 be the complex position in the phase-quadrature plane,  34  and (u, v) be the two complex (unit) vectors defining the local basis at z0, then the local basis (ul, vl) at the  point  1  z0 is given by  �  �  �  ul  = −u |z0|2  z2  0  vl  = −v |z0|2  z2  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (43)  The major advantage of defining the local basis (ul, vl) with the elliptic transform, is that it becomes a global  basis: when the reference point  1  z0 moves because of a change in the acoustic path value or the attenuation  value, the basis is simply translated in the plane but the vectors (ul, vl) do not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' One can find the global  basis (ul, vl) for a given resonance and use it to find the local basis (u, v) at every point in the phase-quadrature  plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We will see in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1 how the global basis (ul, vl) can be easily used to suppress temperature and  pressure drifts during second sound attenuation measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  We finally explain how the elliptic method can be used to measure the bulk attenuation coefficient ξ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We  have seen in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1 that the standard methods to find ξ0 are based on the measure of the half-width ∆f  and on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' As was said previously, the classical method can only be applied to resonances satisfying  (ξ0D) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' It is a global method, in the sense that one has to sweep the frequency to measure a large part of  the resonant curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The method is only accurate provided the resonance does not deviate too much from an  ideal Fabry–Perot resonance, which is often not satisfied for the first modes of second sound tweezers (see for  example the first mode of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The alternative method consists in expanding (Xl, Yl) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (41) to  leading order in f − f0  �  Xl  ∼ sinh(ξ0D)  A  ,  Yl  ∼ 2π(f−f0)D  c2  cosh(ξ0D)  A  ,  (44)  where f0 is the resonant frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (44), we get  Yl(f)  Xl(f) ∼  2πD  tanh (ξ0D) c2  (f − f0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (45)  Yl(f)  Xl(f) is proportional to f in the vicinity of f0, with the proportionality factor  2πD  tanh(ξ0D)c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The attenuation  coefficient ξ0 can be found by a linear fit of the function Yl  Xl provided D and c2 are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4  Applications of the elliptic method  The motivation to develop and use the elliptic method has come from experimental constrains: in a large super-  fluid experiment, it can be very difficult to control the values of mean temperature and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This is even  more the case if the superfluid experiment dissipates energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' One then expects a drift of the thermodynamics  conditions during the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Regarding second sound resonators, the critical parameter is the second  sound velocity c2, because variations lead to uncontrolled acoustic path shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The elliptic method has been  designed to easily filter those variations from experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This includes filtering temperature and pres-  sure drifts (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1) and the vibration of the tweezers arms (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3), and properly extract the quantum  vortex lines fluctuations (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1  Suppression of temperature and pressure drifts  This section presents an example of the elliptic method implementation for second sound tweezers, to find the  relation between ⟨ξV LD⟩ and the mean velocity U in the presence of a superfluid flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  As before, we note (X, Y ) the temperature signal obtained from the second sound tweezers in the phase-  quadrature plane, and (Xl, Yl) the cartesian coordinates obtained by the complex inversion given by  �  �  �  Xl  = Re  �  1  X+iY  �  ,  Yl  = Im  �  1  X+iY  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (46)  The coordinates (Xl, Yl) will be called “elliptic coordinates” for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 30 displays experimental data  obtained from second sound tweezers of size L = 1 mm, in a saturated bath at mean temperature T0 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='14 K,  and for different mean flow velocities 0 < U < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' At such a temperature close to the superfluid transition,  it was difficult to regulate the mean temperature and therefore the second sound velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Uncontrolled acoustic  path variations can be observed, for example in the red points of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The first step consists in sweeping the second sound frequency f in the vicinity of the resonant frequency f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The data (X, Y ) obtained, displayed by the black curve of the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 30, form a part of the Kennelly  circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' As explained in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3, the elliptic coordinates (Xl, Yl) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (46) form a straight line (see the  35  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3  X (mK)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='05  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  Y (mK)  2  4  6  8  4  3  2  1  0  1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  U (m/s)  Attenuation  axis  ul  vl  Acoustic path  axis  Figure 30: Experiment: measurement of a resonance with second sound tweezers at T0 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='14 K, for different  values of the mean flow velocity U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The left panel presents the data in the phase-quadrature plane, and the  right panel presents the same data after the complex inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The red points are obtained for a fixed value  f = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='35 kHz, and sweeping the flow mean velocity between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Using a linear fit, it is then straightforward to obtain the unit vector vl parallel to the  line defining the acoustic path axis, and the orthogonal vector ul defining the attenuation axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We then call  Zl = (Xl, Yl) the vector of the elliptic coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The attenuation coefficient ξ0 can be found by the relation  (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (45))  Zl(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='vl  Zl(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='ul  ∼  2πD  tanh (ξ0D) c2  (f − f0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 30 displays experimental resonant curves obtained for non-zero mean velocities U > 0 , to illustrate the  robustness of the elliptic method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' However, we emphasize that only the resonant curve with U = 0 is necessary  to find the global basis (ul, vl) in the plane of elliptic coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The second step consists in fixing the second sound frequency to f0 and vary the mean velocity U to look  at the resonance attenuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The experimental data are displayed by the red points in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' In can be  seen in the figure that the bulk attenuation is accompanied by a systematic acoustic path deviation as the  mean velocity increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' For mean velocities U ≈ 1 m/s, energy dissipation in the experiment leads to a data  dispersion along the acoustic path direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' To properly recover the mean VLD attenuation coefficient ⟨ξV LD⟩  , we use the elliptic coordinates Zl = (Xl, Yl) , and we project it on the attenuation axis ul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We get from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (44)  Zl(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='ul  Zl(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='ul  = sinh ((ξ0 + ⟨ξV LD⟩) D)  sinh (ξ0D)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  And ⟨ξV LD⟩ is then given by  ⟨ξV LD⟩ = 1  Dasinh  �Zl(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='ul  Zl(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='ul  sinh (ξ0D)  �  − ξ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (47)  We note that the previous expression remains accurate even if the second sound frequency chosen for the  measurement is close but not exactly equal to the resonant frequency f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The average VLD attenuation can  then be converted to the average projected vortex line density ⟨L⊥⟩ using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2  Measure of vortex line density fluctuations  Second sound tweezers are designed to directly probe the small scale vortex line density fluctuations, not only  its average value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The method to probe fluctuations slightly differs from the method used to probe the average  value explained in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The average VLD value can be directly computed using the complex inversion of  experimental data, but this is no longer possible for its fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Indeed, the tweezers signal have different  sources of noise, like e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' thermal white noise, interfering frequencies, electromagnetic bursts, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Those  signals can usually be considered as independent additive noises in the signal data, and easily filtered out or  attenuated by an appropriate post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' On the contrary, the complex inversion is a non-linear transfor-  mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Using the latter on noisy data can lead to an overestimation of the signal fluctuations closest to zero,  36  1/z-2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  X (V)  x 10-6  16  12  8  4  Y (V)  x 10-7  0 m/s  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='20 m/s  Elliptic transformation  acoustic path axes  attenuation axes  Figure 31:  perturb the additivity of noise sources and make them much more difficult to filter out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We thus choose to  compute the VLD fluctuations only using linear transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The first step is similar to that of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We sweep the second sound frequency f close to the resonant  frequency f0, in order to measure a part of the Kennelly circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We then transform this Kennelly circle into a  straight line using the complex inversion, and we find the global basis (ul, vl) in the plane of elliptic coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  A fit of the Kennelly circle, and its transformation into a straight line can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This experimental  step has to be done just before the fluctuations measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  We then record the signal fluctuations (X(t), Y (t)), for different values of the flow mean velocity U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  31 displays the fluctuating signal for U = 0 and U = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2 m/s in the form of clouds of data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' It can be in  particular observed that the U = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2 m/s data are shifted compared to the U = 0 m/s data because of both  an average attenuation and an acoustic path shift (see sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Let us define ⟨Z(t)⟩ = ⟨X(t)⟩ + i ⟨Y (t)⟩ the  average complex position in the phase-quadrature plane, for a given value of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (43), the local  basis (u, v) of the attenuation and acoustic path axes can be computed from the global elliptic basis (ul, vl) by  �  �  �  u  = −ul  ⟨Z⟩2  |⟨Z⟩|2  v  = −vl  ⟨Z⟩2  |⟨Z⟩|2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (48)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  31 shows that the local basis (u, v) depends on ⟨Z⟩ and thus on the value of U: for different mean  velocities, the bases are rotated from one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' If there is a significant drift of the mean signal value during  the measurement (as can e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' be observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 22), the local basis will also depend on time, with a typical  timescale that should be much larger than the fluctuation timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The acoustic path fluctuations and the attenuation fluctuations can then be recovered using a projection on  the (u, v) basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' More precisely, let x be the average acoustic path value and δξ = ξ − ⟨ξ⟩ be a small fluctuation  of the attenuation coefficient, a leading order expansion of expression Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (42) shows that  T ≈  A  sinh (ix + ⟨ξ⟩ D) − δξ  AD  sinh (ix + ⟨ξ⟩ D) tanh (ix + ⟨ξ⟩ D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  (49)  We then do the approximation ⟨Z⟩ ≈ A/ sinh (ix + (ξ0 + ⟨ξV LD⟩) D) which is equivalent to neglecting non-linear  corrections in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We finally get  δξ(t) = 1  Du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (Z(t) − ⟨Z⟩) ×  ����  tanh (ix + (ξ0 + ⟨ξV LD⟩) D)  ⟨Z⟩  ���� ,  (50)  where the value of ξ0 can be found with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (45) and ⟨ξV LD⟩ with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The use of the elliptic method is illustrated by Figure 32, which reports the probability density function of  the fluctuations of the quantum vortex density in a nearly isotropic superfluid turbulent flow The data of this  37  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  2  p (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' shift)  s /<s>  vortex line density fluct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  velocity fluctuations  Figure 32: Example of the probability density function of vortex line density (dashed line) and velocity (contin-  uous line) measured by second sound tweezers in a superfluid turbulent flow [WVR21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' In abscissa, each signal  s is normalized by its mean value < s >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The nearly Gaussian velocity statistics and skewed vorticity statistics  are reminiscent of those in classical turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  plot were reported together with spectra of vorticity fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' For details about the setup and analysis of  these results, see [WVR21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3  Filtering the vibration of the plates  One possible source of noise for second sound resonator measurement is the vibration of the plates arms whenever  U ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The signature of those vibrations can be very clearly identified in the form of two thin peaks in the  fluctuations power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Those two peaks are located at the two arms resonant frequencies: their exact  values can vary for different tweezers, but we always observe them around f ≈ 1 kHz (see sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Fortunately, the tweezers arms vibrations correspond to a variation of the gap D, and thus to acoustic path  fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 33 display a part of the tweezers fluctuations power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The fluctuations are projected  along the attenuation axis (blue curve) and along the acoustic path axis (red curve), following the method  presented in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The two peaks located at f ≈ 825 Hz and f ≈ 1050 Hz are identified on the acoustic  path axis fluctuations power spectrum, whereas the same peaks are damped by many orders of magnitude on  the attenuation axis fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Using the power spectrum of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 33, we can estimate the order of magnitude  of the gap standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We find  ��  (δD)2�  ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5 µm, and  �  ⟨(δD)2⟩  D  ≈ 4 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This confirms that the  arms vibrations have a negligible impact on the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5  Velocity measurements  As shown in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5, the second sound tweezers geometry can be optimized to sense specifically velocity rather  than vortex density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' For this purpose, one trick consists in shifting one plate with respect to the other in the flow  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Figure 34 shows three second sound tweezers and one anemometer that is based on the same principle  as Pitot tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' All sensors are positioned across a nearly isotropic superfluid flow bounded by a cylindrical pipe  not shown here- (for details on this set-up, see [RCSR17a]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The figure insert is a close view of the tip of the  left-side tweezers, which is dedicated to velocity measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This shift of one plate versus the other in the  downstream direction is clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The projection of the anemometer-tweezers signal in the complex plane is not performed along orthogonal  axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' These axes are determined with an in-situ calibration, ramping the mean velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The complementary  “Pitot tube” signal is used to calibrate the axis in units of m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The duration of velocity ramp is chosen to be  much larger than the time resolution of the Pitot tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  As an illustration, Figure 32 presents the probability density function of the velocity fluctuations measured  by the anemometer tweezers, together with vortex line density fluctuations recorded by the other tweezers during  the same experimental run (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' As expected in nearly homogeneous and isotropic quantum turbulence  [SCLR12], the velocity statistics are close to a Gaussian when probed at scales significantly larger than the  intervortex distance, which is the case here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Velocity spectra derived from the same dataset are reported in  [WVR21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  38  700  800  900  1000  1100  f (Hz)  10-18  10-17  10-16  P(f)  attenuation axis fluctuations  acoustic path axis fluctuations  Figure 33: Experiment: a part of second sound tweezers fluctuation power spectrum, with T0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='65 K,  U = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2 m/s and for a tweezers gap D = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='320 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The tweezers arms’ resonances can be clearly identified in  the acoustic path fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Figure 34: Example of an arrangement of three second sound tweezers dedicated to velocity (x1, left side) and  vorticity (x2, right side) time series acquisitions, together with a miniature total head pressure tube (loosely  labelled "Pitot tube" on the bottom left side of the picture) used for velocity calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  All probes are  mounted on a ring connecting two 76mm-inner-diameter coaxial pipes (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='1 of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' [WVR21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The insert  is a close-up view of the shifted plates of the anemometer tweezers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  39  Thermometer  Flow  Heater  VLD probing  second sound  tweezers  Second sound  tweezers  anemometer  Miniature Pitot  tube5  Summary and Perspectives  This study has covered three independent topics : the comprehensive analytical modelling of second-sound  resonators with a cavity allowing a throughflow of superfluid (section 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' new mathematical methods to process  the signal provided by such resonators (section 4) and the miniaturization of immersed second-sound resonators  allowing time and/or space resolved flow sensing (section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' These so-called second-sound tweezers have been  used throughout the manuscript to demonstrate the strength and limits of the modelling and analysis methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Two observations remains unexplained: the origin of some noise on the acoustic path length and a second  order oscillations of the resonator spectral response in quiescent 4He, named the “daisy effect” (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Fortunately, both effects don’t impair the measurement of the flow vorticity or velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Some results that were not anticipated when this study was initiated, are worth recalling and discussing:  the possibility to operate tweezers by over-driving the second-sound standing wave beyond an intrinsic  turbulent transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' In this non-linear mode, the probe becomes sensitive to velocity, which is interpreted  as a signature of the sweeping of the local vortex tangle by the outer flow (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This operating  mode is somehow analogous of hot film and hot wire anemometry in classical fluid, where sensitivity  to velocity is due to the more-or-less pronounced sweeping of the thermal boundary layer around an  overheated thermometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  in the linear regime (standing wave of small amplitude), the probe can be sized and operated to be either  mostly sensitive to quantum vortices or mostly sensitive to the velocity of the throughflow (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  This prediction is verified experimentally by comparing statistics in turbulent flows (section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The  spatial and time resolution of tweezers operated as anemometers -both in non-linear and linear mode-  is close or better than the alternative miniaturized anemometers working in He-II, that is hot-wires  [DBM+15, DRS+21], cantilevers[SMR12, RCSR17b], Pitot tubes [SMR12, RCSR17b] and total head-  pressure probes[MT98, WVR21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  in the absence of throughflow, the full spectral response of the resonators, and in particular its quality  factors, can be accurately determined simply taking into account the loss by diffraction and misalignment  of the reflecting plates of the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  In other words, the other sources of dissipation have negligible  contributions in the range of conditions explored here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  This is no longer the case in a presence of a  throughflow carrying quantum vortices since the latter can significantly contribute to the total dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  We have not explored experimentally the production and detection of second-sound by mechanical means,  which implies cavity with rough surfaces (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' millipore or nucleopore membranes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' In this case, the effect  of vortices pinned on the surface may no longer be negligible ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' for a discussion see [DLL80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  the possibility to sense the variations of the vortex line density or velocity without knowledge on variations  of the second sound velocity (or acoustical path).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' More generally, the elliptic method allows a mathematical  decoupling of both effects by a projection method in the (inverse) complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This results is of major  practical interest in flows where the second-sound velocity is not accurately controlled due to residual  temperature variations or thermal gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' This situation can occur for instance in flows sequentially  driven at various levels of forcing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' to explore a Reynolds number dependence), in inhomogeneous  dissipative flows and in flows close to the lambda superfluid transitions where the second-sound velocities  strongly depends of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Two applications of second-sound tweezers have been illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Measurement within a turbulent boundary  layer (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 25) are possible thanks to the small size of probe, and measurements of time series in the bulk  of quantum turbulent flow (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' 32) are possible thanks to both the time and space resolutions of the probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Among other applications the probe can map the velocity or the vorticity field of an inhomogeneous flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' A  mapping of vorticity in a counterflow jet has been recently done and will be reported elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Another application of tweezers would be to probe simultaneously the temperature fluctuations and those of  either velocity or vorticity, for instance to explore their correlations in turbulent counterflows or even co-flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  Indeed, the tweezers thermometer provides a direct measurement of temperature in a bandwidth spanning from  zero frequency up to a fraction of the frequency of the second sound standing wave, and this signal could be  acquired without impairing the measurement of the second sound standing wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Alternating measurements in  the linear and non-linear modes is also interesting to explore locally both velocity and vorticity in a given flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  A last example of application is to operate a double-tweezers made of a heating plate between two thermometer  plates, or vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Such a stack can be used to probe joint statistics of vorticity on one side, and velocity on  the other side, or this arrangement can be used alternatively to probe transverse gradient of either vorticity or  velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  40  Acknowledgements  We warmly thank our colleague Benoît Chabaud for support during cryogenic tests and operation of the TOUPIE  wind-tunnel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We acknowledge and thank the staff of the PTA and Nanofab clean-rooms in Grenoble, where  microfabrication was done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Data of figure 25 have been acquired in the SHREK facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We thank all the  members of the SHREK collaboration, in particular Michel Bon Mardion and Bernard Rousset for the specific  operation very near the superfluid transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' We acknowledge the indirect but key contribution of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Elbakyan  regarding the comprehensiveness of the bibliography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  The probe design, fabrication, cryogenic operation and theoretical analysis took place over 7 years, and was  possible thanks to following research grants: ANR Ecouturb grant (ANR-16-CE30-0016), ANR QUTE-HPC  grant (18-CE46-0013- 03) and EU Horizon 2020 Research and Innovation Program "the European Microkelvin  Platform (EMP)" (824109).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  This research was funded in part by the Agence nationale de la recherche (ANR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' A CC-BY public copyright  license has been applied by the authors to the present document and will be applied to all subsequent versions  up to the Author Accepted Manuscript arising from this submission, in accordance with the grant’s open access  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  References  [Bal07]  S Balibar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' The discovery of superfluidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Journal of Low Temperature Physics, 146(5-6):441–470,  2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content='  [BLR17]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
+page_content=' Bertolaccini, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
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+page_content='-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE5T4oBgHgl3EQfRQ5h/content/2301.05519v1.pdf'}
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diff --git a/AdE1T4oBgHgl3EQfpAU_/content/tmp_files/2301.03326v1.pdf.txt b/AdE1T4oBgHgl3EQfpAU_/content/tmp_files/2301.03326v1.pdf.txt
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@@ -0,0 +1,419 @@
+Simulating the radio emission of dark matter for new
+high-resolution observations with MeerKAT
+M Sarkis and G Beck
+School of Physics, University of the Witwatersrand, Private Bag 3, WITS-2050, Johannesburg,
+South Africa
+E-mail: michael.sarkis@students.wits.ac.za
+Abstract.
+Recent work has shown that searches for diffuse radio emission by MeerKAT - and
+eventually the SKA - are well suited to provide some of the strongest constraints yet on dark
+matter annihilations. To make full use of the observations by these facilities, accurate simulations
+of the expected dark matter abundance and diffusion mechanisms in these astrophysical objects
+are required. However, because of the computational costs involved, various mathematical and
+numerical techniques have been developed to perform the calculations in a feasible manner.
+Here we provide the first quantitative comparison between methods that are commonly used in
+the literature, and outline the applicability of each one in various simulation scenarios. These
+considerations are becoming ever more important as the hunt for dark matter continues into a
+new era of precision radio observations.
+1. Introduction
+Despite decades of work, indirect Dark Matter (DM) searches – those that look for emission from
+the annihilation and decay products of DM particles – are yet to find a signal that can be solely
+attributed to DM. Until such a detection is made, and as our observing capabilities improve
+with newer and more sophisticated telescopes, we continue to methodically move through the
+parameter spaces of candidate DM models and eliminate those that conflict with the data. The
+recent public release of the MeerKAT Galaxy Cluster Legacy Survey data [1], together with recent
+studies that show the competitiveness of using DM radio emission for indirect detection [2, 3, 4],
+provides strong motivation for a renewed and continued effort in radio DM searches. In this work
+we take a brief but detailed look at the various theoretical aspects involved in the modelling
+of the radio emission from DM, and comment on how the choice of model will likely play an
+important role in indirect searches with high-resolution instruments.
+Our analysis includes simulations of the DM host environments for two source targets, the
+Coma galaxy cluster and the M31 galaxy, and a calculation of the synchrotron emission resulting
+from the annihilation of Weakly Interacting Massive Particles (WIMPs) therein. We model our
+DM halos with a set of reasonable source parameters and find the emission after solving the
+electron propagation equation in each environment. The methods of solving this equation are
+a major focus point of this work, as the choice of technique used can lead to a non-negligible
+change in the observed emission, particularly in smaller source targets where diffusion effects are
+significant. With < 10 arcsecond resolution capabilities, observations with MeerKAT (and soon
+the SKA) are for the first time able to probe the inner regions of these targets, which is where
+the strongest constraints on DM can be found. Therefore, accurate spatial modelling of these
+targets is essential for us to make full use of the new data.
+arXiv:2301.03326v1  [astro-ph.CO]  9 Jan 2023
+
+2. Modelling
+The two source targets in this work, the Coma galaxy cluster and the M31 galaxy, were chosen
+for their well-characterised properties in the literature. Of particular importance are the profiles
+of their magnetic fields and thermal gas densities; as these quantities appear in the modelling
+process (but are often underspecified), the uncertainty of the final solution depends strongly
+on the treatment of these factors [5]. However, since the simulation of the halo environment
+is not the central focus of this work (and for the sake of brevity), we refer the reader to the
+following sources for details regarding the parameters in the Coma cluster [6, 7] and in the M31
+galaxy [8, 9].
+In each halo environment, the emission of synchrotron radiation will be determined by the
+spatial and energy equilibrium distribution of charged annihilation products, ψ(x, E). In this
+work the products considered are electrons and positrons. The evolution of these distributions
+over time is then given by the following propagation equation, which includes the dominant
+effects of energy losses and spatial diffusion:
+∂ψ(x, E)
+∂t
+= ∇ ·
+�
+D(x, E)∇ψ(x, E)
+�
++ ∂
+∂E
+�
+b(x, E)ψ(x, E)
+�
++ Q(x, E).
+(1)
+Here D, b and Q are the diffusion, energy-loss and DM annihilation source functions respectively,
+and the determination of the exact forms of these functions follows the methods laid out in [5].
+2.1. Solving the propagation equation
+We determine the equilibrium electron distribution ψ using two independent techniques. The
+first, referred to here as the ‘Green’s Function (GF) method’ [2, 10], uses a Green’s function
+with simplified forms of D and b to solve Eq. 1 semi-analytically. The second, referred to as the
+‘Alternating Direction Implicit (ADI) method’ [11, 12], uses a numerical approach to solve Eq. 1
+iteratively. In both methods we consider the halo environment to be spherically symmetric, so
+that x may be replaced by r in Eq. 1. We also note here that we have assumed a simplified
+form of D, which would be a tensor in a more general case. As our methodology closely follows
+the above-mentioned literature, we only summarise these methods and point out any major
+differences in the following sections.
+GF method
+If the forms of the diffusion and energy-loss functions are simplified so that they have
+no spatial dependence, a solution to Eq. 1 can be found directly with the use of Green’s functions
+and image charges. However, these simplifications often have an impact on the calculated
+emission (for a review on this topic, see [5]). In this work we use non-weighted averages for the
+magnetic field and thermal gas densities, found using an averaging scale radius that matches
+the scale radius of the DM halo. This choice encapsulates the region in the halo that contains
+the majority of WIMP annihilations – and thus best represents the spatial structure of the
+halo – while allowing us to forgo any explicit spatial dependence in Eq. 1. Now, the equilibrium
+distribution of electrons in the halo can be calculated using
+ψ(r, E) =
+1
+b(E)
+� mχ
+E
+dE′G(r, ∆v)Q(r, E′) ,
+(2)
+with mχ as the WIMP mass and the Green’s function (G) given by
+G(r, ∆v) =
+1
+√
+4π∆v
+∞
+�
+n=−∞
+(−1)n
+� rmax
+0
+dr′ r′
+rn
+�
+�exp
+�
+−(r′ − rn)2
+4∆v
+�
+− exp
+�
+−(r′ + rn)2
+4∆v
+��
+� Q(r′)
+Q(r) .
+(3)
+
+Here rmax is the maximum radius for any diffusion processes and rn = (−1)nr + 2nrmax is the
+location of the nth image charge. The quantity ∆v is calculated as
+∆v = v(E) − v(E′) ,
+(4)
+where
+v(E) =
+� mχ
+E
+dx D(x)
+b(x) .
+(5)
+ADI method
+In this method, we discretise Eq. 1 and solve for the equilibrium distribution
+iteratively. Since the ADI method retains the radial dependence in the diffusion and energy loss
+functions (where the GF method does not), the problem becomes 2-dimensional in energy and
+space. Using a traditional finite-difference technique in this scenario could be computationally
+expensive, which is why we opt for a method that uses so-called ‘operator splitting’ to treat
+each dimension separately and divide the problem into smaller, more manageable pieces. Thus,
+during each step of the method, we use a general form of the 1-dimensional Crank-Nicolson
+(CN), scheme (see, for instance, [13]) which is a finite-differencing technique that includes the
+average of second-order implicit and explicit terms in the updating equation, thereby leveraging
+the unconditional stability of a fully implicit method while maintaining second-order accuracy in
+space and time. This scheme is relatively easy to solve, as the updating equation turns out to be
+a set of linear equations with tridiagonal coefficient matrices. We write this, as in [11, 12], as
+− α1
+2 ψn+1
+x−1 +
+�
+1 + α2
+2
+�
+ψn+1
+x
+− α3
+2 ψn+1
+x+1 = α1
+2 ψn
+x−1 +
+�
+1 − α2
+2
+�
+ψn
+x + α3
+2 ψn
+x+1 + Qx∆t.
+(6)
+Here n is the temporal grid index (with the spacing between indices given by ∆t) and x represents
+either the energy or spatial grid index. The forms of the α coefficients, which encapsulate the
+diffusion and energy loss effects, need to be found by discretising the relevant operators from
+Eq. 1. The scheme we have used for this is as follows:
+1
+r2
+∂
+∂r
+�
+r2D∂ψ
+∂r
+�
+−−−−−−−−→
+discretisation C−2
+˜r
+�
+�ψi+1 − ψi−1
+2∆˜r
+�
+log(10)D + ∂D
+∂˜r
+������
+˜r=˜ri
++ ψi+1 − 2ψi + ψi−1
+∆˜r2
+D|˜r=˜ri
+�
+(7)
+for the radial operator and
+∂
+∂E (bψ) −−−−−−−−→
+discretisation C−1
+˜E
+�b| ˜E= ˜Ej(ψj+1 − ψj)
+∆ ˜E
+�
+(8)
+for energy operator, where C˜r = (r0 log(10)10˜ri), C ˜E = (E0 log(10)10 ˜Ej) and ∆˜r, ∆ ˜E represent
+the radial and energy grid spacings, respectively. We use i and j to denote positions in the
+radial and energy grids, and have omitted the temporal indices as these forms will apply to both
+implicit and explicit terms in the same way. The vertical bars denote that the functions which
+they are attached to are evaluated at the given grid index. We have also made the variable
+transformations ˜r = log10(r/r0) and ˜E = log10(E/E0) (similarly to [12], except with base 10
+instead of e), which allows us to more accurately track the electron distribution in our grids
+when the involved processes operate over a wide range of physical scales. Finally, note that in
+the case of energy losses, we only consider upstream differencing.
+
+The α values can now be found by taking Eqs. 7 and 8 and equating coefficients with Eq. 6;
+once these are found, the updating equation can be solved with some matrix solution algorithm.
+If we represent the discretisation schemes shown above by the symbol Ψ, the overall iterative
+solution can be summarised with the steps
+ψn+1/2 = Ψ ˜E(ψn)
+ψn+1 = Ψ˜r(ψn+1/2) ,
+(9)
+which are repeatedly solved (using Eq. 6) until the value of ψ has converged to the equilibrium
+value. The other minutiae of this method, including initial and boundary conditions, convergence
+criteria and stability considerations, can be found in [11, 12].
+2.2. Synchrotron emission
+Once found via the GF or ADI methods, the equilibrium distribution is used to calculate the
+radio emissivity, given by
+jsync(ν, r) =
+� mχ
+0
+dE ψe±(E, r)Psync(ν, E, r) ,
+(10)
+where ν is the synchrotron frequency, ψe± is the sum of electron and positron equilibrium
+distributions and Psync is the power emitted by an electron with an energy of E (this is calculated
+as in [2]). The emissivity is then used to calculate the two main results in this work. Firstly, the
+azimuthally averaged surface brightness curves,
+Isync(ν, r, Θ, ∆Ω) =
+�
+∆Ω
+dΩ
+�
+l.o.s.
+dl jsync(ν, l)
+4π
+,
+(11)
+where l.o.s. is the line-of-sight to a point in the halo at radius r, which makes an angle of Θ with
+the centre of the halo, and ∆Ω is the solid angle over which the surface brightness is calculated.
+In this work we show results for a single representative frequency of ν = 0.5 GHz. Secondly, we
+calculate the integrated flux density by
+Ssync(ν, R) =
+� R
+0
+d3r′ jsync(ν, r′)
+4πd2
+L
+,
+(12)
+where the emissivity is integrated over the region enclosed by R and dL is the luminosity distance
+to the target. For the results shown in this work we consider R to be the virial radius of the halo.
+3. Results
+Here we provide the details of the simulations we have performed, and show the results for two
+observables: the radio surface brightness (Eq. 11) and integrated flux (Eq. 12). We use a set of
+reasonable source parameters for the halo environments that respect observational constraints,
+and aim to use WIMP parameter values that are representative of the many viable candidates.
+We thus consider a large range of particle masses, from 10 to 1000 GeV, and use a set of four
+annihilation channels, {bb, e+e−, µ+µ−, τ +τ −}. Since the focus of this work is on a comparison
+between the two solution methods, particurly in the way that they differ with various source
+targets, we show the results side-by-side and in the same manner for both targets. In Fig. 1 we
+show the surface brightness curves for the Coma cluster (left-hand panels) and M31 (right-hand
+panels), and Fig. 2 shows the integrated fluxes from the same targets for a range of frequencies.
+
+10−8
+10−4
+100
+b¯b
+e+e−
+10−2
+10−1
+100
+101
+10−8
+10−4
+100
+µ+µ−
+10−2
+10−1
+100
+101
+τ +τ −
+Angular radius from halo centre (arcmin)
+Surface Brightness (Jy/arcmin2)
+10−7
+10−4
+10−1
+b¯b
+e+e−
+100
+102
+10−7
+10−4
+10−1
+µ+µ−
+100
+102
+τ +τ −
+Angular radius from halo centre (arcmin)
+χ Mass
+10 GeV
+1000 GeV
+Method
+ADI
+GF
+Figure 1. Surface brightness curves for the Coma galaxy cluster (left-hand panels) and the M31
+galaxy (right-hand panels). Each of the four panels show different annihilation channels, given
+by the label in the top right of each plot. The ADI and GF methods are represented by the
+red and blue colours respectively, and the region in which the results overlap are given by the
+combination of these (the purple colour). These shaded regions represent the full mass range of
+the WIMPs (from 10 to 1000 GeV), and the domain of each panel runs up until the halo’s virial
+radius R (in angular units).
+600
+800
+1000
+1200
+1400
+Frequency (MHz)
+10−3
+10−2
+Flux (Jy)
+Channels
+bb
+µ+µ−
+τ +τ −
+e+e−
+Methods
+ADI
+Greens
+600
+800
+1000
+1200
+1400
+Frequency (MHz)
+10−3
+10−2
+10−1
+100
+Channels
+bb
+µ+µ−
+τ +τ −
+e+e−
+Methods
+ADI
+Greens
+Figure 2. The integrated fluxes, calculated using Eq. 12, for the Coma galaxy cluster (left) and
+the M31 galaxy (right). The different linestyles represent the two solution methods presented in
+Sec. 2.1, and each colour indicates the use of a different annihilation channel.
+4. Discussion and conclusions
+In Figs. 1, we see generally good agreement between the GF and ADI methods, which can be
+inferred from the significant amount of overlap between the curves in each panel. Noticeably
+however, we see more disagreement (less overlap) between the methods in the M31 galaxy than
+we do for the Coma galaxy cluster. Our explanation for this lies in the mathematical techniques
+employed by each method, and how they each treat the spatial dependence of the diffusion
+function in particular. In the galaxy cluster environment of Coma, diffusion effects are negligible
+on sufficiently large scales [10, 5], whereas in the physically smaller galaxy, diffusion effects
+start to influence the surface brightness distribution at all relevant scales. Since the GF and
+ADI methods leverage a spatially independent and dependent diffusion function (respectively),
+the resulting equilibrium distributions will tend to differ in the environments where the length
+
+scales in question do not greatly exceed the diffusion length, as is the case for M31. This trend
+is also seen in the fluxes from Fig. 2, which show a clear disagreement in all channels for the
+M31 galaxy, and relative agreement in all channels in the Coma cluster. Based on these results
+and the comparison of target environments presented in [5], we also expect that smaller target
+environments (like the dwarf spheroidal satellite galaxies of the Milky Way) would show further
+disagreement between the solution methods, as diffusion effects would be even more significant
+in these environments.
+The other notable result we see from these simulations is that the methods differ on small
+scales, even in the large Coma cluster. This is significant, as the inner regions of the DM halos
+are where we would observe the strongest emission. With high-resolution radio interferometers
+allowing us to resolve these smaller scales, our models could be tested against the strongest
+possible DM emission, allowing us to find more stringent constraints on DM properties than
+previously possible. In this regard, the surface brightness curves displayed here would be especially
+valuable results when determining new observational limits, as their emission profiles are highly
+dependent on the spatial structure of the DM halo.
+With the impressive spatial resolution of telescopes like MeerKAT and the SKA, we are now
+able to probe the inner regions of these DM halos – regions which have formerly been hidden
+from our view. The need for accurate modelling techniques is thus more necessary than ever
+before, and the considerations presented in this work should help inform the modelling choices
+made in future radio searches for DM.
+Acknowledgments
+This work is based on the research that was supported by the National Research Foundation
+of South Africa (Bursary No. 112332). G.B. acknowledges support from a National Research
+Foundation of South Africa Thuthuka grant no. 117969.
+References
+[1] Knowles K, Cotton W D, Rudnick L et al. 2022 Astronomy & Astrophysics 657 A56
+[2] Beck G 2019 Galaxies 7 16
+[3] Regis M, Reynoso-Cordova J, Filipovi´c M D et al. 2021 Journal of Cosmology and Astroparticle Physics 2021
+046
+[4] Chan M 2021 Galaxies 9 11
+[5] Sarkis M and Beck G 2022 The Proceedings of SAIP2021 ed Prinsloo A (UJ) pp 316–322 ISBN 978-0-620-
+97693-0
+[6] Bonafede A, Feretti L, Murgia M et al. 2010 Astronomy and Astrophysics 513 A30
+[7] �Lokas E L and Mamon G A 2003 Monthly Notices of the Royal Astronomical Society 343 401–412
+[8] Ruiz-Granados B, Rubi˜no-Mart´ın J A, Florido E et al. 2010 The Astrophysical Journal 723 L44–L48
+[9] Tamm A, Tempel E, Tenjes P et al. 2012 Astronomy & Astrophysics 546 A4
+[10] Colafrancesco S, Profumo S and Ullio P 2006 Astronomy & Astrophysics 455 21–43
+[11] Strong A W and Moskalenko I V 1998 The Astrophysical Journal 509 212–228
+[12] Regis M, Richter L, Colafrancesco S et al. 2015 Monthly Notices of the Royal Astronomical Society 448
+3747–3765
+[13] Press W H, Teukolsky S A and Vetterling W T 2007 Numerical Recipes: The Art of Scientific Computing 3rd
+ed (Cambridge University Press) ISBN 978-0-521-88068-8
+
diff --git a/AdE1T4oBgHgl3EQfpAU_/content/tmp_files/load_file.txt b/AdE1T4oBgHgl3EQfpAU_/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf,len=138
+page_content='Simulating the radio emission of dark matter for new  high-resolution observations with MeerKAT  M Sarkis and G Beck  School of Physics, University of the Witwatersrand, Private Bag 3, WITS-2050, Johannesburg,  South Africa  E-mail: michael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='sarkis@students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='wits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='za  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  Recent work has shown that searches for diffuse radio emission by MeerKAT - and  eventually the SKA - are well suited to provide some of the strongest constraints yet on dark  matter annihilations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' To make full use of the observations by these facilities, accurate simulations  of the expected dark matter abundance and diffusion mechanisms in these astrophysical objects  are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' However, because of the computational costs involved, various mathematical and  numerical techniques have been developed to perform the calculations in a feasible manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  Here we provide the first quantitative comparison between methods that are commonly used in  the literature, and outline the applicability of each one in various simulation scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' These  considerations are becoming ever more important as the hunt for dark matter continues into a  new era of precision radio observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Introduction  Despite decades of work, indirect Dark Matter (DM) searches – those that look for emission from  the annihilation and decay products of DM particles – are yet to find a signal that can be solely  attributed to DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Until such a detection is made, and as our observing capabilities improve  with newer and more sophisticated telescopes, we continue to methodically move through the  parameter spaces of candidate DM models and eliminate those that conflict with the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' The  recent public release of the MeerKAT Galaxy Cluster Legacy Survey data [1], together with recent  studies that show the competitiveness of using DM radio emission for indirect detection [2, 3, 4],  provides strong motivation for a renewed and continued effort in radio DM searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' In this work  we take a brief but detailed look at the various theoretical aspects involved in the modelling  of the radio emission from DM, and comment on how the choice of model will likely play an  important role in indirect searches with high-resolution instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  Our analysis includes simulations of the DM host environments for two source targets, the  Coma galaxy cluster and the M31 galaxy, and a calculation of the synchrotron emission resulting  from the annihilation of Weakly Interacting Massive Particles (WIMPs) therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' We model our  DM halos with a set of reasonable source parameters and find the emission after solving the  electron propagation equation in each environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' The methods of solving this equation are  a major focus point of this work, as the choice of technique used can lead to a non-negligible  change in the observed emission, particularly in smaller source targets where diffusion effects are  significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' With < 10 arcsecond resolution capabilities, observations with MeerKAT (and soon  the SKA) are for the first time able to probe the inner regions of these targets, which is where  the strongest constraints on DM can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Therefore, accurate spatial modelling of these  targets is essential for us to make full use of the new data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='03326v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='CO]  9 Jan 2023  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Modelling  The two source targets in this work, the Coma galaxy cluster and the M31 galaxy, were chosen  for their well-characterised properties in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Of particular importance are the profiles  of their magnetic fields and thermal gas densities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' as these quantities appear in the modelling  process (but are often underspecified), the uncertainty of the final solution depends strongly  on the treatment of these factors [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' However, since the simulation of the halo environment  is not the central focus of this work (and for the sake of brevity), we refer the reader to the  following sources for details regarding the parameters in the Coma cluster [6, 7] and in the M31  galaxy [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  In each halo environment, the emission of synchrotron radiation will be determined by the  spatial and energy equilibrium distribution of charged annihilation products, ψ(x, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' In this  work the products considered are electrons and positrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' The evolution of these distributions  over time is then given by the following propagation equation, which includes the dominant  effects of energy losses and spatial diffusion:  ∂ψ(x, E)  ∂t  = ∇ ·  �  D(x, E)∇ψ(x, E)  �  + ∂  ∂E  �  b(x, E)ψ(x, E)  �  + Q(x, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  (1)  Here D, b and Q are the diffusion, energy-loss and DM annihilation source functions respectively,  and the determination of the exact forms of these functions follows the methods laid out in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Solving the propagation equation  We determine the equilibrium electron distribution ψ using two independent techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' The  first, referred to here as the ‘Green’s Function (GF) method’ [2, 10], uses a Green’s function  with simplified forms of D and b to solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 1 semi-analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' The second, referred to as the  ‘Alternating Direction Implicit (ADI) method’ [11, 12], uses a numerical approach to solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 1  iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' In both methods we consider the halo environment to be spherically symmetric, so  that x may be replaced by r in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' We also note here that we have assumed a simplified  form of D, which would be a tensor in a more general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' As our methodology closely follows  the above-mentioned literature, we only summarise these methods and point out any major  differences in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  GF method  If the forms of the diffusion and energy-loss functions are simplified so that they have  no spatial dependence, a solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 1 can be found directly with the use of Green’s functions  and image charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' However, these simplifications often have an impact on the calculated  emission (for a review on this topic, see [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' In this work we use non-weighted averages for the  magnetic field and thermal gas densities, found using an averaging scale radius that matches  the scale radius of the DM halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' This choice encapsulates the region in the halo that contains  the majority of WIMP annihilations – and thus best represents the spatial structure of the  halo – while allowing us to forgo any explicit spatial dependence in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Now, the equilibrium  distribution of electrons in the halo can be calculated using  ψ(r, E) =  1  b(E)  � mχ  E  dE′G(r, ∆v)Q(r, E′) ,  (2)  with mχ as the WIMP mass and the Green’s function (G) given by  G(r, ∆v) =  1  √  4π∆v  ∞  �  n=−∞  (−1)n  � rmax  0  dr′ r′  rn  �  �exp  �  −(r′ − rn)2  4∆v  �  − exp  �  −(r′ + rn)2  4∆v  ��  � Q(r′)  Q(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  (3)  Here rmax is the maximum radius for any diffusion processes and rn = (−1)nr + 2nrmax is the  location of the nth image charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' The quantity ∆v is calculated as  ∆v = v(E) − v(E′) ,  (4)  where  v(E) =  � mχ  E  dx D(x)  b(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  (5)  ADI method  In this method, we discretise Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 1 and solve for the equilibrium distribution  iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Since the ADI method retains the radial dependence in the diffusion and energy loss  functions (where the GF method does not), the problem becomes 2-dimensional in energy and  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Using a traditional finite-difference technique in this scenario could be computationally  expensive, which is why we opt for a method that uses so-called ‘operator splitting’ to treat  each dimension separately and divide the problem into smaller, more manageable pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Thus,  during each step of the method, we use a general form of the 1-dimensional Crank-Nicolson  (CN), scheme (see, for instance, [13]) which is a finite-differencing technique that includes the  average of second-order implicit and explicit terms in the updating equation, thereby leveraging  the unconditional stability of a fully implicit method while maintaining second-order accuracy in  space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' This scheme is relatively easy to solve, as the updating equation turns out to be  a set of linear equations with tridiagonal coefficient matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' We write this, as in [11, 12], as  − α1  2 ψn+1  x−1 +  �  1 + α2  2  �  ψn+1  x  − α3  2 ψn+1  x+1 = α1  2 ψn  x−1 +  �  1 − α2  2  �  ψn  x + α3  2 ψn  x+1 + Qx∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  (6)  Here n is the temporal grid index (with the spacing between indices given by ∆t) and x represents  either the energy or spatial grid index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' The forms of the α coefficients, which encapsulate the  diffusion and energy loss effects, need to be found by discretising the relevant operators from  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' The scheme we have used for this is as follows:  1  r2  ∂  ∂r  �  r2D∂ψ  ∂r  �  −−−−−−−−→  discretisation C−2  ˜r  �  �ψi+1 − ψi−1  2∆˜r  �  log(10)D + ∂D  ∂˜r  ������  ˜r=˜ri  + ψi+1 − 2ψi + ψi−1  ∆˜r2  D|˜r=˜ri  �  (7)  for the radial operator and  ∂  ∂E (bψ) −−−−−−−−→  discretisation C−1  ˜E  �b| ˜E= ˜Ej(ψj+1 − ψj)  ∆ ˜E  �  (8)  for energy operator, where C˜r = (r0 log(10)10˜ri), C ˜E = (E0 log(10)10 ˜Ej) and ∆˜r, ∆ ˜E represent  the radial and energy grid spacings, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' We use i and j to denote positions in the  radial and energy grids, and have omitted the temporal indices as these forms will apply to both  implicit and explicit terms in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' The vertical bars denote that the functions which  they are attached to are evaluated at the given grid index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' We have also made the variable  transformations ˜r = log10(r/r0) and ˜E = log10(E/E0) (similarly to [12], except with base 10  instead of e), which allows us to more accurately track the electron distribution in our grids  when the involved processes operate over a wide range of physical scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Finally, note that in  the case of energy losses, we only consider upstream differencing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  The α values can now be found by taking Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 7 and 8 and equating coefficients with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  once these are found, the updating equation can be solved with some matrix solution algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  If we represent the discretisation schemes shown above by the symbol Ψ, the overall iterative  solution can be summarised with the steps  ψn+1/2 = Ψ ˜E(ψn)  ψn+1 = Ψ˜r(ψn+1/2) ,  (9)  which are repeatedly solved (using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 6) until the value of ψ has converged to the equilibrium  value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' The other minutiae of this method, including initial and boundary conditions, convergence  criteria and stability considerations, can be found in [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Synchrotron emission  Once found via the GF or ADI methods, the equilibrium distribution is used to calculate the  radio emissivity, given by  jsync(ν, r) =  � mχ  0  dE ψe±(E, r)Psync(ν, E, r) ,  (10)  where ν is the synchrotron frequency, ψe± is the sum of electron and positron equilibrium  distributions and Psync is the power emitted by an electron with an energy of E (this is calculated  as in [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' The emissivity is then used to calculate the two main results in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Firstly, the  azimuthally averaged surface brightness curves,  Isync(ν, r, Θ, ∆Ω) =  �  ∆Ω  dΩ  �  l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  dl jsync(ν, l)  4π  ,  (11)  where l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' is the line-of-sight to a point in the halo at radius r, which makes an angle of Θ with  the centre of the halo, and ∆Ω is the solid angle over which the surface brightness is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  In this work we show results for a single representative frequency of ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='5 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Secondly, we  calculate the integrated flux density by  Ssync(ν, R) =  � R  0  d3r′ jsync(ν, r′)  4πd2  L  ,  (12)  where the emissivity is integrated over the region enclosed by R and dL is the luminosity distance  to the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' For the results shown in this work we consider R to be the virial radius of the halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Results  Here we provide the details of the simulations we have performed, and show the results for two  observables: the radio surface brightness (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 11) and integrated flux (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' We use a set of  reasonable source parameters for the halo environments that respect observational constraints,  and aim to use WIMP parameter values that are representative of the many viable candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  We thus consider a large range of particle masses, from 10 to 1000 GeV, and use a set of four  annihilation channels, {bb, e+e−, µ+µ−, τ +τ −}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Since the focus of this work is on a comparison  between the two solution methods, particurly in the way that they differ with various source  targets, we show the results side-by-side and in the same manner for both targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 1 we  show the surface brightness curves for the Coma cluster (left-hand panels) and M31 (right-hand  panels), and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 2 shows the integrated fluxes from the same targets for a range of frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  10−8  10−4  100  b¯b  e+e−  10−2  10−1  100  101  10−8  10−4  100  µ+µ−  10−2  10−1  100  101  τ +τ −  Angular radius from halo centre (arcmin)  Surface Brightness (Jy/arcmin2)  10−7  10−4  10−1  b¯b  e+e−  100  102  10−7  10−4  10−1  µ+µ−  100  102  τ +τ −  Angular radius from halo centre (arcmin)  χ Mass  10 GeV  1000 GeV  Method  ADI  GF  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Surface brightness curves for the Coma galaxy cluster (left-hand panels) and the M31  galaxy (right-hand panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Each of the four panels show different annihilation channels, given  by the label in the top right of each plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' The ADI and GF methods are represented by the  red and blue colours respectively, and the region in which the results overlap are given by the  combination of these (the purple colour).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' These shaded regions represent the full mass range of  the WIMPs (from 10 to 1000 GeV), and the domain of each panel runs up until the halo’s virial  radius R (in angular units).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  600  800  1000  1200  1400  Frequency (MHz)  10−3  10−2  Flux (Jy)  Channels  bb  µ+µ−  τ +τ −  e+e−  Methods  ADI  Greens  600  800  1000  1200  1400  Frequency (MHz)  10−3  10−2  10−1  100  Channels  bb  µ+µ−  τ +τ −  e+e−  Methods  ADI  Greens  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' The integrated fluxes, calculated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 12, for the Coma galaxy cluster (left) and  the M31 galaxy (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' The different linestyles represent the two solution methods presented in  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='1, and each colour indicates the use of a different annihilation channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Discussion and conclusions  In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 1, we see generally good agreement between the GF and ADI methods, which can be  inferred from the significant amount of overlap between the curves in each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Noticeably  however, we see more disagreement (less overlap) between the methods in the M31 galaxy than  we do for the Coma galaxy cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Our explanation for this lies in the mathematical techniques  employed by each method, and how they each treat the spatial dependence of the diffusion  function in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' In the galaxy cluster environment of Coma, diffusion effects are negligible  on sufficiently large scales [10, 5], whereas in the physically smaller galaxy, diffusion effects  start to influence the surface brightness distribution at all relevant scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Since the GF and  ADI methods leverage a spatially independent and dependent diffusion function (respectively),  the resulting equilibrium distributions will tend to differ in the environments where the length  scales in question do not greatly exceed the diffusion length, as is the case for M31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' This trend  is also seen in the fluxes from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 2, which show a clear disagreement in all channels for the  M31 galaxy, and relative agreement in all channels in the Coma cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Based on these results  and the comparison of target environments presented in [5], we also expect that smaller target  environments (like the dwarf spheroidal satellite galaxies of the Milky Way) would show further  disagreement between the solution methods, as diffusion effects would be even more significant  in these environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  The other notable result we see from these simulations is that the methods differ on small  scales, even in the large Coma cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' This is significant, as the inner regions of the DM halos  are where we would observe the strongest emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' With high-resolution radio interferometers  allowing us to resolve these smaller scales, our models could be tested against the strongest  possible DM emission, allowing us to find more stringent constraints on DM properties than  previously possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' In this regard, the surface brightness curves displayed here would be especially  valuable results when determining new observational limits, as their emission profiles are highly  dependent on the spatial structure of the DM halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  With the impressive spatial resolution of telescopes like MeerKAT and the SKA, we are now  able to probe the inner regions of these DM halos – regions which have formerly been hidden  from our view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' The need for accurate modelling techniques is thus more necessary than ever  before, and the considerations presented in this work should help inform the modelling choices  made in future radio searches for DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  Acknowledgments  This work is based on the research that was supported by the National Research Foundation  of South Africa (Bursary No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 112332).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' acknowledges support from a National Research  Foundation of South Africa Thuthuka grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 117969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  References  [1] Knowles K, Cotton W D, Rudnick L et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 2022 Astronomy & Astrophysics 657 A56  [2] Beck G 2019 Galaxies 7 16  [3] Regis M, Reynoso-Cordova J, Filipovi´c M D et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 2021 Journal of Cosmology and Astroparticle Physics 2021  046  [4] Chan M 2021 Galaxies 9 11  [5] Sarkis M and Beck G 2022 The Proceedings of SAIP2021 ed Prinsloo A (UJ) pp 316–322 ISBN 978-0-620-  97693-0  [6] Bonafede A, Feretti L, Murgia M et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 2010 Astronomy and Astrophysics 513 A30  [7] �Lokas E L and Mamon G A 2003 Monthly Notices of the Royal Astronomical Society 343 401–412  [8] Ruiz-Granados B, Rubi˜no-Mart´ın J A, Florido E et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 2010 The Astrophysical Journal 723 L44–L48  [9] Tamm A, Tempel E, Tenjes P et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 2012 Astronomy & Astrophysics 546 A4  [10] Colafrancesco S, Profumo S and Ullio P 2006 Astronomy & Astrophysics 455 21–43  [11] Strong A W and Moskalenko I V 1998 The Astrophysical Journal 509 212–228  [12] Regis M, Richter L, Colafrancesco S et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 2015 Monthly Notices of the Royal Astronomical Society 448  3747–3765  [13] Press W H, Teukolsky S A and Vetterling W T 2007 Numerical Recipes: The Art of Scientific Computing 3rd  ed (Cambridge University Press) ISBN 978-0-521-88068-8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
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+Adversarial Attacks on Neural Models of Code via
+Code Difference Reduction
+Zhao Tian
+College of Intelligence and
+Computing, Tianjin University
+Tianjin, China
+tianzhao@tju.edu.cn
+Junjie Chen†
+College of Intelligence and
+Computing, Tianjin University
+Tianjin, China
+junjiechen@tju.edu.cn
+Zhi Jin
+Key Lab of High Confidence Software
+Technologies, Peking University
+Beijing, China
+zhijin@pku.edu.cn
+Abstract—Deep learning has been widely used to solve various
+code-based tasks by building deep code models based on a
+large number of code snippets. However, deep code models
+are still vulnerable to adversarial attacks. As source code is
+discrete and has to strictly stick to the grammar and semantics
+constraints, the adversarial attack techniques in other domains
+are not applicable. Moreover, the attack techniques specific to
+deep code models suffer from the effectiveness issue due to
+the enormous attack space. In this work, we propose a novel
+adversarial attack technique (i.e., CODA). Its key idea is to
+use the code differences between the target input and reference
+inputs (that have small code differences but different prediction
+results with the target one) to guide the generation of adversarial
+examples. It considers both structure differences and identifier
+differences to preserve the original semantics. Hence, the attack
+space can be largely reduced as the one constituted by the two
+kinds of code differences, and thus the attack process can be
+largely improved by designing corresponding equivalent structure
+transformations and identifier renaming transformations. Our
+experiments on 10 deep code models (i.e., two pre-trained models
+with five code-based tasks) demonstrate the effectiveness and
+efficiency of CODA, the naturalness of its generated examples,
+and its capability of defending against attacks after adversarial
+fine-tuning. For example, CODA improves the state-of-the-art
+techniques (i.e., CARROT and ALERT) by 79.25% and 72.20%
+on average in terms of the attack success rate, respectively.
+I. INTRODUCTION
+In recent years, deep learning (DL) has been widely used
+to solve code-based software engineering tasks, such as code
+clone detection [1], vulnerability prediction [2], and code com-
+pletion [3], by building DL models based on a large amount of
+training code snippets (also called deep code models). Indeed,
+deep code models have achieved notable performance and
+largely promoted the process of software development and
+maintenance [4]–[7]. In particular, some industrial products on
+deep code models have been released and received extensive
+attention, such as AlphaCode [8] and Codex [9].
+Like DL models in other areas (e.g., image processing [10]
+and speech recognition [11]), the robustness of deep code
+models is also critical [12]. However, the existing adversarial
+attack techniques proposed in other areas are not applicable to
+deep code models. This is because these techniques perturb an
+input in a continuous space for altering the model prediction
+result, while the inputs (i.e., source code) for deep code models
+†Junjie Chen is the corresponding author.
+are discrete. Moreover, source code has to strictly stick to
+the grammar and semantics constraints, i.e., the adversarial
+example generated from an original input should have no
+grammar errors and preserve the original semantics.
+Indeed, some adversarial attack techniques specific to deep
+code models have been proposed recently, such as MHM [13],
+CARROT [12], and ALERT [14]. In general, they share two
+main steps: (1) designing a series of semantic-preserving code
+transformation rules (e.g., identifier renaming or dead code
+insertion), and (2) searching ingredients from the space defined
+by the rules (e.g., a valid identifier name is an ingredient for
+the rule of identifier renaming) for transforming an input to a
+semantic-preserving adversarial example. For example, CAR-
+ROT designs two semantic-preserving code transformation
+rules (i.e., identifier renaming and dead code insertion), and
+uses the hill-climbing algorithm to search for the ingredients
+from the entire space with the guidance of gradients and
+changes of model prediction results. ALERT considers the rule
+of identifier renaming, and uses the naturalness (i.e., natural
+semantics of code) and changes of model prediction results to
+guide the ingredient search process from the entire space.
+Although some of them have been demonstrated to be
+effective to some degree, these existing techniques still suffer
+from major limitations:
+• The ingredient space defined by the code transformation
+rules is enormous. For example, for the rule of identifier
+renaming, all valid identifier names could be the ingredi-
+ents for renaming the target identifier. Hence, searching
+for the ingredients that can help attack the target model
+successfully is challenging. The existing techniques tend
+to utilize the changes of model prediction results after
+performing semantic-preserving transformations on the
+target input to guide the search process, which is very
+likely to fall into local optimum in the enormous space
+and thus limits their attack effectiveness.
+• Frequently invoking the target model can negatively af-
+fect the efficiency of adversarial attack techniques, as
+model invocation is the most costly part during the
+attack process [12]. Also, when the model is deployed
+remotely, frequent model invocations could be identified
+as malicious attacks and thus lead to blocking access to
+the model. However, the existing techniques often involve
+1
+arXiv:2301.02412v1  [cs.CR]  6 Jan 2023
+
+frequent model invocations due to calculating gradients
+or guiding the search direction via model prediction.
+• Developers care about the natural semantics of code since
+it is helpful to assist human comprehension [15]. Hence,
+guaranteeing the naturalness of generated adversarial ex-
+amples (i.e., source code in our task) is important. How-
+ever, all the existing techniques (except ALERT [14]) do
+not consider this factor. For example, CARROT designs
+the rule of dead code insertion, but it may largely damage
+the naturalness of the generated examples (especially
+when a large amount of dead code is inserted).
+Overall, a more effective adversarial attack technique spe-
+cific to deep code models should enhance the attack effective-
+ness by improving the ingredient search process, and guarantee
+the naturalness of generated adversarial examples as much as
+possible and the times of model invocations as few as possible.
+Our work does propose such a technique, called CODA (COde
+Difference guided Attacking).
+To improve the attack effectiveness, the key idea of CODA
+is to use the inputs, which have small code differences with
+the target input but have different prediction results, to largely
+reduce the ingredient space. For ease of presentation, we call
+such inputs reference inputs. Actually, reference inputs can
+be regarded as invalid successfully-attacking adversarial ex-
+amples generated from the target input, where “invalid” refers
+to altering the original semantics and “successfully-attacking”
+refers to producing different prediction results. Over the target
+input, the code differences brought by reference inputs con-
+tribute to the invalid but successful attack to a large extent.
+Hence, if we extract the ingredients from the code differences
+to support semantic-preserving transformations on the target
+input, their code differences can be gradually reduced without
+altering the original semantics, and thus a valid successfully-
+attacking adversarial example is likely to be generated. In
+this way, the ingredient space is effectively reduced as the
+one constituted by only code differences between reference
+inputs and the target input, and thus the search process
+can be largely improved. Please note that taking reference
+inputs (especially code differences brought by them) as the
+guidance for generating adversarial examples is an innovative
+perspective, which closely utilizes the unique characteristics
+of deep code models (e.g., source code is discrete).
+To preserve the semantics of the target input during the
+attack process, CODA considers code structure differences
+and identifier differences, and thus extracts the ingredients to
+support equivalent structure transformations and identifier re-
+naming transformations. Equivalent structure transformations
+(e.g., transforming a for loop to an equivalent while loop)
+do not affect the naturalness of generated examples, and thus
+CODA first applies this kind of transformations to reduce
+code differences for generating adversarial examples. Then,
+identifier renaming transformations are applied to further
+reduce code differences to improve the attack effectiveness.
+To ensure the naturalness of generated examples by this
+kind of transformations, CODA measures semantic similarity
+between identifiers for guiding iterative transformations. In
+particular, CODA just involves necessary model invocations
+to check whether the generated example attacks successfully,
+without extra gradient calculation and a large amount of model
+prediction for guiding the search process.
+We conducted an extensive study to evaluate CODA based
+on two popular pre-trained models (i.e., CodeBERT [6] and
+GraphCodeBERT [7]) and five code-based tasks (i.e., vul-
+nerability prediction [2], clone detection [16], authorship
+attribution [17], functionality classification [18], and defect
+prediction [12]). In total, we used 10 subjects. Our results
+demonstrate the effectiveness and efficiency of CODA. For
+example, on average across the ten subjects, CODA improves
+the two state-of-the-art adversarial attack techniques specific
+to deep code models (i.e., CARROT [12] and ALERT [14])
+by 79.25% and 72.20% respectively in terms of the attack
+success rate. The time spent by CODA on completing the
+attack process for the ten subjects is just 39.59 hours, while
+those by CARROT and ALERT are 159.19 hours and 198.89
+hours, respectively. Also, we investigated the value of the
+generated adversarial examples by using them to improve
+the robustness of the target model via an adversarial fine-
+tuning strategy. The results show that the models after fine-
+tuning with the examples generated by CODA can successfully
+defend against attacks from 63.64%, 66.96%, and 76.68%
+of adversarial examples generated by CARROT, ALERT, and
+CODA on average, respectively. Besides, we conducted a user
+study to confirm the naturalness of the generated examples by
+CODA and an ablation experiment to confirm the contribution
+of each main component in CODA.
+To sum up, our work makes the four major contributions:
+• Novel Perspective. We propose a novel perspective of
+utilizing code differences between reference inputs and
+the target input to guide the adversarial attack process
+for deep code models.
+• Technique Implementation. We implement CODA fol-
+lowing the novel perspective by measuring code structure
+and identifier differences and designing the corresponding
+semantic-preserving code transformation rules.
+• Performance Evaluation. We conducted an extensive
+study on two popular pre-trained models and five code-
+based tasks, demonstrating the effectiveness and effi-
+ciency of CODA over two state-of-the-art techniques.
+• Public Artifact. We released all the experimental data
+and our source code at the project homepage [19] for
+experiment replication, future research, and practical use.
+II. BACKGROUND AND MOTIVATION
+In this section, we first introduce the background of deep
+code models (Section II-A), define our problem (Section II-B),
+and motivate our key idea with an example (Section II-C).
+A. Deep Code Models
+In the area of software engineering, DL has been widely-
+used to process source code [2], [3], [16], [20]. In particular,
+some popular pre-trained DL models have been constructed
+based on a large number of code snippets, among which
+2
+
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+void f1(int a[], int n){
+  int i; int j; int k;
+  for (i = 0; i < n; i++) {
+    for (j = 0; j < ((n - i) - 1); j++) {
+      if (a[j] > a[j + 1]){
+        k = a[j];
+        a[j] = a[j + 1];
+        a[j + 1] = k;
+      }
+    }
+  }
+}
+int f2(int t[], int len){
+  int i; int j;
+  i = 0; j = 0;
+  while (len != 0) {
+    t[i] = len % 10;
+    len /= 10;
+    i = i + 1;
+  } 
+  while (j < i){
+    if (t[j] != t[(i - j) - 1]) return 0;
+    j = j + 1;
+  }
+  return 1;
+}
+void f3(int t[], int len){
+  int i; int j; int k;
+  i = 0;
+  while (i < len) {
+    j = 0;
+    while (j < ((len - i) - 1)) {
+      if (t[j] > t[j + 1]){
+        k = t[j];
+        t[j] = t[j + 1];
+        t[j + 1] = k;
+      } j = j + 1;
+    } i = i + 1;
+  }
+}
+Ground-truth Label: sort
+Prediction Result: sort (96.52%)
+Ground-truth Label: palindrome
+Prediction Result: palindrome (99.98%)
+Ground-truth Label: sort
+Prediction Result: palindrome (90.88%)
+Fig. 1. An illustrating example (the target input f1, a reference input f2, and a successfully-attacking adversarial example f3 generated from f1)
+CodeBERT [6] and GraphCodeBERT [7] are two state-of-the-
+art pre-trained models. CodeBERT learns features from bi-
+modal data in the form of programming languages and natural
+languages, while GraphCodeBERT takes into consideration the
+code structure and data flow information. Same as the existing
+work [14], we used them in our evaluation (Section IV).
+These
+pre-trained
+models
+have
+brought
+breakthrough
+changes to many downstream code-based tasks [21], including
+both classification tasks and generation tasks, by fine-tuning
+them on the datasets of the corresponding tasks. The former
+makes classification based on the given code snippets (e.g.,
+clone detection [16] and vulnerability prediction [2]), while the
+latter produces a sequence of information based on code snip-
+pets or natural language descriptions (e.g., code completion [3]
+and code summarization [22]). Following most of the existing
+work on attacking deep code models [12]–[14], our work also
+focuses on the classification tasks and takes the generation
+tasks as our future work. In particular, in our study, we adopted
+all the tasks used in the studies of evaluating the state-of-the-
+art attack techniques (i.e., CARROT [12] and ALERT [14]),
+i.e., five classification tasks including vulnerability prediction,
+clone detection, authorship attribution, functionality classifica-
+tion, and defect prediction.
+B. Problem Definition
+Given a code snippet x that is processed as the required
+format by the target deep code model M (e.g., abstract syntax
+trees required by code2seq [4], control-flow graphs required
+by DGCNN [23], or data-flow graphs required by GraphCode-
+BERT [7]), M can predict a probability vector for x, each
+element in which represents the probability classifying x to
+the corresponding class. The class with the largest probability
+is the final prediction result of M for x. If the prediction result
+is different from the ground-truth label (denoted as y) of x, it
+means that M makes a wrong prediction on x; otherwise, M
+makes a correct prediction.
+Although deep code models can achieve great performance
+on the given test sets, they may be vulnerable to adversarial
+examples [12]–[14]. The goal of our work is to generate
+successfully-attacking adversarial examples as much as pos-
+sible, so as to improve the model robustness. As source code
+is discrete and has to stick to the grammar and semantics con-
+straints, the existing adversarial example generation techniques
+proposed in other domains are not applicable.
+The existing attack techniques specific to deep code models
+always generate adversarial examples from a target input by
+performing a series of semantic-preserving code transforma-
+tions [12]–[14], which is also followed by our work. For ease
+of understanding, we formally define our target problem is to
+find {x′|x′ ∈ ϵ ∧ y = M(x) ̸= M(x′)} from a target input x
+for the target model M. Here, ϵ refers to the universal set of
+code snippets that satisfy the grammar constraints and preserve
+the semantics of x. y = M(x) means that we just regard the
+test inputs on which M makes correct predictions as target
+inputs, where M(x) refers to the prediction result of M on
+x. M(x) ̸= M(x′) means that x′ successfully attacks M, that
+is, it is a successfully-attacking adversarial example generated
+from x. Besides, an effective attack technique should be also
+efficient to find x′ and ensure the naturalness of x′ (i.e., natural
+to human comprehension [14]), which are indeed carefully
+considered by the proposed technique in our work.
+C. Motivating Example
+We then use a real-world example (simplified for ease of
+illustration) to help motivate our key idea: utilizing the code
+differences between reference inputs and the target input to
+guide the generation of adversarial examples. In Figure 1, the
+first code snippet f1 is the target input from the test set of
+the functionality classification task [18], and the two state-
+of-the-art techniques (i.e., CARROT [12] and ALERT [14])
+do not generate successfully-attacking adversarial examples
+from it since they can fall into local optimum in the enormous
+ingredient space. In this figure, the second code snippet f2 is
+a reference input from the training set of this task, which has
+the different label with f1.
+In fact, f2 can be regarded as an invalid successfully-
+attacking example from f1, as they are semantically inconsis-
+tent but have different prediction results. The code differences
+between f1 and f2 mainly contribute to this phenomenon. From
+this perspective, to generate a valid successfully-attacking
+adversarial example (denoted as f3) from f1, we should per-
+form semantic-preserving code transformations on f1, and the
+transformations should reduce the code differences between f1
+and f2 in order to alter the prediction result of the target model
+on f1. That is, the ingredients supporting these transformations
+3
+
+Initial
+Snippet
+Adversarial
+Snippet
+Input
+Output
+Training 
+Data
+Input
+Target
+Model
+Test
+Test
+Report
+Attack
+Success ?
+Fig. 2. Overview of CODA.
+should be extracted from the code differences brought by
+f2. With this intuition, by performing equivalent structure
+transformations on f1 (i.e., transforming for loops to while
+loops, where while loops are the used loop structure in f2)
+and identifier remaining transformations (i.e., renaming a and
+n to t and len respectively, where t and len are the used
+identifier names in f2), f3 is generated as shown in the third
+code snippet in Figure 1 and indeed attacks successfully,
+i.e., making a wrong prediction (palindrome) with a high
+confidence (90.88%).
+Based on the code differences between the target input and
+the reference input, the ingredient space is largely reduced. For
+example, the ingredient space defined by identifier renaming
+transformations is reduced from all valid identifier names
+(i.e., almost infinite) to the identifier names occurring in the
+reference input but not in the target input (i.e., only two
+identifier names in this simplified example). Hence, it can help
+improve the ingredient search process and thus improve the
+attack effectiveness. On the other hand, too small ingredient
+space could also lose too many ingredients useful to successful
+attacks, and thus we will select a set of reference inputs (rather
+than only one reference input) for guiding the attack process
+in order to balance the size of the ingredient space and the
+number of useful ingredients.
+III. APPROACH
+A. Overview
+In this work, we propose a novel perspective to attack
+deep code models more effectively and more efficiently, which
+utilizes the code differences between reference inputs and the
+target input to guide the generation of adversarial examples.
+From this perspective, we design an effective attack technique,
+called CODA. Specifically, the code differences brought by
+reference inputs provide effective ingredients for altering the
+prediction result of the target input by transforming it with
+these ingredients, which can contribute to successful attacks in
+CODA. However, as the semantics of reference inputs and the
+target input are different, the ingredients from some kinds of
+code differences can alter the original semantics, which is not
+allowed by the adversarial attack for deep code models. Hence,
+in CODA, we consider the structure differences and identifier
+differences for measuring code differences between them,
+which can preserve the original semantics during the attack
+process. In this way, the ingredient space can be effectively
+reduced as the one constituted by the two kinds of code
+differences between reference inputs and the target input, and
+thus the ingredient search process (for generating adversarial
+examples) can be largely improved.
+In fact, not all the inputs that have different prediction
+results with the target one, can be regarded as effective
+reference inputs for improving the adversarial attack process.
+In other words, different inputs could have different degrees
+of capabilities for reducing the ingredient search space and
+providing effective ingredients for altering the prediction result
+of the target input. Therefore, the first step in CODA is
+to select effective reference inputs for the target input in
+order to improve the attack effectiveness as much as possible
+(to be presented in Section III-B). Based on the selected
+reference inputs, CODA then measures the structure differ-
+ences and identifier differences over the target input, which
+support extracting the ingredients for two corresponding kinds
+of semantic-preserving code transformations (i.e., equivalent
+structure transformations and identifier renaming transforma-
+tions). With the guidance of reducing their code differences
+based on the two kinds of transformations, the target input
+could be effectively transformed to a successfully-attacking
+adversarial example. As equivalent structure transformations
+do not affect the naturalness of generated examples, CODA
+first applies this kind of transformations to reduce the code
+differences for improving the attack effectiveness (to be pre-
+sented in Section III-C). Then, we apply identifier renaming
+transformations to further reduce the code differences for
+improving the generation of successfully-attacking adversarial
+examples (to be presented in Section III-D). In particular,
+CODA measures the semantic similarity between identifiers
+to guarantee the naturalness of generated examples.
+Figure 2 shows the overview of CODA. In a nutshell, by
+successively applying equivalent structure transformations and
+identifier renaming transformations to the target input with
+the ingredient space defined by the code differences between
+the selected reference inputs and the target one, adversarial
+examples can be generated towards the direction of reducing
+the code differences without altering the original semantics. In
+this way, the prediction result of the target input is more likely
+to be changed, leading to a successfully-attacking adversarial
+example. Due to the smaller ingredient search space (but
+including effective ingredients) and the clearer attack direction,
+the attack effectiveness could be largely improved by CODA.
+B. Reference Inputs Selection
+The goal of reference inputs is to largely reduce the in-
+gredient space. Also, the reduced space should include the
+ingredients that are effective to transform the target input
+to a successfully-attacking adversarial example. In this way,
+the adversarial attack process can be largely improved by
+searching for effective ingredients more efficiently.
+4
+
+TABLE I
+DESCRIPTIONS OF EQUIVALENT STRUCTURE TRANSFORMATIONS
+Transformation
+Description
+Example Before Transformation
+Example After Transformation
+R1-loop
+equivalent transformation among for structure
+for ( i=0; i<9; i++ ) {
+i=0; while ( i<9 ) {
+and while structure
+Body; }
+Body; i++; }
+R2-branch
+equivalent transformation between if-else(-if)
+if ( A ) { BodyA; }
+if ( A ) { BodyA; }
+structure and if-if structure
+else if ( B ) { BodyB; }
+if ( !A && B ) { BodyB; }
+R3-calculation
+equivalent numerical calculation transformation, e.g.,
+i += 1;
+i = i + 1;
+++, --, +=, -=, *=, /=, %=, <<=, >>=, &=, |= , ˆ =
+R4-constant
+equivalent transformation between a constant and
+println("Hello, World!");
+String i = "Hello, World!";
+a variable assigned by the same constant
+println(i);
+Although all the inputs that have different prediction results
+with the target one can provide ingredients for altering the pre-
+diction result of the target one after transformations, their capa-
+bilities for successful attacks could be different. To transform
+the target input to a successfully-attacking adversarial example
+with fewer perturbations, CODA should select the reference
+inputs, which can provide the ingredients that are more likely
+to conduct successful attacks for the target input. Similar to the
+existing work [24]–[27], we assume that the prediction result
+of the target input is more likely to be changed from its original
+class denoted as ci (with the largest probability predicted
+by the target model) to the class with the second largest
+probability (denoted as cj). Hence, the ingredients in the inputs
+belonging to cj are more likely to attack successfully on the
+target input, and thus CODA selects the inputs belonging to
+cj as the initial set of reference inputs. Please note that all
+the reference inputs are selected from the training set to avoid
+introducing the contents beyond the cognitive scope of the
+target model. Meanwhile, we only consider the training inputs
+whose prediction results are consistent with their ground-truth
+labels in order to avoid introducing noise.
+However, the number of inputs belonging to the same class
+(i.e., cj as above) could be large, and thus the ingredient space
+constituted by code differences between them and the target
+input could be also large. Hence, to further reduce the ingre-
+dient space for more effective adversarial example generation,
+CODA selects a subset of inputs with high similarity to the
+target input from the initial set of reference inputs, as the
+final set of reference inputs used by CODA. This is because
+smaller code differences can effectively limit the number of
+ingredients, leading to smaller ingredient space. CODA does
+not select only one reference input, as too small ingredient
+space could incur a high risk of missing too many ingredients
+contributing to successful attacks. That is, CODA selects a
+small set of reference inputs following the above two steps of
+selection to balance the ingredient space size and the amount
+of ingredients contributing to successful attacks in the space.
+We further introduce how to measure the similarity be-
+tween the target input (denoted as t) and a reference input
+(denoted as r) for the second step of selection in CODA. In
+general, we can adopt some pre-trained models to represent
+the code as a vector and then measure code similarity by
+calculating the vector distance, like many existing studies [5],
+[6], [28]. However, as presented in Section III-A, CODA
+first applies equivalent structure transformations (rather than
+identifier renaming transformations) to reduce code differences
+for adversarial attacks, as this kind of transformations does not
+affect the naturalness of generated examples. Moreover, the
+identifiers used in different code snippets are usually different
+due to the enormous identifier space, which may lead to the
+low similarity between various code snippets. Hence, when
+measuring code similarity, CODA eliminates the influence
+of identifiers by replacing them with the placeholder <unk>.
+Specifically, CODA first represents t and r after placeholder
+replacement as vectors respectively based on CodeBERT [6]
+(one of the most widely-used pre-trained models [29]–[31]),
+and then calculates the cosine similarity between the two
+vectors. As the descending order of the calculated similarity,
+CODA selects Top-N reference inputs for the follow-up ad-
+versarial attack process. Please note that to make the selection
+process efficient, we randomly sampled U inputs from the
+initial set for the second step of selection. We will investigate
+the influence of both U and N on CODA in Section VI.
+C. Equivalent Structure Transformation
+Based on the small set of selected reference inputs, CODA
+then extracts ingredients from the space defined by their
+brought code differences over the target input. That is, CODA
+transforms the target input to an adversarial example towards
+the direction of reducing code differences. CODA first reduces
+structure differences by applying equivalent structure transfor-
+mations to the target input as they do not affect the naturalness
+of generated examples.
+To preserve the semantics of the target input, we design four
+categories of equivalent structure transformations in CODA
+inspired by the existing work in metamorphic testing and
+code refactoring [32], [33]. In particular, we systematically
+consider all common kinds of code structures, i.e., loop struc-
+tures, branch structures, and sequential structures (including
+numerical calculation and constant usage). We explain the four
+categories in detail in Table I, each of which is also illustrated
+with an example. For each category of transformations, it
+may include several specific rules. For example, the rules
+of transformation on += and transformation on -= belong to
+the category of R3-calculation, and the rules of transforming
+for loop to while loop and transforming while loop to for
+loop belong to the category of R1-loop. In total, CODA has
+20 specific rules for the four categories of transformations.
+Please note that not all the rules are applicable to the code
+5
+
+programmed by any programming language. For example, ++
+and -- in R3-calculation are not supported by Python. Also,
+in R4-constant, the newly-defined variable cannot be the same
+as the existing variables in the code; otherwise, it may incur
+grammar errors and alter the original semantics. Due to the
+space limit, we put more details about all these specific rules
+at our project homepage [19].
+Then, we illustrate how to apply each rule for reducing code
+differences. Each rule involves two structures, i.e., the one
+before transformation (sb) and the one after transformation
+(sa). CODA first counts the occurring times of sb and sa in
+the set of selected reference inputs (denoted as nb and na),
+and then calculates their occurring distribution, i.e.,
+nb
+nb+na
+and
+na
+nb+na . Further, CODA applies each rule in a probabilistic
+way to reduce the occurring distribution differences in terms
+of sb and sa between reference inputs and the target input. In
+this way, the structure differences in terms of sb and sa can
+be reduced effectively. More specifically, for each occurrence
+of sb in the target input, CODA applies this rule with the
+probability of
+na
+nb+na , also indicating that it can be retained
+with the probability of
+nb
+nb+na .
+In this step, CODA obtains M inputs from the target input,
+each of which is generated by applying all the applicable rules
+in the above probabilistic way, and then selects the input with
+the highest average similarity (also measured by the method
+described in Section III-B) to the selected reference inputs as
+the one for the follow-up adversarial attack process.
+D. Identifier Renaming Transformation
+To facilitate the generation of successfully-attacking ad-
+versarial examples, CODA then applies identifier renaming
+transformations to further reduce code differences. Inspired by
+the existing work [12]–[14], identifier renaming transformation
+in CODA refers to replacing the name of an identifier in the
+target input with the name of an identifier in the selected
+reference inputs. For ease of presentation, we denote the set of
+identifiers in the target input as Vt and the set of identifiers in
+the selected reference inputs as Vr. To preserve the semantics
+of the target input and guarantee the grammatical correctness
+of the generated example, CODA ensures that the identifier
+used for replacement does not exist in the target input.
+Then, we illustrate how to apply this kind of transformations
+to the input obtained from the last step (i.e., equivalent
+structure transformations). As demonstrated by the existing
+work [12]–[14], renaming identifiers is effective to generate
+successfully-attacking adversarial examples, but can negatively
+affect the naturalness of generated examples. To ensure the
+naturalness of generated examples, CODA considers the se-
+mantic similarity between identifiers and designs an iterative
+transformation process like ALERT [14]. Specifically, CODA
+measures the semantic similarity between each identifier in
+Vt and each identifier in Vr by representing each identifier as
+a vector via word embedding. Here, CODA builds the pre-
+trained language model with the FastText algorithm [34] and
+calculates the cosine similarity between vectors to measure
+their semantic similarity. Then, CODA prioritizes each pair
+TABLE II
+STATISTICS OF OUR USED SUBJECTS
+Task
+Train/Validate/Test
+Class
+Language
+Model
+Acc.
+Vulnerability
+21,854/2,732/2,732
+2
+C
+CB
+63.76%
+Prediction
+GCB
+63.65%
+Clone
+90,102/4,000/4,000
+2
+Java
+CB
+96.97%
+Detection
+GCB
+97.36%
+Authorship
+528/–/132
+66
+Python
+CB
+90.35%
+Attribution
+GCB
+89.48%
+Functionality
+41,581/–/10,395
+104
+C
+CB
+98.18%
+Classification
+GCB
+98.66%
+Defect
+27,058/–/6,764
+4
+C/C++
+CB
+84.37%
+Prediction
+GCB
+83.98%
+* CB is short for CodeBERT and GCB is short for GraphCodeBERT.
+of identifiers as the descending order of their semantic sim-
+ilarity, and iteratively applies this transformation based on
+each pair of identifiers in the ranking list, which ensures
+that more natural transformations can be first performed.
+After each iteration, CODA invokes the target model to
+check whether a successfully-attacking adversarial example
+is generated. The iterative attack process terminates until a
+successfully-attacking adversarial example is generated or all
+the pairs are used by this transformation. Please note that
+CODA ensures that the pair of identifiers will not introduce
+repetitive identifiers in the generated example in each iteration;
+otherwise, this pair will be discarded.
+Overall, CODA only invokes the target model when check-
+ing if a successfully-attacking adversarial example is gen-
+erated. They are necessary model invocations for this task.
+Hence, CODA can largely reduce the number of model
+invocations compared with the existing techniques (e.g.,
+ALERT [14]), which is confirmed by our study (Section V-A).
+IV. EVALUATION DESIGN
+In the study, we address four research questions (RQs):
+• RQ1: How does CODA perform in terms of effectiveness
+and efficiency compared with state-of-the-art techniques?
+• RQ2: Are the adversarial examples generated by CODA
+natural for humans?
+• RQ3: Are the adversarial examples generated by CODA
+useful to improve the robustness of deep code models?
+• RQ4: Does each main component contribute to the over-
+all effectiveness of CODA?
+A. Subjects
+1) Datasets and Tasks: To sufficiently evaluate CODA,
+we consider all the five code-based tasks in the studies of
+evaluating state-of-the-art techniques (i.e., CARROT [12] and
+ALERT [14]). The statistics of datasets is shown at the first
+four columns in Table II, each of which represents the task, the
+number of inputs in the training/validation/test set, the number
+of classes for the classification task, and the programming
+language for the inputs.
+The task of vulnerability prediction aims to predict whether
+a given code snippet has vulnerabilities. Its used dataset is ex-
+tracted from two C projects (i.e., FFmpeg [35] and Qemu [36])
+by Zhou et al. [2] and has been integrated as part of the
+CodeXGLUE benchmark [30]. The task of clone detection
+6
+
+aims to detect whether two given code snippets are equivalent
+in semantics. Its used dataset is from BigCloneBench [37],
+the most widely-used dataset for clone detection. The existing
+work [14] randomly sampled 90,102/4,000/4,000 inputs from
+the benchmark for training/validation/testing, to make the
+experiment at a computationally friendly scale. In our study,
+we used the same dataset. The task of authorship attribution
+aims to identify the author of a given code snippet. Its used
+dataset is the Google Code Jam (GCJ) dataset [17]. The task
+of functionality classification aims to classify the functionality
+of a given code snippet. If code snippets solve the same
+problem, they are regarded to have the same functionality [18].
+Its used dataset is the Open Judge (OJ) benchmark [38],
+which has been also integrated as part of the CodeXGLUE
+benchmark [30]. The task of defect prediction aims to predict
+whether a given code snippet is defective and its defect type.
+Its used dataset is the CodeChef dataset [39], which is labeled
+by the execution results on the CodeChef platform (i.e., four
+defect types: no defect, wrong answer, timeout, runtime error).
+2) Models: Following the existing work [14], we adopted
+two state-of-the-art pre-trained models for code-based tasks,
+i.e., CodeBERT [6] and GraphCodeBERT [7], and then fine-
+tuned them on the five tasks based on the corresponding
+datasets, respectively. In total, we obtained 10 deep code
+models as the subjects. The last two columns in Table II show
+the used pre-trained model and the accuracy of the deep code
+model after fine-tuning, respectively. When fine-tuning Code-
+BERT and GraphCodeBERT on these tasks (except Graph-
+CodeBERT on functionality classification and defect predic-
+tion), we used the same hyper-parameter settings provided by
+the existing work [12], [14]. As there is no instruction on the
+hyper-parameter settings for fine-tuning GraphCodeBERT on
+functionality classification and defect prediction, we used the
+same settings as the one used by authorship attribution (they
+are all multi-class classification tasks). Indeed, the achieved
+model performance outperforms that achieved by the models
+(e.g., TBCNN [38] and CodeBERT [6]) used in the existing
+work [12] on the same datasets [12], indicating that the
+transferred hyper-parameter settings are reasonable.
+Overall, the subjects used in our study are indeed diverse,
+involving different tasks, different pre-trained models, different
+numbers of classes, different programming languages, etc. It is
+very helpful to sufficiently evaluate the performance of CODA.
+B. Compared Techniques
+In the study, we compared CODA with two state-of-the-art
+techniques attacking deep code models, i.e., CARROT [12]
+and ALERT [14], which have been introduced in Section I (the
+third paragraph). We adopted their implementations and the
+recommended parameter settings provided by the correspond-
+ing papers [12], [14]. As the original version of CARROT can
+only support to attack C/C++ code, we extended it to attack
+Python and Java code for sufficient comparison.
+C. Implementations
+We implemented CODA in Python and adopted tree-
+sitter [40] to extract identifiers from code following the exist-
+ing work [14]. We set the parameters in CODA by conducting
+a preliminary experiment, i.e., U = 256, N = 64, and
+M = 64. We discuss the influence of the settings of main
+parameters on CODA in Section VI. We released our code
+and experimental data at our project homepage [19]. All the
+experiments were conducted on a server with an Ubuntu 20.04
+system with Intel(R) Xeon(R) Silver 4214 @ 2.20GHz CPU,
+256GB memory, and NVIDIA GeForce RTX 2080 Ti GPU.
+V. RESULTS AND ANALYSIS
+A. RQ1: Effectiveness and Efficiency
+1) Setup: For each deep code model, we applied CODA,
+CARROT, ALERT to generate adversarial examples from each
+target input in the test set, respectively. We measured their
+effectiveness and efficiency based on the following metrics.
+To reduce the influence of randomness, we repeated all the
+experiments (including those for other RQs) 10 times, and
+reported the average results.
+Following the existing work [14], we adopted the at-
+tack success rate (ASR) to measure the effectiveness of
+each technique. ASR is the percentage of the target inputs
+from which an attack technique can generate a successfully-
+attacking adversarial example. Larger ASR values mean better
+attack effectiveness. Also, it is important to measure whether
+the prediction confidence (i.e., the probability of being the
+ground-truth class of the target input) is decreased by the gen-
+erated examples (although there is no successfully-attacking
+adversarial example generated from a target input). Hence, we
+also calculated prediction confidence decrement (PCD) to
+measure the effectiveness of each technique. PCD is calculated
+by the prediction confidence of the target input minus the min-
+imum prediction confidence of the set of generated examples
+from the target input. If the former is smaller than the latter,
+we regard PCD to be 0, indicating that the generated examples
+cannot decrease the prediction confidence of the target input.
+Larger PCD values mean better attack effectiveness.
+In addition, following the existing work [12], [14], we used
+the time spent on the overall attack process (i.e., completing
+the adversarial example generation process from all the target
+inputs) and the average number of model invocations for
+generating examples from one target input, to measure
+the efficiency of each technique. Less time and fewer model
+invocations mean higher efficiency.
+2) Results: Table III shows the comparison results among
+CARROT, ALERT, and CODA in terms of ASR. From this
+table, CODA always outperforms CARROT and ALERT on
+all the tasks based on both CodeBERT and GraphCodeBERT,
+demonstrating the stable attack effectiveness of CODA. On
+average, CODA improves 70.11% and 89.83% higher ASR
+than CARROT and ALERT across all the five tasks on
+CodeBERT, and improves 89.34% and 57.67% higher ASR
+on GraphCodeBERT, respectively.
+Figures 3(a) and 3(b) show the comparison results among
+the three techniques in terms of PCD on CodeBERT and
+GraphCodeBERT, respectively. From these figures, the upper
+quartile, median, and lower quartile of CODA are always
+7
+
+TABLE III
+EFFECTIVENESS COMPARISON IN TERMS OF ASR
+Task
+CodeBERT
+GraphCodeBERT
+CARROT
+ALERT
+CODA
+CARROT
+ALERT
+CODA
+Vulnerability Prediction
+33.72%
+53.62%
+89.58%
+37.40%
+76.95%
+94.72%
+Clone Detection
+20.78%
+27.79%
+44.65%
+3.50%
+7.96%
+27.37%
+Authorship Attribution
+44.44%
+35.78%
+79.05%
+31.68%
+61.47%
+92.00%
+Functionality Classification
+44.15%
+10.04%
+56.74%
+42.76%
+11.22%
+57.44%
+Defect Prediction
+71.59%
+65.15%
+95.18%
+79.08%
+75.87%
+96.58%
+Average
+42.94%
+38.48%
+73.04%
+38.88%
+46.69%
+73.62%
+(a) PCD on attacking CodeBERT
+(b) PCD on attacking GraphCodeBERT
+Fig. 3. Comparison in terms of prediction confidence decrement
+(a) Model invocation times on attacking CodeBERT
+(b) Model invocation times on attacking GraphCodeBERT
+Fig. 4. Comparison in terms of model invocation times (y-axis refers to the normalized values following the existing work [14])
+larger than (or equal to) those of both CARROT and ALERT
+regardless of the tasks and the pre-trained models, demon-
+strating that CODA produces more significant attacks for
+decreasing prediction confidence of target inputs. For example,
+on CodeBERT, the average improvements of CODA over
+CARROT and ALERT are 101.88% and 520.65% across all
+the tasks in terms of average PCD, respectively. Similarly, on
+GraphCodeBERT, the average improvements of CODA over
+CARROT and ALERT are 76.35% and 560.15%, respectively.
+Besides, CARROT, ALERT, and CODA take 159.19 hours,
+198.89 hours, and 39.59 hours to complete the entire attack
+process on all five tasks, respectively. Further, we measured
+the number of model invocations for each target input during
+the attack process, whose results are shown in Figure 4(a)
+and 4(b). From these figures, CODA performs fewer model
+invocations than both CARROT and ALERT regardless of the
+tasks and pre-trained models. On average, CODA performs
+65.73% and 78.58% fewer model invocations than CARROT
+and ALERT across all the tasks on CodeBERT, and 34.07%
+and 75.31% fewer model invocations on GraphCodeBERT,
+respectively. The results demonstrate that CODA has the
+significantly highest efficiency among the three techniques.
+Answer to RQ1: CODA spends less time with fewer
+model invocations on completing the entire attack
+process, but generates more successfully-attacking ex-
+amples with more significant prediction confidence
+decrement on all the subjects, than the state-of-the-art
+techniques (i.e., CARROT and ALERT).
+B. RQ2: Naturalness of Adversarial Examples
+1) Setup: It is important to check whether the generated
+adversarial examples are natural to human judges [14], [41].
+Here, we conducted a user study to compare the naturalness
+of examples generated by CODA, CARROT, and ALERT, and
+our user study shares the same design as the one conducted
+by the existing work [14]:
+Data Preparation. For each subject, we randomly sampled
+10 target inputs, and then for each technique we randomly
+sampled an adversarial example from the set of examples
+generated from each sampled target input. That is, for each
+sampled target input, we construct three pairs of code snippets,
+each of which contains the target input and an adversarial
+example generated by CODA, CARROT, or ALERT. In total,
+we obtained 300 pairs of code snippets for the user study due
+to 10 subjects × 3 techniques.
+Participants. Same as the existing work [14], the user study
+also involves four non-author participants, each of whom has
+8
+
+Fig. 5. Average score to evaluate naturalness of examples per participant
+a Bachelor/Master degree in Computer Science with at least
+five years of programming experience.
+Process. For objective evaluation, we did not tell partici-
+pants which technique generates the adversarial example in a
+pair of code snippets. Also, we highlighted the changes in each
+pair of code snippets for facilitating manual evaluation. Then,
+each participant individually evaluated each pair by evaluating
+to what extent the changes are natural to the code context
+and the changed identifiers preserve the original semantics,
+following the existing work [14]. Specifically, participants
+gave a score for each pair based on a 5-point Likert scale [42]
+(1 means strongly disagree and 5 means strongly agree)
+following the existing work [14], [43].
+2) Results: Figure 5 shows the average score of the ad-
+versarial examples generated by each technique for each
+participant. From this figure, the conclusions from different
+participants are consistent: the naturalness of the adversarial
+examples generated by CODA and ALERT is closely high
+(round 4.50 on average), and significantly higher than that by
+CARROT (just 2.89 on average). ALERT is a naturalness-
+aware technique, whose core contribution is to ensure the
+naturalness of generated examples, but CODA achieves similar
+naturalness scores to it, demonstrating that CODA can also
+generate highly natural adversarial examples.
+Answer to RQ2: The adversarial examples gener-
+ated by CODA are natural closely to the state-of-the-
+art naturalness-aware attack technique (i.e., ALERT),
+which is consistently confirmed by participants.
+C. RQ3: Model Robustness Improvement
+1) Setup: We studied the value of generated adversarial
+examples by using them to improve the robustness of the target
+model via an adversarial fine-tuning strategy. For each subject,
+we divided the test set into two equal parts (S1 and S2),
+so as to avoid data leakage between the adversarial training
+set and the evaluation set constructed by the same technique.
+Specifically, we applied each technique to generate examples
+from S1, and selected one generated adversarial example
+for each target input, i.e., the one that successfully attacks
+the model or achieves the largest decrement on prediction
+confidence (if no successfully-attack example is generated).
+The selected examples were integrated with the training set to
+form the adversarial training set, which is used for fine-tuning
+the model. Thus, for a given subject, the size of the adversarial
+training set constructed by each technique is the same.
+After obtaining a fine-tuned model for each subject with
+each technique, we evaluated it on the evaluation set of
+the successfully-attacking examples generated from S2 by
+CODA, CARROT, ALERT, respectively. Then, we measured
+the accuracy of the fine-tuned model on the three evaluation
+sets to measure its ability of defending against attacks.
+2) Results: Table IV shows the effectiveness of improving
+the model robustness with the generated examples by the
+studied techniques, respectively. The first row represents the
+evaluation set constructed by the corresponding technique,
+while the second row represents the adversarial training set
+constructed by the corresponding technique. The value in each
+cell represents the ratio of the adversarial examples in the
+evaluation set that can be defended by the fine-tuned model
+based on the adversarial training set. We found on most sub-
+jects, CODA improves the model robustness to defend against
+attacks from the largest ratio of adversarial examples generated
+by CODA, CARROT, ALERT, respectively. On average, the
+models fine-tuned by CODA can defend against attacks from
+63.64%, 66.96%, 76.68% of successfully-attacking examples
+generated by CARROT, ALERT, CODA respectively, with
+the improvement of 6.35%, 25.69%, 42.67% over those by
+CARROT and 32.65%, 11.70%, 25.99% over those by ALERT
+respectively. Besides, the results of attack defense between
+different techniques indicate that the examples generated by
+CODA could subsume those by CARROT and ALERT to
+a large extent. We also applied each fine-tuned model to
+the corresponding test set, and found its accuracy is almost
+consistent with the original accuracy (all the absolute accuracy
+differences are less than 1%). The results demonstrate that
+CODA is more helpful to improve the model robustness than
+CARROT and ALERT without damaging the original model
+performance. In four evaluation sets (constructed by ALERT or
+CARROT), CODA performs worse than ALERT or CARROT,
+as the adversarial training set and the evaluation set generated
+by the same technique could be more similar.
+Answer to RQ3: CODA helps improve the model ro-
+bustness more effectively than CARROT and ALERT,
+in terms of defending against attacks from the adver-
+sarial examples generated by itself as well as the adver-
+sarial examples generated by the other two techniques.
+D. RQ4: Contribution of Each Main Component
+1) Setup: We studied the contribution of each main compo-
+nent in CODA, i.e., reference inputs selection (RIS), equivalent
+structure transformations (EST), and identifier renaming trans-
+formations (IRT). We constructed three variants of CODA:
+• w/o RIS: we replaced RIS with the method that randomly
+selects N inputs from training data as reference inputs.
+• w/o EST: we removed EST from CODA, i.e., it directly
+performs identifier renaming transformations after select-
+ing reference inputs.
+• w/o IRT: we removed IRT from CODA, i.e., it directly
+checks whether a successfully-attacking example is gen-
+erated after equivalent structure transformations.
+9
+
+TABLE IV
+ROBUSTNESS IMPROVEMENT OF THE TARGET MODELS AFTER ADVERSARIAL FINE-TUNING
+Task
+Model
+CARROT
+ALERT
+CODA
+CARROT
+ALERT
+CODA
+CARROT
+ALERT
+CODA
+CARROT
+ALERT
+CODA
+Vulnerability
+CodeBERT
+29.14%
+21.11%
+29.69%
+23.43%
+26.27%
+34.44%
+32.16%
+31.73%
+38.82%
+Prediction
+GraphCodeBERT
+12.37%
+19.59%
+21.65%
+16.33%
+17.35%
+23.71%
+25.77%
+24.74%
+34.02%
+Clone
+CodeBERT
+83.15%
+42.31%
+94.44%
+52.65%
+72.46%
+75.32%
+38.51%
+71.45%
+89.78%
+Detection
+GraphCodeBERT
+75.00%
+66.67%
+77.50%
+79.17%
+84.29%
+92.31%
+35.71%
+57.69%
+92.97%
+Authorship
+CodeBERT
+45.06%
+40.67%
+41.03%
+51.25%
+56.25%
+58.82%
+45.67%
+43.33%
+76.47%
+Attribution
+GraphCodeBERT
+81.75%
+67.08%
+72.40%
+79.41%
+78.67%
+100.00%
+45.59%
+80.39%
+84.75%
+Functionality
+CodeBERT
+83.46%
+72.80%
+81.51%
+70.83%
+71.75%
+79.41%
+78.92%
+71.18%
+95.43%
+Classification
+GraphCodeBERT
+67.53%
+75.19%
+77.27%
+32.04%
+52.62%
+62.98%
+91.22%
+90.81%
+93.08%
+Defect
+CodeBERT
+52.73%
+25.81%
+66.03%
+74.88%
+75.87%
+83.12%
+76.86%
+68.66%
+85.36%
+Prediction
+GraphCodeBERT
+68.20%
+48.54%
+74.88%
+52.73%
+63.91%
+59.45%
+67.08%
+68.66%
+76.14%
+Average
+59.84%
+47.98%
+63.64%
+53.27%
+59.94%
+66.96%
+53.75%
+60.86%
+76.68%
+TABLE V
+ABLATION TEST FOR CODA IN TERMS OF AVERAGE ASR
+Model
+w/o RIS
+w/o EST
+w/o IRT
+CODA
+CodeBERT
+30.83%
+62.73%
+35.14%
+73.04%
+GraphCodeBERT
+29.49%
+62.41%
+26.24%
+73.62%
+TABLE VI
+INFLUENCE OF HYPER-PARAMETER U.
+U
+64
+128
+256
+512
+1024
+CodeBERT
+60.14%
+67.90%
+73.04%
+75.27%
+75.83%
+GraphCodeBERT
+61.92%
+70.16%
+73.62%
+74.98%
+75.69%
+2) Results: Table V shows the average ASR values of each
+technique across all the tasks on CodeBERT and GraphCode-
+BERT, respectively. The results on each task can be found at
+our project homepage [19] due to the space limit. We found
+that CODA outperforms all three variants in terms of average
+ASR with improvements of 17.20%∼143.14%, demonstrating
+the contribution of each main component in CODA. Also, ref-
+erence inputs selection and identifier renaming transformations
+contribute more than equivalent structure transformations. The
+possible reason is that not all the rules of equivalent structure
+transformations can be applicable to all the target inputs,
+but identifier renaming transformations are applicable to all
+the inputs. We can enrich the rules of equivalent structure
+transformations in the future to further improve the attack
+effectiveness.
+Answer to RQ4: All the components of reference in-
+put selection, equivalent structure transformations, and
+identifier renaming transformations make contributions
+to the overall effectiveness of CODA, demonstrating
+the necessity of each of them in CODA.
+VI. THREATS TO VALIDITY
+The main threat to validity lies in the settings of param-
+eters in CODA. Here, we investigated the influence of two
+important parameters in CODA (i.e., U and N introduced in
+Section III-B). They affect the selection of reference inputs.
+Tables VI and VII show the influence of U and N in
+terms of average ASR across all the tasks. As U increases,
+CODA performs better, as incorporating more inputs for the
+second step of selection can increase the possibility of finding
+TABLE VII
+INFLUENCE OF HYPER-PARAMETER N
+N
+1
+4
+16
+32
+64
+128
+CodeBERT
+28.08%
+46.33%
+61.07%
+67.12%
+73.04%
+76.38%
+GraphCodeBERT
+31.84%
+46.46%
+60.40%
+66.12%
+73.62%
+74.93%
+more effective reference inputs. Similarly, as N increases
+within our studied range, more effective ingredients could
+be included, leading to better effectiveness. However, the
+amount of increase in terms of average ASR becomes smaller
+with U and N increasing, and meanwhile incorporating more
+inputs can incur more costs in similarity calculation or code
+transformations. Hence, by balancing the effectiveness and
+efficiency of CODA, we set U to 256 and N to 64 as the
+default settings in CODA for practical use.
+VII. RELATED WORK
+Besides the state-of-the-art techniques compared in our
+study (i.e., CARROT [12] and ALERT [14]), there are some
+other adversarial example generation techniques for deep code
+models. For example, Yefet et al. [44] proposed DAMP, which
+changes variables in the target input by gradient computation.
+It only works for the models using one-hot encoding to process
+code, and thus cannot attack the models based on state-
+of-the-art CodeBERT [6] and GraphCodeBERT [7] due to
+different encoding methods. Zhang et al. [13] proposed MHM,
+which iteratively performs identifier renaming transformations
+to generate adversarial examples based on the Metropolis-
+Hastings [45]–[47] algorithm. MHM underperforms CARROT
+and ALERT as presented by the existing studies [12], [14].
+Pour et al. [48] proposed a search-based technique with an
+iterative refactoring-based process. It does not ensure the
+naturalness of generated examples, especially with the rule
+of dead code insertion. These techniques still search for
+effective ingredients in the enormous space, limiting their
+effectiveness. Different from them, our work designs the first
+code-difference-guided attack technique, which can largely
+reduce ingredient space for improving the attack effectiveness.
+There are also many adversarial attack techniques in other
+domains, such as FGSM [49], JSMA [50], and BIM [10] in
+image processing. However, they are not applicable to attack
+deep code models as source code is discrete and has to strictly
+stick to the grammar and semantics constraints.
+10
+
+VIII. CONCLUSION
+To improve the attack effectiveness to deep code models, we
+propose a novel perspective by exploiting the code differences
+between reference inputs and the target input to guide the
+generation of adversarial examples. From this perspective, we
+design CODA, which reduces the ingredient space as the one
+constituted by structure and identifier differences and designs
+equivalent structure transformations and identifier renaming
+transformations to preserve original semantics. We conducted
+an extensive study on two popular pre-trained models with
+five tasks. The results demonstrate that CODA performs more
+successful attacks with less time than the state-of-the-art
+techniques (i.e., CARROT and ALERT), and confirm the
+naturalness of its generated examples as well as the capability
+of improving the model robustness.
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf,len=1108
+page_content='Adversarial Attacks on Neural Models of Code via  Code Difference Reduction  Zhao Tian  College of Intelligence and  Computing, Tianjin University  Tianjin, China  tianzhao@tju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='cn  Junjie Chen†  College of Intelligence and  Computing, Tianjin University  Tianjin, China  junjiechen@tju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='cn  Zhi Jin  Key Lab of High Confidence Software  Technologies, Peking University  Beijing, China  zhijin@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='cn  Abstract—Deep learning has been widely used to solve various  code-based tasks by building deep code models based on a  large number of code snippets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' However, deep code models  are still vulnerable to adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' As source code is  discrete and has to strictly stick to the grammar and semantics  constraints, the adversarial attack techniques in other domains  are not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Moreover, the attack techniques specific to  deep code models suffer from the effectiveness issue due to  the enormous attack space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' In this work, we propose a novel  adversarial attack technique (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', CODA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Its key idea is to  use the code differences between the target input and reference  inputs (that have small code differences but different prediction  results with the target one) to guide the generation of adversarial  examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' It considers both structure differences and identifier  differences to preserve the original semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Hence, the attack  space can be largely reduced as the one constituted by the two  kinds of code differences, and thus the attack process can be  largely improved by designing corresponding equivalent structure  transformations and identifier renaming transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Our  experiments on 10 deep code models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', two pre-trained models  with five code-based tasks) demonstrate the effectiveness and  efficiency of CODA, the naturalness of its generated examples,  and its capability of defending against attacks after adversarial  fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' For example, CODA improves the state-of-the-art  techniques (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', CARROT and ALERT) by 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='25% and 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='20%  on average in terms of the attack success rate, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' INTRODUCTION  In recent years, deep learning (DL) has been widely used  to solve code-based software engineering tasks, such as code  clone detection [1], vulnerability prediction [2], and code com-  pletion [3], by building DL models based on a large amount of  training code snippets (also called deep code models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Indeed,  deep code models have achieved notable performance and  largely promoted the process of software development and  maintenance [4]–[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' In particular, some industrial products on  deep code models have been released and received extensive  attention, such as AlphaCode [8] and Codex [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Like DL models in other areas (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', image processing [10]  and speech recognition [11]), the robustness of deep code  models is also critical [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' However, the existing adversarial  attack techniques proposed in other areas are not applicable to  deep code models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' This is because these techniques perturb an  input in a continuous space for altering the model prediction  result, while the inputs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', source code) for deep code models  †Junjie Chen is the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  are discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Moreover, source code has to strictly stick to  the grammar and semantics constraints, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', the adversarial  example generated from an original input should have no  grammar errors and preserve the original semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Indeed, some adversarial attack techniques specific to deep  code models have been proposed recently, such as MHM [13],  CARROT [12], and ALERT [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' In general, they share two  main steps: (1) designing a series of semantic-preserving code  transformation rules (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', identifier renaming or dead code  insertion), and (2) searching ingredients from the space defined  by the rules (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', a valid identifier name is an ingredient for  the rule of identifier renaming) for transforming an input to a  semantic-preserving adversarial example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' For example, CAR-  ROT designs two semantic-preserving code transformation  rules (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', identifier renaming and dead code insertion), and  uses the hill-climbing algorithm to search for the ingredients  from the entire space with the guidance of gradients and  changes of model prediction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' ALERT considers the rule  of identifier renaming, and uses the naturalness (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', natural  semantics of code) and changes of model prediction results to  guide the ingredient search process from the entire space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Although some of them have been demonstrated to be  effective to some degree, these existing techniques still suffer  from major limitations:  The ingredient space defined by the code transformation  rules is enormous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' For example, for the rule of identifier  renaming, all valid identifier names could be the ingredi-  ents for renaming the target identifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Hence, searching  for the ingredients that can help attack the target model  successfully is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' The existing techniques tend  to utilize the changes of model prediction results after  performing semantic-preserving transformations on the  target input to guide the search process, which is very  likely to fall into local optimum in the enormous space  and thus limits their attack effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Frequently invoking the target model can negatively af-  fect the efficiency of adversarial attack techniques, as  model invocation is the most costly part during the  attack process [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Also, when the model is deployed  remotely, frequent model invocations could be identified  as malicious attacks and thus lead to blocking access to  the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' However, the existing techniques often involve  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='02412v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='CR]  6 Jan 2023  frequent model invocations due to calculating gradients  or guiding the search direction via model prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Developers care about the natural semantics of code since  it is helpful to assist human comprehension [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Hence,  guaranteeing the naturalness of generated adversarial ex-  amples (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', source code in our task) is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' How-  ever, all the existing techniques (except ALERT [14]) do  not consider this factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' For example, CARROT designs  the rule of dead code insertion, but it may largely damage  the naturalness of the generated examples (especially  when a large amount of dead code is inserted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Overall, a more effective adversarial attack technique spe-  cific to deep code models should enhance the attack effective-  ness by improving the ingredient search process, and guarantee  the naturalness of generated adversarial examples as much as  possible and the times of model invocations as few as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Our work does propose such a technique, called CODA (COde  Difference guided Attacking).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  To improve the attack effectiveness, the key idea of CODA  is to use the inputs, which have small code differences with  the target input but have different prediction results, to largely  reduce the ingredient space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' For ease of presentation, we call  such inputs reference inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Actually, reference inputs can  be regarded as invalid successfully-attacking adversarial ex-  amples generated from the target input, where “invalid” refers  to altering the original semantics and “successfully-attacking”  refers to producing different prediction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Over the target  input, the code differences brought by reference inputs con-  tribute to the invalid but successful attack to a large extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Hence, if we extract the ingredients from the code differences  to support semantic-preserving transformations on the target  input, their code differences can be gradually reduced without  altering the original semantics, and thus a valid successfully-  attacking adversarial example is likely to be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' In  this way, the ingredient space is effectively reduced as the  one constituted by only code differences between reference  inputs and the target input, and thus the search process  can be largely improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Please note that taking reference  inputs (especially code differences brought by them) as the  guidance for generating adversarial examples is an innovative  perspective, which closely utilizes the unique characteristics  of deep code models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', source code is discrete).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  To preserve the semantics of the target input during the  attack process, CODA considers code structure differences  and identifier differences, and thus extracts the ingredients to  support equivalent structure transformations and identifier re-  naming transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Equivalent structure transformations  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', transforming a for loop to an equivalent while loop)  do not affect the naturalness of generated examples, and thus  CODA first applies this kind of transformations to reduce  code differences for generating adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Then,  identifier renaming transformations are applied to further  reduce code differences to improve the attack effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  To ensure the naturalness of generated examples by this  kind of transformations, CODA measures semantic similarity  between identifiers for guiding iterative transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' In  particular, CODA just involves necessary model invocations  to check whether the generated example attacks successfully,  without extra gradient calculation and a large amount of model  prediction for guiding the search process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  We conducted an extensive study to evaluate CODA based  on two popular pre-trained models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', CodeBERT [6] and  GraphCodeBERT [7]) and five code-based tasks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', vul-  nerability prediction [2], clone detection [16], authorship  attribution [17], functionality classification [18], and defect  prediction [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' In total, we used 10 subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Our results  demonstrate the effectiveness and efficiency of CODA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' For  example, on average across the ten subjects, CODA improves  the two state-of-the-art adversarial attack techniques specific  to deep code models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', CARROT [12] and ALERT [14])  by 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='25% and 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='20% respectively in terms of the attack  success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' The time spent by CODA on completing the  attack process for the ten subjects is just 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='59 hours, while  those by CARROT and ALERT are 159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='19 hours and 198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='89  hours, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Also, we investigated the value of the  generated adversarial examples by using them to improve  the robustness of the target model via an adversarial fine-  tuning strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' The results show that the models after fine-  tuning with the examples generated by CODA can successfully  defend against attacks from 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='64%, 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='96%, and 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='68%  of adversarial examples generated by CARROT, ALERT, and  CODA on average, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Besides, we conducted a user  study to confirm the naturalness of the generated examples by  CODA and an ablation experiment to confirm the contribution  of each main component in CODA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  To sum up, our work makes the four major contributions:  Novel Perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' We propose a novel perspective of  utilizing code differences between reference inputs and  the target input to guide the adversarial attack process  for deep code models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Technique Implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' We implement CODA fol-  lowing the novel perspective by measuring code structure  and identifier differences and designing the corresponding  semantic-preserving code transformation rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Performance Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' We conducted an extensive  study on two popular pre-trained models and five code-  based tasks, demonstrating the effectiveness and effi-  ciency of CODA over two state-of-the-art techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Public Artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' We released all the experimental data  and our source code at the project homepage [19] for  experiment replication, future research, and practical use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' BACKGROUND AND MOTIVATION  In this section, we first introduce the background of deep  code models (Section II-A), define our problem (Section II-B),  and motivate our key idea with an example (Section II-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Deep Code Models  In the area of software engineering, DL has been widely-  used to process source code [2], [3], [16], [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' In particular,  some popular pre-trained DL models have been constructed  based on a large number of code snippets, among which  2  1  2  3  4  5  6  7  8  9  10  11  12  13  14  1  2  3  4  5  6  7  8  9  10  11  12  13  14  1  2  3  4  5  6  7  8  9  10  11  12  13  14  void f1(int a[], int n){  int i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' int j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' int k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  for (i = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' i < n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' i++) {  for (j = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' j < ((n - i) - 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' j++) {  if (a[j] > a[j + 1]){  k = a[j];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  a[j] = a[j + 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  a[j + 1] = k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  }  }  }  }  int f2(int t[], int len){  int i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' int j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  i = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' j = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  while (len !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='= 0) {  t[i] = len % 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  len /= 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  i = i + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  }  while (j < i){  if (t[j] !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='= t[(i - j) - 1]) return 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  j = j + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  }  return 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  }  void f3(int t[], int len){  int i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' int j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' int k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  i = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  while (i < len) {  j = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  while (j < ((len - i) - 1)) {  if (t[j] > t[j + 1]){  k = t[j];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  t[j] = t[j + 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  t[j + 1] = k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  } j = j + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  } i = i + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  }  }  Ground-truth Label: sort  Prediction Result: sort (96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='52%)  Ground-truth Label: palindrome  Prediction Result: palindrome (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='98%)  Ground-truth Label: sort  Prediction Result: palindrome (90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='88%)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' An illustrating example (the target input f1, a reference input f2, and a successfully-attacking adversarial example f3 generated from f1)  CodeBERT [6] and GraphCodeBERT [7] are two state-of-the-  art pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' CodeBERT learns features from bi-  modal data in the form of programming languages and natural  languages, while GraphCodeBERT takes into consideration the  code structure and data flow information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Same as the existing  work [14], we used them in our evaluation (Section IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  These  pre-trained  models  have  brought  breakthrough  changes to many downstream code-based tasks [21], including  both classification tasks and generation tasks, by fine-tuning  them on the datasets of the corresponding tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' The former  makes classification based on the given code snippets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=',  clone detection [16] and vulnerability prediction [2]), while the  latter produces a sequence of information based on code snip-  pets or natural language descriptions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', code completion [3]  and code summarization [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Following most of the existing  work on attacking deep code models [12]–[14], our work also  focuses on the classification tasks and takes the generation  tasks as our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' In particular, in our study, we adopted  all the tasks used in the studies of evaluating the state-of-the-  art attack techniques (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', CARROT [12] and ALERT [14]),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', five classification tasks including vulnerability prediction,  clone detection, authorship attribution, functionality classifica-  tion, and defect prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Problem Definition  Given a code snippet x that is processed as the required  format by the target deep code model M (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', abstract syntax  trees required by code2seq [4], control-flow graphs required  by DGCNN [23], or data-flow graphs required by GraphCode-  BERT [7]), M can predict a probability vector for x, each  element in which represents the probability classifying x to  the corresponding class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' The class with the largest probability  is the final prediction result of M for x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' If the prediction result  is different from the ground-truth label (denoted as y) of x, it  means that M makes a wrong prediction on x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' otherwise, M  makes a correct prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Although deep code models can achieve great performance  on the given test sets, they may be vulnerable to adversarial  examples [12]–[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' The goal of our work is to generate  successfully-attacking adversarial examples as much as pos-  sible, so as to improve the model robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' As source code  is discrete and has to stick to the grammar and semantics con-  straints, the existing adversarial example generation techniques  proposed in other domains are not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  The existing attack techniques specific to deep code models  always generate adversarial examples from a target input by  performing a series of semantic-preserving code transforma-  tions [12]–[14], which is also followed by our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' For ease  of understanding, we formally define our target problem is to  find {x′|x′ ∈ ϵ ∧ y = M(x) ̸= M(x′)} from a target input x  for the target model M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Here, ϵ refers to the universal set of  code snippets that satisfy the grammar constraints and preserve  the semantics of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' y = M(x) means that we just regard the  test inputs on which M makes correct predictions as target  inputs, where M(x) refers to the prediction result of M on  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' M(x) ̸= M(x′) means that x′ successfully attacks M, that  is, it is a successfully-attacking adversarial example generated  from x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Besides, an effective attack technique should be also  efficient to find x′ and ensure the naturalness of x′ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', natural  to human comprehension [14]), which are indeed carefully  considered by the proposed technique in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Motivating Example  We then use a real-world example (simplified for ease of  illustration) to help motivate our key idea: utilizing the code  differences between reference inputs and the target input to  guide the generation of adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' In Figure 1, the  first code snippet f1 is the target input from the test set of  the functionality classification task [18], and the two state-  of-the-art techniques (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', CARROT [12] and ALERT [14])  do not generate successfully-attacking adversarial examples  from it since they can fall into local optimum in the enormous  ingredient space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' In this figure, the second code snippet f2 is  a reference input from the training set of this task, which has  the different label with f1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  In fact, f2 can be regarded as an invalid successfully-  attacking example from f1, as they are semantically inconsis-  tent but have different prediction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' The code differences  between f1 and f2 mainly contribute to this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' From  this perspective, to generate a valid successfully-attacking  adversarial example (denoted as f3) from f1, we should per-  form semantic-preserving code transformations on f1, and the  transformations should reduce the code differences between f1  and f2 in order to alter the prediction result of the target model  on f1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' That is, the ingredients supporting these transformations  3  Initial  Snippet  Adversarial  Snippet  Input  Output  Training  Data  Input  Target  Model  Test  Test  Report  Attack  Success ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Overview of CODA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  should be extracted from the code differences brought by  f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' With this intuition, by performing equivalent structure  transformations on f1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', transforming for loops to while  loops, where while loops are the used loop structure in f2)  and identifier remaining transformations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', renaming a and  n to t and len respectively, where t and len are the used  identifier names in f2), f3 is generated as shown in the third  code snippet in Figure 1 and indeed attacks successfully,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', making a wrong prediction (palindrome) with a high  confidence (90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='88%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Based on the code differences between the target input and  the reference input, the ingredient space is largely reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' For  example, the ingredient space defined by identifier renaming  transformations is reduced from all valid identifier names  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', almost infinite) to the identifier names occurring in the  reference input but not in the target input (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', only two  identifier names in this simplified example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Hence, it can help  improve the ingredient search process and thus improve the  attack effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' On the other hand, too small ingredient  space could also lose too many ingredients useful to successful  attacks, and thus we will select a set of reference inputs (rather  than only one reference input) for guiding the attack process  in order to balance the size of the ingredient space and the  number of useful ingredients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' APPROACH  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Overview  In this work, we propose a novel perspective to attack  deep code models more effectively and more efficiently, which  utilizes the code differences between reference inputs and the  target input to guide the generation of adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  From this perspective, we design an effective attack technique,  called CODA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Specifically, the code differences brought by  reference inputs provide effective ingredients for altering the  prediction result of the target input by transforming it with  these ingredients, which can contribute to successful attacks in  CODA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' However, as the semantics of reference inputs and the  target input are different, the ingredients from some kinds of  code differences can alter the original semantics, which is not  allowed by the adversarial attack for deep code models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Hence,  in CODA, we consider the structure differences and identifier  differences for measuring code differences between them,  which can preserve the original semantics during the attack  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' In this way, the ingredient space can be effectively  reduced as the one constituted by the two kinds of code  differences between reference inputs and the target input, and  thus the ingredient search process (for generating adversarial  examples) can be largely improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  In fact, not all the inputs that have different prediction  results with the target one, can be regarded as effective  reference inputs for improving the adversarial attack process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  In other words, different inputs could have different degrees  of capabilities for reducing the ingredient search space and  providing effective ingredients for altering the prediction result  of the target input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Therefore, the first step in CODA is  to select effective reference inputs for the target input in  order to improve the attack effectiveness as much as possible  (to be presented in Section III-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Based on the selected  reference inputs, CODA then measures the structure differ-  ences and identifier differences over the target input, which  support extracting the ingredients for two corresponding kinds  of semantic-preserving code transformations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', equivalent  structure transformations and identifier renaming transforma-  tions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' With the guidance of reducing their code differences  based on the two kinds of transformations, the target input  could be effectively transformed to a successfully-attacking  adversarial example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' As equivalent structure transformations  do not affect the naturalness of generated examples, CODA  first applies this kind of transformations to reduce the code  differences for improving the attack effectiveness (to be pre-  sented in Section III-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Then, we apply identifier renaming  transformations to further reduce the code differences for  improving the generation of successfully-attacking adversarial  examples (to be presented in Section III-D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' In particular,  CODA measures the semantic similarity between identifiers  to guarantee the naturalness of generated examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Figure 2 shows the overview of CODA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' In a nutshell, by  successively applying equivalent structure transformations and  identifier renaming transformations to the target input with  the ingredient space defined by the code differences between  the selected reference inputs and the target one, adversarial  examples can be generated towards the direction of reducing  the code differences without altering the original semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' In  this way, the prediction result of the target input is more likely  to be changed, leading to a successfully-attacking adversarial  example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Due to the smaller ingredient search space (but  including effective ingredients) and the clearer attack direction,  the attack effectiveness could be largely improved by CODA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Reference Inputs Selection  The goal of reference inputs is to largely reduce the in-  gredient space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Also, the reduced space should include the  ingredients that are effective to transform the target input  to a successfully-attacking adversarial example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' In this way,  the adversarial attack process can be largely improved by  searching for effective ingredients more efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  4  TABLE I  DESCRIPTIONS OF EQUIVALENT STRUCTURE TRANSFORMATIONS  Transformation  Description  Example Before Transformation  Example After Transformation  R1-loop  equivalent transformation among for structure  for ( i=0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' i<9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' i++ ) {  i=0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' while ( i<9 ) {  and while structure  Body;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' }  Body;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' i++;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' }  R2-branch  equivalent transformation between if-else(-if)  if ( A ) { BodyA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' }  if ( A ) { BodyA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' }  structure and if-if structure  else if ( B ) { BodyB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' }  if ( !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='A && B ) { BodyB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' }  R3-calculation  equivalent numerical calculation transformation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=',  i += 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  i = i + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  ++, --, +=, -=, *=, /=, %=, <<=, >>=, &=, |= , ˆ =  R4-constant  equivalent transformation between a constant and  println("Hello, World!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' ");' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  String i = "Hello, World!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' ";' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  a variable assigned by the same constant  println(i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Although all the inputs that have different prediction results  with the target one can provide ingredients for altering the pre-  diction result of the target one after transformations, their capa-  bilities for successful attacks could be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' To transform  the target input to a successfully-attacking adversarial example  with fewer perturbations, CODA should select the reference  inputs, which can provide the ingredients that are more likely  to conduct successful attacks for the target input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Similar to the  existing work [24]–[27], we assume that the prediction result  of the target input is more likely to be changed from its original  class denoted as ci (with the largest probability predicted  by the target model) to the class with the second largest  probability (denoted as cj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Hence, the ingredients in the inputs  belonging to cj are more likely to attack successfully on the  target input, and thus CODA selects the inputs belonging to  cj as the initial set of reference inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Please note that all  the reference inputs are selected from the training set to avoid  introducing the contents beyond the cognitive scope of the  target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Meanwhile, we only consider the training inputs  whose prediction results are consistent with their ground-truth  labels in order to avoid introducing noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  However, the number of inputs belonging to the same class  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', cj as above) could be large, and thus the ingredient space  constituted by code differences between them and the target  input could be also large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Hence, to further reduce the ingre-  dient space for more effective adversarial example generation,  CODA selects a subset of inputs with high similarity to the  target input from the initial set of reference inputs, as the  final set of reference inputs used by CODA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' This is because  smaller code differences can effectively limit the number of  ingredients, leading to smaller ingredient space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' CODA does  not select only one reference input, as too small ingredient  space could incur a high risk of missing too many ingredients  contributing to successful attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' That is, CODA selects a  small set of reference inputs following the above two steps of  selection to balance the ingredient space size and the amount  of ingredients contributing to successful attacks in the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  We further introduce how to measure the similarity be-  tween the target input (denoted as t) and a reference input  (denoted as r) for the second step of selection in CODA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' In  general, we can adopt some pre-trained models to represent  the code as a vector and then measure code similarity by  calculating the vector distance, like many existing studies [5],  [6], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' However, as presented in Section III-A, CODA  first applies equivalent structure transformations (rather than  identifier renaming transformations) to reduce code differences  for adversarial attacks, as this kind of transformations does not  affect the naturalness of generated examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Moreover, the  identifiers used in different code snippets are usually different  due to the enormous identifier space, which may lead to the  low similarity between various code snippets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Hence, when  measuring code similarity, CODA eliminates the influence  of identifiers by replacing them with the placeholder <unk>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Specifically, CODA first represents t and r after placeholder  replacement as vectors respectively based on CodeBERT [6]  (one of the most widely-used pre-trained models [29]–[31]),  and then calculates the cosine similarity between the two  vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' As the descending order of the calculated similarity,  CODA selects Top-N reference inputs for the follow-up ad-  versarial attack process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Please note that to make the selection  process efficient, we randomly sampled U inputs from the  initial set for the second step of selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' We will investigate  the influence of both U and N on CODA in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Equivalent Structure Transformation  Based on the small set of selected reference inputs, CODA  then extracts ingredients from the space defined by their  brought code differences over the target input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' That is, CODA  transforms the target input to an adversarial example towards  the direction of reducing code differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' CODA first reduces  structure differences by applying equivalent structure transfor-  mations to the target input as they do not affect the naturalness  of generated examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  To preserve the semantics of the target input, we design four  categories of equivalent structure transformations in CODA  inspired by the existing work in metamorphic testing and  code refactoring [32], [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' In particular, we systematically  consider all common kinds of code structures, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', loop struc-  tures, branch structures, and sequential structures (including  numerical calculation and constant usage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' We explain the four  categories in detail in Table I, each of which is also illustrated  with an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' For each category of transformations, it  may include several specific rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' For example, the rules  of transformation on += and transformation on -= belong to  the category of R3-calculation, and the rules of transforming  for loop to while loop and transforming while loop to for  loop belong to the category of R1-loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' In total, CODA has  20 specific rules for the four categories of transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Please note that not all the rules are applicable to the code  5  programmed by any programming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' For example, ++  and -- in R3-calculation are not supported by Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Also,  in R4-constant, the newly-defined variable cannot be the same  as the existing variables in the code;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' otherwise, it may incur  grammar errors and alter the original semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Due to the  space limit, we put more details about all these specific rules  at our project homepage [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Then, we illustrate how to apply each rule for reducing code  differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Each rule involves two structures, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', the one  before transformation (sb) and the one after transformation  (sa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' CODA first counts the occurring times of sb and sa in  the set of selected reference inputs (denoted as nb and na),  and then calculates their occurring distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=',  nb  nb+na  and  na  nb+na .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Further, CODA applies each rule in a probabilistic  way to reduce the occurring distribution differences in terms  of sb and sa between reference inputs and the target input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' In  this way, the structure differences in terms of sb and sa can  be reduced effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' More specifically, for each occurrence  of sb in the target input, CODA applies this rule with the  probability of  na  nb+na , also indicating that it can be retained  with the probability of  nb  nb+na .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  In this step, CODA obtains M inputs from the target input,  each of which is generated by applying all the applicable rules  in the above probabilistic way, and then selects the input with  the highest average similarity (also measured by the method  described in Section III-B) to the selected reference inputs as  the one for the follow-up adversarial attack process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Identifier Renaming Transformation  To facilitate the generation of successfully-attacking ad-  versarial examples, CODA then applies identifier renaming  transformations to further reduce code differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Inspired by  the existing work [12]–[14], identifier renaming transformation  in CODA refers to replacing the name of an identifier in the  target input with the name of an identifier in the selected  reference inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' For ease of presentation, we denote the set of  identifiers in the target input as Vt and the set of identifiers in  the selected reference inputs as Vr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' To preserve the semantics  of the target input and guarantee the grammatical correctness  of the generated example, CODA ensures that the identifier  used for replacement does not exist in the target input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Then, we illustrate how to apply this kind of transformations  to the input obtained from the last step (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', equivalent  structure transformations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' As demonstrated by the existing  work [12]–[14], renaming identifiers is effective to generate  successfully-attacking adversarial examples, but can negatively  affect the naturalness of generated examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' To ensure the  naturalness of generated examples, CODA considers the se-  mantic similarity between identifiers and designs an iterative  transformation process like ALERT [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Specifically, CODA  measures the semantic similarity between each identifier in  Vt and each identifier in Vr by representing each identifier as  a vector via word embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Here, CODA builds the pre-  trained language model with the FastText algorithm [34] and  calculates the cosine similarity between vectors to measure  their semantic similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Then, CODA prioritizes each pair  TABLE II  STATISTICS OF OUR USED SUBJECTS  Task  Train/Validate/Test  Class  Language  Model  Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Vulnerability  21,854/2,732/2,732  2  C  CB  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='76%  Prediction  GCB  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='65%  Clone  90,102/4,000/4,000  2  Java  CB  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='97%  Detection  GCB  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='36%  Authorship  528/–/132  66  Python  CB  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='35%  Attribution  GCB  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='48%  Functionality  41,581/–/10,395  104  C  CB  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='18%  Classification  GCB  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='66%  Defect  27,058/–/6,764  4  C/C++  CB  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='37%  Prediction  GCB  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='98%  CB is short for CodeBERT and GCB is short for GraphCodeBERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  of identifiers as the descending order of their semantic sim-  ilarity, and iteratively applies this transformation based on  each pair of identifiers in the ranking list, which ensures  that more natural transformations can be first performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  After each iteration, CODA invokes the target model to  check whether a successfully-attacking adversarial example  is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' The iterative attack process terminates until a  successfully-attacking adversarial example is generated or all  the pairs are used by this transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Please note that  CODA ensures that the pair of identifiers will not introduce  repetitive identifiers in the generated example in each iteration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  otherwise, this pair will be discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Overall, CODA only invokes the target model when check-  ing if a successfully-attacking adversarial example is gen-  erated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' They are necessary model invocations for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Hence, CODA can largely reduce the number of model  invocations compared with the existing techniques (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=',  ALERT [14]), which is confirmed by our study (Section V-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' EVALUATION DESIGN  In the study, we address four research questions (RQs):  RQ1: How does CODA perform in terms of effectiveness  and efficiency compared with state-of-the-art techniques?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  RQ2: Are the adversarial examples generated by CODA  natural for humans?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  RQ3: Are the adversarial examples generated by CODA  useful to improve the robustness of deep code models?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  RQ4: Does each main component contribute to the over-  all effectiveness of CODA?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Subjects  1) Datasets and Tasks: To sufficiently evaluate CODA,  we consider all the five code-based tasks in the studies of  evaluating state-of-the-art techniques (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', CARROT [12] and  ALERT [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' The statistics of datasets is shown at the first  four columns in Table II, each of which represents the task, the  number of inputs in the training/validation/test set, the number  of classes for the classification task, and the programming  language for the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  The task of vulnerability prediction aims to predict whether  a given code snippet has vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Its used dataset is ex-  tracted from two C projects (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', FFmpeg [35] and Qemu [36])  by Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' [2] and has been integrated as part of the  CodeXGLUE benchmark [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' The task of clone detection  6  aims to detect whether two given code snippets are equivalent  in semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Its used dataset is from BigCloneBench [37],  the most widely-used dataset for clone detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' The existing  work [14] randomly sampled 90,102/4,000/4,000 inputs from  the benchmark for training/validation/testing, to make the  experiment at a computationally friendly scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' In our study,  we used the same dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' The task of authorship attribution  aims to identify the author of a given code snippet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Its used  dataset is the Google Code Jam (GCJ) dataset [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' The task  of functionality classification aims to classify the functionality  of a given code snippet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' If code snippets solve the same  problem, they are regarded to have the same functionality [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Its used dataset is the Open Judge (OJ) benchmark [38],  which has been also integrated as part of the CodeXGLUE  benchmark [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' The task of defect prediction aims to predict  whether a given code snippet is defective and its defect type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Its used dataset is the CodeChef dataset [39], which is labeled  by the execution results on the CodeChef platform (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', four  defect types: no defect, wrong answer, timeout, runtime error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  2) Models: Following the existing work [14], we adopted  two state-of-the-art pre-trained models for code-based tasks,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', CodeBERT [6] and GraphCodeBERT [7], and then fine-  tuned them on the five tasks based on the corresponding  datasets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' In total, we obtained 10 deep code  models as the subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' The last two columns in Table II show  the used pre-trained model and the accuracy of the deep code  model after fine-tuning, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' When fine-tuning Code-  BERT and GraphCodeBERT on these tasks (except Graph-  CodeBERT on functionality classification and defect predic-  tion), we used the same hyper-parameter settings provided by  the existing work [12], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' As there is no instruction on the  hyper-parameter settings for fine-tuning GraphCodeBERT on  functionality classification and defect prediction, we used the  same settings as the one used by authorship attribution (they  are all multi-class classification tasks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Indeed, the achieved  model performance outperforms that achieved by the models  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', TBCNN [38] and CodeBERT [6]) used in the existing  work [12] on the same datasets [12], indicating that the  transferred hyper-parameter settings are reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Overall, the subjects used in our study are indeed diverse,  involving different tasks, different pre-trained models, different  numbers of classes, different programming languages, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' It is  very helpful to sufficiently evaluate the performance of CODA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Compared Techniques  In the study, we compared CODA with two state-of-the-art  techniques attacking deep code models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', CARROT [12]  and ALERT [14], which have been introduced in Section I (the  third paragraph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' We adopted their implementations and the  recommended parameter settings provided by the correspond-  ing papers [12], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' As the original version of CARROT can  only support to attack C/C++ code, we extended it to attack  Python and Java code for sufficient comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Implementations  We implemented CODA in Python and adopted tree-  sitter [40] to extract identifiers from code following the exist-  ing work [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' We set the parameters in CODA by conducting  a preliminary experiment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', U = 256, N = 64, and  M = 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' We discuss the influence of the settings of main  parameters on CODA in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' We released our code  and experimental data at our project homepage [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' All the  experiments were conducted on a server with an Ubuntu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='04  system with Intel(R) Xeon(R) Silver 4214 @ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='20GHz CPU,  256GB memory, and NVIDIA GeForce RTX 2080 Ti GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' RESULTS AND ANALYSIS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' RQ1: Effectiveness and Efficiency  1) Setup: For each deep code model, we applied CODA,  CARROT, ALERT to generate adversarial examples from each  target input in the test set, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' We measured their  effectiveness and efficiency based on the following metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  To reduce the influence of randomness, we repeated all the  experiments (including those for other RQs) 10 times, and  reported the average results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Following the existing work [14], we adopted the at-  tack success rate (ASR) to measure the effectiveness of  each technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' ASR is the percentage of the target inputs  from which an attack technique can generate a successfully-  attacking adversarial example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Larger ASR values mean better  attack effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Also, it is important to measure whether  the prediction confidence (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', the probability of being the  ground-truth class of the target input) is decreased by the gen-  erated examples (although there is no successfully-attacking  adversarial example generated from a target input).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Hence, we  also calculated prediction confidence decrement (PCD) to  measure the effectiveness of each technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' PCD is calculated  by the prediction confidence of the target input minus the min-  imum prediction confidence of the set of generated examples  from the target input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' If the former is smaller than the latter,  we regard PCD to be 0, indicating that the generated examples  cannot decrease the prediction confidence of the target input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Larger PCD values mean better attack effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  In addition, following the existing work [12], [14], we used  the time spent on the overall attack process (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', completing  the adversarial example generation process from all the target  inputs) and the average number of model invocations for  generating examples from one target input, to measure  the efficiency of each technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Less time and fewer model  invocations mean higher efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  2) Results: Table III shows the comparison results among  CARROT, ALERT, and CODA in terms of ASR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' From this  table, CODA always outperforms CARROT and ALERT on  all the tasks based on both CodeBERT and GraphCodeBERT,  demonstrating the stable attack effectiveness of CODA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' On  average, CODA improves 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='11% and 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='83% higher ASR  than CARROT and ALERT across all the five tasks on  CodeBERT, and improves 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='34% and 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='67% higher ASR  on GraphCodeBERT, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Figures 3(a) and 3(b) show the comparison results among  the three techniques in terms of PCD on CodeBERT and  GraphCodeBERT, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' From these figures, the upper  quartile, median, and lower quartile of CODA are always  7  TABLE III  EFFECTIVENESS COMPARISON IN TERMS OF ASR  Task  CodeBERT  GraphCodeBERT  CARROT  ALERT  CODA  CARROT  ALERT  CODA  Vulnerability Prediction  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='72%  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='62%  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
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+page_content='44%  Defect Prediction  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
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+page_content='58%  Average  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='94%  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='48%  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='04%  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='88%  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='69%  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='62%  (a) PCD on attacking CodeBERT  (b) PCD on attacking GraphCodeBERT  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Comparison in terms of prediction confidence decrement  (a) Model invocation times on attacking CodeBERT  (b) Model invocation times on attacking GraphCodeBERT  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Comparison in terms of model invocation times (y-axis refers to the normalized values following the existing work [14])  larger than (or equal to) those of both CARROT and ALERT  regardless of the tasks and the pre-trained models, demon-  strating that CODA produces more significant attacks for  decreasing prediction confidence of target inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' For example,  on CodeBERT, the average improvements of CODA over  CARROT and ALERT are 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='88% and 520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='65% across all  the tasks in terms of average PCD, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Similarly, on  GraphCodeBERT, the average improvements of CODA over  CARROT and ALERT are 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='35% and 560.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='15%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Besides, CARROT, ALERT, and CODA take 159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='19 hours,  198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='89 hours, and 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='59 hours to complete the entire attack  process on all five tasks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Further, we measured  the number of model invocations for each target input during  the attack process, whose results are shown in Figure 4(a)  and 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' From these figures, CODA performs fewer model  invocations than both CARROT and ALERT regardless of the  tasks and pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' On average, CODA performs  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='73% and 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='58% fewer model invocations than CARROT  and ALERT across all the tasks on CodeBERT, and 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='07%  and 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='31% fewer model invocations on GraphCodeBERT,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' The results demonstrate that CODA has the  significantly highest efficiency among the three techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Answer to RQ1: CODA spends less time with fewer  model invocations on completing the entire attack  process, but generates more successfully-attacking ex-  amples with more significant prediction confidence  decrement on all the subjects, than the state-of-the-art  techniques (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', CARROT and ALERT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' RQ2: Naturalness of Adversarial Examples  1) Setup: It is important to check whether the generated  adversarial examples are natural to human judges [14], [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Here, we conducted a user study to compare the naturalness  of examples generated by CODA, CARROT, and ALERT, and  our user study shares the same design as the one conducted  by the existing work [14]:  Data Preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' For each subject, we randomly sampled  10 target inputs, and then for each technique we randomly  sampled an adversarial example from the set of examples  generated from each sampled target input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' That is, for each  sampled target input, we construct three pairs of code snippets,  each of which contains the target input and an adversarial  example generated by CODA, CARROT, or ALERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' In total,  we obtained 300 pairs of code snippets for the user study due  to 10 subjects × 3 techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Same as the existing work [14], the user study  also involves four non-author participants, each of whom has  8  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Average score to evaluate naturalness of examples per participant  a Bachelor/Master degree in Computer Science with at least  five years of programming experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' For objective evaluation, we did not tell partici-  pants which technique generates the adversarial example in a  pair of code snippets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Also, we highlighted the changes in each  pair of code snippets for facilitating manual evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Then,  each participant individually evaluated each pair by evaluating  to what extent the changes are natural to the code context  and the changed identifiers preserve the original semantics,  following the existing work [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Specifically, participants  gave a score for each pair based on a 5-point Likert scale [42]  (1 means strongly disagree and 5 means strongly agree)  following the existing work [14], [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  2) Results: Figure 5 shows the average score of the ad-  versarial examples generated by each technique for each  participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' From this figure, the conclusions from different  participants are consistent: the naturalness of the adversarial  examples generated by CODA and ALERT is closely high  (round 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='50 on average), and significantly higher than that by  CARROT (just 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='89 on average).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' ALERT is a naturalness-  aware technique, whose core contribution is to ensure the  naturalness of generated examples, but CODA achieves similar  naturalness scores to it, demonstrating that CODA can also  generate highly natural adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Answer to RQ2: The adversarial examples gener-  ated by CODA are natural closely to the state-of-the-  art naturalness-aware attack technique (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', ALERT),  which is consistently confirmed by participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' RQ3: Model Robustness Improvement  1) Setup: We studied the value of generated adversarial  examples by using them to improve the robustness of the target  model via an adversarial fine-tuning strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' For each subject,  we divided the test set into two equal parts (S1 and S2),  so as to avoid data leakage between the adversarial training  set and the evaluation set constructed by the same technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Specifically, we applied each technique to generate examples  from S1, and selected one generated adversarial example  for each target input, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', the one that successfully attacks  the model or achieves the largest decrement on prediction  confidence (if no successfully-attack example is generated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  The selected examples were integrated with the training set to  form the adversarial training set, which is used for fine-tuning  the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Thus, for a given subject, the size of the adversarial  training set constructed by each technique is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  After obtaining a fine-tuned model for each subject with  each technique, we evaluated it on the evaluation set of  the successfully-attacking examples generated from S2 by  CODA, CARROT, ALERT, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Then, we measured  the accuracy of the fine-tuned model on the three evaluation  sets to measure its ability of defending against attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  2) Results: Table IV shows the effectiveness of improving  the model robustness with the generated examples by the  studied techniques, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' The first row represents the  evaluation set constructed by the corresponding technique,  while the second row represents the adversarial training set  constructed by the corresponding technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' The value in each  cell represents the ratio of the adversarial examples in the  evaluation set that can be defended by the fine-tuned model  based on the adversarial training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' We found on most sub-  jects, CODA improves the model robustness to defend against  attacks from the largest ratio of adversarial examples generated  by CODA, CARROT, ALERT, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' On average, the  models fine-tuned by CODA can defend against attacks from  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='64%, 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='96%, 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='68% of successfully-attacking examples  generated by CARROT, ALERT, CODA respectively, with  the improvement of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='35%, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='69%, 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='67% over those by  CARROT and 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='65%, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='70%, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='99% over those by ALERT  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Besides, the results of attack defense between  different techniques indicate that the examples generated by  CODA could subsume those by CARROT and ALERT to  a large extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' We also applied each fine-tuned model to  the corresponding test set, and found its accuracy is almost  consistent with the original accuracy (all the absolute accuracy  differences are less than 1%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' The results demonstrate that  CODA is more helpful to improve the model robustness than  CARROT and ALERT without damaging the original model  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' In four evaluation sets (constructed by ALERT or  CARROT), CODA performs worse than ALERT or CARROT,  as the adversarial training set and the evaluation set generated  by the same technique could be more similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Answer to RQ3: CODA helps improve the model ro-  bustness more effectively than CARROT and ALERT,  in terms of defending against attacks from the adver-  sarial examples generated by itself as well as the adver-  sarial examples generated by the other two techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' RQ4: Contribution of Each Main Component  1) Setup: We studied the contribution of each main compo-  nent in CODA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', reference inputs selection (RIS), equivalent  structure transformations (EST), and identifier renaming trans-  formations (IRT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' We constructed three variants of CODA:  w/o RIS: we replaced RIS with the method that randomly  selects N inputs from training data as reference inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  w/o EST: we removed EST from CODA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', it directly  performs identifier renaming transformations after select-  ing reference inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  w/o IRT: we removed IRT from CODA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', it directly  checks whether a successfully-attacking example is gen-  erated after equivalent structure transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  9  TABLE IV  ROBUSTNESS IMPROVEMENT OF THE TARGET MODELS AFTER ADVERSARIAL FINE-TUNING  Task  Model  CARROT  ALERT  CODA  CARROT  ALERT  CODA  CARROT  ALERT  CODA  CARROT  ALERT  CODA  Vulnerability  CodeBERT  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
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+page_content='86%  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='68%  TABLE V  ABLATION TEST FOR CODA IN TERMS OF AVERAGE ASR  Model  w/o RIS  w/o EST  w/o IRT  CODA  CodeBERT  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='83%  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='73%  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='14%  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='04%  GraphCodeBERT  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='49%  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='41%  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='24%  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='62%  TABLE VI  INFLUENCE OF HYPER-PARAMETER U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  U  64  128  256  512  1024  CodeBERT  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='14%  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='90%  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='04%  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='27%  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='83%  GraphCodeBERT  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='92%  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='16%  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='62%  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='98%  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='69%  2) Results: Table V shows the average ASR values of each  technique across all the tasks on CodeBERT and GraphCode-  BERT, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' The results on each task can be found at  our project homepage [19] due to the space limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' We found  that CODA outperforms all three variants in terms of average  ASR with improvements of 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='20%∼143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='14%, demonstrating  the contribution of each main component in CODA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Also, ref-  erence inputs selection and identifier renaming transformations  contribute more than equivalent structure transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' The  possible reason is that not all the rules of equivalent structure  transformations can be applicable to all the target inputs,  but identifier renaming transformations are applicable to all  the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' We can enrich the rules of equivalent structure  transformations in the future to further improve the attack  effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Answer to RQ4: All the components of reference in-  put selection, equivalent structure transformations, and  identifier renaming transformations make contributions  to the overall effectiveness of CODA, demonstrating  the necessity of each of them in CODA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' THREATS TO VALIDITY  The main threat to validity lies in the settings of param-  eters in CODA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Here, we investigated the influence of two  important parameters in CODA (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', U and N introduced in  Section III-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' They affect the selection of reference inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Tables VI and VII show the influence of U and N in  terms of average ASR across all the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' As U increases,  CODA performs better, as incorporating more inputs for the  second step of selection can increase the possibility of finding  TABLE VII  INFLUENCE OF HYPER-PARAMETER N  N  1  4  16  32  64  128  CodeBERT  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='08%  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='33%  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='07%  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='12%  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='04%  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='38%  GraphCodeBERT  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='84%  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='46%  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='40%  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='12%  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='62%  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='93%  more effective reference inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Similarly, as N increases  within our studied range, more effective ingredients could  be included, leading to better effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' However, the  amount of increase in terms of average ASR becomes smaller  with U and N increasing, and meanwhile incorporating more  inputs can incur more costs in similarity calculation or code  transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Hence, by balancing the effectiveness and  efficiency of CODA, we set U to 256 and N to 64 as the  default settings in CODA for practical use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' RELATED WORK  Besides the state-of-the-art techniques compared in our  study (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', CARROT [12] and ALERT [14]), there are some  other adversarial example generation techniques for deep code  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' For example, Yefet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' [44] proposed DAMP, which  changes variables in the target input by gradient computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  It only works for the models using one-hot encoding to process  code, and thus cannot attack the models based on state-  of-the-art CodeBERT [6] and GraphCodeBERT [7] due to  different encoding methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' [13] proposed MHM,  which iteratively performs identifier renaming transformations  to generate adversarial examples based on the Metropolis-  Hastings [45]–[47] algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' MHM underperforms CARROT  and ALERT as presented by the existing studies [12], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  Pour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' [48] proposed a search-based technique with an  iterative refactoring-based process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' It does not ensure the  naturalness of generated examples, especially with the rule  of dead code insertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' These techniques still search for  effective ingredients in the enormous space, limiting their  effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' Different from them, our work designs the first  code-difference-guided attack technique, which can largely  reduce ingredient space for improving the attack effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  There are also many adversarial attack techniques in other  domains, such as FGSM [49], JSMA [50], and BIM [10] in  image processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' However, they are not applicable to attack  deep code models as source code is discrete and has to strictly  stick to the grammar and semantics constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='  10  VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' CONCLUSION  To improve the attack effectiveness to deep code models, we  propose a novel perspective by exploiting the code differences  between reference inputs and the target input to guide the  generation of adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' From this perspective, we  design CODA, which reduces the ingredient space as the one  constituted by structure and identifier differences and designs  equivalent structure transformations and identifier renaming  transformations to preserve original semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' We conducted  an extensive study on two popular pre-trained models with  five tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=' The results demonstrate that CODA performs more  successful attacks with less time than the state-of-the-art  techniques (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
+page_content=', CARROT and ALERT), and confirm the  naturalness of its generated examples as well as the capability  of improving the model robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfgAGB/content/2301.02412v1.pdf'}
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+MLMC techniques for discontinuous functions
+Michael B. Giles
+Abstract The Multilevel Monte Carlo (MLMC) approach usually works well when
+estimating the expected value of a quantity which is a Lipschitz function of inter-
+mediate quantities, but if it is a discontinuous function it can lead to a much slower
+decay in the variance of the MLMC correction. This article reviews the literature
+on techniques which can be used to overcome this challenge in a variety of different
+contexts, and discusses recent developments using either a branching diffusion or
+adaptive sampling.
+1 Introduction
+The Multilevel Monte Carlo (MLMC) method is based on the telescoping sum
+E[ �𝑃𝐿] = E[ �𝑃0] +
+𝐿
+∑︁
+ℓ=1
+E[ �𝑃ℓ−�𝑃ℓ−1]
+where �𝑃ℓ represents an approximation to an output quantity of interest 𝑃 on level ℓ,
+with the weak error
+���E[ �𝑃ℓ−𝑃]
+��� and MLMC variance V[ �𝑃ℓ−�𝑃ℓ−1], both decreasing
+as the level ℓ increases, but with the corresponding computational cost per sample
+increasing.
+If �𝑌ℓ has expected value E[ �𝑃ℓ−�𝑃ℓ−1], with variance 𝑉ℓ and cost 𝐶ℓ, then for a
+given target RMS error 𝜀, the number of levels 𝐿 and the number of independent
+samples on each level can be optimised [13, 14] to give an overall cost which is
+approximately equal to 𝜀−2
+� 𝐿
+∑︁
+ℓ=0
+√︁
+𝐶ℓ𝑉ℓ
+�2
+.
+Michael B. Giles
+University of Oxford Mathematical Institute, Woodstock Rd, Oxford OX2 6GG, UK
+e-mail: mike.giles@maths.ox.ac.uk
+1
+arXiv:2301.02882v1  [math.NA]  7 Jan 2023
+
+2
+Michael B. Giles
+If 𝐶ℓ𝑉ℓ → 0 as ℓ → ∞, then the cost is dominated by the first term from level 0,
+and the cost is approximately 𝜀−2𝐶0𝑉0, so proportional to 𝜀−2.
+If 𝐶ℓ𝑉ℓ → const as ℓ → ∞, then the contributions to the cost are spread almost
+equally across all levels and the cost is approximately 𝜀−2𝐿2𝐶𝐿𝑉𝐿, proportional to
+𝜀−2| log 𝜀|2 if E[ �𝑃ℓ−𝑃] decays exponentially with ℓ.
+Even worse, if 𝐶ℓ𝑉ℓ → ∞ then the cost is dominated by the contribution from the
+finest level and so is approximately 𝜀−2𝐶𝐿𝑉𝐿 which is 𝑂(𝜀−2−(𝛾−𝛽)/𝛼) if E[ �𝑃ℓ−𝑃] ∼
+2−𝛼ℓ, 𝑉ℓ ∼ 2−𝛽ℓ and 𝐶ℓ ∼ 2𝛾ℓ.
+In most MLMC applications, 𝑃 is a smooth function of some intermediate solution
+quantities, such as the solution of an SDE, a PDE with stochastic coefficients or
+initial/boundary data, or an estimate of an inner conditional expectation. Under
+these circumstances we usually have 𝛽 ≥ 𝛾 and so the MLMC complexity is 𝑂(𝜀−2)
+or 𝑂(𝜀−2| log 𝜀|2).
+This article is concerned with the small but important class of applications where
+𝑃 is a discontinuous function of the intermediate quantities, and because of this the
+MLMC variance 𝑉ℓ can decay much more slowly, leading to the complexity falling
+into the third category of being 𝑂(𝜀−2−(𝛾−𝛽)/𝛼).
+The good news is that there has been considerable research within the MLMC
+community to address this challenge. This article surveys the variety of methods
+which have been developed in the hope that this can aid researchers meeting similar
+challenges in future applications.
+To illustrate things, we begin by detailing two specific application challenges
+which have motivated much of this research. We then discuss the many approaches
+which have been developed, several of which have borrowed ideas from the literature
+on computing sensitivities (the “greeks” in mathematical finance literature) of the
+form
+𝜕
+𝜕𝛼E [ 𝑓 (𝜔, 𝛼)]
+using the pathwise sensitivity approach [24] (also known as Infinitesimal Perturba-
+tion Analysis, IPA for short [30]) or Likelihood Ratio Method [31], or from methods
+from improving integrand smoothness to improve the rate of convergence for QMC
+integration [1, 6].
+1.1 Challenge 1: nested expectation
+Suppose 𝑓 is a scalar function and we want to estimate the nested expectation
+E [ 𝑓 (E[𝑍|𝑋]) ], where the outer expectation is with respect to a random variable
+𝑋 and we will assume that the inner conditional expectation E[𝑍|𝑋] has a bounded
+density near zero.
+A very simple MLMC treatment 1 uses 𝑀ℓ = 2ℓ𝑀0 inner samples on level ℓ, so
+estimators on level 0 and the higher levels are simply
+1 Note that if 𝑓 is smooth, or at least Lipschitz, then it is better to use an “antithetic” estimator
+[8, 14, 15, 18], but this does not give a better order of convergence when 𝑓 is discontinuous.
+
+MLMC techniques for discontinuous functions
+3
+�𝑌0 = 𝑓 (𝑍
+(0,𝑀0)),
+�𝑌ℓ = 𝑓 (𝑍
+(ℓ,𝑀ℓ)) − 𝑓 (𝑍
+(ℓ,𝑀ℓ−1)),
+where 𝑍
+(ℓ,𝑀ℓ) and 𝑍
+(ℓ,𝑀ℓ−1) represent independent averages of 𝑀ℓ and 𝑀ℓ−1 inde-
+pendent samples of 𝑍, all conditional on the same value of 𝑋 [14, 15].
+If V[𝑍|𝑋] is finite, and 𝑓 is Lipschitz with constant 𝐿 𝑓 , then
+E
+��
+𝑓 (𝑍
+(ℓ,𝑀ℓ)) − 𝑓 (E[𝑍|𝑋])
+�2
+| 𝑋
+�
+≤ 𝐿2
+𝑓 E
+��
+𝑍
+(ℓ,𝑀ℓ) − E[𝑍|𝑋]
+�2
+| 𝑋
+�
+= 𝐿2
+𝑓 𝑀−1
+ℓ V[𝑍|𝑋],
+and hence E[�𝑌2
+ℓ |𝑋] ≤ 4 𝐿2
+𝑓 (𝑀−1
+ℓ
++ 𝑀−1
+ℓ−1)V[𝑍|𝑋] for ℓ>0. If V[𝑍|𝑋] is uniformly
+bounded it follows that 𝑉ℓ = 𝑂(𝑀−1
+ℓ ). If the cost of each conditional sample of 𝑍 is
+𝑂(1) then 𝐶ℓ = 𝑂(𝑀ℓ) and hence the complexity is 𝑂(𝜀−2| log 𝜀|2).
+Unfortunately, the situation is significantly poorer when 𝑓 is the Heaviside step
+function 𝐻 defined by 𝐻(𝑥)=0 if 𝑥<0, and 𝐻(𝑥)=1 if 𝑥≥0. This occurs in many
+applications because P [E[𝑍|𝑋] > 𝐾] = E [𝐻(E[𝑍|𝑋] − 𝐾)] , so it corresponds to
+the probability of a conditional expectation exceeding some threshold 𝐾, which is a
+very important quantity in risk calculations.
+If 𝐾=0 and 𝐸[𝑍|𝑋] has a bounded density near zero then there is an 𝑂(𝑀−1/2
+ℓ
+)
+probability that |𝐸[𝑍|𝑋] | = 𝑂(𝑀−1/2
+ℓ
+), which is the circumstance under which there
+is an 𝑂(1) probability that �𝑌ℓ = ±1 due to 𝑍
+(ℓ,𝑀ℓ) being positive and 𝑍
+(ℓ,𝑀ℓ−1) being
+negative, or vice versa. Hence 𝑉ℓ ≈ 𝑂(𝑀−1/2
+ℓ
+) and the complexity is approximately
+𝑂(𝜀−5/2) [19].
+This challenge is the primary motivation for Section 7, and also arises in the
+context of Section 3.
+1.2 Challenge 2: discontinuous payoff function
+In the case of a scalar SDE
+d𝑆𝑡 = 𝑎(𝑆𝑡) d𝑡 + 𝑏(𝑆𝑡) d𝑊𝑡,
+(1)
+with an output quantity of interest 𝑃 ≡ 𝑓 (𝑆𝑇 ), the standard estimator is
+�𝑌ℓ = �𝑃ℓ − �𝑃ℓ−1
+where the same Brownian motion 𝑊𝑡 is used to calculate both �𝑃ℓ and �𝑃ℓ−1, but with
+different uniform timesteps ℎℓ and ℎℓ−1.
+If 𝑓 is Lipschitz with constant 𝐿 𝑓 , then
+𝑉ℓ ≤ E
+�
+( �𝑃ℓ − �𝑃ℓ−1)2�
+≤ 𝐿2
+𝑓 E
+�
+(�𝑆ℓ − �𝑆ℓ−1)2�
+
+4
+Michael B. Giles
+where �𝑆ℓ is the level ℓ numerical approximation to 𝑆𝑇 . Hence, based on standard
+strong convergence results [28] we have 𝑉ℓ = 𝑂(ℎℓ) for an Euler-Maruyama dis-
+cretisation of the SDE, and 𝑉ℓ = 𝑂(ℎ2
+ℓ) for the first order Milstein discretisation.
+The cost 𝐶ℓ is 𝑂(ℎ−1
+ℓ ) in both cases, giving MLMC complexities of 𝑂(𝜀−2| log 𝜀|2)
+and 𝑂(𝜀−2), respectively,
+In mathematical finance, a digital call option payoff is 0 or 1, depending on
+whether 𝑆𝑇 is below or above the strike 𝐾, so the payoff function can be written
+as 𝑓 (𝑆𝑇 ) = 𝐻(𝑆𝑇 −𝐾). The MLMC problem is that a small difference between the
+coarse and fine paths can give a payoff difference of ±1 if the two paths straddle the
+strike, i.e. are on different sides of the strike.
+When using the Euler-Maruyama approximation of the SDE, �𝑆ℓ−�𝑆ℓ−1 = 𝑂(ℎ1/2
+ℓ ).
+Speaking loosely (see [4, 21] for the rigorous analysis) an 𝑂(ℎ1/2
+ℓ ) fraction of
+fine/coarse pairs straddle the strike, so 𝑉ℓ = 𝑂(ℎ1/2
+ℓ ) and hence the complexity is
+𝑂(𝜀−5/2).
+Similarly, using the Milstein approximation gives �𝑆ℓ−�𝑆ℓ−1 = 𝑂(ℎℓ) so 𝑉ℓ =
+𝑂(ℎℓ). This is clearly better, and gives a complexity which is 𝑂(𝜀−2| log 𝜀|2), but
+there is still the problem that most MLMC samples 𝑌ℓ are zero on the finer levels, so
+the kurtosis is 𝑂(ℎ−1
+ℓ ) which causes problems in practice in estimating 𝑉ℓ accurately
+to determine the number of samples 𝑁ℓ to use on level ℓ. In addition, there is the
+difficulty that the Milstein discretisation of multi-dimensional SDEs often requires
+the simulation of Lévy areas, though this problem can be addressed through the use
+of an antithetic estimator [23].
+This challenge is the primary motivation for Sections 2, 4, 5 and 6, also also arises
+in Sections 3 and 7.
+2 Explicit smoothing
+The pathwise sensitivity analysis (or IPA) approach to compute the parameter sen-
+sitivities known as Greeks in mathematical finance [24] requires that the payoff
+function 𝑓 is continuous and piecewise smooth. This is clearly a problem with digi-
+tal options, and one standard approach is to smooth the payoff function by replacing
+the Heaviside step function 𝐻 with a smoothed approximation 𝐻𝛿(𝑥) ≡ 𝑔(𝑥/𝛿),
+with 𝑔(𝑥) → 0 as 𝑥 → −∞ and 𝑔(𝑥) → 1 as 𝑥 → +∞, so the discontinuity is
+smoothed over a width of 𝑂(𝛿).
+For financial reasons, the preference is often to use a one-sided smoothing, such
+as the piecewise linear approximation shown in yellow in Figure 1. This one-sided
+approximation introduces a weak error, or bias, which is 𝑂(𝛿). If it is used for
+MLMC, then 𝐻′
+𝛿(𝑆𝑇 ) = 𝛿−1 for the 𝑂(𝛿) fraction of the paths which end up in the
+ramp region, and therefore 𝑉ℓ = 𝑂(𝛿 × (𝛿−1)2) = 𝑂(𝛿−1). Hence the optimal choice
+of 𝛿 involves a tradeoff between bias and variance.
+The bias can be reduced by making the smoothing anti-symmetric about 𝑥 = 0 so
+that 𝐻𝛿(𝑥) − 𝐻(𝑥) = −(𝐻𝛿(−𝑥) − 𝐻(−𝑥)), for example by choosing 𝑔(𝑥) ≡ Φ(𝑥)
+
+MLMC techniques for discontinuous functions
+5
+as illustrated in orange in Figure 1. If 𝑆𝑇 has the smooth probability density 𝜌(𝑆)
+then the weak error is
+∫ ∞
+−∞
+(𝐻𝛿(𝑆−𝐾) − 𝐻(𝑆−𝐾)) 𝜌(𝑆) d𝑆 = 𝛿
+∫ ∞
+−∞
+(𝑔(𝑥) − 𝐻(𝑥)) 𝜌(𝐾+𝑥𝛿) d𝑥
+and a Taylor series expansion of 𝜌(𝐾+𝑥𝛿) results in the asymptotic error expansion
+𝑎1𝜌(𝐾) 𝛿 + 𝑎2𝜌′(𝐾) 𝛿2 + 𝑎3𝜌′′(𝐾) 𝛿3 + 𝑎4𝜌′′′(𝐾) 𝛿4 + 𝑂(𝛿5)
+where
+𝑎𝑘 =
+∫ ∞
+−∞
+𝑥𝑘−1 (𝑔(𝑥) − 𝐻(𝑥)) d𝑥.
+If 𝑔(𝑥) − 𝐻(𝑥) = − (𝑔(−𝑥) − 𝐻(−𝑥)) then 𝑎1 = 𝑎3 = 0, and
+𝑎2 = 2
+∫ ∞
+0
+𝑥(𝑔(𝑥) − 1) d𝑥,
+𝑎4 = 2
+∫ ∞
+0
+𝑥3(𝑔(𝑥) − 1) d𝑥.
+If 𝑔(𝑥) is monotonic, then 𝑎2 ≠ 0, but by considering non-monotonic functions
+such as 𝑔(𝑥) = (4/3) Φ(𝑥) − (1/3) Φ(2𝑥) it is possible to set 𝑎2 = 0 making the
+weak error 𝑂(𝛿4). Hence, to achieve 𝑂(𝜀) accuracy overall we need 𝛿=𝑂(𝜀1/4), and
+then on the coarsest levels 𝑉ℓ = 𝑂(𝛿−1) = 𝑂(𝜀−1/4) so the overall complexity is
+approximately 𝑂(𝜀−2−1/4) in the best cases where the overall cost is dominated by
+the cost on the coarsest levels.
+Giles, Nagapetyan & Ritter [22] used explicit smoothing for estimating cumulative
+distribution functions (CDFs). For a scalar random variable 𝑋, to estimate 𝐶(𝑥) =
+P(𝑋 < 𝑥) = E[𝐻(𝑥−𝑋)], the approach they adopted was to use MLMC to estimate
+𝐶(𝑥 𝑗) for a set of spline points 𝑥 𝑗 and then interpolate these values with a cubic
+spline. Overall, their method balanced three weak errors, the SDE discretisation error
+on the finest level, the smoothing error due to 𝐻𝛿, and the cubic spline interpolation
+error, in addition to the MLMC sampling error.
+0.5
+1
+1.5
+ST
+0
+0.2
+0.4
+0.6
+0.8
+1
+f(S T)
+Fig. 1 Two explicitly smoothed versions of the Heaviside step function for a digital call option
+
+6
+Michael B. Giles
+3 Integration/differentiation and Malliavin calculus
+Krumscheid & Nobile [29] used a slightly different approach for estimating CDFs,
+particularly in the context of risk estimation. Starting from the identity
+d
+d𝑥 E [ max(0, 𝑥−𝑆𝑇 ) ] = E[ 𝐻(𝑥−𝑆𝑇 ) ]
+they used MLMC to estimate E[ max(0, 𝑥 𝑗−𝑆𝑇 ) ] for a set of spline points 𝑥 𝑗,
+interpolated these with a cubic spline, and and then differentiated the spline to
+obtain an approximation to the CDF 𝐶(𝑥). This avoids the extra weak error due to
+smoothing the Heaviside function, but differentiating the cubic spline amplifies the
+noise in the spline data.
+On a similar note, Altmayer & Neuenkirch [2] used Malliavin calculus integration
+by parts to treat discontinuous payoffs based on solutions of the Heston stochastic
+volatility SDE. They observed that asymptotically this improves the MLMC vari-
+ance on the finer levels, but it increases the variance on coarse levels. To address
+this, they split the payoff into a smooth part which they treated with the standard
+MLMC approach, and a compact-support discontinuous part for which they used the
+Malliavin MLMC.
+Malliavin calculus was originally developed for computing sensitivities, so this
+is another example of the literature on sensitivity calculations being exploited to
+develop improved MLMC algorithms.
+4 Conditional expectation
+When using the first order Milstein discretisation for an SDE, one way to improve
+the MLMC variance for digital options is to switch to the Euler-Maruyama approxi-
+mation for the final timestep, and then take the conditional expectation with respect
+to the final fine path Brownian increment Δ𝑊 [12, 17].
+For the fine path approximation of the scalar SDE (1) with 𝑁 timesteps of size
+ℎℓ, the path value 𝑆𝑇 at the final time 𝑇 is given by
+�𝑆 𝑓
+𝑇 = �𝑆 𝑓
+𝑇 −ℎℓ + 𝑎(�𝑆 𝑓
+𝑇 −ℎℓ) ℎℓ + 𝑏(�𝑆 𝑓
+𝑇 −ℎℓ) Δ𝑊𝑁 ,
+and therefore the conditional expected value for the digital call option is
+�𝑃 𝑓
+ℓ
+= E
+�
+𝐻(�𝑆 𝑓
+𝑇 −𝐾) | �𝑆 𝑓
+𝑇 −ℎℓ
+�
+= Φ ��
+�
+�𝑆 𝑓
+𝑇 −ℎℓ + 𝑎(�𝑆 𝑓
+𝑇 −ℎℓ) ℎℓ − 𝐾
+𝑏(�𝑆 𝑓
+𝑇 −ℎℓ) √ℎℓ
+��
+�
+.
+Similarly, for the coarse path with coarse timestep ℎℓ−1 = 2 ℎℓ, the Brownian incre-
+ment for the final coarse timestep is the sum of the last two Brownian increments for
+the fine path, Δ𝑊𝑁 −1+Δ𝑊𝑁 , and therefore
+
+MLMC techniques for discontinuous functions
+7
+�𝑆𝑐
+𝑇 = �𝑆𝑐
+𝑇 −ℎℓ−1 + 𝑎(�𝑆𝑐
+𝑇 −ℎℓ−1) ℎℓ−1 + 𝑏(�𝑆𝑐
+𝑇 −ℎℓ−1) (Δ𝑊𝑁 −1+Δ𝑊𝑁 ) ,
+from which we obtain
+�𝑃𝑐
+ℓ−1 = E
+�
+𝐻(�𝑆𝑐
+𝑇 −𝐾) | �𝑆𝑐
+𝑇 −ℎℓ−1, Δ𝑊𝑁 −1
+�
+= Φ
+� �𝑆𝑐
+𝑇 −ℎℓ−1 + 𝑎(�𝑆𝑐
+𝑇 −ℎℓ−1) ℎℓ−1 + 𝑏(�𝑆𝑐
+𝑇 −ℎℓ−1) Δ𝑊𝑁 −1 − 𝐾
+𝑏(�𝑆𝑐
+𝑇 −ℎℓ−1) √ℎℓ
+�
+.
+With �𝑌ℓ ≡ �𝑃ℓ−�𝑃ℓ−1, numerical analysis [17] proves that 𝑉ℓ ≈ 𝑂(ℎ3/2
+ℓ ) so the
+MLMC complexity is 𝑂(𝜀−2). Heuristically, this is because there is an 𝑂(ℎ1/2
+ℓ )
+probability of paths being within 𝑂(ℎ1/2
+ℓ ) of the strike 𝐾, and for these
+�𝑆 𝑓
+𝑇 −ℎℓ−1 − �𝑆𝑐
+𝑇 −ℎℓ−1 = 𝑂(ℎℓ),
+𝜕 �𝑃
+𝜕�𝑆
+= 𝑂(ℎ−1/2
+ℓ
+)
+=⇒
+�𝑃ℓ − �𝑃ℓ−1 = 𝑂(ℎ1/2
+ℓ ),
+so 𝑉ℓ ≈ 𝑂(ℎ1/2
+ℓ
+× (ℎ1/2
+ℓ )2) = 𝑂(ℎ3/2
+ℓ ). In addition, the kurtosis is improved to
+𝑂(ℎ−1/2
+ℓ
+).
+Unfortunately, the conditional expectation approach does not help when the Euler-
+Maruyama discretisation is used for the entire path since �𝑆 𝑓
+𝑇 −ℎℓ−1−�𝑆𝑐
+𝑇 −ℎℓ−1 = 𝑂(ℎ1/2
+ℓ )
+and so �𝑃ℓ − �𝑃ℓ−1 = 𝑂(1)
+The use of this kind of conditional expectation is a standard technique for smooth-
+ing the payoff to enable IPA/pathwise sensitivity calculations [24]. Another example
+is a down-and-out barrier option, where the option is knocked out if the path drops
+below a certain value. In this case the payoff can be smoothed by computing the
+probability of this happening, conditional on the computed path approximations at
+discrete timesteps [24]. Again, this works well for MLMC when using the first or-
+der Milstein discretisation [12, 17], but it does not help with the Euler-Maruyama
+discretisation.
+A different kind of conditional expectation smoothing was introduced by Achtsis,
+Cools & Nuyens [1] and Bayer, Siebenmorgen & Tempone [6] to improve the
+convergence of QMC computations, and then used by Bayer, Ben Hammouda &
+Tempone [5] to improve the MLMC variance for digital options.
+In its simplest form, they split the random inputs for the numerical simulation into
+a scalar 𝑍 and the remainder 𝑍𝑟, and express the desired MLMC level ℓ expectation
+as
+E[ �𝑃ℓ−�𝑃ℓ−1] = E
+�
+E[ �𝑃ℓ−�𝑃ℓ−1 | 𝑍𝑟]
+�
+and observe that in many financial applications it is possible to perform this split
+in a way such that the conditional expectations E[ �𝑃ℓ | 𝑍𝑟], E[ �𝑃ℓ−1 | 𝑍𝑟] are smooth
+functions of 𝑍𝑟, and can be evaluated analytically or very accurately by 1D numerical
+quadrature when there is just a single discontinuity with respect to changes in 𝑍.
+
+8
+Michael B. Giles
+For a scalar SDE, 𝑍 could be the terminal value of the driving Brownian motion,
+in which case 𝑍𝑟 would represent the other Normal random variables required for a
+Brownian Bridge construction of the Brownian increments.
+5 Change of measure
+Another approach to treating digital options using the Milstein discretisation is to
+use a change of measure [9, 14], which has connections to the Likelihood Ratio
+Method (LRM) that is used for sensitivity analysis [31].
+For both the fine and coarse paths, we have conditional Gaussian distributions for
+�𝑆𝑇 , with slightly different means and variances, as illustrated in Figure 2. We can
+therefore perform a change of measure to the same Gaussian distribution with mean
+𝜇 and variance 𝜎2, also illustrated in Figure 2, and then pick the same sample �𝑆𝑇
+for both paths from this common Gaussian distribution.
+The resulting MLMC estimator is
+�𝑌ℓ = 𝑓 (�𝑆𝑇 ) (𝑅ℓ − 𝑅ℓ−1)
+where 𝑅ℓ, 𝑅ℓ−1 are the respective Radon-Nikodym derivatives for the fine and coarse
+paths. For the scalar SDE (1), 𝑅ℓ is
+𝑅ℓ =
+𝜎
+𝑏(�𝑆 𝑓
+𝑇 −ℎℓ)√ℎℓ
+exp
+�
+− (�𝑆𝑇 − �𝑆 𝑓
+𝑇 −ℎℓ − 𝑎(�𝑆 𝑓
+𝑇 −ℎℓ) ℎℓ)2 / (2 𝑏2(�𝑆 𝑓
+𝑇 −ℎℓ) ℎℓ)
+�
+exp
+�
+− (�𝑆𝑇 −𝜇)2 / (2𝜎2)
+�
+and 𝑅ℓ−1 is defined similarly. It can be shown that the difference 𝑅ℓ − 𝑅ℓ−1 is
+approximately 𝑂(ℎ1/2
+ℓ ), which implies that 𝑉ℓ ≈ 𝑂(ℎℓ). To improve the variance we
+note that the conditional expected value of Radon-Nikodym derivatives is always 1,
+0.6
+0.8
+1
+1.2
+1.4
+ST
+0
+0.1
+0.2
+0.3
+0.4
+p(ST)
+Fig. 2 Coarse and fine path conditional Gaussian distributions, plus third common distribution
+
+MLMC techniques for discontinuous functions
+9
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+t
+1
+1.5
+2
+S
+Fig. 3 Illustration of coarse and fine jump-diffusion paths with jumps before and after 𝑇 = 1.
+i.e. E[𝑅ℓ | �𝑆 𝑓
+𝑇 −ℎℓ] = E[𝑅ℓ−1 | �𝑆𝑐
+𝑇 −ℎℓ−1, Δ𝑊𝑁 −1] = 1, and therefore we can change
+the definition of �𝑌ℓ to
+�𝑌ℓ =
+�
+𝑓 (�𝑆𝑇 ) − 𝑓 (𝜇)
+�
+(𝑅ℓ − 𝑅ℓ−1)
+without changing its expected value. This estimator is now non-zero only when �𝑆𝑇
+and 𝜇 are on opposite sides of the strike 𝐾, which occurs for an 𝑂(ℎ1/2
+ℓ ) fraction of
+coarse/fine paths. Hence the new MLMC variance 𝑉ℓ is approximately 𝑂(ℎ3/2
+ℓ ), as
+with the use of the analytic conditional expectation.
+The benefit of this approach is that it works well in multiple dimensions when it is
+often not possible to evaluate the analytic conditional expectation [9, 14]. However,
+again it does not help with the full path Euler-Maruyama discretisation because that
+gives 𝑅ℓ − 𝑅ℓ−1 = 𝑂(1).
+An earlier use of a change of measure in an MLMC computation was by Xia
+[33, 34] for a Merton-style jump-diffusion SDE with a path-dependent jump rate
+𝜆(𝑆, 𝑡). The challenge in this application, as illustrated in Figure 3, is that the
+coarse and fine paths will jump at different times, and one might jump just before
+the final time 𝑇, and the other just after, leading to a large jump in the computed
+value of 𝑓 (𝑆𝑇 ). The path-dependent jump rate was treated by using the thinning
+technique of Glasserman & Merener [25], over-sampling possible jump times using
+a uniform rate 𝜆𝑠𝑢𝑝 > 𝜆(𝑆, 𝑡) and then using an acceptance/rejection step to select
+the real jump times. Xia modified this with a change of measure to ensure the same
+acceptance/rejection decision for both the fine and coarse paths so that they both
+jump at the same time. This leads to an estimator of the form
+�𝑌ℓ = �𝑃ℓ 𝑅ℓ − �𝑃ℓ−1 𝑅ℓ−1.
+
+10
+Michael B. Giles
+When combined with a first order Milstein discretisation of the SDE between the
+jump times, this gives 𝑉ℓ = 𝑂(ℎ2
+ℓ) for Lipschitz payoff functions such as a standard
+put or call option [33, 34].
+6 Splitting
+Returning to the challenge of digital options arising from the solution of an SDE,
+a third approach is to use path-splitting to generate an unbiased estimate of the
+conditional expectation introduced in Section 4 [14].
+This is a variant of the general splitting technique [3]. As illustrated in Figure 4,
+it involves performing a standard fine path simulation up until one timestep before
+the final time 𝑇, and then performing multiple independent simulations of the final
+timestep, averaging the payoff for each of these to get an approximation of the
+conditional expectation. The same is done for the coarse path except that each of the
+splits uses the same Δ𝑊𝑁 −1 that was used for the second to last fine path timestep.
+Since the computational cost of the path up to the splitting time is 𝑂(ℎ−1
+ℓ ), it means
+that up to 𝑂(ℎ−1
+ℓ ) splits can be used without increasing the path cost significantly. If
+𝑀ℓ splits are used, then the standard splitting variance analysis gives
+V[�𝑌ℓ] = V
+�
+E[ �𝑃ℓ−�𝑃ℓ−1 | {Δ𝑊𝑛}𝑛<𝑁 ]
+�
++ 𝑀−1
+ℓ E
+�
+V[ �𝑃ℓ−�𝑃ℓ−1 | {Δ𝑊𝑛}𝑛<𝑁 ]
+�
+.
+As discussed previously V
+�
+E[ �𝑃ℓ−�𝑃ℓ−1 | {Δ𝑊𝑛}𝑛<𝑁 ]
+�
+= 𝑂(ℎ3/2
+ℓ ), and similarly it
+can be argued that E
+�
+V[ �𝑃ℓ−�𝑃ℓ−1 | {Δ𝑊𝑛}𝑛<𝑁 ]
+�
+= 𝑂(ℎℓ). Therefore choosing 𝑀ℓ
+to lie between 𝑂(ℎ−1
+ℓ ) and 𝑂(ℎ−1/2
+ℓ
+) ensures the benefits of the splitting are obtained
+without significantly increasing the computational cost per sample.
+0
+0.2
+0.4
+0.6
+0.8
+1
+t
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+St
+Fig. 4 Path splitting in final timestep to estimate conditional expectation
+
+MLMC techniques for discontinuous functions
+11
+As an additional bonus, one can use the more accurate Milstein discretisation
+for the final timestep, instead of switching to the Euler-Maruyama discretisation.
+Burgos [9, 10] gives more details of the analysis, and also used the same approach
+for pathwise sensitivity analysis for a variety of financial options.
+Giles & Bernal [16] also used splitting for Feynman-Kac functionals arising for
+stopped diffusions, SDE simulations which terminate when the solution path leaves
+a prescribed domain. The issue here is that when a fine path exits, there is an 𝑂(ℎ1/2
+ℓ )
+probability that the corresponding coarse path does not leave until much later. This
+is addressed by estimating a conditional expectation by splitting the coarse path into
+𝑂(ℎ−1/2
+ℓ
+) independent sub-simulations which continue until each of them leaves the
+domain. 𝑉ℓ is improved from 𝑂(ℎ1/2
+ℓ ) to approximately 𝑂(ℎℓ) without a significant
+increase in the cost per sample, and finally the MLMC complexity achieved is
+𝑂(𝜀−2| log 𝜀|3).
+None of the three methods introduced so far (conditional expectation, change of
+measure, splitting) helps when using the Euler-Maruyama discretisation. For this, a
+new method has recently been developed by Giles & Haji-Ali [20].
+It again uses splitting, but inspired by the simulation of branching diffusions, it
+considers splits at multiple deterministic times, as illustrated in Figure 5 which shows
+the logical structure of a set of split paths. Here we are considering a simulation
+on the unit time interval. A single pair of fine/coarse paths is calculated up to time
+𝑡 = 1/2, with the number of fine timesteps being 1
+2 ℎ−1
+ℓ . This simulation is then
+split into two separate independent simulations up to time 𝑡 = 3/4, with the two
+simulations between them accounting for an additional 1
+2 ℎ−1
+ℓ
+fine timesteps. There
+are further splits at 𝑡 = 3/4, then at 𝑡 = 7/8, and so on, with the final split when there
+is just one coarse timestep left.
+The total number of fine timesteps simulated is 𝑂(ℎ−1
+ℓ | log ℎℓ|) so the computa-
+tional cost is only slightly increased compared to the original method with a single
+pair of fine/coarse paths. �𝑌ℓ is defined to be the average of the values �𝑃ℓ−�𝑃ℓ−1 for
+each of the final paths, and it can be proved that its variance is 𝑂(ℎℓ), the same
+0
+0.2
+0.4
+0.6
+0.8
+1
+t
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+Fig. 5 Repeated path splitting to estimate conditional expectation
+
+12
+Michael B. Giles
+asymptotic order of convergence as for Lipschitz payoff functions [20]. The kurtosis
+is also improved, so this technique fully addresses the challenge of using MLMC
+with the Euler-Maruyama discretisation to estimate digital option values.
+7 Adaptive sampling
+We return now to the challenge of estimating the nested expectation E [ 𝐻 (E[𝑍|𝑋]) ]
+and we note that we only need an accurate estimate of the inner conditional expecta-
+tion E[𝑍|𝑋] when it is near zero. This observation is the basis for the development
+of adaptive sampling by Broadie, Du & Moallemi [7] within a standard Monte Carlo
+procedure. This was then extended to adaptive sampling combined with MLMC by
+Giles & Haji-Ali [19] by defining the number of inner samples 𝑀ℓ on level ℓ to be
+•
+𝑀ℓ = 2ℓ𝑀0 inner samples when |E[𝑍|𝑋]| ≫
+√︁
+V[𝑍|𝑋]/(2ℓ𝑀0)
+This is the smallest number of samples used on level ℓ.
+√︁
+V[𝑍|𝑋]/(2ℓ𝑀0) is
+the standard deviation of the Monte Carlo estimate for E[𝑍|𝑋], so the inequality
+means that this number of samples is sufficient to be very sure that the estimate
+has the correct sign.
+•
+𝑀ℓ = 4ℓ𝑀0 inner samples when |E[𝑍|𝑋]| = 𝑂(
+√︁
+V[𝑍|𝑋]/(4ℓ𝑀0))
+This is the maximum number of samples used on level ℓ. In this case, the estimate
+of E[𝑍|𝑋] may have the incorrect sign, but this will only happen when |E[𝑍|𝑋]| =
+𝑂(2−ℓ) which occurs with probability 𝑂(2−ℓ). Likewise, the total cost of the
+higher number of samples in this region is 𝑂(2−ℓ × 4ℓ) = 𝑂(2ℓ), so it does not
+significantly increase the overall average cost.
+•
+2ℓ𝑀0 < 𝑀ℓ < 4ℓ𝑀0 for intermediate values
+In this region the number of samples is chosen to be very sure that the estimate
+of E[𝑍|𝑋] has the correct sign, and at the same time the total cost is 𝑂(2ℓ).
+Overall, this adaptive sampling approach leads to 𝐶ℓ ∼ 2ℓ, 𝑉ℓ ∼ 2−ℓ and hence
+a complexity of roughly 𝑂(𝜀−2) [19]. However, the kurtosis is 𝑂(2ℓ) since only an
+𝑂(2−ℓ) fraction of the outer samples give non-zero values for �𝑌ℓ.
+Haji-Ali, Spence & Teckentrup [27] have further extended this to estimate quan-
+tities of the form
+P[𝐺 ∈ Ω] ≡ E[1𝐺∈Ω]
+where 𝐺 is a 𝑑-dimensional random variable which cannot be sampled directly.
+In their paper they consider in particular the two challenges in this article. In the
+context of the digital option with the Euler-Maruyama discretisation on the unit time
+interval, the adaptive sampling varies the timestep used on level ℓ so that
+•
+ℎℓ = 2−ℓ when |�𝑆ℓ − 𝐾| is large compared to the strong error in the path approx-
+imation
+•
+ℎℓ = 4−ℓ when |�𝑆ℓ − 𝐾| is of the same order as the strong error
+
+MLMC techniques for discontinuous functions
+13
+•
+2−ℓ < ℎℓ < 4−ℓ for intermediate values
+A Brownian bridge construction is used when the timestep needs to be refined as
+part of the adaptation procedure from its initial value ℎℓ = 2−ℓ. The adaptation again
+leads to 𝐶ℓ ∼ 2ℓ, 𝑉ℓ ∼ 2−ℓ and hence a complexity of roughly 𝑂(𝜀−2), but there is
+again a high kurtosis [27].
+In earlier research, Elfverson, Hellman & Malqvist [11] considered estimation of
+E[𝐻(𝑋)] where 𝑋 cannot be sampled exactly but there is a sequence of approxi-
+mations 𝑋′
+0, 𝑋′
+1, 𝑋′
+2, . . . 𝑋 of increasing accuracy and increasing cost. Motivated by
+PDE applications with a well-behaved truncation error so that there are uniform
+geometric bounds on |𝑋′
+𝑗 − 𝑋|, level ℓ in their method uses
+�𝑋ℓ = 𝑋′
+𝑗,
+𝑗 = min{ℓ, min 𝑗 : | �𝑋′
+𝑗 − 𝑋| < |𝑋|}
+and achieves similarly good MLMC benefits. This idea is essentially the same as in
+the work of Haji-Ali et al but requiring a uniform bound on |𝑋′
+𝑗−𝑋| is significantly
+more restrictive than the bounds on E[ |𝑋′
+𝑗−𝑋|𝑞] for some 𝑞>2 required by Haji-Ali
+et al.
+A final comment is that the analysis of Haji-Ali, Spence & Teckentrup can be gen-
+eralised to a product of an indicator function and a Lipschitz function, E[1𝐺∈Ω 𝑓 (𝑆)],
+and so can handle barrier options. Furthermore, Haji-Ali & Spence have extended
+the adaptive sampling methodology to an extremely challenging triply-nested expec-
+tation which arises in mathematical finance [26]. By incorporating the randomised
+MLMC treatment of Rhee & Glynn [32] to handle the time discretisation of the
+underlying SDEs as well as the sampling for the inner conditional expectations, they
+achieve an overall complexity of approximately 𝑂(𝜀−2) which is very impressive for
+such a difficult application.
+-10
+-5
+0
+5
+10
+E[Z|X]
+0
+0.5
+1
+1.5
+2
+2.5
+Fig. 6 Error distributions for two conditional expectations with i) few samples being needed to
+ensure the correct sign (left), and ii) many samples being insufficient to ensure the correct sign
+(centre). The blue line represents the Heaviside step function.
+
+14
+Michael B. Giles
+8 Conclusions
+It is worth repeating that in most MLMC applications the output quantity of interest
+is a Lipschitz function of the intermediate simulation quantities, so good strong
+convergence for the intermediate quantities leads automatically to a good rate of
+convergence of the MLMC variance 𝑉ℓ.
+For those applications in which the function is discontinuous, this article shows
+there is an extensive literature with a variety of different approaches to improve the
+MLMC variance and try to recover the optimal 𝑂(𝜀−2) complexity. It is notable that
+many of these methods have adapted ideas from Monte Carlo sensitivity analysis
+which also has problems with discontinuous functionals. It is hoped that this survey
+will assist future researchers facing similar challenges in other new application areas.
+Acknowledgements This paper is based on research with many students, postdocs and other
+collaborators and I am grateful to all of them. Funding for the research has been provided by the UK
+Engineering and Physical Sciences Research Council through grants EP/E031455/1, EP/H05183X/1
+and EP/P020720/2 as well as the Hong Kong Innovation and Technology Commission (InnoHK
+Project CIMDA). The paper was written while visiting the Oden Institute at UT Austin, and I thank
+my hosts for their warm hospitality.
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+
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+page_content='MLMC techniques for discontinuous functions  Michael B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Giles  Abstract The Multilevel Monte Carlo (MLMC) approach usually works well when  estimating the expected value of a quantity which is a Lipschitz function of inter-  mediate quantities, but if it is a discontinuous function it can lead to a much slower  decay in the variance of the MLMC correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' This article reviews the literature  on techniques which can be used to overcome this challenge in a variety of different  contexts, and discusses recent developments using either a branching diffusion or  adaptive sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  1 Introduction  The Multilevel Monte Carlo (MLMC) method is based on the telescoping sum  E[ �𝑃𝐿] = E[ �𝑃0] +  𝐿  ∑︁  ℓ=1  E[ �𝑃ℓ−�𝑃ℓ−1]  where �𝑃ℓ represents an approximation to an output quantity of interest 𝑃 on level ℓ,  with the weak error  ���E[ �𝑃ℓ−𝑃]  ��� and MLMC variance V[ �𝑃ℓ−�𝑃ℓ−1], both decreasing  as the level ℓ increases, but with the corresponding computational cost per sample  increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  If �𝑌ℓ has expected value E[ �𝑃ℓ−�𝑃ℓ−1], with variance 𝑉ℓ and cost 𝐶ℓ, then for a  given target RMS error 𝜀, the number of levels 𝐿 and the number of independent  samples on each level can be optimised [13, 14] to give an overall cost which is  approximately equal to 𝜀−2  � 𝐿  ∑︁  ℓ=0  √︁  𝐶ℓ𝑉ℓ  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  Michael B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Giles  University of Oxford Mathematical Institute, Woodstock Rd, Oxford OX2 6GG, UK  e-mail: mike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='giles@maths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='uk  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='02882v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='NA]  7 Jan 2023  2  Michael B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Giles  If 𝐶ℓ𝑉ℓ → 0 as ℓ → ∞, then the cost is dominated by the first term from level 0,  and the cost is approximately 𝜀−2𝐶0𝑉0, so proportional to 𝜀−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  If 𝐶ℓ𝑉ℓ → const as ℓ → ∞, then the contributions to the cost are spread almost  equally across all levels and the cost is approximately 𝜀−2𝐿2𝐶𝐿𝑉𝐿, proportional to  𝜀−2| log 𝜀|2 if E[ �𝑃ℓ−𝑃] decays exponentially with ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  Even worse, if 𝐶ℓ𝑉ℓ → ∞ then the cost is dominated by the contribution from the  finest level and so is approximately 𝜀−2𝐶𝐿𝑉𝐿 which is 𝑂(𝜀−2−(𝛾−𝛽)/𝛼) if E[ �𝑃ℓ−𝑃] ∼  2−𝛼ℓ, 𝑉ℓ ∼ 2−𝛽ℓ and 𝐶ℓ ∼ 2𝛾ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  In most MLMC applications, 𝑃 is a smooth function of some intermediate solution  quantities, such as the solution of an SDE, a PDE with stochastic coefficients or  initial/boundary data, or an estimate of an inner conditional expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Under  these circumstances we usually have 𝛽 ≥ 𝛾 and so the MLMC complexity is 𝑂(𝜀−2)  or 𝑂(𝜀−2| log 𝜀|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  This article is concerned with the small but important class of applications where  𝑃 is a discontinuous function of the intermediate quantities, and because of this the  MLMC variance 𝑉ℓ can decay much more slowly, leading to the complexity falling  into the third category of being 𝑂(𝜀−2−(𝛾−𝛽)/𝛼).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  The good news is that there has been considerable research within the MLMC  community to address this challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' This article surveys the variety of methods  which have been developed in the hope that this can aid researchers meeting similar  challenges in future applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  To illustrate things, we begin by detailing two specific application challenges  which have motivated much of this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' We then discuss the many approaches  which have been developed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' several of which have borrowed ideas from the literature  on computing sensitivities (the “greeks” in mathematical finance literature) of the  form  𝜕  𝜕𝛼E [ 𝑓 (𝜔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' 𝛼)]  using the pathwise sensitivity approach [24] (also known as Infinitesimal Perturba-  tion Analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' IPA for short [30]) or Likelihood Ratio Method [31],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' or from methods  from improving integrand smoothness to improve the rate of convergence for QMC  integration [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='1 Challenge 1: nested expectation  Suppose 𝑓 is a scalar function and we want to estimate the nested expectation  E [ 𝑓 (E[𝑍|𝑋]) ], where the outer expectation is with respect to a random variable  𝑋 and we will assume that the inner conditional expectation E[𝑍|𝑋] has a bounded  density near zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  A very simple MLMC treatment 1 uses 𝑀ℓ = 2ℓ𝑀0 inner samples on level ℓ, so  estimators on level 0 and the higher levels are simply  1 Note that if 𝑓 is smooth, or at least Lipschitz, then it is better to use an “antithetic” estimator  [8, 14, 15, 18], but this does not give a better order of convergence when 𝑓 is discontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  MLMC techniques for discontinuous functions  3  �𝑌0 = 𝑓 (𝑍  (0,𝑀0)),  �𝑌ℓ = 𝑓 (𝑍  (ℓ,𝑀ℓ)) − 𝑓 (𝑍  (ℓ,𝑀ℓ−1)),  where 𝑍  (ℓ,𝑀ℓ) and 𝑍  (ℓ,𝑀ℓ−1) represent independent averages of 𝑀ℓ and 𝑀ℓ−1 inde-  pendent samples of 𝑍, all conditional on the same value of 𝑋 [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  If V[𝑍|𝑋] is finite, and 𝑓 is Lipschitz with constant 𝐿 𝑓 , then  E  ��  𝑓 (𝑍  (ℓ,𝑀ℓ)) − 𝑓 (E[𝑍|𝑋])  �2  | 𝑋  �  ≤ 𝐿2  𝑓 E  ��  𝑍  (ℓ,𝑀ℓ) − E[𝑍|𝑋]  �2  | 𝑋  �  = 𝐿2  𝑓 𝑀−1  ℓ V[𝑍|𝑋],  and hence E[�𝑌2  ℓ |𝑋] ≤ 4 𝐿2  𝑓 (𝑀−1  ℓ  + 𝑀−1  ℓ−1)V[𝑍|𝑋] for ℓ>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' If V[𝑍|𝑋] is uniformly  bounded it follows that 𝑉ℓ = 𝑂(𝑀−1  ℓ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' If the cost of each conditional sample of 𝑍 is  𝑂(1) then 𝐶ℓ = 𝑂(𝑀ℓ) and hence the complexity is 𝑂(𝜀−2| log 𝜀|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  Unfortunately, the situation is significantly poorer when 𝑓 is the Heaviside step  function 𝐻 defined by 𝐻(𝑥)=0 if 𝑥<0, and 𝐻(𝑥)=1 if 𝑥≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' This occurs in many  applications because P [E[𝑍|𝑋] > 𝐾] = E [𝐻(E[𝑍|𝑋] − 𝐾)] , so it corresponds to  the probability of a conditional expectation exceeding some threshold 𝐾, which is a  very important quantity in risk calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  If 𝐾=0 and 𝐸[𝑍|𝑋] has a bounded density near zero then there is an 𝑂(𝑀−1/2  ℓ  )  probability that |𝐸[𝑍|𝑋] | = 𝑂(𝑀−1/2  ℓ  ), which is the circumstance under which there  is an 𝑂(1) probability that �𝑌ℓ = ±1 due to 𝑍  (ℓ,𝑀ℓ) being positive and 𝑍  (ℓ,𝑀ℓ−1) being  negative, or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Hence 𝑉ℓ ≈ 𝑂(𝑀−1/2  ℓ  ) and the complexity is approximately  𝑂(𝜀−5/2) [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  This challenge is the primary motivation for Section 7, and also arises in the  context of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='2 Challenge 2: discontinuous payoff function  In the case of a scalar SDE  d𝑆𝑡 = 𝑎(𝑆𝑡) d𝑡 + 𝑏(𝑆𝑡) d𝑊𝑡,  (1)  with an output quantity of interest 𝑃 ≡ 𝑓 (𝑆𝑇 ), the standard estimator is  �𝑌ℓ = �𝑃ℓ − �𝑃ℓ−1  where the same Brownian motion 𝑊𝑡 is used to calculate both �𝑃ℓ and �𝑃ℓ−1, but with  different uniform timesteps ℎℓ and ℎℓ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  If 𝑓 is Lipschitz with constant 𝐿 𝑓 , then  𝑉ℓ ≤ E  �  ( �𝑃ℓ − �𝑃ℓ−1)2�  ≤ 𝐿2  𝑓 E  �  (�𝑆ℓ − �𝑆ℓ−1)2�  4  Michael B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Giles  where �𝑆ℓ is the level ℓ numerical approximation to 𝑆𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Hence, based on standard  strong convergence results [28] we have 𝑉ℓ = 𝑂(ℎℓ) for an Euler-Maruyama dis-  cretisation of the SDE, and 𝑉ℓ = 𝑂(ℎ2  ℓ) for the first order Milstein discretisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  The cost 𝐶ℓ is 𝑂(ℎ−1  ℓ ) in both cases, giving MLMC complexities of 𝑂(𝜀−2| log 𝜀|2)  and 𝑂(𝜀−2), respectively,  In mathematical finance, a digital call option payoff is 0 or 1, depending on  whether 𝑆𝑇 is below or above the strike 𝐾, so the payoff function can be written  as 𝑓 (𝑆𝑇 ) = 𝐻(𝑆𝑇 −𝐾).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' The MLMC problem is that a small difference between the  coarse and fine paths can give a payoff difference of ±1 if the two paths straddle the  strike, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' are on different sides of the strike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  When using the Euler-Maruyama approximation of the SDE, �𝑆ℓ−�𝑆ℓ−1 = 𝑂(ℎ1/2  ℓ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  Speaking loosely (see [4, 21] for the rigorous analysis) an 𝑂(ℎ1/2  ℓ ) fraction of  fine/coarse pairs straddle the strike, so 𝑉ℓ = 𝑂(ℎ1/2  ℓ ) and hence the complexity is  𝑂(𝜀−5/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  Similarly, using the Milstein approximation gives �𝑆ℓ−�𝑆ℓ−1 = 𝑂(ℎℓ) so 𝑉ℓ =  𝑂(ℎℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' This is clearly better, and gives a complexity which is 𝑂(𝜀−2| log 𝜀|2), but  there is still the problem that most MLMC samples 𝑌ℓ are zero on the finer levels, so  the kurtosis is 𝑂(ℎ−1  ℓ ) which causes problems in practice in estimating 𝑉ℓ accurately  to determine the number of samples 𝑁ℓ to use on level ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' In addition, there is the  difficulty that the Milstein discretisation of multi-dimensional SDEs often requires  the simulation of Lévy areas, though this problem can be addressed through the use  of an antithetic estimator [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  This challenge is the primary motivation for Sections 2, 4, 5 and 6, also also arises  in Sections 3 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  2 Explicit smoothing  The pathwise sensitivity analysis (or IPA) approach to compute the parameter sen-  sitivities known as Greeks in mathematical finance [24] requires that the payoff  function 𝑓 is continuous and piecewise smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' This is clearly a problem with digi-  tal options, and one standard approach is to smooth the payoff function by replacing  the Heaviside step function 𝐻 with a smoothed approximation 𝐻𝛿(𝑥) ≡ 𝑔(𝑥/𝛿),  with 𝑔(𝑥) → 0 as 𝑥 → −∞ and 𝑔(𝑥) → 1 as 𝑥 → +∞, so the discontinuity is  smoothed over a width of 𝑂(𝛿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  For financial reasons, the preference is often to use a one-sided smoothing, such  as the piecewise linear approximation shown in yellow in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' This one-sided  approximation introduces a weak error, or bias, which is 𝑂(𝛿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' If it is used for  MLMC, then 𝐻′  𝛿(𝑆𝑇 ) = 𝛿−1 for the 𝑂(𝛿) fraction of the paths which end up in the  ramp region, and therefore 𝑉ℓ = 𝑂(𝛿 × (𝛿−1)2) = 𝑂(𝛿−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Hence the optimal choice  of 𝛿 involves a tradeoff between bias and variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  The bias can be reduced by making the smoothing anti-symmetric about 𝑥 = 0 so  that 𝐻𝛿(𝑥) − 𝐻(𝑥) = −(𝐻𝛿(−𝑥) − 𝐻(−𝑥)), for example by choosing 𝑔(𝑥) ≡ Φ(𝑥)  MLMC techniques for discontinuous functions  5  as illustrated in orange in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' If 𝑆𝑇 has the smooth probability density 𝜌(𝑆)  then the weak error is  ∫ ∞  −∞  (𝐻𝛿(𝑆−𝐾) − 𝐻(𝑆−𝐾)) 𝜌(𝑆) d𝑆 = 𝛿  ∫ ∞  −∞  (𝑔(𝑥) − 𝐻(𝑥)) 𝜌(𝐾+𝑥𝛿) d𝑥  and a Taylor series expansion of 𝜌(𝐾+𝑥𝛿) results in the asymptotic error expansion  𝑎1𝜌(𝐾) 𝛿 + 𝑎2𝜌′(𝐾) 𝛿2 + 𝑎3𝜌′′(𝐾) 𝛿3 + 𝑎4𝜌′′′(𝐾) 𝛿4 + 𝑂(𝛿5)  where  𝑎𝑘 =  ∫ ∞  −∞  𝑥𝑘−1 (𝑔(𝑥) − 𝐻(𝑥)) d𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  If 𝑔(𝑥) − 𝐻(𝑥) = − (𝑔(−𝑥) − 𝐻(−𝑥)) then 𝑎1 = 𝑎3 = 0, and  𝑎2 = 2  ∫ ∞  0  𝑥(𝑔(𝑥) − 1) d𝑥,  𝑎4 = 2  ∫ ∞  0  𝑥3(𝑔(𝑥) − 1) d𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  If 𝑔(𝑥) is monotonic, then 𝑎2 ≠ 0, but by considering non-monotonic functions  such as 𝑔(𝑥) = (4/3) Φ(𝑥) − (1/3) Φ(2𝑥) it is possible to set 𝑎2 = 0 making the  weak error 𝑂(𝛿4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Hence, to achieve 𝑂(𝜀) accuracy overall we need 𝛿=𝑂(𝜀1/4), and  then on the coarsest levels 𝑉ℓ = 𝑂(𝛿−1) = 𝑂(𝜀−1/4) so the overall complexity is  approximately 𝑂(𝜀−2−1/4) in the best cases where the overall cost is dominated by  the cost on the coarsest levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  Giles, Nagapetyan & Ritter [22] used explicit smoothing for estimating cumulative  distribution functions (CDFs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' For a scalar random variable 𝑋, to estimate 𝐶(𝑥) =  P(𝑋 < 𝑥) = E[𝐻(𝑥−𝑋)], the approach they adopted was to use MLMC to estimate  𝐶(𝑥 𝑗) for a set of spline points 𝑥 𝑗 and then interpolate these values with a cubic  spline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Overall, their method balanced three weak errors, the SDE discretisation error  on the finest level, the smoothing error due to 𝐻𝛿, and the cubic spline interpolation  error, in addition to the MLMC sampling error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='5  ST  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='8  1  f(S T)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' 1 Two explicitly smoothed versions of the Heaviside step function for a digital call option  6  Michael B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Giles  3 Integration/differentiation and Malliavin calculus  Krumscheid & Nobile [29] used a slightly different approach for estimating CDFs,  particularly in the context of risk estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Starting from the identity  d  d𝑥 E [ max(0, 𝑥−𝑆𝑇 ) ] = E[ 𝐻(𝑥−𝑆𝑇 ) ]  they used MLMC to estimate E[ max(0, 𝑥 𝑗−𝑆𝑇 ) ] for a set of spline points 𝑥 𝑗,  interpolated these with a cubic spline, and and then differentiated the spline to  obtain an approximation to the CDF 𝐶(𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' This avoids the extra weak error due to  smoothing the Heaviside function, but differentiating the cubic spline amplifies the  noise in the spline data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  On a similar note, Altmayer & Neuenkirch [2] used Malliavin calculus integration  by parts to treat discontinuous payoffs based on solutions of the Heston stochastic  volatility SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' They observed that asymptotically this improves the MLMC vari-  ance on the finer levels, but it increases the variance on coarse levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' To address  this, they split the payoff into a smooth part which they treated with the standard  MLMC approach, and a compact-support discontinuous part for which they used the  Malliavin MLMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  Malliavin calculus was originally developed for computing sensitivities, so this  is another example of the literature on sensitivity calculations being exploited to  develop improved MLMC algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  4 Conditional expectation  When using the first order Milstein discretisation for an SDE, one way to improve  the MLMC variance for digital options is to switch to the Euler-Maruyama approxi-  mation for the final timestep, and then take the conditional expectation with respect  to the final fine path Brownian increment Δ𝑊 [12, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  For the fine path approximation of the scalar SDE (1) with 𝑁 timesteps of size  ℎℓ, the path value 𝑆𝑇 at the final time 𝑇 is given by  �𝑆 𝑓  𝑇 = �𝑆 𝑓  𝑇 −ℎℓ + 𝑎(�𝑆 𝑓  𝑇 −ℎℓ) ℎℓ + 𝑏(�𝑆 𝑓  𝑇 −ℎℓ) Δ𝑊𝑁 ,  and therefore the conditional expected value for the digital call option is  �𝑃 𝑓  ℓ  = E  �  𝐻(�𝑆 𝑓  𝑇 −𝐾) | �𝑆 𝑓  𝑇 −ℎℓ  �  = Φ ��  �  �𝑆 𝑓  𝑇 −ℎℓ + 𝑎(�𝑆 𝑓  𝑇 −ℎℓ) ℎℓ − 𝐾  𝑏(�𝑆 𝑓  𝑇 −ℎℓ) √ℎℓ  ��  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' for the coarse path with coarse timestep ℎℓ−1 = 2 ℎℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' the Brownian incre-  ment for the final coarse timestep is the sum of the last two Brownian increments for  the fine path,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Δ𝑊𝑁 −1+Δ𝑊𝑁 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' and therefore  MLMC techniques for discontinuous functions  7  �𝑆𝑐  𝑇 = �𝑆𝑐  𝑇 −ℎℓ−1 + 𝑎(�𝑆𝑐  𝑇 −ℎℓ−1) ℎℓ−1 + 𝑏(�𝑆𝑐  𝑇 −ℎℓ−1) (Δ𝑊𝑁 −1+Δ𝑊𝑁 ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  from which we obtain  �𝑃𝑐  ℓ−1 = E  �  𝐻(�𝑆𝑐  𝑇 −𝐾) | �𝑆𝑐  𝑇 −ℎℓ−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Δ𝑊𝑁 −1  �  = Φ  � �𝑆𝑐  𝑇 −ℎℓ−1 + 𝑎(�𝑆𝑐  𝑇 −ℎℓ−1) ℎℓ−1 + 𝑏(�𝑆𝑐  𝑇 −ℎℓ−1) Δ𝑊𝑁 −1 − 𝐾  𝑏(�𝑆𝑐  𝑇 −ℎℓ−1) √ℎℓ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  With �𝑌ℓ ≡ �𝑃ℓ−�𝑃ℓ−1, numerical analysis [17] proves that 𝑉ℓ ≈ 𝑂(ℎ3/2  ℓ ) so the  MLMC complexity is 𝑂(𝜀−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Heuristically, this is because there is an 𝑂(ℎ1/2  ℓ )  probability of paths being within 𝑂(ℎ1/2  ℓ ) of the strike 𝐾, and for these  �𝑆 𝑓  𝑇 −ℎℓ−1 − �𝑆𝑐  𝑇 −ℎℓ−1 = 𝑂(ℎℓ),  𝜕 �𝑃  𝜕�𝑆  = 𝑂(ℎ−1/2  ℓ  )  =⇒  �𝑃ℓ − �𝑃ℓ−1 = 𝑂(ℎ1/2  ℓ ),  so 𝑉ℓ ≈ 𝑂(ℎ1/2  ℓ  × (ℎ1/2  ℓ )2) = 𝑂(ℎ3/2  ℓ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' In addition, the kurtosis is improved to  𝑂(ℎ−1/2  ℓ  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  Unfortunately, the conditional expectation approach does not help when the Euler-  Maruyama discretisation is used for the entire path since �𝑆 𝑓  𝑇 −ℎℓ−1−�𝑆𝑐  𝑇 −ℎℓ−1 = 𝑂(ℎ1/2  ℓ )  and so �𝑃ℓ − �𝑃ℓ−1 = 𝑂(1)  The use of this kind of conditional expectation is a standard technique for smooth-  ing the payoff to enable IPA/pathwise sensitivity calculations [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Another example  is a down-and-out barrier option, where the option is knocked out if the path drops  below a certain value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' In this case the payoff can be smoothed by computing the  probability of this happening, conditional on the computed path approximations at  discrete timesteps [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Again, this works well for MLMC when using the first or-  der Milstein discretisation [12, 17], but it does not help with the Euler-Maruyama  discretisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  A different kind of conditional expectation smoothing was introduced by Achtsis,  Cools & Nuyens [1] and Bayer, Siebenmorgen & Tempone [6] to improve the  convergence of QMC computations, and then used by Bayer, Ben Hammouda &  Tempone [5] to improve the MLMC variance for digital options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  In its simplest form,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' they split the random inputs for the numerical simulation into  a scalar 𝑍 and the remainder 𝑍𝑟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' and express the desired MLMC level ℓ expectation  as  E[ �𝑃ℓ−�𝑃ℓ−1] = E  �  E[ �𝑃ℓ−�𝑃ℓ−1 | 𝑍𝑟]  �  and observe that in many financial applications it is possible to perform this split  in a way such that the conditional expectations E[ �𝑃ℓ | 𝑍𝑟],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' E[ �𝑃ℓ−1 | 𝑍𝑟] are smooth  functions of 𝑍𝑟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' and can be evaluated analytically or very accurately by 1D numerical  quadrature when there is just a single discontinuity with respect to changes in 𝑍.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  8  Michael B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Giles  For a scalar SDE, 𝑍 could be the terminal value of the driving Brownian motion,  in which case 𝑍𝑟 would represent the other Normal random variables required for a  Brownian Bridge construction of the Brownian increments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  5 Change of measure  Another approach to treating digital options using the Milstein discretisation is to  use a change of measure [9, 14], which has connections to the Likelihood Ratio  Method (LRM) that is used for sensitivity analysis [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  For both the fine and coarse paths, we have conditional Gaussian distributions for  �𝑆𝑇 , with slightly different means and variances, as illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' We can  therefore perform a change of measure to the same Gaussian distribution with mean  𝜇 and variance 𝜎2, also illustrated in Figure 2, and then pick the same sample �𝑆𝑇  for both paths from this common Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  The resulting MLMC estimator is  �𝑌ℓ = 𝑓 (�𝑆𝑇 ) (𝑅ℓ − 𝑅ℓ−1)  where 𝑅ℓ, 𝑅ℓ−1 are the respective Radon-Nikodym derivatives for the fine and coarse  paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' For the scalar SDE (1), 𝑅ℓ is  𝑅ℓ =  𝜎  𝑏(�𝑆 𝑓  𝑇 −ℎℓ)√ℎℓ  exp  �  − (�𝑆𝑇 − �𝑆 𝑓  𝑇 −ℎℓ − 𝑎(�𝑆 𝑓  𝑇 −ℎℓ) ℎℓ)2 / (2 𝑏2(�𝑆 𝑓  𝑇 −ℎℓ) ℎℓ)  �  exp  �  − (�𝑆𝑇 −𝜇)2 / (2𝜎2)  �  and 𝑅ℓ−1 is defined similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' It can be shown that the difference 𝑅ℓ − 𝑅ℓ−1 is  approximately 𝑂(ℎ1/2  ℓ ), which implies that 𝑉ℓ ≈ 𝑂(ℎℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' To improve the variance we  note that the conditional expected value of Radon-Nikodym derivatives is always 1,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='4  ST  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='4  p(ST)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' 2 Coarse and fine path conditional Gaussian distributions, plus third common distribution  MLMC techniques for discontinuous functions  9  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='2  t  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='5  2  S  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' 3 Illustration of coarse and fine jump-diffusion paths with jumps before and after 𝑇 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' E[𝑅ℓ | �𝑆 𝑓  𝑇 −ℎℓ] = E[𝑅ℓ−1 | �𝑆𝑐  𝑇 −ℎℓ−1, Δ𝑊𝑁 −1] = 1, and therefore we can change  the definition of �𝑌ℓ to  �𝑌ℓ =  �  𝑓 (�𝑆𝑇 ) − 𝑓 (𝜇)  �  (𝑅ℓ − 𝑅ℓ−1)  without changing its expected value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' This estimator is now non-zero only when �𝑆𝑇  and 𝜇 are on opposite sides of the strike 𝐾, which occurs for an 𝑂(ℎ1/2  ℓ ) fraction of  coarse/fine paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Hence the new MLMC variance 𝑉ℓ is approximately 𝑂(ℎ3/2  ℓ ), as  with the use of the analytic conditional expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  The benefit of this approach is that it works well in multiple dimensions when it is  often not possible to evaluate the analytic conditional expectation [9, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' However,  again it does not help with the full path Euler-Maruyama discretisation because that  gives 𝑅ℓ − 𝑅ℓ−1 = 𝑂(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  An earlier use of a change of measure in an MLMC computation was by Xia  [33, 34] for a Merton-style jump-diffusion SDE with a path-dependent jump rate  𝜆(𝑆, 𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' The challenge in this application, as illustrated in Figure 3, is that the  coarse and fine paths will jump at different times, and one might jump just before  the final time 𝑇, and the other just after, leading to a large jump in the computed  value of 𝑓 (𝑆𝑇 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' The path-dependent jump rate was treated by using the thinning  technique of Glasserman & Merener [25], over-sampling possible jump times using  a uniform rate 𝜆𝑠𝑢𝑝 > 𝜆(𝑆, 𝑡) and then using an acceptance/rejection step to select  the real jump times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Xia modified this with a change of measure to ensure the same  acceptance/rejection decision for both the fine and coarse paths so that they both  jump at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' This leads to an estimator of the form  �𝑌ℓ = �𝑃ℓ 𝑅ℓ − �𝑃ℓ−1 𝑅ℓ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  10  Michael B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Giles  When combined with a first order Milstein discretisation of the SDE between the  jump times, this gives 𝑉ℓ = 𝑂(ℎ2  ℓ) for Lipschitz payoff functions such as a standard  put or call option [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  6 Splitting  Returning to the challenge of digital options arising from the solution of an SDE,  a third approach is to use path-splitting to generate an unbiased estimate of the  conditional expectation introduced in Section 4 [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  This is a variant of the general splitting technique [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' As illustrated in Figure 4,  it involves performing a standard fine path simulation up until one timestep before  the final time 𝑇, and then performing multiple independent simulations of the final  timestep, averaging the payoff for each of these to get an approximation of the  conditional expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' The same is done for the coarse path except that each of the  splits uses the same Δ𝑊𝑁 −1 that was used for the second to last fine path timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  Since the computational cost of the path up to the splitting time is 𝑂(ℎ−1  ℓ ), it means  that up to 𝑂(ℎ−1  ℓ ) splits can be used without increasing the path cost significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' If  𝑀ℓ splits are used, then the standard splitting variance analysis gives  V[�𝑌ℓ] = V  �  E[ �𝑃ℓ−�𝑃ℓ−1 | {Δ𝑊𝑛}𝑛<𝑁 ]  �  + 𝑀−1  ℓ E  �  V[ �𝑃ℓ−�𝑃ℓ−1 | {Δ𝑊𝑛}𝑛<𝑁 ]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  As discussed previously V  �  E[ �𝑃ℓ−�𝑃ℓ−1 | {Δ𝑊𝑛}𝑛<𝑁 ]  �  = 𝑂(ℎ3/2  ℓ ), and similarly it  can be argued that E  �  V[ �𝑃ℓ−�𝑃ℓ−1 | {Δ𝑊𝑛}𝑛<𝑁 ]  �  = 𝑂(ℎℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Therefore choosing 𝑀ℓ  to lie between 𝑂(ℎ−1  ℓ ) and 𝑂(ℎ−1/2  ℓ  ) ensures the benefits of the splitting are obtained  without significantly increasing the computational cost per sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='8  1  t  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='8  St  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' 4 Path splitting in final timestep to estimate conditional expectation  MLMC techniques for discontinuous functions  11  As an additional bonus, one can use the more accurate Milstein discretisation  for the final timestep, instead of switching to the Euler-Maruyama discretisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  Burgos [9, 10] gives more details of the analysis, and also used the same approach  for pathwise sensitivity analysis for a variety of financial options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  Giles & Bernal [16] also used splitting for Feynman-Kac functionals arising for  stopped diffusions, SDE simulations which terminate when the solution path leaves  a prescribed domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' The issue here is that when a fine path exits, there is an 𝑂(ℎ1/2  ℓ )  probability that the corresponding coarse path does not leave until much later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' This  is addressed by estimating a conditional expectation by splitting the coarse path into  𝑂(ℎ−1/2  ℓ  ) independent sub-simulations which continue until each of them leaves the  domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' 𝑉ℓ is improved from 𝑂(ℎ1/2  ℓ ) to approximately 𝑂(ℎℓ) without a significant  increase in the cost per sample, and finally the MLMC complexity achieved is  𝑂(𝜀−2| log 𝜀|3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  None of the three methods introduced so far (conditional expectation, change of  measure, splitting) helps when using the Euler-Maruyama discretisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' For this, a  new method has recently been developed by Giles & Haji-Ali [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  It again uses splitting, but inspired by the simulation of branching diffusions, it  considers splits at multiple deterministic times, as illustrated in Figure 5 which shows  the logical structure of a set of split paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Here we are considering a simulation  on the unit time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' A single pair of fine/coarse paths is calculated up to time  𝑡 = 1/2, with the number of fine timesteps being 1  2 ℎ−1  ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' This simulation is then  split into two separate independent simulations up to time 𝑡 = 3/4, with the two  simulations between them accounting for an additional 1  2 ℎ−1  ℓ  fine timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' There  are further splits at 𝑡 = 3/4, then at 𝑡 = 7/8, and so on, with the final split when there  is just one coarse timestep left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  The total number of fine timesteps simulated is 𝑂(ℎ−1  ℓ | log ℎℓ|) so the computa-  tional cost is only slightly increased compared to the original method with a single  pair of fine/coarse paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' �𝑌ℓ is defined to be the average of the values �𝑃ℓ−�𝑃ℓ−1 for  each of the final paths, and it can be proved that its variance is 𝑂(ℎℓ), the same  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='8  1  t  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='2  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' 5 Repeated path splitting to estimate conditional expectation  12  Michael B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Giles  asymptotic order of convergence as for Lipschitz payoff functions [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' The kurtosis  is also improved, so this technique fully addresses the challenge of using MLMC  with the Euler-Maruyama discretisation to estimate digital option values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  7 Adaptive sampling  We return now to the challenge of estimating the nested expectation E [ 𝐻 (E[𝑍|𝑋]) ]  and we note that we only need an accurate estimate of the inner conditional expecta-  tion E[𝑍|𝑋] when it is near zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' This observation is the basis for the development  of adaptive sampling by Broadie, Du & Moallemi [7] within a standard Monte Carlo  procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' This was then extended to adaptive sampling combined with MLMC by  Giles & Haji-Ali [19] by defining the number of inner samples 𝑀ℓ on level ℓ to be 𝑀ℓ = 2ℓ𝑀0 inner samples when |E[𝑍|𝑋]| ≫  √︁  V[𝑍|𝑋]/(2ℓ𝑀0)  This is the smallest number of samples used on level ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  √︁  V[𝑍|𝑋]/(2ℓ𝑀0) is  the standard deviation of the Monte Carlo estimate for E[𝑍|𝑋], so the inequality  means that this number of samples is sufficient to be very sure that the estimate  has the correct sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' 𝑀ℓ = 4ℓ𝑀0 inner samples when |E[𝑍|𝑋]| = 𝑂(  √︁  V[𝑍|𝑋]/(4ℓ𝑀0))  This is the maximum number of samples used on level ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' In this case, the estimate  of E[𝑍|𝑋] may have the incorrect sign, but this will only happen when |E[𝑍|𝑋]| =  𝑂(2−ℓ) which occurs with probability 𝑂(2−ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Likewise, the total cost of the  higher number of samples in this region is 𝑂(2−ℓ × 4ℓ) = 𝑂(2ℓ), so it does not  significantly increase the overall average cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' 2ℓ𝑀0 < 𝑀ℓ < 4ℓ𝑀0 for intermediate values  In this region the number of samples is chosen to be very sure that the estimate  of E[𝑍|𝑋] has the correct sign, and at the same time the total cost is 𝑂(2ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  Overall, this adaptive sampling approach leads to 𝐶ℓ ∼ 2ℓ, 𝑉ℓ ∼ 2−ℓ and hence  a complexity of roughly 𝑂(𝜀−2) [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' However, the kurtosis is 𝑂(2ℓ) since only an  𝑂(2−ℓ) fraction of the outer samples give non-zero values for �𝑌ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  Haji-Ali, Spence & Teckentrup [27] have further extended this to estimate quan-  tities of the form  P[𝐺 ∈ Ω] ≡ E[1𝐺∈Ω]  where 𝐺 is a 𝑑-dimensional random variable which cannot be sampled directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  In their paper they consider in particular the two challenges in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' In the  context of the digital option with the Euler-Maruyama discretisation on the unit time  interval,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' the adaptive sampling varies the timestep used on level ℓ so that ℎℓ = 2−ℓ when |�𝑆ℓ − 𝐾| is large compared to the strong error in the path approx-  imation ℎℓ = 4−ℓ when |�𝑆ℓ − 𝐾| is of the same order as the strong error  MLMC techniques for discontinuous functions  13 2−ℓ < ℎℓ < 4−ℓ for intermediate values  A Brownian bridge construction is used when the timestep needs to be refined as  part of the adaptation procedure from its initial value ℎℓ = 2−ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' The adaptation again  leads to 𝐶ℓ ∼ 2ℓ, 𝑉ℓ ∼ 2−ℓ and hence a complexity of roughly 𝑂(𝜀−2), but there is  again a high kurtosis [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  In earlier research, Elfverson, Hellman & Malqvist [11] considered estimation of  E[𝐻(𝑋)] where 𝑋 cannot be sampled exactly but there is a sequence of approxi-  mations 𝑋′  0, 𝑋′  1, 𝑋′  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' 𝑋 of increasing accuracy and increasing cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Motivated by  PDE applications with a well-behaved truncation error so that there are uniform  geometric bounds on |𝑋′  𝑗 − 𝑋|, level ℓ in their method uses  �𝑋ℓ = 𝑋′  𝑗,  𝑗 = min{ℓ, min 𝑗 : | �𝑋′  𝑗 − 𝑋| < |𝑋|}  and achieves similarly good MLMC benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' This idea is essentially the same as in  the work of Haji-Ali et al but requiring a uniform bound on |𝑋′  𝑗−𝑋| is significantly  more restrictive than the bounds on E[ |𝑋′  𝑗−𝑋|𝑞] for some 𝑞>2 required by Haji-Ali  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  A final comment is that the analysis of Haji-Ali, Spence & Teckentrup can be gen-  eralised to a product of an indicator function and a Lipschitz function, E[1𝐺∈Ω 𝑓 (𝑆)],  and so can handle barrier options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Furthermore, Haji-Ali & Spence have extended  the adaptive sampling methodology to an extremely challenging triply-nested expec-  tation which arises in mathematical finance [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' By incorporating the randomised  MLMC treatment of Rhee & Glynn [32] to handle the time discretisation of the  underlying SDEs as well as the sampling for the inner conditional expectations, they  achieve an overall complexity of approximately 𝑂(𝜀−2) which is very impressive for  such a difficult application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  10  5  0  5  10  E[Z|X]  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='5  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' 6 Error distributions for two conditional expectations with i) few samples being needed to  ensure the correct sign (left), and ii) many samples being insufficient to ensure the correct sign  (centre).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' The blue line represents the Heaviside step function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  14  Michael B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Giles  8 Conclusions  It is worth repeating that in most MLMC applications the output quantity of interest  is a Lipschitz function of the intermediate simulation quantities, so good strong  convergence for the intermediate quantities leads automatically to a good rate of  convergence of the MLMC variance 𝑉ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  For those applications in which the function is discontinuous, this article shows  there is an extensive literature with a variety of different approaches to improve the  MLMC variance and try to recover the optimal 𝑂(𝜀−2) complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' It is notable that  many of these methods have adapted ideas from Monte Carlo sensitivity analysis  which also has problems with discontinuous functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' It is hoped that this survey  will assist future researchers facing similar challenges in other new application areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  Acknowledgements This paper is based on research with many students, postdocs and other  collaborators and I am grateful to all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Funding for the research has been provided by the UK  Engineering and Physical Sciences Research Council through grants EP/E031455/1, EP/H05183X/1  and EP/P020720/2 as well as the Hong Kong Innovation and Technology Commission (InnoHK  Project CIMDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' The paper was written while visiting the Oden Institute at UT Austin, and I thank  my hosts for their warm hospitality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
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+page_content=' Giles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
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+page_content=' Operations Research, 56(3):607–617,  2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
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+page_content=' Giles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Multilevel Monte Carlo methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
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+page_content=' Giles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' MLMC for nested expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
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+page_content=' Multilevel estimation of expected exit times and other functionals of  stopped diffusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
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+page_content=' Unbiased estimation with square root convergence for SDE  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Operations Research, 63(5):1026–1043, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
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+page_content=' Multilevel Monte Carlo for jump processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' DPhil thesis, University of Oxford, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content='  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
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+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Giles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Multilevel path simulation for jump-diffusion SDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' In L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Plaskota  and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Woźniakowski, editors, Monte Carlo and Quasi-Monte Carlo Methods 2010, pages  695–708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
+page_content=' Springer, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'}
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+arXiv:2301.04457v1  [gr-qc]  11 Jan 2023
+Ghost and Laplacian Instabilities in Teleparallel Horndeski Gravity
+Salvatore Capozziello,1, 2, 3, ∗ Maria Caruana,4, 5, † Jackson Levi Said,4, 5, ‡ and Joseph Sultana6, §
+1Dipartimento di Fisica ”E. Pancini”, Universit`a degli Studi di Napoli,
+”Federico II”, Complesso Universitario Monte S. Angelo,
+Via Cinthia 9 Edificio G, 80126 Napoli, Italy
+2Istituto Nazionale di Fisica Nucleare (INFN),
+Sezione di Napoli Complesso Universitario Monte S. Angelo,
+Via Cinthia 9 Edificio G, 80126 Napoli, Italy
+3Scuola Superiore Meridionale, Largo San Marcellino 10, 80138 Napoli, Italy
+4Institute of Space Sciences and Astronomy, University of Malta, Msida, Malta
+5Department of Physics, University of Malta, Msida, Malta
+6Department of Mathematics, University of Malta, Msida, Malta
+Teleparallel geometry offers a platform on which to build up theories of gravity where torsion
+rather than curvature mediates gravitational interaction. The teleparallel analogue of Horndeski
+gravity is an approach to teleparallel geometry where scalar-tensor theories are considered in this
+torsional framework. Being teleparallel gravity of lower order in dynamics, this turns out to be more
+general than metric Horndeski gravity. In other words, the class of teleparallel Horndeski gravity
+models is much broader than the standard metric one. In this work, we explore constraints on this
+wide range of models coming from ghost and Laplacian instabilities. The aim is to limit pathological
+branches of the theory by fundamental considerations. It is possible to conclude that a very large
+class of models results physically viable.
+I.
+INTRODUCTION
+General relativity (GR) has been the gravitational foundation of cosmology for over a century. Its latest
+embodiment appears in a picture wherein the evolutionary processes of the Universe are described in the
+framework of the so-called ΛCDM model [1, 2]. Supported by overwhelming observational evidences, this
+model describes a Universe started with a big bang, then driven through an inflationary phase and other well
+known epochs to eventually produce an accelerating Universe at late-times [3, 4]. In the ΛCDM model, the
+late time acceleration expansion is driven by the cosmological constant or some form of dark energy. Despite
+a lot of foundational works, internal consistency issues persist in this respect [5–7]. Recently, observational
+challenges to the standard model has arisen in the form of cosmological tensions with statistically significant
+differences between predictions of expansion from early time data [8–10] and measurements from late time
+[11, 12]. These tensions continue to increase with new survey data [13–15], and may permeate into other
+sectors of cosmology besides expansion [16–18]. This situation leads to take into account potential alternatives
+to the standard cosmological ΛCDM model in the context of possible modifications to the gravitational sector.
+A solution to the problem of cosmic tensions, as well as of other longstanding issues, can be offered by
+alternative theories of gravity, in particular by scalar-tensor gravity. Scalar fields improve GR in view of Mach
+principle and could naturally address several issues in cosmology and astrophysics like inflation, dark matter
+and dark energy [19]. The most general scalar-tensor theory of gravity, where scalar fields are minimally and
+non-minimally coupled to curvature, is the so-called Horndeski gravity [20]. It is the most general theory
+of gravity producing second order field equations. This is advantageous because nature appears to produce,
+generally, second order equations of motion. Furthermore, higher order theories can produce Ostrogradsky
+instabilities making second order theories more attractive from a fundamental perspective [21–23]. Horndeski
+gravity is formulated by four free functions of the scalar field and its kinetic term [24, 25]. This offers a
+rich phenomenology on which to produce dark energy models but also dark matter and inflation. However,
+recent multimessenger observations by the LIGO collaboration, in the gravitational event GW170817 [26], and
+measurements of the companion electromagnetic counterpart, namely GRB170817A [27], has placed stringent
+constraints on the speed of propagation of gravitational waves (GW) to within deviations of at most one part
+∗ capozziello@na.infn.it
+† maria.caruana.16@um.edu.mt
+‡ jackson.said@um.edu.mt
+§ joseph.sultana@um.edu.mt
+
+2
+in 1015. Considering these constraints, many branches of Horndeski gravity have been disqualified in regular
+curvature-based gravity [28–30]. However, Horndeski gravity has a number of interesting generalizations
+[31] where higher order terms, avoiding the Ostrogradsky instabilities, can be incorporated into the theory.
+Another possibility is to reconsider the geometric foundations of curvature on which Horndeski gravity is
+built. Curvature can be expressed through the Levi-Civita connection
+◦Γ
+λ
+µν obtained from the metric [1].
+Here an over-circle represents quantities determined by the Levi-Civita connection.
+In teleparallel gravity (TG), the teleparallel connection (Γλ
+µν) replaces the Levi-Civita connection and so
+dynamics of gravity is replaced from curvature to torsion [32–35]. The teleparallel connection is curvatureless
+and satisfies metricity. For this reason, the teleparallel Ricci scalar turns out to identically vanish, i.e. R = 0
+(this is not to say that the standard Ricci scalar is zero, being, in general,
+◦R ̸= 0). On the other hand, TG
+naturally produces a torsion scalar (T ) [33], which turns out to be equal to the curvature-based Ricci scalar
+up to a boundary term (B). For this reason, the linear torsion scalar produces the Teleparallel equivalent of
+General Relativity (TEGR) which is dynamically equivalent to GR in the classical regime. This distinction
+between the torsion scalar, producing second order terms in the equations of motion, and the boundary
+term, producing fourth order terms in the equations of motion (when the boundary term is nonlinear in the
+action), gives rise to a much richer landscape of theories on which to build cosmological models, if compared
+with the standard Einstein-Hilbert action.
+Another way to conceptualize this feature is through the teleparallel analogue of the Lovelock theorem
+[36–38] which produces a much wider range of actions with second order field equations. Equivalences and
+differences of these representations are discussed in Ref. [39].
+Similar to GR, TEGR features many modifications such as f(T ) gravity [34, 40–46] which is generically
+second order in nature, as well as f(T, B) gravity [47–55] which now features fourth order contributions
+similar to f(
+◦R) gravity [19, 56–62]. There have also been various formulations of scalar–tensor gravity. See,
+e.g. Refs.[63–65].
+Equipped with the TG formalism, one can consider the teleparallel analogue of Horndeski gravity [38] where
+the generically lower order nature of TG turns out to produce a much richer structure to be developed. This
+feature directly comes from the teleparallel analogue of the Lovelock theorem. This larger framework of
+models means that there are many more classes of theories that now satisfy the speed of GWs constraint
+[66, 67] as required from the above mentioned multimessenger analysis [68].
+This class of gravitational
+models also satisfies the parameterized post-Newtonian observational constraints [69] for a large number of
+models. The teleparallel analogue of Horndeski gravity also produces a number of interesting cosmologies
+such as those which are well-tempered [70, 71], as well as those that are derived from Noether symmetry
+considerations [72–76]. In this way, all models that were previously disqualified in regular Horndeski may be
+revived in this new formalism based on TG.
+In this work, we perform a stability analysis of the teleparallel analogue of Horndeski gravity through
+considerations on a Minkowski spacetime background. The Minkowski background is an ideal arena to probe
+the fundamental behavior of theories with large classes of models since any astrophysical or cosmological
+setting must first be stable on a Minkowski spacetime. In our analysis, we consider both background and
+leading order perturbations in the context of a scalar-vector-tensor decomposition. This is important to fully
+decompose any potentially problematic part of the theory in detail. In the TG setting, this is more intricate
+since the metric tensor (gµν) is replaced as the fundamental dynamical variable of the field equations with
+the tetrad (eA
+µ) and the spin connection (ωA
+Bµ), which are discussed in detail later on. In all cases, we
+focus our study on the ghost and gradient instabilities, which can wreak havoc on gravitational models since
+this may, respectively, produce scalar field kinetic terms with the wrong sign and unbounded propagation
+speeds of particular perturbations, which can lead to unphysical models.
+The structure of the manuscript is the following: in Sec. II, we summarize TG and its teleparallel analogue
+of Horndeski gravity; this then leads to the Minkowski background equations of motion in Sec. III. Scalar–
+vector–tensor perturbations are discussed in Sec. IV. From these perturbations, we are able to explore
+potential instabilities in Sec. V where our main results are reported. Discussion and conclusions are drawn
+in Sec. VI. In this work, we use geometric units and the signature (−, +, +, +).
+
+3
+II.
+TELEPARALLEL HORNDESKI GRAVITY
+GR, a curvature-based theory, is constructed on torsionless Levi-Civita connection
+◦Γλ
+µν, where, as said
+above, the overhead circle (◦) represents quantities built with this connection.
+This leads to numerous
+theories of gravity beyond GR as the gravitational field can be expressed through the Riemann tensor and
+its contractions such as the Ricci tensor and the Riemann scalar [59], the latter produces the Einstein-
+Hilbert action [1]. On the other hand, TG incorporates the teleparallel connection, dubbed as teleparallel
+connection Γλ
+µν, leading to a theory satisfying the curvature-less and metricity conditions [32–35], resulting
+in a torsionful theory. Later in this section, it will be shown that even though the Ricci scalar R vanishes,
+due to the symmetry of the connection, the curvature-ful connection gives a non-zero value for the Riemann
+tensor. Hence, the Ricci scalar
+◦R can be defined through teleparallel quantities.
+The fundamental dynamical object of GR is the metric tensor gµν, but within TG, the metric is expressed
+through the tetrad eA
+µ, which acts as a transformation between the local and general manifold spaces. The
+tetrad and the inertial spin connection ωB
+Cν become the fundamental objects [33] while creating a link
+between the general manifold denoted through Greek indices to the local Minkowski manifold denoted by
+Latin indices. Thus, the relationship between Minkowski and general spacetimes is given by [77]
+gµν = ηAB eA
+µ eB
+ν ,
+ηAB = gµν E
+µ
+A E
+ν
+B ,
+(1)
+where E
+µ
+A
+is the inverse tetrad which satisfies orthogonality conditions
+eA
+µ E
+µ
+B
+= δA
+B ,
+eA
+µ E
+ν
+A
+= δν
+µ .
+(2)
+The teleparallel connection is defined through the TG variables as [34, 35]
+Γλ
+µν = E
+λ
+A
+�
+∂νeA
+µ + ωA
+BνeB
+µ
+�
+.
+(3)
+Note, the quantities without an overhead circle will correspond to those objects that are related to teleparallel
+gravity or calculated on its connection. The condition
+∂[µωA
+|B|ν] + ωA
+C[µωC
+|B|ν] ≡ 0 ,
+(4)
+results in a flat spin connection [32], where square brackets denote the antisymmetric operator, and can
+equivalently be used to determine the components of the spin connection. The local Lorentz transformation
+(LLT), wherein ΛA
+B represents Lorentz boosts and rotations, is used to define the spin connection ωA
+Bµ =
+ΛA
+C ∂µΛ
+C
+B
+[33]. It plays a role in the field equations since an infinite number of tetrad choices could satisfy
+Eq. (1) for a particular metric, thus the spin connection is used to counterbalance inertial effects. This
+ensures the theory, in this case TG, remains covariant [78]. Moreover, there exists a Lorentz frame such that
+the spin connection is set to zero. It is referred to as the Weitzenb¨ock gauge [35, 79]. Hence, from this point
+onwards, the spin connection will be dropped following the application of the aforementioned gauge.
+Similar to how the Riemann tensor, constructed on the Levi-Civita connection, is associated to the cur-
+vature property of GR, the analogy of this for TG is the torsion tensor defined by the teleparallel connec-
+tion [35, 80]
+T A
+µν = ΓA
+νµ − ΓA
+µν .
+(5)
+While curvature in GR is defined as a deformation of spacetime, the torsion tensor in TG represents the
+field strength of gravitation that transforms covariantly under LLTs and diffeomorphisms [78]. Another
+important aspect of TG is that the torsion tensor can be decomposed in irreducible parts [81–83], namely
+aµ = 1
+6ǫµνλρ T νλρ ,
+(6)
+vµ = T λ
+λν ,
+(7)
+tλµν = 1
+2(Tλµν + Tµλν) + 1
+6(gνλvµ + gνµvλ) − 1
+3gλµvν ,
+(8)
+
+4
+giving the axial, vectorial and tensorial pieces, respectively, and ǫABCD being the four-dimensional Levi-
+Civita tensor. Hence, the respective scalar invariants are given by [47, 82]
+Tax = aµaµ = − 1
+18Tλµν(T λµν − 2T µλν) ,
+(9)
+Tvec = vµvµ = T λ
+λµT
+ρµ
+ρ
+,
+(10)
+Tten = tλµνtλµν = 1
+2Tλµν(T λµν + T µλν) − 1
+2T λ
+λµT ρµ
+ρ
+,
+(11)
+which, under parity transformation, are invariant scalars. On the other hand, terms such as P1 = vµaµ and
+P2 = ǫµνσρtλµνtλ
+ρσ are excluded due to parity-violation [38, 81]. The combination of the scalar invariants
+leads to the torsion scalar
+T = 3
+2Tax + 2
+3Tten − 2
+3Tvec = 1
+2
+�
+E
+λ
+A gρµE
+ν
+B
+− 2E
+ρ
+B gλµE
+ν
+A + 1
+2ηABgµρgνλ
+�
+T A
+µνT B
+ρλ .
+(12)
+An identical result can be obtained through contraction between the torsion tensor and the superpotential
+S
+µν
+A
+which represents the potential relation of the gravitational energy-momentum tensor [80, 84, 85]:
+S
+µν
+A
+= 1
+2
+�
+Kµν
+A − E
+ν
+A T αµ
+α + E
+µ
+A T αν
+α
+�
+,
+(13)
+where
+Kλ
+µν = Γλ
+µν −
+◦Γλ
+µν = 1
+2
+�
+T
+λ
+µ ν + T
+λ
+ν µ − T λ
+µν
+�
+(14)
+is the contorsion tensor relating TG and GR. These quantities can be used to form the torsion scalar [34],
+written as
+T = S
+µν
+A
+T A
+µν ,
+(15)
+which can be shown to be equivalent to the result obtained in Eq. (12). The Ricci scalar dependent on
+the teleparallel connection, which vanishes, can be related, using the contorsion tensor, to the regular Ricci
+scalar. This results in a relationship between the Levi-Civita and the teleparallel connections defined Ricci
+scalar [47, 77]
+R =
+◦R + T − B = 0 ,
+(16)
+where B = 2
+e∂µ(e T λ µ
+λ ) = 2
+◦∇µT λ µ
+λ
+is a boundary term and e = det(eA
+µ) = √−g is the tetrad determinant.
+Hence, the curvature-ful Ricci scalar is non-vanishing being
+◦R = −T + B .
+(17)
+The total divergence term found in B accounts for the fourth order derivative contributions to the field
+equations in modified theories of gravity [47, 54]. This is embodied within the Ricci scalar in GR [57, 59].
+Thus, the Teleparallel Equivalent of General Relativity (TEGR) [86, 87] is defined as the linear appearance
+of the torsion scalar since both Lagrangians are equal up to a boundary term.
+The equivalence principle in GR allows one to raise local Lorentz frames from a Minkowski metric to
+a general metric tensor, and additionally, partial derivatives are raised to covariant derivatives defined by
+the Levi-Civita connection [1]. This procedure, referred to as the minimal coupling prescription, if applied
+within TG, is preserved for additional fields. Minkowski tetrads are raised to arbitrary ones and tangent
+space partial derivatives are exchanged for covariant derivatives based on Levi-Civita connection [33, 88]
+∂µ →
+◦∇µ .
+(18)
+Thus, by having both gravitational and scalar fields well developed, it is possible to look at the teleparallel
+analogue of Horndeski gravity, referred to at times as Bahamonde-Dialektopoulos-Levi Said (BDLS) the-
+ory [38, 66, 69]. The construction of BDLS theory depends on the following criteria: (1) field equations are
+
+5
+at most second order with respect to the tetrad and scalar; (2) as previously mentioned, scalar invariants do
+not violate parity; (3) contractions of the torsion tensor are at most quadratic [38]. All of these requirements
+allow for an adequate extension of the standard metric Horndeski gravity. Note that Lovelock’s theorem
+states that Einstein fields equations are the only second-order field equations from a Lagrangian density
+constructed through a four-dimensional metric. TG allows for the weakening of Lovelock [36, 37] theory as
+an additional scalar field φ is introduced, giving rise to further terms in the gravitational action.
+Starting off with the non-minimal coupling of scalar and torsion field, the linear contraction is given by [38]
+I2 = vµ ◦∇µφ ,
+(19)
+being the scalar I1 = tλµν
+◦∇λφ
+◦∇µφ
+◦∇µφ = 0, due to the complete symmetry of the tensor decomposition,
+Furthermore I3 = aλ
+◦∇λφ violates the parity condition due to an odd number of axial parts. Moreover, the
+quadratic contractions of this nature are given by [38]
+J1 = aµaν ◦∇µφ
+◦∇νφ ,
+(20)
+J3 = vλtλµν ◦∇µφ
+◦∇νφ ,
+(21)
+J5 = tλµνt α
+λ ν
+◦∇µφ
+◦∇αφ ,
+(22)
+J6 = tλµνt αβ
+λ
+◦∇µφ
+◦∇νφ
+◦∇αφ
+◦∇βφ ,
+(23)
+J8 = tλµνt
+α
+λµ
+◦∇νφ
+◦∇αφ ,
+(24)
+J10 = ǫµ
+νλρaνtαρλ ◦∇µφ
+◦∇αφ ,
+(25)
+while other possibilities are eliminated as they have already been included:
+J2 = vµvν ◦∇µφ
+◦∇νφ = I2
+2 ,
+J4 = vµtλµν ◦∇λφ
+◦∇νφ = J3 ,
+J7 = tλµνtαβ
+λ
+◦∇µφ
+◦∇νφ
+◦∇αφ
+◦∇βφ = −2J6 ,
+(26)
+while J9 = tλµνtαβγ
+◦∇λφ
+◦∇µφ
+◦∇νφ
+◦∇αφ
+◦∇βφ
+◦∇γφ = 0 due to the total symmetry of the tensor irreducible part.
+Hence, BDLS action is described by [38]
+SBDLS =
+1
+2κ2
+�
+d4x e LTele +
+1
+2κ2
+5
+�
+i=2
+�
+d4x e Li +
+�
+d4x e Lm ,
+(27)
+where
+LTele = GTele(φ, X, T, Tax, Tvec, I2, J1, J3.J5, J6, J8, J10) .
+(28)
+Here GTele is an arbitrary function, X := − 1
+2∂µφ∂µφ and
+L2 := G2(φ, X) ,
+(29)
+L3 := −G3(φ, X)
+◦□φ ,
+(30)
+L4 := G4(φ, X)(−T + B) + G4,X(φ, X)[(
+◦□φ)2 −
+◦∇µ
+◦∇νφ
+◦∇µ ◦∇νφ] ,
+(31)
+L5 := G5(φ, X)
+◦Gµν
+◦∇µ
+◦∇νφ
+− 1
+6G5,X(φ, X)[(
+◦□φ)3 + 2
+◦∇µ
+◦∇νφ
+◦∇ν
+◦∇αφ
+◦∇α
+◦∇µφ − 3
+◦∇µ
+◦∇νφ
+◦∇µ ◦∇νφ
+◦□φ] ,
+(32)
+which are identical to the standard metric Horndeski Lagrangians [20] but calculated using the tetrad.
+Finally, Lm is the matter Lagrangian in Jordan conformal frame,
+◦Gµν is the Einstein tensor, and κ2 = 8πG
+where G is the gravitational constant.
+III.
+MINKOWSKI BACKGROUND EQUATIONS IN TELEPARALLEL HORNDESKI GRAVITY
+Let us explore now perturbations on a Minkowski background within the gravitational theory of teleparallel
+analogue of Horndeski. Firstly, we tackle the background equations for a general flat Friedman–Lemaˆıtre–Robertson–Walker
+
+6
+(FLRW) metric to obtain the necessary constraints. Then, this is followed by the analysis for perturbation
+theory up to second order which is needed for our approach since we consider the Euler-Lagrange equations
+to get the cosmological equations of motion.
+The flat FLRW metric in Cartesian coordinates is given by
+ds2 = −N(t)2dt2 + a(t)2(dx2 + dy2 + dx2) ,
+(33)
+where N(t) is the lapse function and a(t) is the scale factor. As previously mentioned, this follows the metric
+signature that is mostly positive. The tetrad can be written in diagonal form as [35]
+eA
+µ = diag(N(t), a(t), a(t), a(t)) ,
+(34)
+which is compatible with the Weitzenb¨ock gauge giving a vanishing spin connection. Hence, the action in
+Eq. (27) can be re-expressed in terms of these background quantities. Friedman equations are then obtained
+by varying with respect to the lapse function and scale factor. Additionally, the scalar field Klein-Gordon
+equation can be obtained by varying with respect to the scalar field [38].
+Since we will be working in
+Minkowski background, the limits N(t) → 1 and a(t) → 1 are taken. Hence, the constraints obtained from
+the Friedman equation and scalar field variation in Minkowski background are given by
+0 = −G2 − GTele + 2XG2,X − 2XG3,φ + 2XGTele,X ,
+(35)
+0 = G2 + GTele − 2XG3,φ + 4XG4,φφ + 2¨φG4,φ − 2X ¨φG3,X + 4X ¨φG4,φX − d
+dt( ˙φGTele,I2) ,
+(36)
+0 = G2,φ + GTele,φ − 2XG2,φX + 2XG3,φφ − 2XGTele,φX − G2,X ¨φ + 2G3,φ ¨φ − GTele,X ¨φ
+− 2X ¨φGTele,X − 2X ¨φG2,XX + 2X ¨φG3,φX − 2X ¨φGTele,XX ,
+(37)
+where all scalar invariants are background quantities and comma ( , ) denotes partial derivative. Moreover,
+the scalar field equation can also be expressed in terms of a Klein-Gordon equation as shown in Ref. [38].
+IV.
+SECOND ORDER PERTURBED ACTION WITH SCALAR-VECTOR-TENSOR
+DECOMPOSITION
+Here, we calculate the perturbed action up to second order terms which are necessary for the Euler-
+Lagrange method to find the equations of motion of the system. By taking perturbations about a Minkowski
+background for both tetrad and scalar field, we can build the action up to second order. The tetrad and
+scalar perturbation are respectively given by
+eA
+µ → eA
+µ + ǫ δeA
+µ = δA
+µ + ǫ δeA
+µ ,
+(38)
+φ → φ + ǫ δφ ,
+(39)
+where ǫ is the perturbation parameter representing the perturbation order of background Minkowski tetrad.
+It is given by a four-dimensional identity matrix given by the Kronecker delta δA
+µ . It is sufficient to expand
+the scalar field up to first order since higher order terms do not contribute. Additionally, it should be noted
+that the background scalar field can be taken to be as a function of time and apply the unitary gauge such
+that
+φ → φ(t) .
+(40)
+Moreover, the perturbations of arbitrary functions present in the Lagrangian are obtained by performing a
+Taylor expansion up to second order such that
+Gi(φ, X) = Gi + ǫ Gi,XX(1) + ǫ2 �
+1
+2Gi,XX(X(1))2 + Gi,XX(2)�
+,
+(41)
+GTele(φ, X, T, Tax, Tvec, I2, J1, J3, J5, J6, J8, J10)
+= GTele + ǫ(GTele,XX(1) + GTele,I2I(1)
+2 ) + 1
+2ǫ2(GTele,XX(X(1))2 + GTele,I2I2(I(1)
+2 )2)
++ ǫ2�
+GTele,XX(2) + GTele,T T + GTele,TaxTax + GTele,TvecTvec + GTele,I2I(2)
+2
+�
+,
+(42)
+
+7
+where, for i = {2, 3, 4, 5}, j = {X, XX} and k = {X, T, Tax, Tvec, XX, I2I2}, Gi,j and GTele,k are background
+functions, such that XX and I2I2 are the second order derivatives with respect to X and I2. The numbered
+superscripts represent the order of perturbation of the scalar invariant.
+In this section, we consider a scalar-vector-tensor (SVT) decomposition of the tetrad based on the formal-
+ism applied in Ref. [89] for first order perturbations:
+δeA
+µ :=
+
+
+ϕ
+−(∂iβ + βi)
+δI
+i (∂ib + bi)
+δIi �
+−ψδij + ∂i∂jh + ∂ihj + ∂jhi + 1
+2hij + ǫijk(∂kσ + σk)
+�
+
+ ,
+(43)
+where {ϕ, β, b, ψ, h} are scalars and σ is a pseudoscalar of 1 degree of freedom (DoF) each, {βi, bi, hi} are
+vectors and σi is a pseudovector of 1 DoFs each, and hij is the tensor mode of 2 DoFs, for a total of of 16
+DoFs. The tensor modes are symmetric hij = h(ij), traceless δijhij = 0 and divergenceless ∂ihij = 0. See
+also Ref. [90, 91]. The divergenceless property also applies for vectors and pseudovectors such that ∂iαi = 0
+where α = {β, b, h, σ}. Unlike Ref. [92, 93], the pseudoscalar and pseudovector are included to account for the
+anti-symmetry of the tetrad. The mid-range Latin indices are spacial coordinates: {I, J, K, . . .} for spacial
+inner bundle and {i, j, k, . . .} for spacial spacetime manifold. Note, δij is the spacial Minkowski metric such
+that δij = −ηij. In turn, the first order metric perturbation from Eq. (1) is given by
+δgµν =
+
+
+2ϕ
+∂iB + Bi
+∂iB + Bi
+2
+�
+−ψδij + ∂i∂jh + ∂ihj + ∂jhi + 1
+2hij
+�
+
+ ,
+(44)
+where B = −β + b and Bi = −βi + bi. The off-diagonals are identical hence verifying the symmetry of the
+metric. The pseudoscalar and pseudovectors no longer play a role, and thus, the number of DoFs reduces to
+10.
+The gauge transformation through the coordinate change [94–96] is
+˜xµ → xµ + ξµ ,
+(45)
+where ξµ is a vector field applied to study the gauge transformation of perturbative quantities. Such a
+coordinate transformation can be extended to include orders higher than second one, but it is sufficient to
+consider up to this point. Thus, the transformations for first tetrad are given by [97]
+δ˜eA
+µ = δeA
+µ + Lξ(1)eA (0)
+µ
+,
+(46)
+where Lξ is the Lie derivative along ξµ wherein ξµ = (ξ0, ξi + δij∂jξ) further splits and once again obeys
+divergencelessness as ∂iξi = 0. The analysis of each combination of temporal and spatial parts of the tetrad
+indicates that ψ, σ, βi and hij are gauge invariant in Minkowski background while the rest have the following
+gauge transformations
+˜ϕ = ϕ − ˙ξ0 ,
+β = β − ξ0 ,
+˜b = b − ˙ξ ,
+˜h = h − ξ ,
+˜bi = bi + ˙ξi ,
+˜hi = hi + 1
+2ξi ,
+˜σi = σi − 1
+2ǫijk∂jξk .
+(47)
+This shows that since the pseudoscalar σ is gauge-invariant, it can be treated separately than the rest of
+the scalar modes, but the same cannot be said for the pseudovector. In fact, the pseudovector σi can be
+expressed in terms of hi (and its derivative in terms of bi). This implies that the vector and pseudovector
+cannot be decomposed [98].
+Next, we construct groups of non-gauge invariant quantities: {ϕ, β}, {b, h} and {bi, hi, σi}. For gauge
+choice, a quantity from each group is set to zero [98]. In particular, we will choose β = 0, h = 0 and σi = 0.
+V.
+GHOST AND LAPLACIAN INSTABILITIES IN MINKOWSKI BACKGROUND
+Ghost instabilities stem from a negative kinetic term in the action associated to a propagating degree of
+freedom. A ghost mode can be determined by expanding the action up to second order perturbations about
+
+8
+a background. Therefore, for a Lagrangian of the form L = ˙⃗χtA ˙⃗χ + . . ., we impose that the eigenvalues of
+A should be positive to eliminate ghost modes. The procedure is applied in what follows. The second order
+perturbation of the action is obtained by separately applying the scalar, vector and tensor field perturbations
+given by Eq. (43). The dynamical fields in the action are identified, while a system of equations is obtained
+by varying the action with respect to the auxiliary fields. This system of equations is substituted back into
+the action to eliminate the non-dynamic fields [99, 100].
+In order to obtain a gauge invariant action, the action is varied with respect to the temporal and spatial
+part of the vector field associated with the coordinate transformation, and imposing that this vanishes. Then,
+a diagonalized kinetic matrix is constructed and a constraint is generated for each entry. In general, these
+constraints could be time dependent: this feature arises from the background spacetime and the background
+quantities of fields. When looking at the propagating speeds of these modes, a positive definite value should
+be applied in order to ensure that the perturbation does not lead to an exponential growth [101] i.e. c2 > 0,
+referred to as gradient or Laplacian instability. Hence, both ghost and gradient instabilities are checked for
+each mode in order to obtain the constraints which lead to a stable model.
+A.
+Tensor Perturbations
+The tensor mode perturbations, presented in Eq. (43), can be extended as
+eA
+µ →
+
+1
+0
+0 δij + 1
+2ǫhij − 1
+8ǫ2hikhk
+j
+
+ ,
+(48)
+analogous to the Arnowitt-Deser-Misner (ADM) for tensor modes [100]. The action can be extended to
+second order perturbations such that
+S(2)
+T
+= 1
+2
+�
+d4x
+�
+MT ˙hij ˙hij − NT ∂mhij∂mhij + Phijhij�
+(49)
+where
+MT = G4 − 2XG4,X + XG5,φ − GTele,T + 1
+2XGTele,J5 + 2XGTele,J8 ,
+(50)
+NT = G4 − X(G5,φ + ¨φG5,X) − GTele,T ,
+(51)
+PT = − 1
+2
+d
+dt( ˙φGTele,I2) .
+(52)
+For the sake of simplicity, we switch to Fourier space such that
+S(2)
+T
+= 1
+2
+�
+dt d3k
+(2π)3
+�
+MT ˙hij ˙hij +
+�
+−k2NT + PT
+�
+hijhij .
+�
+(53)
+This result is obtained after applying integration by parts, removing the surface terms and applying the
+background Eqs. (35-37). By imposing MT > 0, the theory is ghost free in tensor modes. When considering
+a constant background scalar φ, this implies that G4 − GTele,T > 0 which corresponds to the condition
+G4 −GTele,T ̸= 0 imposed in Ref. [102] in order to ensure that there are tensor mode DoFs propagating. The
+propagation speed of tensor modes is given by
+c2
+T = NT
+MT
+=
+G4 − X(G5,φ + ¨φG5,X) − GTele,T
+G4 − 2XG4,X + XG5,φ − GTele,T + 1
+2XGTele,J5 + 2XGTele,J8
+,
+(54)
+for which c2
+T > 0 is required to ensure gradient stability. Hence, MT > 0 and NT > 0 result in ghost
+and gradient stability, respectively. The same result would by obtained when eliminating I2 contribution
+from the GTele function. Following the observations of gravitational wave signal GW170817 [103] and its
+electromagnetic counterpart GRB170817A [27], it is interesting to calculate the deviation from speed of light
+propagation such that the excess speed is given by
+αT = c2
+T − 1 =
+X(2G4,X − 2G5,φ − ¨φG5,X − 1
+2GTele,J5 − 2GTele,J8)
+G4 − 2XG4,X + XG5,φ − GTele,T + 1
+2XGTele,J5 + 2XGTele,J8
+,
+(55)
+
+9
+which corresponds to the result obtained through the GW propagation equation [66], a value which is highly
+constrained such that a graviton mass would be minute.
+Additionally, from the general result of teleparallel Horndeski analogue, given by Eq. (48), one may obtain
+the results of well-studied theories from literature, as summarized in Table I. See also Ref.[73] for the
+metric cases derived from Noether symmetries. As an extension analogous to standard Horndeski theory,
+the stability conditions can be obtained by setting GTele = 0. The ghost modes are excluded for when
+the constant of the kinetic term is G4 − 2XG4,X + XG5,X > 0 while gradient stability is obtained for
+G4 − X(G5,φ + ¨φG5,X) > 0, in agreement with Refs. [101, 104–106] when taking the appropriate limits
+to Minkowski spacetime. An example of a subclass of Horndeski gravity is the Brans-Dicke theory, where
+G2 = 2wBDX
+φ
+and G4 = φ for which wBD is the Dicke coupling constant [107], other constants are set to
+vanish and ghost instabilities are avoided for φ > 0. By considering the generalized Brans Dicke theory,
+ghost instabilities are avoided for a positive value of a function of the scalar field, F(φ) > 0 [108]. Another
+example is f(
+◦R) where G2 = f(φ) − φf ′(φ), G4 = f ′(φ), the rest of the constants are zero, implying that
+ghost stability is achieved for f ′(φ) > 0. A final subcase of Horndeski gravity considered here is GR. In this
+case, the only non-vanishing constant is G4 = 1 for which there are no ghost modes since MT = 1 > 0. All
+of these subcases do not result in any gradient instabilities as cT = 1 > 0.
+Next, we look at cases that arise due to the inclusion of teleparallel terms. For a purely teleparallel theory
+such as f(T ) gravity, all terms are set to vanish except for GTele,T = f(T ). This implies in −f ′(T ) > 0 to
+avoid ghost modes similar to the result obtained in Ref. [98]. An equivalent result is obtained for scalar-
+tensor theory with a Lagrangian of the form L = f(φ, T ) + XP(φ), an extension of f(T ) [100, 109]. Once
+again, gradient instabilities are not an issue. Finally, the case where only I2 contributions are present is
+considered. In this case, all constants are going to vanish while GTele,I2 ̸= 0 leads to a non-dynamical degree
+of freedom.
+Theory
+Case
+MT
+NT
+Horndeski
+GTele = 0
+G4 − 2XG4,X + ¨φXG5,X G4 − X(G5,φ + ¨φG5,X)
+Generalized
+Brans-Dicke
+GTele = G5 = 0, G2 = B(φ)X
+G3 = 2ξ(φ)X, G4 = 1
+2F(φ)
+1
+2F(φ)
+1
+2F(φ)
+Brans-Dicke
+GTele = G3 = G5 = 0
+G2 = 2wBDX
+φ
+, G4 = φ
+φ
+φ
+f(
+◦R)
+GTele = G3 = G5 = 0
+G2 = f(φ) − φf ′(φ), G4 = f ′(φ)
+f ′(φ)
+f ′(φ)
+General Relativity
+GTele = G2 = G3 = G5 = 0
+G4 = 1
+1
+1
+Teleparallel
+or f(T )
+G2 = G3 = G4 = G5 = 0,
+GTele = f(T )
+−f ′(T )
+−f ′(T )
+f(φ, T ) + XP(φ)
+G3 = G4 = G5 = 0
+G2 = XP(φ), GTele = f(φ, T )
+−f,T (φ, T )
+−f,T (φ, T )
+GTele,I2 only
+G2 = G3 = G4 = G5 = 0
+GTele,T = 0
+no propagating mode
+TABLE I. List of literature models with the respective ghost MT and gradient stability NT conditions are positive
+definite, while the propagation speed is cT = NT/MT for tensor modes. The models include Horndeski theory [20,
+101, 104, 105], Generalized Brans-Dicke [108] and Brans-Dicke [107], f(˚
+R) theory, General Relativity, f(T ) theory [98],
+f(φ, T ) theory [100] and the case where the action is dependent on I2 only.
+
+10
+B.
+Vector Perturbations
+Next, we consider the vector perturbation of the tetrad (43) with the application of gauge fixing which
+eliminates the pseudovector and any coupling with it such that
+eA
+µ →
+
+
+1
+−ǫβi
+ǫδI
+i bi δIi(δij + ǫ(∂ihj + ∂jhi))
+
+ .
+(56)
+This result is a case of only vector modes in this portion of decomposition. Extending the action to second
+order vector perturbations, we obtain
+S(2)
+V
+=
+�
+dt d3k
+(2π)3
+�
+4k4Ahihi + E ˙βi ˙βi
++ k2(4B ˙hi( ˙hi − bi) + Cbibi + Dβiβi + 4Fhiβi + 4Ghi ˙βi − 2Hβibi)
+�
+,
+(57)
+where
+A = GTele,Tvec + 1
+9X
+�
+−GTele,J8 + XGTele,J6 − 5
+2GTele,J5 − 3GTele,J3
+�
+,
+B = G4 − 2XG4,X + XG5,φ − GTele,T + X
+�
+2GTele,J8 + 1
+2GTele,J5
+�
+,
+C = B + 1
+2XGTele,J5 + 2
+3XGTele,J10 + 2
+9GTele,Tax ,
+D = C + XGTele,J5 − 2XGTele,J8 − 2XGTele,J10 ,
+E = 4A − 3GTele,Tvec + 2XGTele,J3 ,
+F = 1
+2 ˙φGTele,I2 − d
+dt (G4 − 2XG4,X + XG5,φ) .
+G = 2A − B − 3GTele,Tvec + 1
+3XGTele,J3 − 2XGTele,J8 + 5
+2XGTele,J5 ,
+H = C − 2XGTele,J8 − 4
+9GTele,Tax ,
+(58)
+While the decomposition along unitary gauge ensures that there are no higher order time derivatives, one
+can see that the action contains higher order spatial derivatives [110]. Here, we will set the coefficients of
+these derivatives to vanish, but it should be noted that a general analysis for each constraint should be
+carried out before solving the next one and leading to a very complicate solution for the action. By imposing
+the conditions A = 0 and B = 0, the latter condition accounting for the higher order mix of temporal and
+spatial derivatives [111], the action reduces to
+S(2)
+V
+=
+�
+dt d3k
+(2π)3
+�
+E ˙βi ˙βi + k2(Cbibi + Dβiβi + 4Fhiβi + 4Ghi ˙βi − 2Hβibi)
+�
+,
+(59)
+where bi and hi are auxiliary vector modes. By varying with respect to each of these non-dynamical modes,
+the respective field equations are given by
+2k2(Cbi − Hβi) = 0 ,
+and
+4k2Fβi = 0 .
+(60)
+By solving for each auxiliary field and plugging it back in into Eq. (59), we get
+S(2)
+V
+=
+�
+dt d3k
+(2π)3
+�
+E ˙βi ˙βi − k2 �
+H2
+C − D
+�
+βiβi
+�
+,
+(61)
+wherein the action is expressed in terms of the dynamical non-gauge invariant mode βi. Thus the ghost and
+Laplacian conditions are given respectively by
+MV = E > 0 ,
+and NV = H2
+C − D > 0 .
+(62)
+suggesting that there is a propagating vector mode with speed
+cV = H2 − CD
+CE
+.
+(63)
+This propagating mode generally steps from the introduction of the Ji terms in the BDLS action (27)
+representing the quadratic contractions of the scalar and torsion fields.
+
+11
+a.
+Vanishing Ji terms:
+When considering the case were J1 = J3 = J5 = J6 = J10 = 0, the action (61)
+reduces to
+Svanishing J terms
+V
+=
+�
+dt d3k
+(2π)3 k2
+�
+C − H2
+C
+�
+βiβi ,
+(64)
+exhibiting no dynamical modes i.e. no propagating vector mode. In the case of only teleparallel function,
+the same conditions apply with vanishing Horndeski terms in each constant.
+The subcases of standard
+Horndeski [31, 104], f(
+◦R) [99, 112], Generalized Brans-Dicke[108] and f(T ) [98] fall under this category of
+non-viable propagating vector mode.
+b.
+C = 0
+: The Laplacian constraint, given in Eq. (62), is undefined for C = 0. By considering this
+particular case, the action results in
+SC=0
+V
+=
+�
+dt d3k
+(2π)3 E ˙βi ˙βi ,
+(65)
+to which the Laplacian condition is violated since it is null. On the other hand, imposing only E = 0, the
+ghost instability is generated since this value should be a positive definite value.
+c.
+Ji terms only:
+On the other hand, when considering only the contribution of the quadratic contrac-
+tions, the only surviving term is that with coefficient E leading to a ghost stability condition but leading
+to Laplacian instability, similar to the C = 0 case. This implies the importance of having a combination of
+the Ji terms along with the rest of the teleparallel scalar invariants to maintain stability conditions and a
+possible propagating vector mode.
+d.
+Constant Background Scalar Field:
+With regards to the scalar field, the unitary gauge is applied to
+the perturbative part of the scalar, while, at background level, it is a function of time only. Alternatively,
+if one considers a constant scalar field φ = c, then the entire action is vanishing such that no vector modes
+propagate, thus it corresponds to the results in Ref. [67] for the DoF and polarisation modes in the vector
+portion of the decomposition.
+C.
+Scalar Perturbations
+Let us now consider the scalar modes. Since there is no mixing between the scalar and pseudoscalar modes,
+we will treat them separately. The tetrad perturbation using scalar modes is given by
+eA
+µ →
+
+1 + ϕ
+0
+δI
+i ∂ib (1 − ψδIiδij .
+
+
+(66)
+As already stated, for the gauge invariant quantities, given by Eq. (47), the gauge choice is β = h = 0. The
+pseudoscalar will be analyzed separately. It should be noted that this situation is identical to the Arnowitt-
+Deser-Misner (ADM) decomposition used in torsion-based theories [100]. It leads to the same metric as that
+applied to curvature-based theories [101]. On the other hand, the choice β = b = 0 results in the Newtonian
+(longitudinal) gauge. Substituting the tetrad perturbation to the action, followed by integration by parts
+and application of the background equations (35-37), the second order perturbation of the action is given by
+S(2)
+S
+=
+�
+dt d3k
+(2π)3
+�
+¯
+Aϕ2 + 6 ¯Dψ2 − 6 ¯F ˙ψ2 − 6 ¯Gϕ ˙ψ
+− k2(− ¯Bϕ2 − 2 ¯Eψ2 + 4 ¯Hϕψ − 2 ¯Gϕb − 4 ¯F ˙ψb) + k4 ¯Cb2�
+(67)
+
+12
+where
+¯
+A = XG2,X + 2X2G2,XX − 2XG3,φ − 2X2G3,φX + XGTele,X + 2X2GTele,XX ,
+¯B = GTele,Tvec + 2
+9X (2GTele,J8 + 2XGTele,J6 − 5GTele,J5 + 3GTele,J3) ,
+¯C = −GTele,Tvec + XGTele,I2I2 + 1
+3X(4GTele,J8 + GTele,J5) ,
+¯D = d
+dt( ˙φGTele,I2) ,
+¯E = G4 − X(G5,φ − ¨φG5,X) − GTele,T + 2GTele,Tvec + 1
+9X (−2GTele,J8 + 2XGTele,J6 − 5GTele,J5 − 6GTele,J3) ,
+¯F = G4 − 2XG4,X + XG5,φ − GTele,T + 3
+2(GTele,Tvec − XGTele,I2I2) ,
+¯G = − ˙φXG3,X + ˙φG4,φ + 2 ˙φXG4,φX + 1
+2 ˙φGTele,I2 + ˙φXGTele,XI2 ,
+¯H = G4 − 2XG4,X + XG5,φ − GTele,T + GTele,Tvec + 1
+9X(2GTele,J8 − 2XGTele,J6 + 5GTele,J5 + 3
+2GTele,J3) .
+(68)
+Here k is the co-vector. As shown in coefficient of ¯C in Eq. (67), there is a higher derivative contribution,
+hence it is set to vanish [110]. It is clear that variables ϕ and b are non-dynamical with respect to the time
+component. By varying with respect to each of these auxiliary fields, one obtains a set of equations
+0 = ( ¯
+A + k2 ¯B)ϕ − 3 ¯G ˙ψ + k2(2 ¯Hψ − ¯Gb) ,
+(69)
+0 = ¯Gϕ + 2 ¯F ˙ψ .
+(70)
+By solving these equations of motion for the modes ϕ and b, it leads to an action expressed in terms of the
+dynamical mode only, that is
+S(2)
+S
+=
+�
+dt d3k
+(2π)3
+�
+4k2 ¯B ¯F
+2
+¯G
+2
+˙ψ2 +
+�
+4 ¯
+A ¯F
+2
+¯G
+2
++ 6 ¯F
+�
+˙ψ2 − k2
+�
+−2 ¯E + 4 d
+dt
+� ¯F ¯H
+¯G
+��
+ψ2 + 6 ¯Dψ2
+�
+.
+(71)
+Taking into account the mix of spatial and temporal higher-order derivatives, the first term of the action
+vanishes [111]. This implies that either ¯F = 0 (which renders the whole action without a dynamical mode),
+or ¯B = 0. By applying the latter condition, the action can be expressed in terms of a gauge invariant quantity
+within a Minkowski background and no higher-order terms. It is
+S(2)
+S
+=
+�
+dt d3k
+(2π)3
+��
+4 ¯
+A ¯F
+2
+¯G
+2
++ 6 ¯F
+�
+˙ψ2 − k2
+�
+−2 ¯E + 4 d
+dt
+� ¯F ¯H
+¯G
+��
+ψ2 + 6 ¯Dψ2
+�
+.
+(72)
+In the case of teleparallel analogue of Horndeski theory, ghost stability and gradient stability are obtained,
+respectively, when
+MS = 4 ¯
+A ¯F
+2
+¯G
+2
++ 6 ¯F > 0 ,
+(73)
+¯
+N S = −2 ¯F + 4 d
+dt
+� ¯F ¯H
+¯G
+�
+> 0 ,
+(74)
+and the propagating speed is
+cS = NS
+MS
+.
+(75)
+Different subcases are reported in Table II. Through the substitution of Eqs (73) and (74), conditions to
+avoid ghost and Laplacian instabilities are realised.
+a.
+Horndeski:
+The standard Horndeski case has identical expressions for ghost and gradient stability
+conditions as those for teleparallel Horndeski (73-74) with the difference that each constant does not have
+teleparallel contributions.
+
+13
+b.
+Generalized Brans-Dicke:
+Considering Generalised Brans-Dicke (GBD) theory means that G2 =
+B(φ)X, G3 = 2Xξ(φ), G4 = 1
+2F(φ) and all other terms are set to zero. When we take the background
+equations
+B − 4Xξ′ = 0 ,
+and
+˙φ2F ′′ + ¨φ(F ′ − 2 ˙φ2ξ) = 0 ,
+(76)
+for ghost stability, we have
+MGBD
+S
+= F[3F ′2 − 4 ˙φ2(Fξ′ + 3ξF ′) + 12 ˙φ4ξ2]
+[F ′ − 2 ˙φ2ξ]2
+> 0 ,
+(77)
+where ′ denotes a derivative with respect to the respective variable of the function. In this case, it is φ, while
+the gradient condition is
+N GBD
+S
+= F[3F ′3 − 2 ˙φ2(5F ′2ξ − 4ξFF ′′ − 2Fξ′F ′) + 4 ˙φ4(F ′ξ2 − 2Fξ′ξ) + 8 ˙φ6ξ3]
+[F ′ − 2 ˙φ2ξ]3
+> 0 .
+(78)
+With these conditions, the speed of propagation is positive. It is
+(cGBD
+S
+)2 = 3F ′3 − 2 ˙φ2(5F ′2ξ − 4ξFF ′′ − 2Fξ′F ′) + 4 ˙φ4(F ′ξ2 − 2Fξ′ξ) + 8 ˙φ6ξ3
+[3F ′2 − 4 ˙φ2(Fξ′ + 3ξF ′) + 12 ˙φ4ξ2][F ′ − 2 ˙φ2ξ]
+,
+(79)
+which correlates with results obtained in Ref. [108] for a Minkowski background.
+c.
+Brans-Dicke:
+As for the Brans-Dicke theory in Ref. [107] with G2 = 2wBDX
+φ
+and G4 = φ, ghosts
+instabilities are avoided if 2φ(3 + 2wBD) > 0.
+Provided that φ > 0, this implies that the Brans-Dicke
+constant is wBD > − 3
+2 [99]. The gradient stability condition is given by 2φ(3 − 2φ¨φ/ ˙φ2) > 0 to ensure
+a positive propagating speed. Moreover, when applying the second Friedman Eq. (36), gradient condition
+reduces to 2φ(3 + wBD) such that wBD > −3 and a unitary propagating speed [113], analogous to ξ = 0,
+reported in Ref. [108], is obtained. The first Friedman Eq. (35) shows that the only non-trivial solution is
+that for wBD = 0, which results in identical ghost and gradient expressions.
+d.
+f(˚
+R):
+The scalar-tensor theory equivalent to f(R) [114] is given by setting G2 = f(φ) − φf ′(φ) and
+G4 = f ′(φ) while the rest of the functions are set to vanish. See also Ref. [73]. In order to have ghost
+stability, it has to be 6f ′(φ) > 0, i.e. f ′(φ) > 0. Once again, upon applying the background Eqs. (35-36),
+the gradient condition reduces to f ′(φ) > 0. In fact, as a subcase of GBD theory, the case ξ = 0 always
+results in a unitary speed propagating mode.
+e.
+GR, f(T ), f(φ, T ):
+When considering the cases of GR, f(T ) and f(φ, T ), the substitution in Eqs˜(73-
+74) results in an undefined expression. In these cases, although taking the appropriate limits of each coeffi-
+cient does lead to a positive ghost condition value, the propagating speed is negative, thus scalar modes are
+not viable. This is expected in GR. Additionally, Ref. [98] shows that the only propagating scalar field is that
+dependent on the perturbation of scalar field φ, while, in this paper, we are applying the unitary gauge. As
+in Ref. [100], f(φ, T ) theory gives rise to a propagating mode when considering cosmological perturbations.
+The same applies for GR with an additional canonical scalar field such that G2 = X − V (φ), G4 = M 2
+Pl/2,
+G3 = G5 = GTele = 0 [101].
+f.
+Teleparallel:
+If one considers only the teleparallel portions of the action, all terms from Eq. (71) are
+retained, with all standard Horndeski contributions set to vanish. This implies that the ghost and Laplacian
+instabilities coincide with the form of Eqs. (73) and (51), respectively, provided that the background equations
+are satisfied.
+D.
+Pseudoscalar Perturbations
+The gauge invariant pseudoscalar σ can be treated separately as
+eA
+µ →
+
+1
+0
+0 δIi(δij + ǫ εijk∂kσ)
+
+ ,
+(80)
+
+14
+Theory
+Case
+MS
+NS
+Horndeski
+GTele = 0
+6 ¯
+F + 4 ¯
+A ¯
+F2
+¯
+G2
+−2 ¯
+E + 4 d
+dt ( ¯
+F2
+¯
+G )
+Generalized
+Brans-Dicke
+GTele = G5 = 0,
+G2 = B(φ)X, G3 = 2ξ(φ)X,
+G4 = 1
+2 F (φ)
+F [3F ′2−4 ˙φ2(F ξ′+3ξF′)+12 ˙φ4ξ2]
+[F ′−2 ˙φ2ξ]2
+F [3F ′3−2 ˙φ2(5F ′2ξ−4ξFF ′′−2F ξ′F ′)+4 ˙φ4(F ′ξ2−2F ξ′ξ)+8 ˙φ6ξ3]
+[F ′−2 ˙φ2ξ]3
+Brans-Dicke
+GTele = G3 = G5 = 0
+G2 = 2wBDX
+φ
+, G4 = φ
+2φ(3 + 2wBD)
+2(3φ −
+¨
+φ
+X ) → 2φ(3 + 2wBD)
+f( ◦
+R)
+GTele = G3 = G5 = 0
+G2 = f(φ) − φf′(φ),
+G4 = f′(φ)
+6f′(φ)
+−2f′(φ) + 4 d
+dt ( f′(φ)2
+˙φf′′(φ) ) → 6f′(φ)
+GR
+GTele = G2 = G3 = G5 = 0
+G4 = 1
+no propagating mode
+f(T )
+G2 = G3 = G4 = G5 = 0,
+GTele = f(T )
+no propagating mode
+f(φ, T ) + XP (φ)
+G3 = G4 = G5 = 0,
+G2 = XP (φ),
+GTele = f(φ, T )
+no propagating mode
+Teleparallel
+G2 = G3 = G4 = G5 = 0
+4 ¯
+A ¯
+F2
+¯
+G2
++ 6 ¯
+F
+−2 ¯
+F + 4 d
+dt
+� ¯
+F ¯
+H
+¯
+G
+�
+TABLE II. List of literature models with the respective ghost MS and gradient stability NS conditions are positive
+definite, and propagation speed cS = NS/MS for scalar modes. The models include Horndeski theory [20, 101, 104,
+105], Generalized Brans-Dicke [108] and Brans-Dicke [107], f(˚
+R) theory , General Relativity, f(T ) theory [98], f(φ, T )
+theory [100]. and an action with only teleparallel terms.
+The action in Fourier space is expanded up to second order
+S(2)
+PS = −
+�
+dt d3k
+(2π)3
+��
+k2 d
+dt( ˙φGTele,I2) + 4
+9k4 (GTele,Tax − 2XGTele,J1)
+�
+σ2
++k2
+�4
+9GTele,Tax + X
+�
+GTele,J5 + 4
+3GTele,J10
+��
+˙σ2
+�
+,
+(81)
+where the first term exhibits higher-order spatial derivatives and the second term exhibits higher order mixed
+derivatives. By imposing that GTele,Tax = 2XGTele,J1 = −X
+� 9
+4GTele,J5 + 3GTele,J10
+�
+, the action reduces to
+S(2)
+PS =
+�
+dt d3k
+(2π)3
+�
+− k2 d
+dt( ˙φGTele,I2)σ2�
+,
+(82)
+where σ is a non-dynamical mode and thus it is a non-propagating mode. This shows that in Minkowski
+background, the additional scalar invariants, provided by teleparallel analogue of Horndeski theory, do not
+contribute to any additional propagating DoF. See also Ref. [98].
+VI.
+DISCUSSION AND CONCLUSIONS
+We have investigated constraints emerging by considering the stability of perturbations about a Minkowski
+background in teleparallel analogue of Horndeski gravity. The approach is performed by exploring the ghost
+instabilities through looking at potentially negative expressions of kinetic term associated with propagating
+DoFs, as well as Laplacian instabilities emerging when propagation speeds of perturbations are not positive.
+This leads to possible exponential growth rates. These considerations are fundamental for the construction
+of robust and self-consistent cosmological models related to the scalar-tensor gravity.
+Specifically, we have discussed teleparallel analogue of Horndeski gravity where the most general scalar-
+tensor action is considered, provided that the Lagrangian terms are not parity violating and that scalars
+are produced by no more than quadratic contractions of torsion scalar. The naturally lower order nature
+of teleparallel gravity induces a much richer landscape of functional models when compared with standard
+metric Horndeski gravity, based on Levi-Civita connection. In other words, the replacement of torsion with
+curvature to lead dynamics turns out to manifest as a generalization of the standard Horndeski theory. Im-
+portantly, this produces a generalization of tensor mode propagation speed which means that the restrictions
+appearing in standard Horndeski gravity, due to the constraints on the gravitational waves speed, do not
+have the same effect here.
+
+15
+Our calculations have proceeded by first deriving the background equations of motion (35–37) which are
+obtained by considering a flat FLRW background in which we take the Minkowski limit. We find this to
+be a convenient approach to determine the system of equations. These are useful equations for reducing
+the expressions of perturbations that ensue. The procedure is described in Sec. IV. We then proceeded to
+directly determine constraints to prevent ghost or Laplacian instabilities starting with tensor modes which
+results in the action in Eq. (53) and tensor propagation speed in Eq. (54). We collected the constraints for
+popular subclasses of BDLS theory in Table I where the known results are obtained for standard Horndeski
+gravity as well as f(
+◦R) and Brans-Dicke gravity, but new conditions are found in other cases. As for the
+vector perturbations, Sec. V B shows that vector modes are indeed propagating for some cases of GTele
+contribution, namely when Ji scalars appear in this function. However, these contributions may be small
+and within observational constraints.
+The scalar perturbations contain the most diverse range of DoFs. This rich structure produces the intricate
+action in Eq. (72), together with the propagation speed in Eq. (75). As in the case of tensor modes, constraints
+for each subclass of popular models is reported in Table II where results for the Minkowski perturbations are
+recovered for standard metric Horndeski gravity together with f(
+◦R) and Brans-Dicke gravity. Interestingly,
+either no constraints are set on some of the purely teleparallel models or only lightly limited cases are set.
+Importantly, these constraints are consistent with the tensor mode constraints on the functional models.
+These results are promising in terms of the viability of BDLS theory. As future work, it is intriguing to
+explore constraints from tachyonic instabilities. These constraints have been connected to Jeans instability
+where exponential growth of perturbations is slowed by the expansion of the Universe and where the matter
+dispersion relation has a negative mass term that renders it to vanish. This could lead to a more general
+consideration of perturbations about a flat FLRW background spacetime which would then be directly linked
+to observations such as those related to the growth of large scale structure. In forthcoming studies, possible
+observational constraints will be scrutinized.
+ACKNOWLEDGMENTS
+This paper is based upon work from COST Action CA21136 Addressing observational tensions in cosmology
+with systematics and fundamental physics (CosmoVerse) supported by COST (European Cooperation in
+Science and Technology). SC acknowledges the Istituto Nazionale di Fisica Nucleare (iniziative specifiche
+QGSKY and MOONLIGHT2). MC acknowledges funding by the Tertiary Education Scholarship Scheme
+(TESS, Malta).
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new file mode 100644
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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='04457v1  [gr-qc]  11 Jan 2023  Ghost and Laplacian Instabilities in Teleparallel Horndeski Gravity  Salvatore Capozziello,1, 2, 3, ∗ Maria Caruana,4, 5, † Jackson Levi Said,4, 5, ‡ and Joseph Sultana6, §  1Dipartimento di Fisica ”E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Pancini”, Universit`a degli Studi di Napoli,  ”Federico II”, Complesso Universitario Monte S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Angelo,  Via Cinthia 9 Edificio G, 80126 Napoli, Italy  2Istituto Nazionale di Fisica Nucleare (INFN),  Sezione di Napoli Complesso Universitario Monte S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Angelo,  Via Cinthia 9 Edificio G, 80126 Napoli, Italy  3Scuola Superiore Meridionale, Largo San Marcellino 10, 80138 Napoli, Italy  4Institute of Space Sciences and Astronomy, University of Malta, Msida, Malta  5Department of Physics, University of Malta, Msida, Malta  6Department of Mathematics, University of Malta, Msida, Malta  Teleparallel geometry offers a platform on which to build up theories of gravity where torsion  rather than curvature mediates gravitational interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The teleparallel analogue of Horndeski  gravity is an approach to teleparallel geometry where scalar-tensor theories are considered in this  torsional framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Being teleparallel gravity of lower order in dynamics, this turns out to be more  general than metric Horndeski gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' In other words, the class of teleparallel Horndeski gravity  models is much broader than the standard metric one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' In this work, we explore constraints on this  wide range of models coming from ghost and Laplacian instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The aim is to limit pathological  branches of the theory by fundamental considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' It is possible to conclude that a very large  class of models results physically viable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  INTRODUCTION  General relativity (GR) has been the gravitational foundation of cosmology for over a century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Its latest  embodiment appears in a picture wherein the evolutionary processes of the Universe are described in the  framework of the so-called ΛCDM model [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Supported by overwhelming observational evidences, this  model describes a Universe started with a big bang, then driven through an inflationary phase and other well  known epochs to eventually produce an accelerating Universe at late-times [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' In the ΛCDM model, the  late time acceleration expansion is driven by the cosmological constant or some form of dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Despite  a lot of foundational works, internal consistency issues persist in this respect [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Recently, observational  challenges to the standard model has arisen in the form of cosmological tensions with statistically significant  differences between predictions of expansion from early time data [8–10] and measurements from late time  [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' These tensions continue to increase with new survey data [13–15], and may permeate into other  sectors of cosmology besides expansion [16–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' This situation leads to take into account potential alternatives  to the standard cosmological ΛCDM model in the context of possible modifications to the gravitational sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  A solution to the problem of cosmic tensions, as well as of other longstanding issues, can be offered by  alternative theories of gravity, in particular by scalar-tensor gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Scalar fields improve GR in view of Mach  principle and could naturally address several issues in cosmology and astrophysics like inflation, dark matter  and dark energy [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The most general scalar-tensor theory of gravity, where scalar fields are minimally and  non-minimally coupled to curvature, is the so-called Horndeski gravity [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' It is the most general theory  of gravity producing second order field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' This is advantageous because nature appears to produce,  generally, second order equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Furthermore, higher order theories can produce Ostrogradsky  instabilities making second order theories more attractive from a fundamental perspective [21–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Horndeski  gravity is formulated by four free functions of the scalar field and its kinetic term [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' This offers a  rich phenomenology on which to produce dark energy models but also dark matter and inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' However,  recent multimessenger observations by the LIGO collaboration, in the gravitational event GW170817 [26], and  measurements of the companion electromagnetic counterpart, namely GRB170817A [27], has placed stringent  constraints on the speed of propagation of gravitational waves (GW) to within deviations of at most one part  ∗ capozziello@na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='it  † maria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='caruana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='16@um.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='mt  ‡ jackson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='said@um.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='mt  § joseph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='sultana@um.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='mt  2  in 1015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Considering these constraints, many branches of Horndeski gravity have been disqualified in regular  curvature-based gravity [28–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' However, Horndeski gravity has a number of interesting generalizations  [31] where higher order terms, avoiding the Ostrogradsky instabilities, can be incorporated into the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  Another possibility is to reconsider the geometric foundations of curvature on which Horndeski gravity is  built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Curvature can be expressed through the Levi-Civita connection  Γ  λ  µν obtained from the metric [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  Here an over-circle represents quantities determined by the Levi-Civita connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  In teleparallel gravity (TG), the teleparallel connection (Γλ  µν) replaces the Levi-Civita connection and so  dynamics of gravity is replaced from curvature to torsion [32–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The teleparallel connection is curvatureless  and satisfies metricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' For this reason, the teleparallel Ricci scalar turns out to identically vanish, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' R = 0  (this is not to say that the standard Ricci scalar is zero, being, in general,  R ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' On the other hand, TG  naturally produces a torsion scalar (T ) [33], which turns out to be equal to the curvature-based Ricci scalar  up to a boundary term (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' For this reason, the linear torsion scalar produces the Teleparallel equivalent of  General Relativity (TEGR) which is dynamically equivalent to GR in the classical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' This distinction  between the torsion scalar, producing second order terms in the equations of motion, and the boundary  term, producing fourth order terms in the equations of motion (when the boundary term is nonlinear in the  action), gives rise to a much richer landscape of theories on which to build cosmological models, if compared  with the standard Einstein-Hilbert action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  Another way to conceptualize this feature is through the teleparallel analogue of the Lovelock theorem  [36–38] which produces a much wider range of actions with second order field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Equivalences and  differences of these representations are discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  Similar to GR, TEGR features many modifications such as f(T ) gravity [34, 40–46] which is generically  second order in nature, as well as f(T, B) gravity [47–55] which now features fourth order contributions  similar to f(  R) gravity [19, 56–62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' There have also been various formulations of scalar–tensor gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' See,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' [63–65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  Equipped with the TG formalism, one can consider the teleparallel analogue of Horndeski gravity [38] where  the generically lower order nature of TG turns out to produce a much richer structure to be developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' This  feature directly comes from the teleparallel analogue of the Lovelock theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' This larger framework of  models means that there are many more classes of theories that now satisfy the speed of GWs constraint  [66, 67] as required from the above mentioned multimessenger analysis [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  This class of gravitational  models also satisfies the parameterized post-Newtonian observational constraints [69] for a large number of  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The teleparallel analogue of Horndeski gravity also produces a number of interesting cosmologies  such as those which are well-tempered [70, 71], as well as those that are derived from Noether symmetry  considerations [72–76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' In this way, all models that were previously disqualified in regular Horndeski may be  revived in this new formalism based on TG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  In this work, we perform a stability analysis of the teleparallel analogue of Horndeski gravity through  considerations on a Minkowski spacetime background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The Minkowski background is an ideal arena to probe  the fundamental behavior of theories with large classes of models since any astrophysical or cosmological  setting must first be stable on a Minkowski spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' In our analysis, we consider both background and  leading order perturbations in the context of a scalar-vector-tensor decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' This is important to fully  decompose any potentially problematic part of the theory in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' In the TG setting, this is more intricate  since the metric tensor (gµν) is replaced as the fundamental dynamical variable of the field equations with  the tetrad (eA  µ) and the spin connection (ωA  Bµ), which are discussed in detail later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' In all cases, we  focus our study on the ghost and gradient instabilities, which can wreak havoc on gravitational models since  this may, respectively, produce scalar field kinetic terms with the wrong sign and unbounded propagation  speeds of particular perturbations, which can lead to unphysical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  The structure of the manuscript is the following: in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' II, we summarize TG and its teleparallel analogue  of Horndeski gravity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' this then leads to the Minkowski background equations of motion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Scalar–  vector–tensor perturbations are discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' From these perturbations, we are able to explore  potential instabilities in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' V where our main results are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Discussion and conclusions are drawn  in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' In this work, we use geometric units and the signature (−, +, +, +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  3  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  TELEPARALLEL HORNDESKI GRAVITY  GR, a curvature-based theory, is constructed on torsionless Levi-Civita connection  Γλ  µν, where, as said  above, the overhead circle (◦) represents quantities built with this connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  This leads to numerous  theories of gravity beyond GR as the gravitational field can be expressed through the Riemann tensor and  its contractions such as the Ricci tensor and the Riemann scalar [59], the latter produces the Einstein-  Hilbert action [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' On the other hand, TG incorporates the teleparallel connection, dubbed as teleparallel  connection Γλ  µν, leading to a theory satisfying the curvature-less and metricity conditions [32–35], resulting  in a torsionful theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Later in this section, it will be shown that even though the Ricci scalar R vanishes,  due to the symmetry of the connection, the curvature-ful connection gives a non-zero value for the Riemann  tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Hence, the Ricci scalar  R can be defined through teleparallel quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  The fundamental dynamical object of GR is the metric tensor gµν, but within TG, the metric is expressed  through the tetrad eA  µ, which acts as a transformation between the local and general manifold spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The  tetrad and the inertial spin connection ωB  Cν become the fundamental objects [33] while creating a link  between the general manifold denoted through Greek indices to the local Minkowski manifold denoted by  Latin indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Thus, the relationship between Minkowski and general spacetimes is given by [77]  gµν = ηAB eA  µ eB  ν ,  ηAB = gµν E  µ  A E  ν  B ,  (1)  where E  µ  A  is the inverse tetrad which satisfies orthogonality conditions  eA  µ E  µ  B  = δA  B ,  eA  µ E  ν  A  = δν  µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (2)  The teleparallel connection is defined through the TG variables as [34, 35]  Γλ  µν = E  λ  A  �  ∂νeA  µ + ωA  BνeB  µ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (3)  Note, the quantities without an overhead circle will correspond to those objects that are related to teleparallel  gravity or calculated on its connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The condition  ∂[µωA  |B|ν] + ωA  C[µωC  |B|ν] ≡ 0 ,  (4)  results in a flat spin connection [32], where square brackets denote the antisymmetric operator, and can  equivalently be used to determine the components of the spin connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The local Lorentz transformation  (LLT), wherein ΛA  B represents Lorentz boosts and rotations, is used to define the spin connection ωA  Bµ =  ΛA  C ∂µΛ  C  B  [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' It plays a role in the field equations since an infinite number of tetrad choices could satisfy  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' (1) for a particular metric, thus the spin connection is used to counterbalance inertial effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' This  ensures the theory, in this case TG, remains covariant [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Moreover, there exists a Lorentz frame such that  the spin connection is set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' It is referred to as the Weitzenb¨ock gauge [35, 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Hence, from this point  onwards, the spin connection will be dropped following the application of the aforementioned gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  Similar to how the Riemann tensor, constructed on the Levi-Civita connection, is associated to the cur-  vature property of GR, the analogy of this for TG is the torsion tensor defined by the teleparallel connec-  tion [35, 80]  T A  µν = ΓA  νµ − ΓA  µν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (5)  While curvature in GR is defined as a deformation of spacetime, the torsion tensor in TG represents the  field strength of gravitation that transforms covariantly under LLTs and diffeomorphisms [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Another  important aspect of TG is that the torsion tensor can be decomposed in irreducible parts [81–83], namely  aµ = 1  6ǫµνλρ T νλρ ,  (6)  vµ = T λ  λν ,  (7)  tλµν = 1  2(Tλµν + Tµλν) + 1  6(gνλvµ + gνµvλ) − 1  3gλµvν ,  (8)  4  giving the axial, vectorial and tensorial pieces, respectively, and ǫABCD being the four-dimensional Levi-  Civita tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Hence, the respective scalar invariants are given by [47, 82]  Tax = aµaµ = − 1  18Tλµν(T λµν − 2T µλν) ,  (9)  Tvec = vµvµ = T λ  λµT  ρµ  ρ  ,  (10)  Tten = tλµνtλµν = 1  2Tλµν(T λµν + T µλν) − 1  2T λ  λµT ρµ  ρ  ,  (11)  which, under parity transformation, are invariant scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' On the other hand, terms such as P1 = vµaµ and  P2 = ǫµνσρtλµνtλ  ρσ are excluded due to parity-violation [38, 81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The combination of the scalar invariants  leads to the torsion scalar  T = 3  2Tax + 2  3Tten − 2  3Tvec = 1  2  �  E  λ  A gρµE  ν  B  − 2E  ρ  B gλµE  ν  A + 1  2ηABgµρgνλ  �  T A  µνT B  ρλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (12)  An identical result can be obtained through contraction between the torsion tensor and the superpotential  S  µν  A  which represents the potential relation of the gravitational energy-momentum tensor [80, 84, 85]:  S  µν  A  = 1  2  �  Kµν  A − E  ν  A T αµ  α + E  µ  A T αν  α  �  ,  (13)  where  Kλ  µν = Γλ  µν −  Γλ  µν = 1  2  �  T  λ  µ ν + T  λ  ν µ − T λ  µν  �  (14)  is the contorsion tensor relating TG and GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' These quantities can be used to form the torsion scalar [34],  written as  T = S  µν  A  T A  µν ,  (15)  which can be shown to be equivalent to the result obtained in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The Ricci scalar dependent on  the teleparallel connection, which vanishes, can be related, using the contorsion tensor, to the regular Ricci  scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' This results in a relationship between the Levi-Civita and the teleparallel connections defined Ricci  scalar [47, 77]  R =  R + T − B = 0 ,  (16)  where B = 2  e∂µ(e T λ µ  λ ) = 2  ∇µT λ µ  λ  is a boundary term and e = det(eA  µ) = √−g is the tetrad determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  Hence, the curvature-ful Ricci scalar is non-vanishing being  R = −T + B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (17)  The total divergence term found in B accounts for the fourth order derivative contributions to the field  equations in modified theories of gravity [47, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' This is embodied within the Ricci scalar in GR [57, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  Thus, the Teleparallel Equivalent of General Relativity (TEGR) [86, 87] is defined as the linear appearance  of the torsion scalar since both Lagrangians are equal up to a boundary term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  The equivalence principle in GR allows one to raise local Lorentz frames from a Minkowski metric to  a general metric tensor, and additionally, partial derivatives are raised to covariant derivatives defined by  the Levi-Civita connection [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' This procedure, referred to as the minimal coupling prescription, if applied  within TG, is preserved for additional fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Minkowski tetrads are raised to arbitrary ones and tangent  space partial derivatives are exchanged for covariant derivatives based on Levi-Civita connection [33, 88]  ∂µ →  ∇µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (18)  Thus, by having both gravitational and scalar fields well developed, it is possible to look at the teleparallel  analogue of Horndeski gravity, referred to at times as Bahamonde-Dialektopoulos-Levi Said (BDLS) the-  ory [38, 66, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The construction of BDLS theory depends on the following criteria: (1) field equations are  5  at most second order with respect to the tetrad and scalar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' (2) as previously mentioned, scalar invariants do  not violate parity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' (3) contractions of the torsion tensor are at most quadratic [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' All of these requirements  allow for an adequate extension of the standard metric Horndeski gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Note that Lovelock’s theorem  states that Einstein fields equations are the only second-order field equations from a Lagrangian density  constructed through a four-dimensional metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' TG allows for the weakening of Lovelock [36, 37] theory as  an additional scalar field φ is introduced, giving rise to further terms in the gravitational action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  Starting off with the non-minimal coupling of scalar and torsion field, the linear contraction is given by [38]  I2 = vµ ◦∇µφ ,  (19)  being the scalar I1 = tλµν  ∇λφ  ∇µφ  ∇µφ = 0, due to the complete symmetry of the tensor decomposition,  Furthermore I3 = aλ  ∇λφ violates the parity condition due to an odd number of axial parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' the  quadratic contractions of this nature are given by [38]  J1 = aµaν ◦∇µφ  ∇νφ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (20)  J3 = vλtλµν ◦∇µφ  ∇νφ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (21)  J5 = tλµνt α  λ ν  ∇µφ  ∇αφ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (22)  J6 = tλµνt αβ  λ  ∇µφ  ∇νφ  ∇αφ  ∇βφ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (23)  J8 = tλµνt  α  λµ  ∇νφ  ∇αφ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (24)  J10 = ǫµ  νλρaνtαρλ ◦∇µφ  ∇αφ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (25)  while other possibilities are eliminated as they have already been included:  J2 = vµvν ◦∇µφ  ∇νφ = I2  2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  J4 = vµtλµν ◦∇λφ  ∇νφ = J3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  J7 = tλµνtαβ  λ  ∇µφ  ∇νφ  ∇αφ  ∇βφ = −2J6 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (26)  while J9 = tλµνtαβγ  ∇λφ  ∇µφ  ∇νφ  ∇αφ  ∇βφ  ∇γφ = 0 due to the total symmetry of the tensor irreducible part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  Hence, BDLS action is described by [38]  SBDLS =  1  2κ2  �  d4x e LTele +  1  2κ2  5  �  i=2  �  d4x e Li +  �  d4x e Lm ,  (27)  where  LTele = GTele(φ, X, T, Tax, Tvec, I2, J1, J3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='J5, J6, J8, J10) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (28)  Here GTele is an arbitrary function, X := − 1  2∂µφ∂µφ and  L2 := G2(φ, X) ,  (29)  L3 := −G3(φ, X)  □φ ,  (30)  L4 := G4(φ, X)(−T + B) + G4,X(φ, X)[(  □φ)2 −  ∇µ  ∇νφ  ∇µ ◦∇νφ] ,  (31)  L5 := G5(φ, X)  Gµν  ∇µ  ∇νφ  − 1  6G5,X(φ, X)[(  □φ)3 + 2  ∇µ  ∇νφ  ∇ν  ∇αφ  ∇α  ∇µφ − 3  ∇µ  ∇νφ  ∇µ ◦∇νφ  □φ] ,  (32)  which are identical to the standard metric Horndeski Lagrangians [20] but calculated using the tetrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  Finally, Lm is the matter Lagrangian in Jordan conformal frame,  Gµν is the Einstein tensor, and κ2 = 8πG  where G is the gravitational constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  MINKOWSKI BACKGROUND EQUATIONS IN TELEPARALLEL HORNDESKI GRAVITY  Let us explore now perturbations on a Minkowski background within the gravitational theory of teleparallel  analogue of Horndeski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Firstly, we tackle the background equations for a general flat Friedman–Lemaˆıtre–Robertson–Walker  6  (FLRW) metric to obtain the necessary constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Then, this is followed by the analysis for perturbation  theory up to second order which is needed for our approach since we consider the Euler-Lagrange equations  to get the cosmological equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  The flat FLRW metric in Cartesian coordinates is given by  ds2 = −N(t)2dt2 + a(t)2(dx2 + dy2 + dx2) ,  (33)  where N(t) is the lapse function and a(t) is the scale factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' As previously mentioned, this follows the metric  signature that is mostly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The tetrad can be written in diagonal form as [35]  eA  µ = diag(N(t), a(t), a(t), a(t)) ,  (34)  which is compatible with the Weitzenb¨ock gauge giving a vanishing spin connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Hence, the action in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' (27) can be re-expressed in terms of these background quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Friedman equations are then obtained  by varying with respect to the lapse function and scale factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Additionally, the scalar field Klein-Gordon  equation can be obtained by varying with respect to the scalar field [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  Since we will be working in  Minkowski background, the limits N(t) → 1 and a(t) → 1 are taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' the constraints obtained from  the Friedman equation and scalar field variation in Minkowski background are given by  0 = −G2 − GTele + 2XG2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='X − 2XG3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='φ + 2XGTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='X ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (35)  0 = G2 + GTele − 2XG3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='φ + 4XG4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='φφ + 2¨φG4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='φ − 2X ¨φG3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='X + 4X ¨φG4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='φX − d  dt( ˙φGTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='I2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (36)  0 = G2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='φ + GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='φ − 2XG2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='φX + 2XG3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='φφ − 2XGTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='φX − G2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='X ¨φ + 2G3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='φ ¨φ − GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='X ¨φ  − 2X ¨φGTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='X − 2X ¨φG2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='XX + 2X ¨φG3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='φX − 2X ¨φGTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='XX ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (37)  where all scalar invariants are background quantities and comma ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' ) denotes partial derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Moreover,  the scalar field equation can also be expressed in terms of a Klein-Gordon equation as shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  SECOND ORDER PERTURBED ACTION WITH SCALAR-VECTOR-TENSOR  DECOMPOSITION  Here, we calculate the perturbed action up to second order terms which are necessary for the Euler-  Lagrange method to find the equations of motion of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' By taking perturbations about a Minkowski  background for both tetrad and scalar field, we can build the action up to second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The tetrad and  scalar perturbation are respectively given by  eA  µ → eA  µ + ǫ δeA  µ = δA  µ + ǫ δeA  µ ,  (38)  φ → φ + ǫ δφ ,  (39)  where ǫ is the perturbation parameter representing the perturbation order of background Minkowski tetrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  It is given by a four-dimensional identity matrix given by the Kronecker delta δA  µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' It is sufficient to expand  the scalar field up to first order since higher order terms do not contribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Additionally, it should be noted  that the background scalar field can be taken to be as a function of time and apply the unitary gauge such  that  φ → φ(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (40)  Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' the perturbations of arbitrary functions present in the Lagrangian are obtained by performing a  Taylor expansion up to second order such that  Gi(φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' X) = Gi + ǫ Gi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='XX(1) + ǫ2 �  1  2Gi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='XX(X(1))2 + Gi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='XX(2)�  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (41)  GTele(φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Tax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Tvec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' I2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' J1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' J3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' J5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' J6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' J8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' J10)  = GTele + ǫ(GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='XX(1) + GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='I2I(1)  2 ) + 1  2ǫ2(GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='XX(X(1))2 + GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='I2I2(I(1)  2 )2)  + ǫ2�  GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='XX(2) + GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='T T + GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='TaxTax + GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='TvecTvec + GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='I2I(2)  2  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (42)  7  where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' for i = {2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' 5},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' j = {X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' XX} and k = {X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Tax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Tvec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' XX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' I2I2},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Gi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='j and GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='k are background  functions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' such that XX and I2I2 are the second order derivatives with respect to X and I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The numbered  superscripts represent the order of perturbation of the scalar invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  In this section, we consider a scalar-vector-tensor (SVT) decomposition of the tetrad based on the formal-  ism applied in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' [89] for first order perturbations:  δeA  µ :=  \uf8ee  \uf8f0  ϕ  −(∂iβ + βi)  δI  i (∂ib + bi)  δIi �  −ψδij + ∂i∂jh + ∂ihj + ∂jhi + 1  2hij + ǫijk(∂kσ + σk)  �  \uf8f9  \uf8fb ,  (43)  where {ϕ, β, b, ψ, h} are scalars and σ is a pseudoscalar of 1 degree of freedom (DoF) each, {βi, bi, hi} are  vectors and σi is a pseudovector of 1 DoFs each, and hij is the tensor mode of 2 DoFs, for a total of of 16  DoFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The tensor modes are symmetric hij = h(ij), traceless δijhij = 0 and divergenceless ∂ihij = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' See  also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' [90, 91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The divergenceless property also applies for vectors and pseudovectors such that ∂iαi = 0  where α = {β, b, h, σ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Unlike Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' [92, 93], the pseudoscalar and pseudovector are included to account for the  anti-symmetry of the tetrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The mid-range Latin indices are spacial coordinates: {I, J, K, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='} for spacial  inner bundle and {i, j, k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='} for spacial spacetime manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Note, δij is the spacial Minkowski metric such  that δij = −ηij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' In turn, the first order metric perturbation from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' (1) is given by  δgµν =  \uf8ee  \uf8f0  2ϕ  ∂iB + Bi  ∂iB + Bi  2  �  −ψδij + ∂i∂jh + ∂ihj + ∂jhi + 1  2hij  �  \uf8f9  \uf8fb ,  (44)  where B = −β + b and Bi = −βi + bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The off-diagonals are identical hence verifying the symmetry of the  metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The pseudoscalar and pseudovectors no longer play a role, and thus, the number of DoFs reduces to  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  The gauge transformation through the coordinate change [94–96] is  ˜xµ → xµ + ξµ ,  (45)  where ξµ is a vector field applied to study the gauge transformation of perturbative quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Such a  coordinate transformation can be extended to include orders higher than second one, but it is sufficient to  consider up to this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Thus, the transformations for first tetrad are given by [97]  δ˜eA  µ = δeA  µ + Lξ(1)eA (0)  µ  ,  (46)  where Lξ is the Lie derivative along ξµ wherein ξµ = (ξ0, ξi + δij∂jξ) further splits and once again obeys  divergencelessness as ∂iξi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The analysis of each combination of temporal and spatial parts of the tetrad  indicates that ψ, σ, βi and hij are gauge invariant in Minkowski background while the rest have the following  gauge transformations  ˜ϕ = ϕ − ˙ξ0 ,  β = β − ξ0 ,  ˜b = b − ˙ξ ,  ˜h = h − ξ ,  ˜bi = bi + ˙ξi ,  ˜hi = hi + 1  2ξi ,  ˜σi = σi − 1  2ǫijk∂jξk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (47)  This shows that since the pseudoscalar σ is gauge-invariant, it can be treated separately than the rest of  the scalar modes, but the same cannot be said for the pseudovector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' In fact, the pseudovector σi can be  expressed in terms of hi (and its derivative in terms of bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' This implies that the vector and pseudovector  cannot be decomposed [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  Next, we construct groups of non-gauge invariant quantities: {ϕ, β}, {b, h} and {bi, hi, σi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' For gauge  choice, a quantity from each group is set to zero [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' In particular, we will choose β = 0, h = 0 and σi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  GHOST AND LAPLACIAN INSTABILITIES IN MINKOWSKI BACKGROUND  Ghost instabilities stem from a negative kinetic term in the action associated to a propagating degree of  freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' A ghost mode can be determined by expanding the action up to second order perturbations about  8  a background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Therefore, for a Lagrangian of the form L = ˙⃗χtA ˙⃗χ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=', we impose that the eigenvalues of  A should be positive to eliminate ghost modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The procedure is applied in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The second order  perturbation of the action is obtained by separately applying the scalar, vector and tensor field perturbations  given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The dynamical fields in the action are identified, while a system of equations is obtained  by varying the action with respect to the auxiliary fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' This system of equations is substituted back into  the action to eliminate the non-dynamic fields [99, 100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  In order to obtain a gauge invariant action, the action is varied with respect to the temporal and spatial  part of the vector field associated with the coordinate transformation, and imposing that this vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Then,  a diagonalized kinetic matrix is constructed and a constraint is generated for each entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' In general, these  constraints could be time dependent: this feature arises from the background spacetime and the background  quantities of fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' When looking at the propagating speeds of these modes, a positive definite value should  be applied in order to ensure that the perturbation does not lead to an exponential growth [101] i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' c2 > 0,  referred to as gradient or Laplacian instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Hence, both ghost and gradient instabilities are checked for  each mode in order to obtain the constraints which lead to a stable model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  Tensor Perturbations  The tensor mode perturbations, presented in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' (43), can be extended as  eA  µ →  \uf8ee  \uf8f01  0  0 δij + 1  2ǫhij − 1  8ǫ2hikhk  j  \uf8f9  \uf8fb ,  (48)  analogous to the Arnowitt-Deser-Misner (ADM) for tensor modes [100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The action can be extended to  second order perturbations such that  S(2)  T  = 1  2  �  d4x  �  MT ˙hij ˙hij − NT ∂mhij∂mhij + Phijhij�  (49)  where  MT = G4 − 2XG4,X + XG5,φ − GTele,T + 1  2XGTele,J5 + 2XGTele,J8 ,  (50)  NT = G4 − X(G5,φ + ¨φG5,X) − GTele,T ,  (51)  PT = − 1  2  d  dt( ˙φGTele,I2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (52)  For the sake of simplicity, we switch to Fourier space such that  S(2)  T  = 1  2  �  dt d3k  (2π)3  �  MT ˙hij ˙hij +  �  −k2NT + PT  �  hijhij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  �  (53)  This result is obtained after applying integration by parts, removing the surface terms and applying the  background Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' (35-37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' By imposing MT > 0, the theory is ghost free in tensor modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' When considering  a constant background scalar φ, this implies that G4 − GTele,T > 0 which corresponds to the condition  G4 −GTele,T ̸= 0 imposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' [102] in order to ensure that there are tensor mode DoFs propagating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The  propagation speed of tensor modes is given by  c2  T = NT  MT  =  G4 − X(G5,φ + ¨φG5,X) − GTele,T  G4 − 2XG4,X + XG5,φ − GTele,T + 1  2XGTele,J5 + 2XGTele,J8  ,  (54)  for which c2  T > 0 is required to ensure gradient stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Hence, MT > 0 and NT > 0 result in ghost  and gradient stability, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The same result would by obtained when eliminating I2 contribution  from the GTele function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Following the observations of gravitational wave signal GW170817 [103] and its  electromagnetic counterpart GRB170817A [27],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' it is interesting to calculate the deviation from speed of light  propagation such that the excess speed is given by  αT = c2  T − 1 =  X(2G4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='X − 2G5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='φ − ¨φG5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='X − 1  2GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='J5 − 2GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='J8)  G4 − 2XG4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='X + XG5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='φ − GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='T + 1  2XGTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='J5 + 2XGTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='J8  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (55)  9  which corresponds to the result obtained through the GW propagation equation [66],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' a value which is highly  constrained such that a graviton mass would be minute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  Additionally, from the general result of teleparallel Horndeski analogue, given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' (48), one may obtain  the results of well-studied theories from literature, as summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' See also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' [73] for the  metric cases derived from Noether symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' As an extension analogous to standard Horndeski theory,  the stability conditions can be obtained by setting GTele = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The ghost modes are excluded for when  the constant of the kinetic term is G4 − 2XG4,X + XG5,X > 0 while gradient stability is obtained for  G4 − X(G5,φ + ¨φG5,X) > 0, in agreement with Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' [101, 104–106] when taking the appropriate limits  to Minkowski spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' An example of a subclass of Horndeski gravity is the Brans-Dicke theory, where  G2 = 2wBDX  φ  and G4 = φ for which wBD is the Dicke coupling constant [107], other constants are set to  vanish and ghost instabilities are avoided for φ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' By considering the generalized Brans Dicke theory,  ghost instabilities are avoided for a positive value of a function of the scalar field, F(φ) > 0 [108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Another  example is f(  R) where G2 = f(φ) − φf ′(φ), G4 = f ′(φ), the rest of the constants are zero, implying that  ghost stability is achieved for f ′(φ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' A final subcase of Horndeski gravity considered here is GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' In this  case, the only non-vanishing constant is G4 = 1 for which there are no ghost modes since MT = 1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' All  of these subcases do not result in any gradient instabilities as cT = 1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  Next, we look at cases that arise due to the inclusion of teleparallel terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' For a purely teleparallel theory  such as f(T ) gravity, all terms are set to vanish except for GTele,T = f(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' This implies in −f ′(T ) > 0 to  avoid ghost modes similar to the result obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' An equivalent result is obtained for scalar-  tensor theory with a Lagrangian of the form L = f(φ, T ) + XP(φ), an extension of f(T ) [100, 109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Once  again, gradient instabilities are not an issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Finally, the case where only I2 contributions are present is  considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' In this case, all constants are going to vanish while GTele,I2 ̸= 0 leads to a non-dynamical degree  of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  Theory  Case  MT  NT  Horndeski  GTele = 0  G4 − 2XG4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='X + ¨φXG5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='X G4 − X(G5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='φ + ¨φG5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='X)  Generalized  Brans-Dicke  GTele = G5 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' G2 = B(φ)X  G3 = 2ξ(φ)X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' G4 = 1  2F(φ)  1  2F(φ)  1  2F(φ)  Brans-Dicke  GTele = G3 = G5 = 0  G2 = 2wBDX  φ  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' G4 = φ  φ  φ  f(  R)  GTele = G3 = G5 = 0  G2 = f(φ) − φf ′(φ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' G4 = f ′(φ)  f ′(φ)  f ′(φ)  General Relativity  GTele = G2 = G3 = G5 = 0  G4 = 1  1  1  Teleparallel  or f(T )  G2 = G3 = G4 = G5 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  GTele = f(T )  −f ′(T )  −f ′(T )  f(φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' T ) + XP(φ)  G3 = G4 = G5 = 0  G2 = XP(φ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' GTele = f(φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' T )  −f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='T (φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' T )  −f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='T (φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' T )  GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='I2 only  G2 = G3 = G4 = G5 = 0  GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='T = 0  no propagating mode  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' List of literature models with the respective ghost MT and gradient stability NT conditions are positive  definite, while the propagation speed is cT = NT/MT for tensor modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The models include Horndeski theory [20,  101, 104, 105], Generalized Brans-Dicke [108] and Brans-Dicke [107], f(˚  R) theory, General Relativity, f(T ) theory [98],  f(φ, T ) theory [100] and the case where the action is dependent on I2 only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  10  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  Vector Perturbations  Next, we consider the vector perturbation of the tetrad (43) with the application of gauge fixing which  eliminates the pseudovector and any coupling with it such that  eA  µ →  \uf8ee  \uf8f0  1  −ǫβi  ǫδI  i bi δIi(δij + ǫ(∂ihj + ∂jhi))  \uf8f9  \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (56)  This result is a case of only vector modes in this portion of decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Extending the action to second  order vector perturbations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' we obtain  S(2)  V  =  �  dt d3k  (2π)3  �  4k4Ahihi + E ˙βi ˙βi  + k2(4B ˙hi( ˙hi − bi) + Cbibi + Dβiβi + 4Fhiβi + 4Ghi ˙βi − 2Hβibi)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (57)  where  A = GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='Tvec + 1  9X  �  −GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='J8 + XGTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='J6 − 5  2GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='J5 − 3GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='J3  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  B = G4 − 2XG4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='X + XG5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='φ − GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='T + X  �  2GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='J8 + 1  2GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='J5  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  C = B + 1  2XGTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='J5 + 2  3XGTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='J10 + 2  9GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='Tax ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  D = C + XGTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='J5 − 2XGTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='J8 − 2XGTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='J10 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  E = 4A − 3GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='Tvec + 2XGTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='J3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  F = 1  2 ˙φGTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='I2 − d  dt (G4 − 2XG4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='X + XG5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='φ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  G = 2A − B − 3GTele,Tvec + 1  3XGTele,J3 − 2XGTele,J8 + 5  2XGTele,J5 ,  H = C − 2XGTele,J8 − 4  9GTele,Tax ,  (58)  While the decomposition along unitary gauge ensures that there are no higher order time derivatives, one  can see that the action contains higher order spatial derivatives [110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Here, we will set the coefficients of  these derivatives to vanish, but it should be noted that a general analysis for each constraint should be  carried out before solving the next one and leading to a very complicate solution for the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' By imposing  the conditions A = 0 and B = 0, the latter condition accounting for the higher order mix of temporal and  spatial derivatives [111], the action reduces to  S(2)  V  =  �  dt d3k  (2π)3  �  E ˙βi ˙βi + k2(Cbibi + Dβiβi + 4Fhiβi + 4Ghi ˙βi − 2Hβibi)  �  ,  (59)  where bi and hi are auxiliary vector modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' By varying with respect to each of these non-dynamical modes,  the respective field equations are given by  2k2(Cbi − Hβi) = 0 ,  and  4k2Fβi = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (60)  By solving for each auxiliary field and plugging it back in into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' (59), we get  S(2)  V  =  �  dt d3k  (2π)3  �  E ˙βi ˙βi − k2 �  H2  C − D  �  βiβi  �  ,  (61)  wherein the action is expressed in terms of the dynamical non-gauge invariant mode βi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Thus the ghost and  Laplacian conditions are given respectively by  MV = E > 0 ,  and NV = H2  C − D > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (62)  suggesting that there is a propagating vector mode with speed  cV = H2 − CD  CE  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (63)  This propagating mode generally steps from the introduction of the Ji terms in the BDLS action (27)  representing the quadratic contractions of the scalar and torsion fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  11  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  Vanishing Ji terms:  When considering the case were J1 = J3 = J5 = J6 = J10 = 0, the action (61)  reduces to  Svanishing J terms  V  =  �  dt d3k  (2π)3 k2  �  C − H2  C  �  βiβi ,  (64)  exhibiting no dynamical modes i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' no propagating vector mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' In the case of only teleparallel function,  the same conditions apply with vanishing Horndeski terms in each constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  The subcases of standard  Horndeski [31, 104], f(  R) [99, 112], Generalized Brans-Dicke[108] and f(T ) [98] fall under this category of  non-viable propagating vector mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  C = 0  : The Laplacian constraint, given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' (62), is undefined for C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' By considering this  particular case, the action results in  SC=0  V  =  �  dt d3k  (2π)3 E ˙βi ˙βi ,  (65)  to which the Laplacian condition is violated since it is null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' On the other hand, imposing only E = 0, the  ghost instability is generated since this value should be a positive definite value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  Ji terms only:  On the other hand, when considering only the contribution of the quadratic contrac-  tions, the only surviving term is that with coefficient E leading to a ghost stability condition but leading  to Laplacian instability, similar to the C = 0 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' This implies the importance of having a combination of  the Ji terms along with the rest of the teleparallel scalar invariants to maintain stability conditions and a  possible propagating vector mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  Constant Background Scalar Field:  With regards to the scalar field, the unitary gauge is applied to  the perturbative part of the scalar, while, at background level, it is a function of time only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Alternatively,  if one considers a constant scalar field φ = c, then the entire action is vanishing such that no vector modes  propagate, thus it corresponds to the results in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' [67] for the DoF and polarisation modes in the vector  portion of the decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  Scalar Perturbations  Let us now consider the scalar modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Since there is no mixing between the scalar and pseudoscalar modes,  we will treat them separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The tetrad perturbation using scalar modes is given by  eA  µ →  \uf8ee  \uf8f01 + ϕ  0  δI  i ∂ib (1 − ψδIiδij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  \uf8f9  \uf8fb  (66)  As already stated, for the gauge invariant quantities, given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' (47), the gauge choice is β = h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The  pseudoscalar will be analyzed separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' It should be noted that this situation is identical to the Arnowitt-  Deser-Misner (ADM) decomposition used in torsion-based theories [100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' It leads to the same metric as that  applied to curvature-based theories [101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' On the other hand, the choice β = b = 0 results in the Newtonian  (longitudinal) gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Substituting the tetrad perturbation to the action,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' followed by integration by parts  and application of the background equations (35-37),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' the second order perturbation of the action is given by  S(2)  S  =  �  dt d3k  (2π)3  �  ¯  Aϕ2 + 6 ¯Dψ2 − 6 ¯F ˙ψ2 − 6 ¯Gϕ ˙ψ  − k2(− ¯Bϕ2 − 2 ¯Eψ2 + 4 ¯Hϕψ − 2 ¯Gϕb − 4 ¯F ˙ψb) + k4 ¯Cb2�  (67)  12  where  ¯  A = XG2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='X + 2X2G2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='XX − 2XG3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='φ − 2X2G3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='φX + XGTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='X + 2X2GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='XX ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  ¯B = GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='Tvec + 2  9X (2GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='J8 + 2XGTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='J6 − 5GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='J5 + 3GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='J3) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  ¯C = −GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='Tvec + XGTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='I2I2 + 1  3X(4GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='J8 + GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='J5) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  ¯D = d  dt( ˙φGTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='I2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  ¯E = G4 − X(G5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='φ − ¨φG5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='X) − GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='T + 2GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='Tvec + 1  9X (−2GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='J8 + 2XGTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='J6 − 5GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='J5 − 6GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='J3) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  ¯F = G4 − 2XG4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='X + XG5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='φ − GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='T + 3  2(GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='Tvec − XGTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='I2I2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  ¯G = − ˙φXG3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='X + ˙φG4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='φ + 2 ˙φXG4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='φX + 1  2 ˙φGTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='I2 + ˙φXGTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='XI2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  ¯H = G4 − 2XG4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='X + XG5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='φ − GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='T + GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='Tvec + 1  9X(2GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='J8 − 2XGTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='J6 + 5GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='J5 + 3  2GTele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='J3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (68)  Here k is the co-vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' As shown in coefficient of ¯C in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' (67), there is a higher derivative contribution,  hence it is set to vanish [110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' It is clear that variables ϕ and b are non-dynamical with respect to the time  component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' By varying with respect to each of these auxiliary fields, one obtains a set of equations  0 = ( ¯  A + k2 ¯B)ϕ − 3 ¯G ˙ψ + k2(2 ¯Hψ − ¯Gb) ,  (69)  0 = ¯Gϕ + 2 ¯F ˙ψ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (70)  By solving these equations of motion for the modes ϕ and b, it leads to an action expressed in terms of the  dynamical mode only, that is  S(2)  S  =  �  dt d3k  (2π)3  �  4k2 ¯B ¯F  2  ¯G  2  ˙ψ2 +  �  4 ¯  A ¯F  2  ¯G  2  + 6 ¯F  �  ˙ψ2 − k2  �  −2 ¯E + 4 d  dt  � ¯F ¯H  ¯G  ��  ψ2 + 6 ¯Dψ2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (71)  Taking into account the mix of spatial and temporal higher-order derivatives, the first term of the action  vanishes [111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' This implies that either ¯F = 0 (which renders the whole action without a dynamical mode),  or ¯B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' By applying the latter condition, the action can be expressed in terms of a gauge invariant quantity  within a Minkowski background and no higher-order terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' It is  S(2)  S  =  �  dt d3k  (2π)3  ��  4 ¯  A ¯F  2  ¯G  2  + 6 ¯F  �  ˙ψ2 − k2  �  −2 ¯E + 4 d  dt  � ¯F ¯H  ¯G  ��  ψ2 + 6 ¯Dψ2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (72)  In the case of teleparallel analogue of Horndeski theory, ghost stability and gradient stability are obtained,  respectively, when  MS = 4 ¯  A ¯F  2  ¯G  2  + 6 ¯F > 0 ,  (73)  ¯  N S = −2 ¯F + 4 d  dt  � ¯F ¯H  ¯G  �  > 0 ,  (74)  and the propagating speed is  cS = NS  MS  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (75)  Different subcases are reported in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Through the substitution of Eqs (73) and (74), conditions to  avoid ghost and Laplacian instabilities are realised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  Horndeski:  The standard Horndeski case has identical expressions for ghost and gradient stability  conditions as those for teleparallel Horndeski (73-74) with the difference that each constant does not have  teleparallel contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  13  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  Generalized Brans-Dicke:  Considering Generalised Brans-Dicke (GBD) theory means that G2 =  B(φ)X, G3 = 2Xξ(φ), G4 = 1  2F(φ) and all other terms are set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' When we take the background  equations  B − 4Xξ′ = 0 ,  and  ˙φ2F ′′ + ¨φ(F ′ − 2 ˙φ2ξ) = 0 ,  (76)  for ghost stability, we have  MGBD  S  = F[3F ′2 − 4 ˙φ2(Fξ′ + 3ξF ′) + 12 ˙φ4ξ2]  [F ′ − 2 ˙φ2ξ]2  > 0 ,  (77)  where ′ denotes a derivative with respect to the respective variable of the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' In this case, it is φ, while  the gradient condition is  N GBD  S  = F[3F ′3 − 2 ˙φ2(5F ′2ξ − 4ξFF ′′ − 2Fξ′F ′) + 4 ˙φ4(F ′ξ2 − 2Fξ′ξ) + 8 ˙φ6ξ3]  [F ′ − 2 ˙φ2ξ]3  > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (78)  With these conditions, the speed of propagation is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' It is  (cGBD  S  )2 = 3F ′3 − 2 ˙φ2(5F ′2ξ − 4ξFF ′′ − 2Fξ′F ′) + 4 ˙φ4(F ′ξ2 − 2Fξ′ξ) + 8 ˙φ6ξ3  [3F ′2 − 4 ˙φ2(Fξ′ + 3ξF ′) + 12 ˙φ4ξ2][F ′ − 2 ˙φ2ξ]  ,  (79)  which correlates with results obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' [108] for a Minkowski background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  Brans-Dicke:  As for the Brans-Dicke theory in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' [107] with G2 = 2wBDX  φ  and G4 = φ, ghosts  instabilities are avoided if 2φ(3 + 2wBD) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  Provided that φ > 0, this implies that the Brans-Dicke  constant is wBD > − 3  2 [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The gradient stability condition is given by 2φ(3 − 2φ¨φ/ ˙φ2) > 0 to ensure  a positive propagating speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Moreover, when applying the second Friedman Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' (36), gradient condition  reduces to 2φ(3 + wBD) such that wBD > −3 and a unitary propagating speed [113], analogous to ξ = 0,  reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' [108], is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The first Friedman Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' (35) shows that the only non-trivial solution is  that for wBD = 0, which results in identical ghost and gradient expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  f(˚  R):  The scalar-tensor theory equivalent to f(R) [114] is given by setting G2 = f(φ) − φf ′(φ) and  G4 = f ′(φ) while the rest of the functions are set to vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' See also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' In order to have ghost  stability, it has to be 6f ′(φ) > 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' f ′(φ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Once again, upon applying the background Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' (35-36),  the gradient condition reduces to f ′(φ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' In fact, as a subcase of GBD theory, the case ξ = 0 always  results in a unitary speed propagating mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  GR, f(T ), f(φ, T ):  When considering the cases of GR, f(T ) and f(φ, T ), the substitution in Eqs˜(73-  74) results in an undefined expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' In these cases, although taking the appropriate limits of each coeffi-  cient does lead to a positive ghost condition value, the propagating speed is negative, thus scalar modes are  not viable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' This is expected in GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Additionally, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' [98] shows that the only propagating scalar field is that  dependent on the perturbation of scalar field φ, while, in this paper, we are applying the unitary gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' As  in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' [100], f(φ, T ) theory gives rise to a propagating mode when considering cosmological perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  The same applies for GR with an additional canonical scalar field such that G2 = X − V (φ), G4 = M 2  Pl/2,  G3 = G5 = GTele = 0 [101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  Teleparallel:  If one considers only the teleparallel portions of the action, all terms from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' (71) are  retained, with all standard Horndeski contributions set to vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' This implies that the ghost and Laplacian  instabilities coincide with the form of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' (73) and (51), respectively, provided that the background equations  are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  Pseudoscalar Perturbations  The gauge invariant pseudoscalar σ can be treated separately as  eA  µ →  \uf8ee  \uf8f01  0  0 δIi(δij + ǫ εijk∂kσ)  \uf8f9  \uf8fb ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  (80)  14  Theory  Case  MS  NS  Horndeski  GTele = 0  6 ¯  F + 4 ¯  A ¯  F2  ¯  G2  −2 ¯  E + 4 d  dt ( ¯  F2  ¯  G )  Generalized  Brans-Dicke  GTele = G5 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  G2 = B(φ)X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' G3 = 2ξ(φ)X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  G4 = 1  2 F (φ)  F [3F ′2−4 ˙φ2(F ξ′+3ξF′)+12 ˙φ4ξ2]  [F ′−2 ˙φ2ξ]2  F [3F ′3−2 ˙φ2(5F ′2ξ−4ξFF ′′−2F ξ′F ′)+4 ˙φ4(F ′ξ2−2F ξ′ξ)+8 ˙φ6ξ3]  [F ′−2 ˙φ2ξ]3  Brans-Dicke  GTele = G3 = G5 = 0  G2 = 2wBDX  φ  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' G4 = φ  2φ(3 + 2wBD)  2(3φ −  ¨  φ  X ) → 2φ(3 + 2wBD)  f( ◦  R)  GTele = G3 = G5 = 0  G2 = f(φ) − φf′(φ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  G4 = f′(φ)  6f′(φ)  −2f′(φ) + 4 d  dt ( f′(φ)2  ˙φf′′(φ) ) → 6f′(φ)  GR  GTele = G2 = G3 = G5 = 0  G4 = 1  no propagating mode  f(T )  G2 = G3 = G4 = G5 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  GTele = f(T )  no propagating mode  f(φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' T ) + XP (φ)  G3 = G4 = G5 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  G2 = XP (φ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  GTele = f(φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' T )  no propagating mode  Teleparallel  G2 = G3 = G4 = G5 = 0  4 ¯  A ¯  F2  ¯  G2  + 6 ¯  F  −2 ¯  F + 4 d  dt  � ¯  F ¯  H  ¯  G  �  TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' List of literature models with the respective ghost MS and gradient stability NS conditions are positive  definite, and propagation speed cS = NS/MS for scalar modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The models include Horndeski theory [20, 101, 104,  105], Generalized Brans-Dicke [108] and Brans-Dicke [107], f(˚  R) theory , General Relativity, f(T ) theory [98], f(φ, T )  theory [100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' and an action with only teleparallel terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  The action in Fourier space is expanded up to second order  S(2)  PS = −  �  dt d3k  (2π)3  ��  k2 d  dt( ˙φGTele,I2) + 4  9k4 (GTele,Tax − 2XGTele,J1)  �  σ2  +k2  �4  9GTele,Tax + X  �  GTele,J5 + 4  3GTele,J10  ��  ˙σ2  �  ,  (81)  where the first term exhibits higher-order spatial derivatives and the second term exhibits higher order mixed  derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' By imposing that GTele,Tax = 2XGTele,J1 = −X  � 9  4GTele,J5 + 3GTele,J10  �  , the action reduces to  S(2)  PS =  �  dt d3k  (2π)3  �  − k2 d  dt( ˙φGTele,I2)σ2�  ,  (82)  where σ is a non-dynamical mode and thus it is a non-propagating mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' This shows that in Minkowski  background, the additional scalar invariants, provided by teleparallel analogue of Horndeski theory, do not  contribute to any additional propagating DoF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' See also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  DISCUSSION AND CONCLUSIONS  We have investigated constraints emerging by considering the stability of perturbations about a Minkowski  background in teleparallel analogue of Horndeski gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The approach is performed by exploring the ghost  instabilities through looking at potentially negative expressions of kinetic term associated with propagating  DoFs, as well as Laplacian instabilities emerging when propagation speeds of perturbations are not positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  This leads to possible exponential growth rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' These considerations are fundamental for the construction  of robust and self-consistent cosmological models related to the scalar-tensor gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  Specifically, we have discussed teleparallel analogue of Horndeski gravity where the most general scalar-  tensor action is considered, provided that the Lagrangian terms are not parity violating and that scalars  are produced by no more than quadratic contractions of torsion scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The naturally lower order nature  of teleparallel gravity induces a much richer landscape of functional models when compared with standard  metric Horndeski gravity, based on Levi-Civita connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' In other words, the replacement of torsion with  curvature to lead dynamics turns out to manifest as a generalization of the standard Horndeski theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Im-  portantly, this produces a generalization of tensor mode propagation speed which means that the restrictions  appearing in standard Horndeski gravity, due to the constraints on the gravitational waves speed, do not  have the same effect here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  15  Our calculations have proceeded by first deriving the background equations of motion (35–37) which are  obtained by considering a flat FLRW background in which we take the Minkowski limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' We find this to  be a convenient approach to determine the system of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' These are useful equations for reducing  the expressions of perturbations that ensue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' The procedure is described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' We then proceeded to  directly determine constraints to prevent ghost or Laplacian instabilities starting with tensor modes which  results in the action in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' (53) and tensor propagation speed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' We collected the constraints for  popular subclasses of BDLS theory in Table I where the known results are obtained for standard Horndeski  gravity as well as f(  R) and Brans-Dicke gravity, but new conditions are found in other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' As for the  vector perturbations, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' V B shows that vector modes are indeed propagating for some cases of GTele  contribution, namely when Ji scalars appear in this function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' However, these contributions may be small  and within observational constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  The scalar perturbations contain the most diverse range of DoFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' This rich structure produces the intricate  action in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' (72), together with the propagation speed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' (75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' As in the case of tensor modes, constraints  for each subclass of popular models is reported in Table II where results for the Minkowski perturbations are  recovered for standard metric Horndeski gravity together with f(  R) and Brans-Dicke gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' Interestingly,  either no constraints are set on some of the purely teleparallel models or only lightly limited cases are set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  Importantly, these constraints are consistent with the tensor mode constraints on the functional models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  These results are promising in terms of the viability of BDLS theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' As future work, it is intriguing to  explore constraints from tachyonic instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' These constraints have been connected to Jeans instability  where exponential growth of perturbations is slowed by the expansion of the Universe and where the matter  dispersion relation has a negative mass term that renders it to vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' This could lead to a more general  consideration of perturbations about a flat FLRW background spacetime which would then be directly linked  to observations such as those related to the growth of large scale structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' In forthcoming studies, possible  observational constraints will be scrutinized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  This paper is based upon work from COST Action CA21136 Addressing observational tensions in cosmology  with systematics and fundamental physics (CosmoVerse) supported by COST (European Cooperation in  Science and Technology).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' SC acknowledges the Istituto Nazionale di Fisica Nucleare (iniziative specifiche  QGSKY and MOONLIGHT2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content=' MC acknowledges funding by the Tertiary Education Scholarship Scheme  (TESS, Malta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
+page_content='  [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE3T4oBgHgl3EQfVQoB/content/2301.04457v1.pdf'}
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+Dynamic Behavior of a Railway Track Under a Moving Wheel
+Load Modelled as a Sinusoidal Pulse
+Mohammed Touati1, Nouzha Lamdouar1 and Chakir Tajani2,∗
+1Civil Department, Mohammadia School of Engineers -Mohammed V University - Morocco
+mohammed.touati87@gmail.com, nlamdouar@gmail.com
+2Department of Mathematics, Polydisciplinary faculty of Larache,
+Abdelmalek Essadi University, Morocco
+chakir tajani@hotmail.fr
+Abstract
+The aim of this paper is to evaluate the train/track induced loads on the substructure
+by modelling the wheel, at each instant, as a moving sinusoidal pulse applied in a very short
+period of time. This assumption has the advantage of being more realistic as it reduces the
+impact of time on the load definition. To that end, mass, stiffness, and dumping matrices
+of an elementary section of track will be determined. As a result, the equations of motion
+of a section of track subjected to a sinusoidal pulse and a rectangular pulse respectively is
+concluded. Two numerical methods of resolution of that equation, depending on the nature
+of the dumping matrix, will be presented. The computation results will be compared in
+order to conclude about the relevance of that load model. This approach is used in order to
+assess the nature and the value of the loads received by the substructure.
+2020 MSC: Primary 74S05, 37M05, 74-10 Secondary 37N30
+Key Words and Phrases: Dynamic properties, Finite elements modelling, Railway
+track dynamics, Sinusoidal pulse load.
+1
+Introduction
+Various theoretical and experimental researches have been performed in order to assess train/track
+induced loads on the substructure. Mohammed Touati and al. [1] determined the loads induced
+by a non-linear 3D multi-body modelled train on the track with taking into account wheel/rail
+contact properties and track irregularities. Yang Xinwen and al. [2] concluded, through a vehicle-
+track-subgrade coupling dynamic theory and finite element method, about the train/track in-
+duced loads on each layer of the substructure. As an experimental study, Al Shaer and al. [3]
+presented the dynamic behavior of a portion of ballasted railway track subjected to cyclic loads
+in substitution of a moving wheelset. In conclusion, the dynamics behavior of the substructure
+is widely studied in the literature [4-8] based on the train/track coupling model.
+Actually, even if modelling a wheel load as a rectangular pulse is a common assumption, real
+measurements don’t show the same shape. In fact, ONCF (Moroccan railway network manager)
+has many tools that record wheel pulse like GOTCHA. This system shows that the shape of
+the load has never been rectangular, but it’s more likely compared to a sinusoidal pulse. Then,
+this paper deals with evaluating train/track induced loads on the substructure by proposing a
+new approach when it comes to modelling the shape of the wheel impact. Indeed, it’s common
+to consider a moving load as a rectangular impulse applied on the nodes of a mesh structure in
+each period of time depending on signal sampling. This paper shows that assuming the wheel
+1
+arXiv:2301.01524v1  [math.NA]  4 Jan 2023
+
+Figure 1:
+Elementary track modelling
+load as a sinusoidal pulse may reduce the impact of the period of time of its application and,
+consequently, minimize the loads induced on the substructure oversized by the common assump-
+tion. In that matter, a finite element model of the track will be presented and the numerical
+results will be compared.
+2
+Track elementary section modeling
+2.1
+Determination of mass, stiffness et dumping matrices
+Let’s assume a portion of ballasted track composed of two elements of rail considered as a contin-
+uous Euler-Bernoulli beam, fixed to two sleepers by a couples of springs/dampers representing
+the railpads. The ballast is modelled as a couples of springs/dampers under each sleeper (Figure
+1).
+The displacement vector is written as:
+U = [u1, θ1, u2, θ2, u3, θ3, uT1, uT2]
+The effective mass and the stiffness matrices of an element of rail [9], are given by:
+Mr = (ρrArL/420)
+�
+�������
+156
+22L
+54
+−13L
+0
+0
+22L
+4L2
+13L
+−3L2
+0
+0
+54
+13L
+312
+0
+54
+−13L
+−13L
+−3L2
+0
+8L2
+13L
+−3L2
+0
+0
+54
+13L
+156
+−22L
+0
+0
+−13L
+−3L2
+−22L
+4L2
+�
+�������
+Kr =
+�
+ErIr/L3�
+�
+�������
+12
+6L
+−12
+6L
+0
+0
+6L
+4L2
+−6L
+2L2
+0
+0
+−12
+−6L
+24
+0
+−12
+6L
+6L
+2L2
+0
+8L2
+−6L
+2L2
+0
+0
+−12
+−6L
+12
+−6L
+0
+0
+6L
+2L2
+−6L
+4L2
+�
+�������
+2
+
+element1
+blement 2
+ua
+2
+kn2
+c
+"
+ur2
+Masse (m-)
+Masse (m-)where ρr is the density of the rail, Ar is the surface of the rail section, Er is Young modulus,
+and Ir is the rail moment of inertia. The dumping matrix of the rail is obtained as a linear
+combination of mass and stiffness matrices by assuming that the displacements u1 and u3 are
+completely dumped by the effect of railpads.
+Therefore, the dumping matrix is written as:
+C∗
+r = a0 · M∗
+r + a1 · K∗
+r
+where,
+M∗
+r = (ρrArL/420)
+�
+���
+4L2
+13L
+−3L2
+0
+13L
+312
+0
+−13L
+−3L2
+0
+8L2
+−3L2
+0
+13L
+−3L2
+4L2
+�
+���
+K∗
+r =
+�
+ErIr/L3�
+�
+���
+4L2
+−6L
+2L2
+0
+−6L
+24
+0
+6L
+2L2
+0
+8L2
+2L2
+0
+6L
+2L2
+4L2
+�
+���
+a0 and a1 are concluded from the equation:
+� a0
+a1
+�
+=
+�
+2ω1ω2/
+�
+ω2
+2 − ω2
+1
+�� �
+ω2
+−ω1
+−1/ω2
+1/ω1
+� � ζ1
+ζ2
+�
+where ωi2, (i = 1, 2) are the eigenvalues associated to the vibration of the rail described by
+the matrices Mr∗ and Kr∗, and ζi, (i = 1, 2) are the dumping ratios according to the first and
+second modes.
+In one hand, the equation of motion of the rail is written as:
+Mr ¨U ∗ + Cr ˙U ∗ + KrU ∗ = F
+(1)
+where Cr is the transformation of the matrix C∗
+r in the base U ∗, and U ∗ is defined by:
+U ∗ = [u1, θ1, u2, θ2, u3, θ3]
+F is given by:
+F =
+�
+�������
+−ks (u1 − uT1) − cs ( ˙u1 − ˙uT1)
+0
+0
+0
+−ks (u3 − uT3) − cs ( ˙u3 − ˙uT3)
+0
+�
+�������
+In the other hand, the equations of motion of the sleepers are written as:
+� mT ¨uT1 = ks (u1 − uT1) + cs ( ˙u1 − ˙uT1) − kbuT1 − cb ˙uT1
+mT ¨uT2 = ks (u3 − uT2) + cs ( ˙u3 − ˙uT2) − kbuT2 − cb ˙uT2
+(2)
+From (1) and (2), we may conclude about the equation of motion of the track elementary
+section as it’s modelled. It’s written as:
+M ¨U + C ˙U + KU = 0
+where M, C and K are the mass, dumping, and the stiffness of the track elementary section
+respectively.
+3
+
+Symbol
+Quantity
+Value
+ρr
+Rail density (kg/m3)
+7850
+Ar
+Rail section surface (cm²)
+76.70
+Er
+Young modulus of the rail (GPa)
+210
+Ir
+Rail moment of inertia (cm4)
+3038.6
+mT
+Sleeper mass (kg)
+90.84
+ks
+Railpad stiffness (MN/m)
+90
+cs
+Railpad damping (kN.s/m)
+30
+kb
+Ballast stiffness (MN/m)
+25.5
+cb
+Ballast damping (kN.s/m)
+40
+ζ
+Rail dumping ratio
+5%
+Table 1: Track properties
+2.2
+Numerical application
+Let’s assume a track elementary section characterized by the data given in table 1 (we can refer
+to ([10], [11], [12]).
+The figure 2 illustrates the evolution of natural frequencies according to vibration modes. It
+shows that:
+• The frequencies of the 1st and 2nd modes correspond to a movement in phase between rail
+and sleepers. It’s equal to 81.62 Hz;
+• The frequency of the 3rd mode corresponds to a movement in opposition of phase between
+rail and sleepers. It’s equal to 381.1 Hz.
+3
+Track response to a rectangular and a sinusoidal pulses
+3.1
+Description of the studied track
+Let’s assume a section of track composed of N track elementary sections subjected to an external
+load F as it’s shown in figure 3.
+The number of degrees of freedom is given by:
+Ndof = 8N − 3(N − 1)
+The displacement vector is written as:
+U =
+�
+���
+...
+uj,k
+...
+�
+���
+where,
+� uj,k = u∗
+j,k
+where
+k ∈ [1, 8]
+if
+j = 1
+uj,k = u∗
+j,k
+where
+k ∈ [3, 4, 5, 6, 8]
+if
+j ̸= 1
+and,
+U ∗
+j = [uj,1, θj,1, uj,2, θj,2, uj,3, θj,3, uj,T1, uj,T2]
+4
+
+Figure 2:
+Natural frequencies of an elementary track section
+Figure 3:
+Track section modelling
+5
+
+4.5
+X104
+3.5
+(ZH)
+3
+2.5
+2
+1.5
+1
+0.5
+0
+0
+0
+0
+1
+2
+3
+4
+5
+6
+7
+8
+6
+10
+NumerodumodeFigure 4:
+Sinusoidal and rectangular pulses over a period of td
+j refers to the element’s number.
+The mass, stiffness and dumping matrices in the base U are obtained by assembling those
+of a track elementary section determined earlier.
+The vector of loads is defined by:
+F =
+�
+���
+...
+fj
+...
+�
+���
+where,
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+N is even
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+N = 2
+� fj = P
+if
+j = 5
+fj = 0
+else
+N ̸= 2
+� fj = P
+if
+j = (5N/2) + 1
+fj = 0
+else
+N is uneven
+� fj = P
+if j = (5(N + 1)/2) − 1
+fj = 0
+else
+P is a rectangular or a sinusoidal load given as:
+• Sinusoidal pulse:
+� P = P0 sin ωt
+if
+t ≤ td
+P = 0
+else
+• Rectangular pulse:
+� P = P0
+if
+t ≤ td
+P = 0
+else
+Its shape is shown in the figure 4.
+6
+
+sinusoidal pulse
+rectangular pulse
+PO
+P
+x=0
+x=td
+X = 2td
+→3.2
+Description of the methods of resolution
+The dynamic behavior of the section of track may be analyzed by modal superposition if the
+dumping matrix verifies orthogonality properties.
+That method is used in particular for an
+undumped system. In that case, the equation of motion is reduced to:
+M ¨U + KU = F
+Let’s assume that ω2
+i are the eigenvalues associated to the track vibration. We note {φi} the
+normalized eigenvectors related to ω2
+i . Therefore, the equation of motion is written as:
+¨Z + diag(ω2
+i )Z = φT F
+(3)
+where diag(ω2
+i ) is a diagonal matrix of the eigenvalues and:
+U = Φ.Z
+The system of equations (3) is uncoupled where each equation is written as:
+¨zi + ω2
+i zi = Φj,iP(t)
+The resolution of that equation is given by DUHAMEL integral:
+zi(t) = (1/ωi)
+� t
+0
+Φj,iP(τ) sin ωi(t − τ)dτ
+Therefore, the solution for a sinusoidal pulse load is given as:
+zi(t) =
+� (Φj,iP0/ω2
+i ).(1/(1 − β2))(sin ωt − β sin ωit)
+if
+t ≤ td
+( ˙zi(td)/ωi) sin ωi(t − td) + zi(td) cos ωi(t − td)
+if
+t ≥ td
+where,
+β = ω/ωi
+and the solution for a rectangular pulse load is given as:
+zi(t) =
+� (Φj,iP0/ω2
+i )(1 − cos ωit)
+if
+t ≤ td
+(Φj,iP0/ω2
+i )(cos ωi(t − td) − cos ωit
+if
+t ≥ td
+The figure 5 shows the response z(t) to a sinusoidal and a rectangular pulse. It’s obvious that
+in the forced phase, the maximum rectangular response is higher than the maximum sinusoidal
+response.
+In general, the dumping matrix doesn’t verify the orthogonality characteristics. Therefore,
+the modal superposition method is substituted by the following method.
+The equation of motion can be written as:
+¨Z + φT Cφ. ˙Z + diag(ω2
+i ).Z = φT F
+(4)
+Where diag(ωi2) and φ are defined earlier. Knowing that:
+˙Z − ˙Z = 0
+(5)
+(4) and (5) could be written as:
+˙Y = D.Y + F ∗
+(6)
+7
+
+Figure 5:
+z(t) response to a rectangular and sinusoidal pulse td/T = 0.75
+where,
+Y =
+� Z
+˙Z
+�
+,
+D = A−1B,
+F ∗ = A−1
+� φT F
+0
+�
+and,
+A =
+� φT Cφ
+I
+I
+0
+�
+,
+B =
+� diag(ω2
+i )
+0
+0
+−I
+�
+Let’s assume that {λi} are the eigenvalues associated to the matrix D. We note {ψi} the
+normalized eigenvectors related to {ωi2}. We define X(t) as:
+Z = ψ.X
+The equation (6) is written as:
+˙X = diag(λi).X + ψ−1F ∗
+(7)
+The system of equations (7) is uncoupled where each equation is written as:
+˙xi(t) = ai.xi(t) + bi.P
+(8)
+where,
+ai = λi
+and
+bi = χi
+and,
+χ = ψ−1
+� φT
+0
+0
+0
+�
+The resolution of the equation (8) gives:
+8
+
+sinusoidal pulse
+rectangularpulse
+2
+-2
+-3
+-4
+X=0
+PI = X
+x = 2td
+PI =X
+X = 4td
+t• Sinusoidal pulse:
+xi(t) =
+�
+biPω
+a2
+i +ω2 eait − aibiP
+a2
+i +ω2 sin ωt −
+biPω
+a2
+i +ω2 cos ωt
+if
+t ≤ td
+xi(td)eai(t−td)
+if
+t ≥ td
+• Rectangular pulse:
+xi(t) =
+� (bP/a)(eait − 1)
+if
+t ≤ td
+xi(td)eai(t−td)
+if
+t ≥ td
+3.3
+Results and discussion
+The figures presented in this section show the numerical resolution of the system of equations
+of a dumped track section subjected to a rectangular and sinusoidal loads. The properties of
+the track are defined in table 1. In figure 6 and figure 7, the sinusoidal pulse is presented in red;
+however, the rectangular pulse is presented in black.
+1. Displacements and rotations of the rail
+2. Displacements of the sleepers
+It’s clear that the maximum values of rail and sleepers movement under rectangular pulse
+are higher than those reached under a sinusoidal pulse. The figure 8 shows the maximum loads
+induced in the substructure. The table 2 shows the repartition of the loads under the sleepers.
+These results have many consequences in the railway field. Actually, we may optimize railway
+infrastructure components for example (like ballast height). Moreover, the study is made by
+considering a static load (10 T). This load is mainly amplified by rail/wheel interaction and
+train speed [1].
+Sleeper
+number
+% of load
+(undumped -
+sinusoidal load)
+% of load
+(dumped -
+sinusoidal load)
+% of load
+(undumped -
+rectangular
+load)
+% of load
+(dumped -
+rectangular
+load)
+13
+4.49%
+1.73%
+5.50%
+-
+14
+11.52%
+6.12%
+13.66%
+6.63%
+15
+23.24%
+15.29%
+26.00%
+16.41%
+16
+30.78%
+22.43%
+34.35%
+23.55%
+17
+23.24%
+15.29%
+26.00%
+16.41%
+18
+11.52%
+6.12%
+13.66%
+6.63%
+19
+4.49%
+1.73%
+5.50%
+-
+Table 2:
+Repartition of the loads under the sleepers (N = 30, td = 0.01s and P = 10 T)
+4
+Conclusion
+Based on the results of the model analysis studied in order to determine the loads induced on
+the substructure, the following conclusions can be drawn:
+9
+
+Figure 6:
+Rail response under sinusoidal and rectangular pulse (N = 4, td = 0.01s and P = 10
+T)
+10
+
+10-
+X10-
+X104
+X10-
+w)
+0
+0.02
+0.04
+0.06
+0.02
+0.04
+0.06
+0
+0.02
+0.04
+0.06
+0
+0.02
+0.04
+0.06
+t (s)
+t (s)
+t (s)
+t (s)
+a) Displacements and rotations of the 1st node
+f) Displacements and rotations of the 6th node
+×10~4
+X104
+(pe))
+X104
+PO'O
+0.06
+0.02
+0.04
+0.06
+t (s)
+t (s)
+b) Displacements and rotations of the 2nd node
+0.02
+0.04
+0.06
+0.02
+0.04
+0.06
+t (s)
+t (s)
+104
+g) Displacements and rotations of the 7th node
+(e)
+X104
+0.02
+0.06
+0
+0.02
+0.04
+0.06
+t (s)
+t (s)
+c) Displacements and rotations of the 3rd node
+0.02
+0.04
+90'0
+0
+0.02
+0.06
+t (s)
+t (s)
+×10+
+X104
+h) Displacements and rotations of the 8th node
+10-4
+10
+5-10
+0.02
+0.04
+0.06
+0.02
+0.04
+0.06
+t (s)
+t (s)
+d) Displacements and rotations of the 4th node
+0.020.04
+0.06
+t (s)
+0.02
+0.04
+0.06
+t (s)
+i) Displacements and rotations of the 9th node
+×10
+X10-19
+-
+0.02
+PO'O
+0.06
+2
+0.02
+0.04
+0.06
+t (s)
+t (s)
+e) Displacements and rotations of the 5th nodeFigure 7:
+Sleeper response under sinusoidal and rectangular pulse (N = 4, td = 0.01s and P =
+10 T)
+Figure 8:
+Loads induced in the substructure (N = 30, td = 0.01s and P = 10 T)
+11
+
+10-4
+×10
+[w)
+-2
+-2
+0
+0.02
+0.04
+0.06
+0
+0.02
+0.04
+0.06
+t (s)
+t (s)
+a) Sleeper 1
+b) Sleeper 2
+×104
+cement (m)
+X104
+-2
+-5
+10
+0
+0.02
+0.04
+0.06
+0
+0.02
+0.04
+0.06
+(s) 1
+t (s)
+c) Sleeper 3
+d) Sleeper 4
+2
+0
+0.02
+0.04
+0.06
+t (s)
+e) Sleeper 50.5
+X104
+-0.5
+-1
+(N) 
+Force
+-1.5
+-2
+-2.5
+Undumped strcture-Sinusoidal pulse
+Dumped strcture -Sinusoidalpulse
+-3
+Undumped strcture - Rectangular pulse
+Dumpedstrcture-Rectangularpulse
+-3.5
+1
+2
+4
+5
+6
+7
+8
+6
+101112
+13141516171819202122232425262728293031
+Sleepernumber• The common modelling of the load applied on the track due to a moving wheel as a
+rectangular pulse acting in the time sample of a force signal generates a higher rate of
+movement in the track and over sizes the loads induced in the substructure than a sinusoidal
+pulse model;
+• Dumping matrix has a major influence on reducing the loads induced in the substructure.
+Therefore, it’s necessary to preserve the quality of the track components while maintaining
+it.
+As an application, we may evaluate the track behavior according to different characteristics
+of the track elements that degrade because of maintenance operations. Indeed, the ballast is
+considered as the most affected element because of operations of damping required for track
+geometry corrections.
+References
+[1] M. Touati, N. Lamdouar and A. Bouyahyaoui: Railway vehicle response under random
+irregularities on a tangent track – nonlinear 3d multi-body modelling. Inter. J. of Mech.
+Eng. and Tech. 9 (2018) 944–956.
+[2] Y. Xiwen, G. Shaojie, Z. Shunhua, S. Yao, M. Xiaoyun: Vertical Vibration Analysis of
+Vehicle-Track-Subgrade Coupled System in High Speed Railway with Dynamic Flexibility
+Method. Transportation Research Procedia 25 (2017) 291–300.
+[3] A. Al Shaer, D. Duhamel, K. Sab, G. Foret, L. Schmitt: Experimental settlement and
+dynamic behavior of a portion of ballasted railway track under high speed trains. J. Sound
+Vib. 316(1-5) (2008) 211–233.
+[4] A. Kaynia, C. Madshus, P. Zackrisson: Ground Vibration from High-speed Trains: Predic-
+tion and Countermeasure, J. of Geotech. and Geo-env. Eng.. 126 (2000) 531–537.
+[5] H. Takemiya: Simulation of Track–ground Vibrations due to High-speed Trains, Proceedings
+of the Eighth International Congress on Sound and Vibration. Hong Kong, China (2000)
+2875–2882.
+[6] C. Madshus, A. Kaynia: High-speed Railway Lines on Soft Ground: Dynamic Behaviour at
+Critical Train Speed, Journal of Sound and Vibration. 231(3) (2000) 689-701.
+[7] S. Qian, C. Ying: A Spatial Time-Varying Coupling Model for Dynamic Analysis of High
+Speed Railway Subgrade, J. of South. Jiao. Univ. (2001) 36(5) 509-513.
+[8] H. Chebli, D. Clouteau and L. Schmitt: Dynamic response of high-speed ballasted rail-
+way tracks: 3D periodic model and in situ measurements. Soil Dynamics and Earthquake
+Engineering, 28(2) (2008) 118-131.
+[9] P. Mario and L. William, Structural dynamics: Theory and Computation, Springer US,
+(2004).
+[10] EN 13481 : Railway applications. Track. Performance requirements for fastening systems.
+Definitions.
+[11] G. Kouroussis, O. Verlinden and C. Conti: Prediction of vibratory nuisances of rail transport
+vehicles. In 6th National Congress of Theoretical and Applied Mechanics, Ghent (Belgium),
+(2003) National Committee for Theoretical and Applied Mechanics.
+12
+
+[12] W. Zhai and X. Sun: A detailed model for investigating vertical interaction between railway
+vehicle and track. Vehicle System Dynamics 23(sup1) (1994) 603-615.
+13
+
diff --git a/JtAzT4oBgHgl3EQfj_0_/content/tmp_files/load_file.txt b/JtAzT4oBgHgl3EQfj_0_/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..6d3e88fc41211e2a5502de1cd2bdf6a3ae5fa88c
--- /dev/null
+++ b/JtAzT4oBgHgl3EQfj_0_/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf,len=401
+page_content='Dynamic Behavior of a Railway Track Under a Moving Wheel  Load Modelled as a Sinusoidal Pulse  Mohammed Touati1, Nouzha Lamdouar1 and Chakir Tajani2,∗  1Civil Department, Mohammadia School of Engineers -Mohammed V University - Morocco  mohammed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='touati87@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='com, nlamdouar@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='com  2Department of Mathematics, Polydisciplinary faculty of Larache,  Abdelmalek Essadi University, Morocco  chakir tajani@hotmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='fr  Abstract  The aim of this paper is to evaluate the train/track induced loads on the substructure  by modelling the wheel, at each instant, as a moving sinusoidal pulse applied in a very short  period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' This assumption has the advantage of being more realistic as it reduces the  impact of time on the load definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' To that end, mass, stiffness, and dumping matrices  of an elementary section of track will be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' As a result, the equations of motion  of a section of track subjected to a sinusoidal pulse and a rectangular pulse respectively is  concluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' Two numerical methods of resolution of that equation, depending on the nature  of the dumping matrix, will be presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' The computation results will be compared in  order to conclude about the relevance of that load model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' This approach is used in order to  assess the nature and the value of the loads received by the substructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  2020 MSC: Primary 74S05, 37M05, 74-10 Secondary 37N30  Key Words and Phrases: Dynamic properties, Finite elements modelling, Railway  track dynamics, Sinusoidal pulse load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  1  Introduction  Various theoretical and experimental researches have been performed in order to assess train/track  induced loads on the substructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' Mohammed Touati and al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' [1] determined the loads induced  by a non-linear 3D multi-body modelled train on the track with taking into account wheel/rail  contact properties and track irregularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' Yang Xinwen and al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' [2] concluded, through a vehicle-  track-subgrade coupling dynamic theory and finite element method, about the train/track in-  duced loads on each layer of the substructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' As an experimental study, Al Shaer and al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' [3]  presented the dynamic behavior of a portion of ballasted railway track subjected to cyclic loads  in substitution of a moving wheelset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' In conclusion, the dynamics behavior of the substructure  is widely studied in the literature [4-8] based on the train/track coupling model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  Actually, even if modelling a wheel load as a rectangular pulse is a common assumption, real  measurements don’t show the same shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' In fact, ONCF (Moroccan railway network manager)  has many tools that record wheel pulse like GOTCHA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' This system shows that the shape of  the load has never been rectangular, but it’s more likely compared to a sinusoidal pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' Then,  this paper deals with evaluating train/track induced loads on the substructure by proposing a  new approach when it comes to modelling the shape of the wheel impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' Indeed, it’s common  to consider a moving load as a rectangular impulse applied on the nodes of a mesh structure in  each period of time depending on signal sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' This paper shows that assuming the wheel  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='01524v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='NA]  4 Jan 2023  Figure 1:  Elementary track modelling  load as a sinusoidal pulse may reduce the impact of the period of time of its application and,  consequently, minimize the loads induced on the substructure oversized by the common assump-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' In that matter, a finite element model of the track will be presented and the numerical  results will be compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  2  Track elementary section modeling  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='1  Determination of mass, stiffness et dumping matrices  Let’s assume a portion of ballasted track composed of two elements of rail considered as a contin-  uous Euler-Bernoulli beam, fixed to two sleepers by a couples of springs/dampers representing  the railpads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' The ballast is modelled as a couples of springs/dampers under each sleeper (Figure  1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  The displacement vector is written as:  U = [u1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' θ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' θ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' u3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' θ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' uT1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' uT2]  The effective mass and the stiffness matrices of an element of rail [9],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
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+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='�������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='element1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='blement 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='ua  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='kn2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='c  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='"  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='ur2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='Masse (m-)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='Masse (m-)where ρr is the density of the rail,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' Ar is the surface of the rail section,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' Er is Young modulus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  and Ir is the rail moment of inertia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' The dumping matrix of the rail is obtained as a linear  combination of mass and stiffness matrices by assuming that the displacements u1 and u3 are  completely dumped by the effect of railpads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' the dumping matrix is written as:  C∗  r = a0 · M∗  r + a1 · K∗  r  where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  M∗  r = (ρrArL/420)  �  ���  4L2  13L  −3L2  0  13L  312  0  −13L  −3L2  0  8L2  −3L2  0  13L  −3L2  4L2  �  ���  K∗  r =  �  ErIr/L3�  �  ���  4L2  −6L  2L2  0  −6L  24  0  6L  2L2  0  8L2  2L2  0  6L  2L2  4L2  �  ���  a0 and a1 are concluded from the equation:  � a0  a1  �  =  �  2ω1ω2/  �  ω2  2 − ω2  1  �� �  ω2  −ω1  −1/ω2  1/ω1  � � ζ1  ζ2  �  where ωi2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' (i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' 2) are the eigenvalues associated to the vibration of the rail described by  the matrices Mr∗ and Kr∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' and ζi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' (i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' 2) are the dumping ratios according to the first and  second modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  In one hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' the equation of motion of the rail is written as:  Mr ¨U ∗ + Cr ˙U ∗ + KrU ∗ = F  (1)  where Cr is the transformation of the matrix C∗  r in the base U ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' and U ∗ is defined by:  U ∗ = [u1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' θ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' θ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' u3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' θ3]  F is given by:  F =  �  �������  −ks (u1 − uT1) − cs ( ˙u1 − ˙uT1)  0  0  0  −ks (u3 − uT3) − cs ( ˙u3 − ˙uT3)  0  �  �������  In the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' the equations of motion of the sleepers are written as:  � mT ¨uT1 = ks (u1 − uT1) + cs ( ˙u1 − ˙uT1) − kbuT1 − cb ˙uT1  mT ¨uT2 = ks (u3 − uT2) + cs ( ˙u3 − ˙uT2) − kbuT2 − cb ˙uT2  (2)  From (1) and (2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' we may conclude about the equation of motion of the track elementary  section as it’s modelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' It’s written as:  M ¨U + C ˙U + KU = 0  where M, C and K are the mass, dumping, and the stiffness of the track elementary section  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  3  Symbol  Quantity  Value  ρr  Rail density (kg/m3)  7850  Ar  Rail section surface (cm²)  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='70  Er  Young modulus of the rail (GPa)  210  Ir  Rail moment of inertia (cm4)  3038.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='6  mT  Sleeper mass (kg)  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='84  ks  Railpad stiffness (MN/m)  90  cs  Railpad damping (kN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='s/m)  30  kb  Ballast stiffness (MN/m)  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='5  cb  Ballast damping (kN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='s/m)  40  ζ  Rail dumping ratio  5%  Table 1: Track properties  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='2  Numerical application  Let’s assume a track elementary section characterized by the data given in table 1 (we can refer  to ([10], [11], [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  The figure 2 illustrates the evolution of natural frequencies according to vibration modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' It  shows that:  The frequencies of the 1st and 2nd modes correspond to a movement in phase between rail  and sleepers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' It’s equal to 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='62 Hz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  The frequency of the 3rd mode corresponds to a movement in opposition of phase between  rail and sleepers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' It’s equal to 381.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='1 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  3  Track response to a rectangular and a sinusoidal pulses  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='1  Description of the studied track  Let’s assume a section of track composed of N track elementary sections subjected to an external  load F as it’s shown in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  The number of degrees of freedom is given by:  Ndof = 8N − 3(N − 1)  The displacement vector is written as:  U =  �  ���  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  uj,k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  �  ���  where,  � uj,k = u∗  j,k  where  k ∈ [1, 8]  if  j = 1  uj,k = u∗  j,k  where  k ∈ [3, 4, 5, 6, 8]  if  j ̸= 1  and,  U ∗  j = [uj,1, θj,1, uj,2, θj,2, uj,3, θj,3, uj,T1, uj,T2]  4  Figure 2:  Natural frequencies of an elementary track section  Figure 3:  Track section modelling  5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='5  X104  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='5  (ZH)  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='5  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='5  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='5  0  0  0  0  1  2  3  4  5  6  7  8  6  10  NumerodumodeFigure 4:  Sinusoidal and rectangular pulses over a period of td  j refers to the element’s number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  The mass, stiffness and dumping matrices in the base U are obtained by assembling those  of a track elementary section determined earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  The vector of loads is defined by:  F =  �  ���  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  fj  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  �  ���  where,  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  N is even  �  �  �  �  �  �  �  �  �  �  �  N = 2  � fj = P  if  j = 5  fj = 0  else  N ̸= 2  � fj = P  if  j = (5N/2) + 1  fj = 0  else  N is uneven  � fj = P  if j = (5(N + 1)/2) − 1  fj = 0  else  P is a rectangular or a sinusoidal load given as:  Sinusoidal pulse:  � P = P0 sin ωt  if  t ≤ td  P = 0  else  Rectangular pulse:  � P = P0  if  t ≤ td  P = 0  else  Its shape is shown in the figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  6  sinusoidal pulse  rectangular pulse  PO  P  x=0  x=td  X = 2td  →3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='2  Description of the methods of resolution  The dynamic behavior of the section of track may be analyzed by modal superposition if the  dumping matrix verifies orthogonality properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  That method is used in particular for an  undumped system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' In that case, the equation of motion is reduced to:  M ¨U + KU = F  Let’s assume that ω2  i are the eigenvalues associated to the track vibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' We note {φi} the  normalized eigenvectors related to ω2  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' Therefore, the equation of motion is written as:  ¨Z + diag(ω2  i )Z = φT F  (3)  where diag(ω2  i ) is a diagonal matrix of the eigenvalues and:  U = Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='Z  The system of equations (3) is uncoupled where each equation is written as:  ¨zi + ω2  i zi = Φj,iP(t)  The resolution of that equation is given by DUHAMEL integral:  zi(t) = (1/ωi)  � t  0  Φj,iP(τ) sin ωi(t − τ)dτ  Therefore, the solution for a sinusoidal pulse load is given as:  zi(t) =  � (Φj,iP0/ω2  i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' (1/(1 − β2))(sin ωt − β sin ωit)  if  t ≤ td  ( ˙zi(td)/ωi) sin ωi(t − td) + zi(td) cos ωi(t − td)  if  t ≥ td  where,  β = ω/ωi  and the solution for a rectangular pulse load is given as:  zi(t) =  � (Φj,iP0/ω2  i )(1 − cos ωit)  if  t ≤ td  (Φj,iP0/ω2  i )(cos ωi(t − td) − cos ωit  if  t ≥ td  The figure 5 shows the response z(t) to a sinusoidal and a rectangular pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' It’s obvious that  in the forced phase, the maximum rectangular response is higher than the maximum sinusoidal  response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  In general, the dumping matrix doesn’t verify the orthogonality characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' Therefore,  the modal superposition method is substituted by the following method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  The equation of motion can be written as:  ¨Z + φT Cφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' ˙Z + diag(ω2  i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='Z = φT F  (4)  Where diag(ωi2) and φ are defined earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' Knowing that:  ˙Z − ˙Z = 0  (5)  (4) and (5) could be written as:  ˙Y = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='Y + F ∗  (6)  7  Figure 5:  z(t) response to a rectangular and sinusoidal pulse td/T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='75  where,  Y =  � Z  ˙Z  �  ,  D = A−1B,  F ∗ = A−1  � φT F  0  �  and,  A =  � φT Cφ  I  I  0  �  ,  B =  � diag(ω2  i )  0  0  −I  �  Let’s assume that {λi} are the eigenvalues associated to the matrix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' We note {ψi} the  normalized eigenvectors related to {ωi2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' We define X(t) as:  Z = ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='X  The equation (6) is written as:  ˙X = diag(λi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='X + ψ−1F ∗  (7)  The system of equations (7) is uncoupled where each equation is written as:  ˙xi(t) = ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='xi(t) + bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='P  (8)  where,  ai = λi  and  bi = χi  and,  χ = ψ−1  � φT  0  0  0  �  The resolution of the equation (8) gives:  8  sinusoidal pulse  rectangularpulse  2  2  3  4  X=0  PI = X  x = 2td  PI =X  X = 4td  t• Sinusoidal pulse:  xi(t) =  �  biPω  a2  i +ω2 eait − aibiP  a2  i +ω2 sin ωt −  biPω  a2  i +ω2 cos ωt  if  t ≤ td  xi(td)eai(t−td)  if  t ≥ td  Rectangular pulse:  xi(t) =  � (bP/a)(eait − 1)  if  t ≤ td  xi(td)eai(t−td)  if  t ≥ td  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='3  Results and discussion  The figures presented in this section show the numerical resolution of the system of equations  of a dumped track section subjected to a rectangular and sinusoidal loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' The properties of  the track are defined in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' In figure 6 and figure 7, the sinusoidal pulse is presented in red;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  however, the rectangular pulse is presented in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' Displacements and rotations of the rail  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' Displacements of the sleepers  It’s clear that the maximum values of rail and sleepers movement under rectangular pulse  are higher than those reached under a sinusoidal pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' The figure 8 shows the maximum loads  induced in the substructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' The table 2 shows the repartition of the loads under the sleepers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  These results have many consequences in the railway field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' Actually, we may optimize railway  infrastructure components for example (like ballast height).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' Moreover, the study is made by  considering a static load (10 T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' This load is mainly amplified by rail/wheel interaction and  train speed [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  Sleeper  number  % of load  (undumped -  sinusoidal load)  % of load  (dumped -  sinusoidal load)  % of load  (undumped -  rectangular  load)  % of load  (dumped -  rectangular  load)  13  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='49%  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='73%  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='50% 14  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='52%  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='12%  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='66%  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='63%  15  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='24%  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='29%  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='00%  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='41%  16  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='78%  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='43%  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='35%  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='55%  17  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='24%  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='29%  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='00%  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='41%  18  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='52%  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='12%  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='66%  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='63%  19  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='49%  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='73%  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='50% Table 2:  Repartition of the loads under the sleepers (N = 30, td = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='01s and P = 10 T)  4  Conclusion  Based on the results of the model analysis studied in order to determine the loads induced on  the substructure, the following conclusions can be drawn:  9  Figure 6:  Rail response under sinusoidal and rectangular pulse (N = 4, td = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='01s and P = 10  T)  10  10-  X10-  X104  X10-  w)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
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+page_content='06  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='06  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
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+page_content="06  t (s)  t (s)  t (s)  t (s)  a) Displacements and rotations of the 1st node  f) Displacements and rotations of the 6th node  ×10~4  X104  (pe))  X104  PO'O  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
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+page_content='06  t (s)  t (s)  b) Displacements and rotations of the 2nd node  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='06  t (s)  t (s)  104  g) Displacements and rotations of the 7th node  (e)  X104  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='06  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='06  t (s)  t (s)  c) Displacements and rotations of the 3rd node  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content="04  90'0  0  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='06  t (s)  t (s)  ×10+  X104  h) Displacements and rotations of the 8th node  10-4  10  5-10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
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+page_content='06  t (s)  t (s)  d) Displacements and rotations of the 4th node  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='06  t (s)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='06  t (s)  i) Displacements and rotations of the 9th node  ×10  X10-19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content="02  PO'O  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='06  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='06  t (s)  t (s)  e) Displacements and rotations of the 5th nodeFigure 7:  Sleeper response under sinusoidal and rectangular pulse (N = 4, td = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='01s and P =  10 T)  Figure 8:  Loads induced in the substructure (N = 30, td = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='01s and P = 10 T)  11  10-4  ×10  [w)  2  2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='06  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='06  t (s)  t (s)  a) Sleeper 1  b) Sleeper 2  ×104  cement (m)  X104  2  5  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='06  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='06  (s) 1  t (s)  c) Sleeper 3  d) Sleeper 4  2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='06  t (s)  e) Sleeper 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='5  X104  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='5  1  (N)  Force  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='5  Undumped strcture-Sinusoidal pulse  Dumped strcture -Sinusoidalpulse  3  Undumped strcture - Rectangular pulse  Dumpedstrcture-Rectangularpulse  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='5  1  2  4  5  6  7  8  6  101112  13141516171819202122232425262728293031  Sleepernumber• The common modelling of the load applied on the track due to a moving wheel as a  rectangular pulse acting in the time sample of a force signal generates a higher rate of  movement in the track and over sizes the loads induced in the substructure than a sinusoidal  pulse model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  Dumping matrix has a major influence on reducing the loads induced in the substructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  Therefore, it’s necessary to preserve the quality of the track components while maintaining  it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  As an application, we may evaluate the track behavior according to different characteristics  of the track elements that degrade because of maintenance operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' Indeed, the ballast is  considered as the most affected element because of operations of damping required for track  geometry corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
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+page_content=' William, Structural dynamics: Theory and Computation, Springer US,  (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  [10] EN 13481 : Railway applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
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+page_content=' Performance requirements for fastening systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  Definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
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+page_content=' In 6th National Congress of Theoretical and Applied Mechanics, Ghent (Belgium),  (2003) National Committee for Theoretical and Applied Mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  12  [12] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content=' Zhai and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
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+page_content=' Vehicle System Dynamics 23(sup1) (1994) 603-615.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
+page_content='  13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfj_0_/content/2301.01524v1.pdf'}
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+arXiv:2301.00120v1  [math.FA]  31 Dec 2022
+LARGE STRUCTURES WITHIN THE CLASS OF SUMMING OPERATORS
+N. G. ALBUQUERQUE* AND L. COLETA**
+Abstract. We investigate lineability/spaceability problems within the setting of multilinear
+summing operators on quasi-Banach sequence spaces. Furthermore, we deal with the contem-
+porary geometric notions of pointwise-lineability and (α, β)-lineability. Among other results,
+we prove that the class of multilinear operators taking values on ℓq (0 < q ≤ ∞) that are ab-
+solutely but not multiple summing is pointwise c-spaceable when non-empty. Spaceability in
+other classes, such as Dunford-Pettis operators and multilinear operators on general summable
+scalar families, is also studied.
+1. Introduction
+The modern terminology of lineability and spaceability was coined by Gurariy, Aron, and
+Seoane in the seminal paper [4]: for a cardinal α, a subset A of a vector space X is said to
+be α-lineable if A ∪ {0} contains a α-dimensional linear subspace of X. If X is, in addition, a
+topological vector space, then A is called α-spaceable if A ∪ {0} contains a closed α-dimensional
+linear subspace of X. These concepts are widely investigated in several research branches. For
+an overview, we refer the reader to recent works in [8, 9], and, to our knowledge, the most
+comprehensive panorama of the matter are given in [5], and [9]. The search for new lineability
+notions yields to several new concepts, and some recent ones can be found in [18] and [22].
+Many authors contributed to investigating lineability aspects of certain classes of summing
+operators. Puglisi and Seoane proved in [26] that, under certain restrictions over a Banach space
+E, the set L(E, ℓ2)\Π1(E, ℓ2) of bounded linear non-absolutely summing operators is lineable.
+Later in [12], Botelho, Diniz and Pellegrino obtained the lineability of K(E; F) \ Πp(E; F), the
+set of compact but not p-summing operators, with p ≥ 1, with some conditions of reflexivity,
+infinite dimension, and unconditional basis on the Banach spaces E, F. More generally, Kitson
+and Timoney in [21] obtained a remarkable and general spaceability result for Fr´echet spaces
+with several applications, among them, the spaceability of K(E, F)\ �
+1≤p<∞ Πp(E, F), whenever
+E is superreflexive and F is infinite dimensional. Hernand´ez, Ruiz, and S´anchez took a step
+further and dealt with the spaceability of operators ideals: we refer to [19] for the spaceability
+of I1(E; F) \ I2(E; F), with I1, I2 operator ideals over certain Banach spaces E, F.
+Alves
+and Turco dealt with the spaceability of sets of p-compact linear maps where the domain and
+codomain are ℓr, with 1 ≤ r < ∞, and c0 was investigated in depth and many results are
+obtained in [2]. In the context of quasi-Banach spaces, in [27] Tom´az proved that L(ℓp, ℓp) \
+�
+1≤s≤r<∞ Π(r,s)(ℓp, ℓp) is maximal lineable for any 0 < p < 1. In the multilinear environment,
+G. Ara´ujo and D. Pellegrino established in [3] that set L(mℓp, K) \ Πms
+(r;s)(mℓp, K) of bounded m-
+linear but not multiple summing forms is maximal spaceable, for m ≥ 2, p ∈ [2, ∞), 1 ≤ s < p∗
+and r <
+2ms
+s+2m−ms. Here, as usual, Πas and Πms stand for the class of absolutely and multiple
+multilinear summing classes, respectively. Also in [17] the authors investigated the lineability of
+non-bounded and non-absolutely summing multilinear operators.
+It is well known from the classical Bohnenblust-Hille inequality and also the Defant-Voigt
+theorem that Πms
+(r;1) (mE; K) ⊊ L (mE; K) = Πas
+(r;1) (mE; K) for all 1 ≤ r <
+2m
+m+1.
+Therefore
+Πas
+(r;1) (mE; K)\Πms
+(r;1) (mE; K) is non-empty and the search for large closed vector spaces naturally
+2020 Mathematics Subject Classification. 15A03, 46G25, 46B87, 47H60.
+Keywords. Multilinear operators, Summing operators, Lineability, Spaceability.
+* Supported by CNPq Grant 312167/2021-01 and Grant 2019/0014 Para´ıba State Research Foundation
+(FAPESQ).
+** Supported by CAPES.
+1
+
+2
+ALBUQUERQUE AND COLETA
+arises in this setting. Similar problems to this one are answered as an application of the result
+we provide. A particular case we prove is that the class of multilinear operators taking values
+on sequence spaces that are absolutely but not multiple summing, for instance, for E1, . . . , Em
+Banach spaces, 1 ≤ r ≤ s < ∞ and q ∈ (0, ∞],
+Πas
+(r,s)(E1, . . . , Em; ℓq) \ Πms
+(r,s)(E1, . . . , Em; ℓq),
+is c-spaceable when it is not empty. In fact, we prove a strong version dealing with the restrictive
+pointwise-lineability/spaceability notion and in the general setting of Λ-multiple summing class
+(see [1, 6, 11, 25] for more details).
+The main goals of this paper are to contribute to the research of lineability and spaceability
+on the class of multilinear operators in two perspectives not often covered: we investigate the
+geometric concepts of pointwise and (α, β)-lineability / spaceability (we discuss it with further
+details in Sections 2 and 5), which are more restrictive than the classical notion; also we deal
+with operators taking value on quasi-Banach sequence spaces. The panorama is quite different
+from Banach spaces when dealing with quasi-Banach spaces, such as non-locally spaces ℓq for
+0 < q < 1. Thus the search for lineability/spaceability techniques in quasi-Banach spaces can
+be an exciting and challenging matter.
+This paper is organized as follows. Section 2 presents preliminary concepts and notations
+used throughout the paper. In Section 3 we obtain the main result of the paper: the point-
+wise c-spaceability of classes of Λ-summing operators (Theorem 3.1) and some applications are
+presented. Section 4 deals with absolutely summing operators that fail to be Dunford-Petis (or
+completely continuous). In Section 5 we prove that some sets of operators taking values on
+spaces of general q-summable scalar families ℓq (I) can be (α, card I)-lineable, for I some infinite
+set and some cardinal α < card I.
+2. Background and preliminaries
+Throughout this paper we deal with vector spaces over the scalar field K, which can be either
+R or C, and we write card R = c and card N = ℵ0. Banach spaces over K shall be denoted by
+capital letter (eventually with indexes) such as E, E1, . . . , Em, F, G, H, unless stated otherwise.
+The topological dual and the closed unit ball of E will be denoted by E′ and BE, respectively.
+We will denote by L (E1, . . . , Em; F) the Banach space of bounded m-linear operators from
+E1 × · · · × Em to F endowed with the usual sup norm.
+A quasi-norm on a vector space X is a non-negative real-valued function ∥ · ∥ : X → [0, ∞)
+satisfying for all x, y ∈ X and λ ∈ K the following properties: ∥x∥ = 0 only if x = 0; ∥λx∥ =
+|λ|∥x∥; and ∥x + y∥ ≤ C(∥x∥ + ∥y∥) with C ≥ 1 an universal constant. If, in addition, the
+map ∥ · ∥ is p-subadditive for some 0 < p ≤ 1, i.e., it satisfies ∥x + y∥p ≤ ∥x∥p + ∥y∥p for all
+x, y ∈ X, ∥ · ∥ is called a p-norm. Clearly we have a norm when C = 1 or p = 1. A quasi-norm
+defines a metrizable vector topology on X whose base of neighborhoods of the origin is given
+by sets of the form {x ∈ X : ∥x∥ < 1/n}, n ∈ N. X is called a quasi-Banach space when it is
+complete for this metric. A p-subadditive quasi-norm induces a metric topology on X given by
+d(x, y) := ∥x − y∥p. A quasi-Banach space with an associated p-norm is also called a p-Banach
+space. A deep result known as the Aoki-Rolewicz theorem (see [20, 24]) guarantees that every
+quasi-normed space is p-normable for some 0 < p ≤ 1, i.e., the space can be endowed with
+an equivalent quasi-norm which is a p-norm. Thus we assume that a quasi-Banach space is
+p-Banach for some 0 < p ≤ 1. Many definitions and classical results in Banach spaces can be
+translated in a natural way to quasi-Banach (p-Banach) spaces, like standard results depending
+on Baire Category Theorem as Open Mapping Theorem. Nevertheless, p-normed spaces are not
+necessarily locally convex. Results which hold on Banach spaces and are based somehow on
+the local convexity (e.g., Hahn-Banach extension property or the Krein-Milman theorem) are
+no longer valid, in general, in that setting. The extension of lineability/spaceability arguments
+from Banach to quasi-Banach spaces is not straightforward. Thus the search for large closed
+infinite dimensional subspaces of p-Banach spaces is quite a delicate issue. Therefore, looking
+for lineability/spaceability techniques that also cover this environment spaces seems interesting.
+For more details on quasi-Banach and p-Banach spaces, we refer to [20].
+
+LARGE STRUCTURES WITHIN THE CLASS OF SUMMING OPERATORS
+3
+Given a positive integer m, i := (i1, . . . , im) stands for a multi-index in Nm, and r :=
+(r1, . . . , rm) ∈ [1, +∞]m a multi-parameter. By ℓr(E) we denote the Banach space that gathers
+all E-valued multi-matrices (xi)i∈Nm ∈ ENm with finite ℓr-norm, that is, when r ∈ [1, +∞)m we
+have
+��(xi)i∈Nm
+��
+ℓr(E) :=
+
+
+
+∞
+�
+i1=1
+
+· · ·
+� ∞
+�
+im=1
+∥xi∥rm
+E
+� rm−1
+rm
+· · ·
+
+
+r1
+r2
+
+
+
+1
+r1
+< ∞.
+We simply write ℓr = ℓr(K) and when no precision is needed ∥·∥ℓr(E) = ∥·∥r. Notice that ℓr(E)
+is a quasi-Banach space when some 0 < rj < 1. Given a non-empty set of indexes Λ ⊂ Nm, by
+(xi)i∈Λ we mean an E-valued multi-matrix indexed over Λ, and its ℓr-norm can be seen as
+��(xi)i∈Λ
+��
+ℓr(E) =
+��(xi · 1Λ(i))i∈Nm
+��
+ℓr(E)
+where 1Λ is the characteristic function of Λ. It is worth pointing out some simple but useful
+monotonicity properties: ∥ · ∥s ≤ ∥ · ∥r whenever 0 < r < s < ∞;
+��(αi)i∈Λ
+��
+r ≤
+��(αi)i∈Γ
+��
+r for
+all scalar multi-matrix (αi)i∈Nm and for (non-empty) indexes sets Λ ⊂ Γ ⊂ Nm; ∥(αi)i∈Λ∥r ≤
+∥(βi)i∈Λ∥r if 0 ≤ αi ≤ βi for all i ∈ Λ.
+A quasi-Banach standard sequence space over a Banach space X is an infinite dimensional
+quasi-Banach space E whose elements are X-valued sequences with the usual operations enjoying
+the following conditions:
+(i) There is a constant C > 0 such that
+∥xj∥X ≤ C∥x∥E,
+for every x = (xj)j∈N ∈ E and all j ∈ N.
+(ii) If x = (xj)j∈N ∈ E and (xnk)k∈N is a subsequence of x, then (xnk)k∈N ∈ E and
+∥(xnk)k∈N∥E ≤ ∥x∥E.
+(iii) If x = (xj)j∈N ∈ E and N′ := {n1 < n2 < n3 < · · · } is an infinite subset of N, then the
+X−valued sequence y = (yj)j∈N defined as
+yj :=
+�
+xi,
+if j = ni,
+0,
+otherwise ,
+belongs to E and ∥y∥E ≤ ∥x∥E.
+It is plain that combining (ii) and (iii) yields ∥y∥E = ∥x∥E. The n-th coordinate of a sequence
+z ∈ E ⊂ XN will be denoted by zn or, when more precision is needed, by z(n). Unless the contrary
+is explicitly established, E will denote a quasi-Banach standard sequence space. Usual sequence
+spaces are (quasi)-Banach standard sequence spaces as, for instance, the classical sequence spaces
+ℓp(E), ℓw
+p (E), ℓu
+p(E) with p ∈ (0, ∞), c0(E), c(E), and the Lorentz space ℓp,q.
+For a deeper
+discussion on general sequence spaces, more details and examples, we refer the reader to [10, 16].
+The proof of the next lemma is straightforward.
+Lemma 2.1. Let r := (r1, . . . , rm) ∈ [1, ∞]m and Λ ⊂ Nm. If (αi)i∈Λ /∈ ℓr(E), then either
+�
+(αi(j))j∈N′
+�
+i∈Λ /∈ ℓr(E)
+or
+�
+(αi(j))j∈N\N′
+�
+i∈Λ /∈ ℓr(E),
+for any N′ ⊂ N with card N′ = card(N \ N′) = ℵ0.
+Recently, Pellegrino and Raposo Jr. in [22] introduced the geometric notion of pointwise-
+lineability. Let V be a vector space and α be a cardinal number with α ≤ dim V . A set A ⊂ V
+is pointwise α-lineable if for each x ∈ A there is a subspace W = Wx,α such that x ∈ W ⊂ A∪{0}
+and dim W = α. If X is a topological vector space and W can be chosen to be closed, A is
+said pointwise α-spaceable. It is common to refer to A as lineable or spaceable when α ≥ ℵ0.
+Pointwise lineability is a more restrictive concept than classical lineability. The latter faces a
+technicality not often covered in usual lineability results: the subspace generated contains every
+initial mother vector. Also, this notion is related with (α, β)-lineability (see Section 5). For
+more details, we refer to [22].
+
+4
+ALBUQUERQUE AND COLETA
+We finish this section with the following result, which will reveal itself useful and it is also
+closely connected to a lineability technique that dates back to [23]. First we recall a basic ideal
+property of a quasi-Banach multi-ideal operators (M, ∥ · ∥M): if S ∈ M(E1, . . . , Em; F), uj ∈
+L(Gj; Ej), j = 1, . . . , m, and t ∈ L(F; H), then t ◦ S ◦ (u1, . . . , uj) ∈ M(G1, . . . , Gm; H) and
+∥t ◦ S ◦ (u1, . . . , uj)∥M ≤ ∥t∥ · ∥S∥M · ∥u1∥ · · · ∥um∥ (see [13] and references therein for more
+details). Also, we say that (Nk)k∈N is a countably disjoint decomposition of the natural numbers
+when we can write N = �
+k∈N Nk with each Nk :=
+�
+n(k)
+1
+< n(k)
+2
+< · · ·
+�
+an infinite subset of N
+and Ni ∩ Nj = ∅ whenever i ̸= j.
+Proposition 2.2. Let M be quasi-Banach operators multi-ideals. Given a countably disjoint
+decomposition (Nk)k∈N of N and a non-trivial operator T ∈ M (E1, . . . , Em; E), we have the
+following.
+(i) For each natural k ≥ 1, the map Tk : E1 × · · · × Em → E defined by
+Tkx (j) :=
+�
+0,
+if j /∈ Nk,
+Tx(i),
+if j = n(k)
+i
+∈ Nk,
+lies in M (E1, . . . , Em; E), and ∥Tkx∥E = ∥Tx∥E for all x ∈ E1 × · · · × Em. Moreover,
+{Tk : k ≥ 2} ∪ {T} is linearly independent.
+(ii) For some 0 < p ≤ 1, such that the map Ψ : ℓp → M (E1, . . . , Em; E) given by
+Ψa := α1T +
+∞
+�
+j=2
+αjTj,
+a = (αj)j∈N ∈ ℓp,
+is well-defined, linear, bounded, and injective.
+Proof. (i) Since T is a non-zero operator, there exist x0 ∈ E1 × · · · × Em and j0 ∈ N such that
+Tx0(j0) ̸= 0, where Tx0(j0) is the j0-th coordinate of the sequence Tx0 ∈ E. For simplicity,
+we shall assume that j0 ∈ N1.
+Using the notation in condition (iii) of standard sequence
+spaces, for each k ∈ N we take N′ = Nk =
+�
+n(k)
+1
+< n(k)
+2
+< · · ·
+�
+, and we define Vk : E → E
+by Vkx := (yj)j∈N for x = (xn)n∈N ∈ E.
+It is clear that Vk is a bounded linear map with
+∥Vk∥ ≤ 1, Tk = Vk ◦ T ∈ M (E1, . . . , Em; E), and ∥Tk∥M ≤ ∥T∥M. Using conditions (ii) and
+(iii) of standard sequence spaces, and the fact that {Tkx}k∈N is a family of sequences in E
+whose supports are pairwise disjoint for any x, it is straightforward to conclude the remaining
+statements.
+(ii) A := M (E1, . . . , Em; E) ⊂ L (E1, . . . , Em; E) is a p-Banach for some p ∈ (0, 1]. For any
+a = (αj)j∈N ∈ ℓp,
+�
+j≥2
+∥αjTj∥p
+A =
+�
+j≥2
+|αj|p∥Tj∥p
+A =
+�
+j≥2
+|αj|p∥T∥p
+A = ∥T∥p
+A
+�
+j≥2
+|αj|p < ∞.
+Hence the series �
+j≥2 αjTj converges in A and, consequently, the map Ψ : ℓp → A is well defined
+and linear. In order to simplify the notation we write Ψa := Ψa for a ∈ ℓp. Now suppose that
+Ψa = 0. Combining Tx0(j0) ̸= 0 with the fact that the sequence Tjx has null coordinates on
+N \ Nj, for any x ∈ E1 × · · · × Em and j ≥ 2,
+0 = Ψax0(j0) = α1Tx0(j0) +
+∞
+�
+j=2
+αjTjx0(j0) = α1Tx0(j0),
+implies that α1 = 0. Since (Tj)j is linearly independent, we have αj = 0 for all j. Therefore
+a = 0, and this concludes that Ψ is injective.
+□
+
+LARGE STRUCTURES WITHIN THE CLASS OF SUMMING OPERATORS
+5
+3. Spaceability of multilinear summing operators
+In this Section the search of large topological structures is focused in a general environment
+of multilinear summing operators. The concept of Λ-multiple summing (or Λ-summing) is a
+notion of multilinear summing operators that encompasses the classical notions of absolutely
+and multiple multilinear summing operators. It was independently introduced in [6, 25] and
+recent developments in this environment can be found, e.g., in [1, 11]. More precisely, let m ∈ N
+be a positive integer, r, s ∈ [1, +∞)m and Λ ⊂ Nm a set of indexes. An m-linear operator
+T : E1 × · · · × Em → F is Λ-(r; s)-summing if there is a constant C > 0 such that
+(3.1)
+��(Txi)i∈Λ
+��
+ℓr(F ) ≤ C
+m
+�
+k=1
+sup
+φk∈BE′
+k
+� N
+�
+i=1
+���φ(x(k)
+i
+)
+���
+sk
+� 1
+sk
+for all N ∈ N and x(k)
+i
+∈ Ek, k = 1, . . . , m, i = 1, . . . , N. For simplicity of notation we write
+Txi := T
+�
+x(1)
+i1 , . . . , x(m)
+im
+�
+.
+The class of all operators that fulfills the previous inequality is
+denoted by ΠΛ
+(r;s) (E1, . . . , Em; F), which is a Banach space endowed with the norm πΛ
+(r;s)(T)
+taken as the infimum of the constants C > 0 satisfying (3.1).
+Notice that, by taking Λ =
+Diag (Nm) := {(n, · · · , n) ∈ Nm : n ∈ N} and Λ = Nm, the Λ-summing class ΠΛ recovers both
+absolutely and multiple multilinear summing classes, respectively It is noteworthy to mention
+that given Λ ⊂ Γ ⊂ Nm, the following inclusions and norm inequalities are easily seen (over
+fixed parameters (r; s) we omit): Πms ⊂ ΠΓ ⊂ ΠΛ ⊂ Πas and πms(·) ≤ πΓ(·) ≤ πΛ(·) ≤ πas(·).
+Moreover, one can deal with the codomain F as a quasi-Banach space and, with standard
+reasoning, ΠΛ
+(r;s) (E1, . . . , Em; F) also is quasi-Banach. For more details and applications in the
+theory of absolutely multiple summing operators we refer the reader to [11] and the references
+therein.
+The main result of this section reads as follows.
+Theorem 3.1. Let r, s, t, u ∈ [1, +∞)m and Λ ⊂ Γ ⊂ Nm sets of indexes. Then
+ΠΛ
+(r,s) (E1, . . . , Em; E) \ ΠΓ
+(t,u) (E1, . . . , Em; E)
+is either empty or pointwise c-spaceable.
+Proof. For simplicity, we write A := �Λ
+(r,s)(E1, . . . , Em; E) and B := �Γ
+(t,u)(E1, . . . , Em; E). Let
+us suppose there exists an operator T ∈ A \ B. We prove first that A \ B is pointwise c-lineable.
+Let us fix x0 ∈ E1 × · · · × Em and j0 ∈ N such that Tx0(j0) ̸= 0. We also consider weakly
+summable sequences z(j) =
+�
+z(j)
+n
+�
+n∈N ∈ ℓw
+uj(Ej), j = 1, . . . , m, such that
+��(Tzi)i∈Γ
+��
+ℓt(E) = ∞.
+Recall the notation Tzi := T
+�
+z(1)
+i1 , . . . , z(m)
+im
+�
+. Lemma 2.1 assures that we can take N1 ⊂ N with
+j0 ∈ N1, card N1 = card(N \ N1) = ℵ0 and also such that
+(3.2)
+���
+�
+(Tzi(n))n∈N1
+�
+i∈Γ
+���
+ℓt(E) = ∞.
+Now we take the following countably infinite decomposition: N\N1 = �
+k≥2 Nk. Then Propo-
+sition 2.2 (i), applied with M = ΠΛ
+(r,s), provides a sequence of (linearly independent) operators
+(Tk)k∈N with ∥Tkx∥E = ∥Tx∥E for all x ∈ E1 × · · · × Em. Clearly this implies ∥Tk∥A = ∥T∥A for
+all k and also (Tk)k∈N belongs to A \ B.
+In order to obtain the c-dimensional space in A \ B that contains T, we use the construction
+and notation of Proposition 2.2 (ii): for some 0 < p ≤ 1, the map Ψ : ℓp → A is well-
+defined, bounded, linear and injective. According to this, Ψ(ℓp) ⊂ A is a c-dimensional space
+that contains T.
+We shall now prove that Ψ(ℓp) \ {0} ⊂ A \ B, that is: Ψa /∈ B, for any
+
+6
+ALBUQUERQUE AND COLETA
+a = (αj)j ∈ ℓp \ {0}. Consider z(j) ∈ ℓw
+uj(Ej), j = 1, . . . , m, as in (3.2). If α1 ̸= 0, using again
+that every image of Tj is a sequence with null coordinates on N \ Nj for j ≥ 2,
+∥Ψazi∥E =
+������
+
+α1Tzi(n) +
+�
+j≥2
+αjTjzi(n)
+
+
+n∈N
+������
+E
+≥
+������
+
+α1Tzi(n) +
+�
+j≥2
+αjTjzi(n)
+
+
+n∈N1
+������
+E
+=
+��(α1Tzi(n))n∈N1
+��
+E
+= |α1|
+��(Tzi(n))n∈N1
+��
+E .
+The monotonicity of the quasi-norm yields
+��(Ψazi)i∈Γ
+��
+ℓt(E) ≥ |α1|
+���
+�
+(Tzi(n))n∈N1
+�
+i∈Γ
+���
+ℓt(E) = ∞,
+with the last inequality being a consequence of (3.2), which gives Ψa /∈ B. We now turn to the
+case α1 = 0, in which we proceed in a similar fashion. Let us consider αi ̸= 0 for some i ∈ N.
+Using the definition of the operators (Tk)k∈N in Proposition 2.2 (i),
+∥Ψazi∥E =
+������
+
+�
+j≥2
+αjTjzi(n)
+
+
+n∈N
+������
+E
+≥
+���
+�
+αiTizi
+�
+n(i)
+k
+��
+k∈N
+���
+E = |αi|
+��(Tzi (k))k∈N
+��
+E ,
+thus
+��(Ψazi)i∈Γ
+��
+ℓt(E) ≥ |αi|
+��(Tzi)i∈Γ
+��
+ℓt(E) = ∞.
+This proves the pointwise c-lineability of A \ B.
+Now we turn to the pointwise c-spaceability of A \ B. The obvious candidate for the closed
+c-dimensional subspace of A is Ψ(ℓp).
+We will prove that Ψ(ℓp) \ {0} ⊂ A \ B.
+Let us fix
+S = limk Ψk ∈ Ψ(ℓp) \ {0}, with Ψk := Ψa(k) and a(k) =
+�
+α(k)
+j
+�
+j ∈ ℓp for all k ∈ N.
+First let us show that there exists the limit limk α(k)
+1 . Notice that, for all b = (βn)n ∈ ℓp, x ∈
+E1 × · · · × Em and j ∈ N1, Ψbx(j) = β1Tx(j), taking x0 ∈ E1 × · · · × Em and j0 ∈ N1 with
+Tx0(j0) ̸= 0, we get
+Sx0(j0) = lim
+k→∞ Ψkx0(j0) =
+�
+lim
+k→∞ α(k)
+1
+�
+Tx0(j0),
+and this guarantees the existence of the aforementioned limit.
+Now let z(j) ∈ ℓw
+uj(Ej), j = 1, . . . , m, as in (3.2). Suppose that limk α(k)
+1
+̸= 0. Notice that, for
+all x ∈ E1 × · · · × Em and all a ∈ ℓp,
+∥Ψax∥E =
+������
+
+α1Tx(i) +
+�
+j≥2
+αjTjx(i)
+
+
+i∈N
+������
+E
+≥
+������
+
+α1Tx(i) +
+�
+j≥2
+αjTjx(i)
+
+
+i∈N1
+������
+E
+= |α1|
+��(Tx(i))i∈N1
+��
+E .
+Then
+∥Szi∥E ≥ lim
+k |α(k)
+1 |
+��(Tzi(i))i∈N1
+��
+E
+yields
+��(Szi)i∈Γ
+��
+ℓt(E) ≥ lim
+k |α(k)
+1 | ·
+��(Tzi)i∈Γ
+��
+ℓt(E) = ∞.
+
+LARGE STRUCTURES WITHIN THE CLASS OF SUMMING OPERATORS
+7
+Let us turn to the case limk α(k)
+1
+= 0. Since S ̸= 0, there are x ∈ E1 × · · · × Em and i0 ∈ N
+such that Sx(i0) ̸= 0. There are unique l, t ∈ N such that i0 = n(l)
+t , thus
+Sx(i0) = lim
+k→∞ Ψkx(i0)
+= lim
+k→∞
+
+α(k)
+1 Tx(i0) +
+�
+j≥2
+α(k)
+j Tjx(i0)
+
+
+= lim
+k→∞
+�
+α(k)
+1 Tx(n(l)
+t ) + α(k)
+l
+Tlx(n(l)
+t )
+�
+= lim
+k→∞ α(k)
+l
+Tx(t),
+and, hence limk→∞ α(k)
+l
+̸= 0. Moreover,
+∥Sx∥E =
+������
+lim
+k→∞
+
+α(k)
+1 Tx +
+�
+j≥2
+α(k)
+j Tjx
+
+
+������
+E
+= lim
+k→∞
+������
+�
+j≥2
+α(k)
+j Tjx
+������
+E
+≥ lim
+k→∞
+������
+
+�
+j≥2
+α(k)
+j Tjx(i)
+
+
+i∈Nl
+������
+E
+= lim
+k→∞
+����
+�
+α(k)
+l
+Tlx(i)
+�
+i∈Nl
+����
+E
+= lim
+k→∞
+���α(k)
+l
+���
+���(Tlx(i))i∈Nl
+���
+E
+= lim
+k→∞
+���α(k)
+l
+���
+��(Tx(i))i∈N
+��
+E
+= lim
+k→∞
+���α(k)
+l
+��� ∥Tx∥E
+for all x ∈ E1 × · · · × Em. In particular, for such zi as in (3.2),
+��(Szi)i∈Γ
+��
+ℓt(E) ≥
+�
+lim
+k→∞
+���α(k)
+l
+���
+�
+·
+��(Tzi)i∈Γ
+��
+ℓt(E) = ∞,
+this leads to S /∈ B and, therefore, the proof is complete.
+□
+We obtain the following result proceeding in the same manner as in the proof of Theorem
+3.1. Furthermore, taking Λ = Diag (Nm) := {(n, · · · , n) ∈ Nm : n ∈ N} and Γ = Nm, we return
+to the classical setting of absolutely and multiple summing multilinear operators.
+Theorem 3.2. Let Λ ⊂ Nm be a set of indexes and r, s ∈ [1, +∞)m. Then
+L (E1, . . . , Em; E) \ ΠΛ
+(r,s) (E1, . . . , Em; E)
+is either empty or pointwise c-spaceable.
+Corollary 3.3. Let q ∈ (0, ∞] and r, s, t, u ∈ [1, +∞)m. Then each one of the sets
+L (E1, . . . , Em; ℓq) \ Πms
+(t,u) (E1, . . . , Em; ℓq) ,
+L (E1, . . . , Em; ℓq) \ Πas
+(t,u) (E1, . . . , Em; ℓq)
+and
+Πas
+(r,s) (E1, . . . , Em; ℓq) \ Πms
+(t,u) (E1, . . . , Em; ℓq)
+is either empty or pointwise c-spaceable.
+
+8
+ALBUQUERQUE AND COLETA
+3.1. Non-absolutely summing operators on classical sequence spaces. We now turn our
+attention to linear bounded operators on the classical quasi-Banach spaces ℓq for 0 < q < 1 that
+fail to be absolutely summing. More precisely, for 0 < q < 1, we investigate large topological
+linear structures within L (ℓq, ℓq)\�
+1≤s≤r<∞ Π(r,s) (ℓq, ℓq) . This matter was recently investigated
+in [27], where the c-lineability of the set mentioned earlier was obtained. We take a step forward,
+providing that the set is c-spaceable. The following result generalizes [27, Theorem 3.1].
+Theorem 3.4. Let 0 < q < 1. Then
+L (ℓq; ℓq) \
+�
+1≤s≤r<∞
+Π(r,s) (ℓq; ℓq)
+is c-spaceable. Moreover, the result is sharp, since it is not valid for q ≥ 1.
+Proof. The starting point is a key result due to Maddox, who proved that the identity map on
+ℓp is never absolutely summing. More precisely, if 0 < q < 1 and 1 ≤ s ≤ r < ∞, then the
+identity map Id : ℓq → ℓq is not (r, s)-absolutely summing (for more details see [27]). Hence we
+can take T := Id ∈ L (ℓq; ℓq) \ �
+1≤s≤r<∞ Π(r,s) (ℓq; ℓq). The argument is similar to the proof of
+Theorem 3.1. We mention just the slight changes needed. In Proposition 2.2 we can take any
+countably disjoint decomposition (Nk)k∈N; and the operator Ψ in item (ii) must be defined as
+Ψa := �∞
+j=1 αjTj for a = (αj)j∈N ∈ ℓp. The last statement was observed in [27] as for q ≥ 1 the
+coincidence Πmax{2,q},1 (ℓq; ℓq) = L (ℓq; ℓq) is well known.
+□
+4. Absolutely summing and Dunford-Pettis operators
+It is well known that the class of Dunford-Pettis operators plays an important role in general
+Banach theory and Operator Theory. An operator T : E → F is called Dunford-Pettis (or
+completely continuous) if Txn converges to Tx (in F), whenever xn converges weakly to x in E.
+From now on the class of Dunford-Pettis operators from Banach spaces E into F will be denoted
+by DP(E, F). Observe that a compact operator must be a Dunford-Pettis operator and, clearly,
+the two notions coincide when E is reflexive. The interested reader can find more information
+and a panorama on the subject in the work [14] and the references therein.
+Bennet, in [7], showed a connection between absolutely (r, s)-summing and Dunford-Pettis
+operators: for any Banach spaces E and F, there exists an absolutely (r, s)-summing operator
+T : E → F, with 1 ≤ r < s < ∞, that does not satisfy the Dunford-Pettis property. With this we
+have a suitable environment to investigate exotic properties such as lineability and spaceability.
+In fact, the next result shows that the set of absolutely (r, s)-summing operators with values on
+ℓq that fails to be Dunford-Pettis is pointwise c-spaceable.
+Theorem 4.1. Let 1 ≤ q ≤ ∞ and 1 ≤ r < s < ∞. Then
+Π(r,s)(E, ℓq) \ DP(E, ℓq)
+is either empty or pointwise c-spaceable.
+Proof. To shorten notation we write A := �
+(r,s)(E; ℓq) and DP := DP(E, ℓq). We fix a non-
+trivial operator T in A \ DP and take a sequence (xn)n be such that it weakly converges to
+x ∈ E but Txn does not converge to Tx in ℓq. Also let x0 ∈ E and j0 ∈ N such that Tx0(j0) ̸= 0,
+that is, the j0-th coordinate of the sequence Tx0 ∈ ℓq is non-zero. Then we can choose ǫ0 > 0
+and N1 ⊂ N with j0 ∈ N1, card N1 = card(N \ N1) = ℵ0 and such that
+(4.1)
+
+�
+l∈N1
+|Txn(l) − Tx(l)|q
+
+
+1
+q
+≥ ǫ0,
+for some n large enough.
+We proceed as in the proof of Theorem 3.1, thus we omit some details. Proposition 2.2 is
+used with M = Π(r,s). Fixed a countably disjoint decomposition N\N1 = �
+j≥2 Nj, the sequence
+of operators (Tk)k∈N from Proposition 2.2 (i) fulfills ∥Tk∥A = ∥T∥A and Tk ∈ A \ DP for all k.
+Since A is Banach, in Proposition 2.2 (ii) we have p = 1 and the map constructed, Ψ : ℓ1 → A, is
+
+LARGE STRUCTURES WITHIN THE CLASS OF SUMMING OPERATORS
+9
+such that Ψ(ℓp) ⊂ A is a c-dimensional space containing T. We just need to prove that Ψa /∈ DP
+for all non-zero a = (αj)∞
+j=1 ∈ ℓ1.
+First, suppose that α1 = 0.
+Since {Tkx}k∈N, are sequences in ℓq with supports pairwise
+disjoint for all x,
+∞
+�
+l=1
+|Ψaxn(l) − Ψax(l)|q =
+∞
+�
+l=1
+������
+∞
+�
+j=2
+αjTjxn(l) − αjTjx(l)
+������
+q
+=
+∞
+�
+l=1
+������
+∞
+�
+j=2
+αjTj(xn − x)(l)
+������
+q
+.
+For each l ∈ N, there are unique t, s ∈ N such that l = n(s)
+t , hence
+∞
+�
+l=1
+������
+∞
+�
+j=1
+αjTj(xn − x)(l)
+������
+q
+=
+∞
+�
+s,t=1
+������
+∞
+�
+j=1
+αjTj(xn − x)
+�
+n(s)
+t
+�
+������
+q
+=
+∞
+�
+s,t=1
+���αsTs(xn − x)
+�
+n(s)
+t
+����
+q
+=
+∞
+�
+s=1
+|αs|q
+∞
+�
+t=1
+|T(xn − x)(t)|q
+≥ ∥a∥q
+ℓq · ǫq
+0,
+where the last inequality follows by (4.1). On the other hand, α1 ̸= 0 implies
+∥Ψaxn − Ψax∥q
+ℓq
+≥
+�
+l∈N1
+������
+α1T(xn − x)(l) +
+∞
+�
+j=2
+αjTj(xn − x)(l)
+������
+q
+= |α1|q ·
+�
+l∈N1
+|T(xn − x)(l)|q
+≥ |α1|q · ǫq
+0.
+Again, the last inequality follows by (4.1). This yields Ψa /∈ DP and, therefore, the pointwise
+c-lineability of A \ DP.
+Now we prove that Ψ(ℓ1) ⊂ (A \ DP) ∪ {0}. Let S = limk Ψa(k) ∈ Ψ(ℓ1) \ {0}, with a(k) =
+�
+α(k)
+j
+�
+j ∈ ℓ1 for all k ∈ N. It remains for us to verify that S /∈ DP. Let us consider ǫ0 > 0 and
+(xn)n∈N, x ∈ E as in (4.1). By the same reasoning from the proof of Theorem 3.1, there exists
+limk α(k)
+1
+and first we suppose that this limit is non-zero. Thus we have
+∥S(xn − x)∥q
+ℓq =
+������
+lim
+k→∞
+
+α(k)
+1 T(xn − x) +
+∞
+�
+j=2
+α(k)
+j Tj(xn − x)
+
+
+������
+q
+ℓq
+=
+�
+l∈N
+lim
+k→∞
+������
+α(k)
+1 T(xn − x)(l) +
+∞
+�
+j=2
+α(k)
+j Tj(xn − x)(l)
+������
+q
+≥
+�
+l∈N1
+lim
+k→∞
+������
+α(k)
+1 T(xn − x)(l) +
+∞
+�
+j=2
+α(k)
+j Tj(xn − x)(l)
+������
+q
+=
+�
+l∈N1
+lim
+k→∞
+���α(k)
+1 T(xn − x)(l)
+���
+q
+= lim
+k→∞ |α(k)
+1 |q ·
+�
+l∈N1
+|T(xn − x)(l)|q
+
+10
+ALBUQUERQUE AND COLETA
+≥ lim
+k→∞ |α(k)
+1 |q · ǫq
+0.
+The argument of the case limk α(k)
+1
+= 0 is similar. Hence S fails the be a Dunford-Petis operator,
+and this concludes the proof.
+□
+5. Summing operators on general summable scalar families
+In this section we deal with operators taking values on ℓq(I), the space of q-summable scalar
+family indexed over a fixed non-empty set I, and q ∈ (0, ∞). Recall that ℓq(I) is the vector
+space of all functions f : I → K such that �
+i∈I |f(i)|q < ∞, with this sum defined by
+�
+i∈I
+|f(i)|q := sup
+��
+i∈F
+|f(i)|q : F is a finite subset of I
+�
+,
+and also recall that ℓq(I) endowed with ∥f∥q =
+��
+i∈I |f(i)|q�1/q is a quasi-Banach space. For
+details about the space ℓq(I) and related results we refer to [24].
+In this setting we investigate the geometric notion of (α, β)-lineability, recently introduced
+by F´avaro, Pellegrino and Tomaz in [18]. More precisely, let α, β, λ be cardinal numbers and
+V be a vector space, with dim V = λ and α < β ≤ λ. A set A ⊂ V is (α, β)-lineable if it is
+α-lineable and for every subspace Wα ⊂ V with Wα ⊂ A ∪ {0} and dim Wα = α, there is a
+subspace Wβ ⊂ V with dim Wβ = β and Wα ⊂ Wβ ⊂ A ∪ {0}. When V is a topological vector
+space and the subspace Wβ can be chosen closed, A is said to be (α, β)-spaceable. Observe that
+this encompasses the lineability notion (take α = 0) and it is clear that pointwise α-lineability
+/ spaceability implies (1, α)-lineability / spaceability. Recent papers provided results in this
+fashion [15, 16, 17, 22]. Next, we obtain a result of this type.
+Proposition 5.1. Let q ∈ (0, ∞], I be a non-empty set, and Ei be infinite-dimensional Banach
+spaces with dim Ei < card I for i = 1, . . . , m. Also consider r, s, t, u ∈ [1, +∞)m and Λ ⊂ Γ ⊂
+Nm sets of indexes. Then
+ΠΛ
+(r,s) (E1, . . . , Em; ℓq (I)) \ ΠΓ
+(t,u) (E1, . . . , Em; ℓq (I))
+is either empty or (α, card I)-lineable for a cardinal α with α < card I.
+Proof. We will write A = ΠΛ
+(r,s) (E1, . . . , Em; ℓq (I)) and B = ΠΓ
+(t,u) (E1, . . . , Em; ℓq (I)) to sim-
+plify the notation. Suppose there is T ∈ A\B and write I = �
+j∈I Ij as a pairwise disjoint union,
+with card Ij = card I for all j ∈ I. For each j ∈ I, we denote Ij :=
+�
+η(j)
+i
+: i ∈ I
+�
+, and thus
+I =
+�
+η(j)
+i
+: i, j ∈ I
+�
+. Then for any ν ∈ I, there are unique i, j ∈ I such that ν = η(j)
+i . Similarly
+to the construction in Proposition 2.2 (i), for all ν ∈ I we define Tν : E1 × · · · × Em → ℓq(I) by
+Tνx
+�
+η(j)
+i
+�
+=
+�
+0,
+if j ̸= ν
+Tx(i),
+if j = ν.
+Thus ∥Tν∥A = ∥T∥A and this implies Tν ∈ A \ B for all ν ∈ I. Moreover, {Tν : ν ∈ I} is linearly
+independent and X := span {Tν : ν ∈ I} ⊂ (A \ B) ∪ {0}. Notice that, since dim X = card I,
+this yields the card I-lineability of A \ B.
+Now we turn to proving that A\B is (α, card I)-lineable for all α < card I. Let V ⊂ A\B∪{0}
+be an arbitrary α-dimensional subspace. Consider
+I := {V x : V ∈ V and x ∈ E1 × · · · × Em} ⊂ ℓq(I).
+and also denote by ∆ the set of indexes formed by all non-zero coordinates of each scalar family
+V x ∈ I ⊂ ℓq(I). Note that the number of such coordinates is not bigger than
+β := card (E1 × · · · × Em) · ℵ0 · card V < card I.
+
+LARGE STRUCTURES WITHIN THE CLASS OF SUMMING OPERATORS
+11
+Consider a new disjoint decomposition of I,
+I =
+
+�
+j∈I
+�Ij
+
+ ∪ ∆,
+with card �Ij = card I for all j ∈ I.
+As before, we have to construct operators Fν : E1 ×
+· · · × Em → ℓq(I) such that the support of the scalar family Fνx ∈ ℓq(I) lies in �Iν, and also
+∥Fνx∥ℓp(I) = ∥Tνx∥ℓq(I), for all x ∈ E1 × · · · × Em. Hence Fν belongs to A \ B for all ν ∈ I and
+the subspace
+W := span {Fν, V : ν ∈ I and V ∈ V}
+fulfills V ⊂ W ⊂ (A \ B) ∪ {0} and dim W = card I.
+□
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+[14] J. Diestel, A survey of results related to the Dunford-Pettis property, Contemp. Math. 2 (1980), 15–60.
+[15] D. Diniz and A. Raposo Jr., A note on the geometry of certain classes of linear operators, Bull. Braz. Math.
+Soc. (N.S.) 52 (2021), 1073–1080.
+[16] D. Diniz, V. F´avaro, D. Pellegrino, A. Raposo Jr., Spaceability of the sets of surjective and injective operators
+between sequence spaces, RACSAM. 114 (2020), no. 4, Paper No. 194, 11 pp.
+[17] V. F´avaro, D. Pellegrino, and P. Rueda, On the size of the set of unbounded multilinear operators between
+Banach spaces, Linear Algebra Appl. 606 (2020), 144–158.
+[18] V. F´avaro, D. Pellegrino, and D. Tom´az, Lineability and spaceability: a new approach, Bull. Braz. Math. Soc.
+(N.S) 51 (2020), no. 1, 27–46.
+[19] F. Hern´andez, C. Ruiz and V. S´anchez, Spaceability and operator ideals, J. Math. Anal. Appl. 431 (2015),
+no. 2, 1035–1044.
+[20] N. J. Kalton, N. T. Peck, and J. W. Roberts, An F-space sampler, London Mathematical Society Lecture
+Note Series, vol. 89. Cambridge University Press, Cambridge (1984).
+[21] D. Kitson and R. M. Timoney, Operator ranges and spaceability, J. Math. Anal. Appl. 378 (2011), no. 2,
+680–686.
+[22] D. Pellegrino and A. Raposo Jr., Pointwise lineability in sequence spaces, Indag. Math. (N.S.) 32 (2021), no.
+2, 536–546.
+[23] D. Pellegrino and E. Teixeira, Norm optimization problem for linear operators in classical Banach spaces,
+Bull. Braz. Math. Soc. (N.S.) 40 (2009), no. 3, 417–431.
+[24] A. Pietsch, Operators Ideals, North-Holland, 1980.
+[25] D. Popa, Operators with the Maurey–Pietsch multiple splitting property, Linear Multilinear Algebra. 68
+(2020), 1201–1217.
+
+12
+ALBUQUERQUE AND COLETA
+[26] D. Puglisi and J. B. Seoane-Sep´ulveda, Bounded linear non-absolutely summing operators, J. Math. Anal.
+Appl. 338 (2008), no. 1, 292–298.
+[27] D. Tom´az, Linear structure in certain subsets of quasi-Banach sequence spaces, Linear Multilinear Algebra.
+67 (2019), no. 8, 1561–1566.
+(N. G. Albuquerque) Departamento de Matem´atica
+Universidade Federal da Para´ıba
+Jo˜ao Pessoa - PB
+58.051-900 (Brazil)
+Email address: nacib.albuquerque@academico.ufpb.br
+(L. Coleta) Departamento de Matem´atica
+Universidade Federal da Para´ıba
+Jo˜ao Pessoa - PB
+58.051-900 (Brazil)
+Email address: lindinescoleta@gmail.com
+
diff --git a/M9AyT4oBgHgl3EQfUPcx/content/tmp_files/load_file.txt b/M9AyT4oBgHgl3EQfUPcx/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf,len=722
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='00120v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='FA]  31 Dec 2022  LARGE STRUCTURES WITHIN THE CLASS OF SUMMING OPERATORS  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' ALBUQUERQUE* AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' COLETA**  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' We investigate lineability/spaceability problems within the setting of multilinear  summing operators on quasi-Banach sequence spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Furthermore, we deal with the contem-  porary geometric notions of pointwise-lineability and (α, β)-lineability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Among other results,  we prove that the class of multilinear operators taking values on ℓq (0 < q ≤ ∞) that are ab-  solutely but not multiple summing is pointwise c-spaceable when non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Spaceability in  other classes, such as Dunford-Pettis operators and multilinear operators on general summable  scalar families, is also studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Introduction  The modern terminology of lineability and spaceability was coined by Gurariy, Aron, and  Seoane in the seminal paper [4]: for a cardinal α, a subset A of a vector space X is said to  be α-lineable if A ∪ {0} contains a α-dimensional linear subspace of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' If X is, in addition, a  topological vector space, then A is called α-spaceable if A ∪ {0} contains a closed α-dimensional  linear subspace of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' These concepts are widely investigated in several research branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' For  an overview, we refer the reader to recent works in [8, 9], and, to our knowledge, the most  comprehensive panorama of the matter are given in [5], and [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' The search for new lineability  notions yields to several new concepts, and some recent ones can be found in [18] and [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Many authors contributed to investigating lineability aspects of certain classes of summing  operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Puglisi and Seoane proved in [26] that, under certain restrictions over a Banach space  E, the set L(E, ℓ2)\\Π1(E, ℓ2) of bounded linear non-absolutely summing operators is lineable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Later in [12], Botelho, Diniz and Pellegrino obtained the lineability of K(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' F) \\ Πp(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' F), the  set of compact but not p-summing operators, with p ≥ 1, with some conditions of reflexivity,  infinite dimension, and unconditional basis on the Banach spaces E, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' More generally, Kitson  and Timoney in [21] obtained a remarkable and general spaceability result for Fr´echet spaces  with several applications, among them, the spaceability of K(E, F)\\ �  1≤p<∞ Πp(E, F), whenever  E is superreflexive and F is infinite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Hernand´ez, Ruiz, and S´anchez took a step  further and dealt with the spaceability of operators ideals: we refer to [19] for the spaceability  of I1(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' F) \\ I2(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' F), with I1, I2 operator ideals over certain Banach spaces E, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Alves  and Turco dealt with the spaceability of sets of p-compact linear maps where the domain and  codomain are ℓr, with 1 ≤ r < ∞, and c0 was investigated in depth and many results are  obtained in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' In the context of quasi-Banach spaces, in [27] Tom´az proved that L(ℓp, ℓp) \\  �  1≤s≤r<∞ Π(r,s)(ℓp, ℓp) is maximal lineable for any 0 < p < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' In the multilinear environment,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Ara´ujo and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Pellegrino established in [3] that set L(mℓp, K) \\ Πms  (r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='s)(mℓp, K) of bounded m-  linear but not multiple summing forms is maximal spaceable, for m ≥ 2, p ∈ [2, ∞), 1 ≤ s < p∗  and r <  2ms  s+2m−ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Here, as usual, Πas and Πms stand for the class of absolutely and multiple  multilinear summing classes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Also in [17] the authors investigated the lineability of  non-bounded and non-absolutely summing multilinear operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  It is well known from the classical Bohnenblust-Hille inequality and also the Defant-Voigt  theorem that Πms  (r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='1) (mE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' K) ⊊ L (mE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' K) = Πas  (r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='1) (mE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' K) for all 1 ≤ r <  2m  m+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Therefore  Πas  (r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='1) (mE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' K)\\Πms  (r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='1) (mE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' K) is non-empty and the search for large closed vector spaces naturally  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' 15A03, 46G25, 46B87, 47H60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Multilinear operators, Summing operators, Lineability, Spaceability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Supported by CNPq Grant 312167/2021-01 and Grant 2019/0014 Para´ıba State Research Foundation  (FAPESQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  ** Supported by CAPES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  1  2  ALBUQUERQUE AND COLETA  arises in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Similar problems to this one are answered as an application of the result  we provide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' A particular case we prove is that the class of multilinear operators taking values  on sequence spaces that are absolutely but not multiple summing, for instance, for E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , Em  Banach spaces, 1 ≤ r ≤ s < ∞ and q ∈ (0, ∞],  Πas  (r,s)(E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , Em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' ℓq) \\ Πms  (r,s)(E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , Em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' ℓq),  is c-spaceable when it is not empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' In fact, we prove a strong version dealing with the restrictive  pointwise-lineability/spaceability notion and in the general setting of Λ-multiple summing class  (see [1, 6, 11, 25] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  The main goals of this paper are to contribute to the research of lineability and spaceability  on the class of multilinear operators in two perspectives not often covered: we investigate the  geometric concepts of pointwise and (α, β)-lineability / spaceability (we discuss it with further  details in Sections 2 and 5), which are more restrictive than the classical notion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' also we deal  with operators taking value on quasi-Banach sequence spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' The panorama is quite different  from Banach spaces when dealing with quasi-Banach spaces, such as non-locally spaces ℓq for  0 < q < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Thus the search for lineability/spaceability techniques in quasi-Banach spaces can  be an exciting and challenging matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Section 2 presents preliminary concepts and notations  used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' In Section 3 we obtain the main result of the paper: the point-  wise c-spaceability of classes of Λ-summing operators (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='1) and some applications are  presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Section 4 deals with absolutely summing operators that fail to be Dunford-Petis (or  completely continuous).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' In Section 5 we prove that some sets of operators taking values on  spaces of general q-summable scalar families ℓq (I) can be (α, card I)-lineable, for I some infinite  set and some cardinal α < card I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Background and preliminaries  Throughout this paper we deal with vector spaces over the scalar field K, which can be either  R or C, and we write card R = c and card N = ℵ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Banach spaces over K shall be denoted by  capital letter (eventually with indexes) such as E, E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , Em, F, G, H, unless stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  The topological dual and the closed unit ball of E will be denoted by E′ and BE, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  We will denote by L (E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , Em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' F) the Banach space of bounded m-linear operators from  E1 × · · · × Em to F endowed with the usual sup norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  A quasi-norm on a vector space X is a non-negative real-valued function ∥ · ∥ : X → [0, ∞)  satisfying for all x, y ∈ X and λ ∈ K the following properties: ∥x∥ = 0 only if x = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' ∥λx∥ =  |λ|∥x∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' and ∥x + y∥ ≤ C(∥x∥ + ∥y∥) with C ≥ 1 an universal constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' If, in addition, the  map ∥ · ∥ is p-subadditive for some 0 < p ≤ 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=', it satisfies ∥x + y∥p ≤ ∥x∥p + ∥y∥p for all  x, y ∈ X, ∥ · ∥ is called a p-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Clearly we have a norm when C = 1 or p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' A quasi-norm  defines a metrizable vector topology on X whose base of neighborhoods of the origin is given  by sets of the form {x ∈ X : ∥x∥ < 1/n}, n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' X is called a quasi-Banach space when it is  complete for this metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' A p-subadditive quasi-norm induces a metric topology on X given by  d(x, y) := ∥x − y∥p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' A quasi-Banach space with an associated p-norm is also called a p-Banach  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' A deep result known as the Aoki-Rolewicz theorem (see [20, 24]) guarantees that every  quasi-normed space is p-normable for some 0 < p ≤ 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=', the space can be endowed with  an equivalent quasi-norm which is a p-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Thus we assume that a quasi-Banach space is  p-Banach for some 0 < p ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Many definitions and classical results in Banach spaces can be  translated in a natural way to quasi-Banach (p-Banach) spaces, like standard results depending  on Baire Category Theorem as Open Mapping Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Nevertheless, p-normed spaces are not  necessarily locally convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Results which hold on Banach spaces and are based somehow on  the local convexity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=', Hahn-Banach extension property or the Krein-Milman theorem) are  no longer valid, in general, in that setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' The extension of lineability/spaceability arguments  from Banach to quasi-Banach spaces is not straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Thus the search for large closed  infinite dimensional subspaces of p-Banach spaces is quite a delicate issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Therefore, looking  for lineability/spaceability techniques that also cover this environment spaces seems interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  For more details on quasi-Banach and p-Banach spaces, we refer to [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  LARGE STRUCTURES WITHIN THE CLASS OF SUMMING OPERATORS  3  Given a positive integer m, i := (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , im) stands for a multi-index in Nm, and r :=  (r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , rm) ∈ [1, +∞]m a multi-parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' By ℓr(E) we denote the Banach space that gathers  all E-valued multi-matrices (xi)i∈Nm ∈ ENm with finite ℓr-norm, that is, when r ∈ [1, +∞)m we  have  ��(xi)i∈Nm  ��  ℓr(E) :=  \uf8eb  \uf8ec  \uf8ed  ∞  �  i1=1  \uf8eb  \uf8ed· · ·  � ∞  �  im=1  ∥xi∥rm  E  � rm−1  rm  · ·  \uf8f6  \uf8f8  r1  r2  \uf8f6  \uf8f7  \uf8f8  1  r1  < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  We simply write ℓr = ℓr(K) and when no precision is needed ∥·∥ℓr(E) = ∥·∥r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Notice that ℓr(E)  is a quasi-Banach space when some 0 < rj < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Given a non-empty set of indexes Λ ⊂ Nm, by  (xi)i∈Λ we mean an E-valued multi-matrix indexed over Λ, and its ℓr-norm can be seen as  ��(xi)i∈Λ  ��  ℓr(E) =  ��(xi · 1Λ(i))i∈Nm  ��  ℓr(E)  where 1Λ is the characteristic function of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' It is worth pointing out some simple but useful  monotonicity properties: ∥ · ∥s ≤ ∥ · ∥r whenever 0 < r < s < ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  ��(αi)i∈Λ  ��  r ≤  ��(αi)i∈Γ  ��  r for  all scalar multi-matrix (αi)i∈Nm and for (non-empty) indexes sets Λ ⊂ Γ ⊂ Nm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' ∥(αi)i∈Λ∥r ≤  ∥(βi)i∈Λ∥r if 0 ≤ αi ≤ βi for all i ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  A quasi-Banach standard sequence space over a Banach space X is an infinite dimensional  quasi-Banach space E whose elements are X-valued sequences with the usual operations enjoying  the following conditions:  (i) There is a constant C > 0 such that  ∥xj∥X ≤ C∥x∥E,  for every x = (xj)j∈N ∈ E and all j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  (ii) If x = (xj)j∈N ∈ E and (xnk)k∈N is a subsequence of x, then (xnk)k∈N ∈ E and  ∥(xnk)k∈N∥E ≤ ∥x∥E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  (iii) If x = (xj)j∈N ∈ E and N′ := {n1 < n2 < n3 < · · · } is an infinite subset of N, then the  X−valued sequence y = (yj)j∈N defined as  yj :=  �  xi,  if j = ni,  0,  otherwise ,  belongs to E and ∥y∥E ≤ ∥x∥E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  It is plain that combining (ii) and (iii) yields ∥y∥E = ∥x∥E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' The n-th coordinate of a sequence  z ∈ E ⊂ XN will be denoted by zn or, when more precision is needed, by z(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Unless the contrary  is explicitly established, E will denote a quasi-Banach standard sequence space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Usual sequence  spaces are (quasi)-Banach standard sequence spaces as, for instance, the classical sequence spaces  ℓp(E), ℓw  p (E), ℓu  p(E) with p ∈ (0, ∞), c0(E), c(E), and the Lorentz space ℓp,q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  For a deeper  discussion on general sequence spaces, more details and examples, we refer the reader to [10, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  The proof of the next lemma is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Let r := (r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , rm) ∈ [1, ∞]m and Λ ⊂ Nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' If (αi)i∈Λ /∈ ℓr(E), then either  �  (αi(j))j∈N′  �  i∈Λ /∈ ℓr(E)  or  �  (αi(j))j∈N\\N′  �  i∈Λ /∈ ℓr(E),  for any N′ ⊂ N with card N′ = card(N \\ N′) = ℵ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Recently, Pellegrino and Raposo Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' in [22] introduced the geometric notion of pointwise-  lineability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Let V be a vector space and α be a cardinal number with α ≤ dim V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' A set A ⊂ V  is pointwise α-lineable if for each x ∈ A there is a subspace W = Wx,α such that x ∈ W ⊂ A∪{0}  and dim W = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' If X is a topological vector space and W can be chosen to be closed, A is  said pointwise α-spaceable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' It is common to refer to A as lineable or spaceable when α ≥ ℵ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Pointwise lineability is a more restrictive concept than classical lineability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' The latter faces a  technicality not often covered in usual lineability results: the subspace generated contains every  initial mother vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Also, this notion is related with (α, β)-lineability (see Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' For  more details, we refer to [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  4  ALBUQUERQUE AND COLETA  We finish this section with the following result, which will reveal itself useful and it is also  closely connected to a lineability technique that dates back to [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' First we recall a basic ideal  property of a quasi-Banach multi-ideal operators (M, ∥ · ∥M): if S ∈ M(E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , Em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' F), uj ∈  L(Gj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Ej), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , m, and t ∈ L(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' H), then t ◦ S ◦ (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , uj) ∈ M(G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , Gm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' H) and  ∥t ◦ S ◦ (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , uj)∥M ≤ ∥t∥ · ∥S∥M · ∥u1∥ · · · ∥um∥ (see [13] and references therein for more  details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Also, we say that (Nk)k∈N is a countably disjoint decomposition of the natural numbers  when we can write N = �  k∈N Nk with each Nk :=  �  n(k)  1  < n(k)  2  < · · ·  �  an infinite subset of N  and Ni ∩ Nj = ∅ whenever i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Let M be quasi-Banach operators multi-ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Given a countably disjoint  decomposition (Nk)k∈N of N and a non-trivial operator T ∈ M (E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , Em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' E), we have the  following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  (i) For each natural k ≥ 1, the map Tk : E1 × · · · × Em → E defined by  Tkx (j) :=  �  0,  if j /∈ Nk,  Tx(i),  if j = n(k)  i  ∈ Nk,  lies in M (E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , Em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' E), and ∥Tkx∥E = ∥Tx∥E for all x ∈ E1 × · · · × Em.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Moreover,  {Tk : k ≥ 2} ∪ {T} is linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  (ii) For some 0 < p ≤ 1, such that the map Ψ : ℓp → M (E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , Em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' E) given by  Ψa := α1T +  ∞  �  j=2  αjTj,  a = (αj)j∈N ∈ ℓp,  is well-defined, linear, bounded, and injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' (i) Since T is a non-zero operator, there exist x0 ∈ E1 × · · · × Em and j0 ∈ N such that  Tx0(j0) ̸= 0, where Tx0(j0) is the j0-th coordinate of the sequence Tx0 ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' For simplicity,  we shall assume that j0 ∈ N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Using the notation in condition (iii) of standard sequence  spaces, for each k ∈ N we take N′ = Nk =  �  n(k)  1  < n(k)  2  < · · ·  �  , and we define Vk : E → E  by Vkx := (yj)j∈N for x = (xn)n∈N ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  It is clear that Vk is a bounded linear map with  ∥Vk∥ ≤ 1, Tk = Vk ◦ T ∈ M (E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , Em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' E), and ∥Tk∥M ≤ ∥T∥M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Using conditions (ii) and  (iii) of standard sequence spaces, and the fact that {Tkx}k∈N is a family of sequences in E  whose supports are pairwise disjoint for any x, it is straightforward to conclude the remaining  statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  (ii) A := M (E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , Em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' E) ⊂ L (E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , Em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' E) is a p-Banach for some p ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' For any  a = (αj)j∈N ∈ ℓp,  �  j≥2  ∥αjTj∥p  A =  �  j≥2  |αj|p∥Tj∥p  A =  �  j≥2  |αj|p∥T∥p  A = ∥T∥p  A  �  j≥2  |αj|p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Hence the series �  j≥2 αjTj converges in A and, consequently, the map Ψ : ℓp → A is well defined  and linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' In order to simplify the notation we write Ψa := Ψa for a ∈ ℓp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Now suppose that  Ψa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Combining Tx0(j0) ̸= 0 with the fact that the sequence Tjx has null coordinates on  N \\ Nj, for any x ∈ E1 × · · · × Em and j ≥ 2,  0 = Ψax0(j0) = α1Tx0(j0) +  ∞  �  j=2  αjTjx0(j0) = α1Tx0(j0),  implies that α1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Since (Tj)j is linearly independent, we have αj = 0 for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Therefore  a = 0, and this concludes that Ψ is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  □  LARGE STRUCTURES WITHIN THE CLASS OF SUMMING OPERATORS  5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Spaceability of multilinear summing operators  In this Section the search of large topological structures is focused in a general environment  of multilinear summing operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' The concept of Λ-multiple summing (or Λ-summing) is a  notion of multilinear summing operators that encompasses the classical notions of absolutely  and multiple multilinear summing operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' It was independently introduced in [6, 25] and  recent developments in this environment can be found, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=', in [1, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' More precisely, let m ∈ N  be a positive integer, r, s ∈ [1, +∞)m and Λ ⊂ Nm a set of indexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' An m-linear operator  T : E1 × · · · × Em → F is Λ-(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' s)-summing if there is a constant C > 0 such that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='1)  ��(Txi)i∈Λ  ��  ℓr(F ) ≤ C  m  �  k=1  sup  φk∈BE′  k  � N  �  i=1  ���φ(x(k)  i  )  ���  sk  � 1  sk  for all N ∈ N and x(k)  i  ∈ Ek, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , m, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' For simplicity of notation we write  Txi := T  �  x(1)  i1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , x(m)  im  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  The class of all operators that fulfills the previous inequality is  denoted by ΠΛ  (r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='s) (E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , Em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' F), which is a Banach space endowed with the norm πΛ  (r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='s)(T)  taken as the infimum of the constants C > 0 satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Notice that, by taking Λ =  Diag (Nm) := {(n, · · · , n) ∈ Nm : n ∈ N} and Λ = Nm, the Λ-summing class ΠΛ recovers both  absolutely and multiple multilinear summing classes, respectively It is noteworthy to mention  that given Λ ⊂ Γ ⊂ Nm, the following inclusions and norm inequalities are easily seen (over  fixed parameters (r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' s) we omit): Πms ⊂ ΠΓ ⊂ ΠΛ ⊂ Πas and πms(·) ≤ πΓ(·) ≤ πΛ(·) ≤ πas(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Moreover, one can deal with the codomain F as a quasi-Banach space and, with standard  reasoning, ΠΛ  (r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='s) (E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , Em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' F) also is quasi-Banach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' For more details and applications in the  theory of absolutely multiple summing operators we refer the reader to [11] and the references  therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  The main result of this section reads as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Let r, s, t, u ∈ [1, +∞)m and Λ ⊂ Γ ⊂ Nm sets of indexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Then  ΠΛ  (r,s) (E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , Em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' E) \\ ΠΓ  (t,u) (E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , Em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' E)  is either empty or pointwise c-spaceable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' For simplicity, we write A := �Λ  (r,s)(E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , Em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' E) and B := �Γ  (t,u)(E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , Em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Let  us suppose there exists an operator T ∈ A \\ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' We prove first that A \\ B is pointwise c-lineable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Let us fix x0 ∈ E1 × · · · × Em and j0 ∈ N such that Tx0(j0) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' We also consider weakly  summable sequences z(j) =  �  z(j)  n  �  n∈N ∈ ℓw  uj(Ej), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , m, such that  ��(Tzi)i∈Γ  ��  ℓt(E) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Recall the notation Tzi := T  �  z(1)  i1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , z(m)  im  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='1 assures that we can take N1 ⊂ N with  j0 ∈ N1, card N1 = card(N \\ N1) = ℵ0 and also such that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='2)  ���  �  (Tzi(n))n∈N1  �  i∈Γ  ���  ℓt(E) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Now we take the following countably infinite decomposition: N\\N1 = �  k≥2 Nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Then Propo-  sition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='2 (i), applied with M = ΠΛ  (r,s), provides a sequence of (linearly independent) operators  (Tk)k∈N with ∥Tkx∥E = ∥Tx∥E for all x ∈ E1 × · · · × Em.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Clearly this implies ∥Tk∥A = ∥T∥A for  all k and also (Tk)k∈N belongs to A \\ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  In order to obtain the c-dimensional space in A \\ B that contains T, we use the construction  and notation of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='2 (ii): for some 0 < p ≤ 1, the map Ψ : ℓp → A is well-  defined, bounded, linear and injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' According to this, Ψ(ℓp) ⊂ A is a c-dimensional space  that contains T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  We shall now prove that Ψ(ℓp) \\ {0} ⊂ A \\ B, that is: Ψa /∈ B, for any  6  ALBUQUERQUE AND COLETA  a = (αj)j ∈ ℓp \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Consider z(j) ∈ ℓw  uj(Ej), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , m, as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' If α1 ̸= 0, using again  that every image of Tj is a sequence with null coordinates on N \\ Nj for j ≥ 2,  ∥Ψazi∥E =  ������  \uf8eb  \uf8edα1Tzi(n) +  �  j≥2  αjTjzi(n)  \uf8f6  \uf8f8  n∈N  ������  E  ≥  ������  \uf8eb  \uf8edα1Tzi(n) +  �  j≥2  αjTjzi(n)  \uf8f6  \uf8f8  n∈N1  ������  E  =  ��(α1Tzi(n))n∈N1  ��  E  = |α1|  ��(Tzi(n))n∈N1  ��  E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  The monotonicity of the quasi-norm yields  ��(Ψazi)i∈Γ  ��  ℓt(E) ≥ |α1|  ���  �  (Tzi(n))n∈N1  �  i∈Γ  ���  ℓt(E) = ∞,  with the last inequality being a consequence of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='2), which gives Ψa /∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' We now turn to the  case α1 = 0, in which we proceed in a similar fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Let us consider αi ̸= 0 for some i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Using the definition of the operators (Tk)k∈N in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='2 (i),  ∥Ψazi∥E =  ������  \uf8eb  \uf8ed�  j≥2  αjTjzi(n)  \uf8f6  \uf8f8  n∈N  ������  E  ≥  ���  �  αiTizi  �  n(i)  k  ��  k∈N  ���  E = |αi|  ��(Tzi (k))k∈N  ��  E ,  thus  ��(Ψazi)i∈Γ  ��  ℓt(E) ≥ |αi|  ��(Tzi)i∈Γ  ��  ℓt(E) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  This proves the pointwise c-lineability of A \\ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Now we turn to the pointwise c-spaceability of A \\ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' The obvious candidate for the closed  c-dimensional subspace of A is Ψ(ℓp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  We will prove that Ψ(ℓp) \\ {0} ⊂ A \\ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Let us fix  S = limk Ψk ∈ Ψ(ℓp) \\ {0}, with Ψk := Ψa(k) and a(k) =  �  α(k)  j  �  j ∈ ℓp for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  First let us show that there exists the limit limk α(k)  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Notice that, for all b = (βn)n ∈ ℓp, x ∈  E1 × · · · × Em and j ∈ N1, Ψbx(j) = β1Tx(j), taking x0 ∈ E1 × · · · × Em and j0 ∈ N1 with  Tx0(j0) ̸= 0, we get  Sx0(j0) = lim  k→∞ Ψkx0(j0) =  �  lim  k→∞ α(k)  1  �  Tx0(j0),  and this guarantees the existence of the aforementioned limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Now let z(j) ∈ ℓw  uj(Ej), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , m, as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Suppose that limk α(k)  1  ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Notice that, for  all x ∈ E1 × · · · × Em and all a ∈ ℓp,  ∥Ψax∥E =  ������  \uf8eb  \uf8edα1Tx(i) +  �  j≥2  αjTjx(i)  \uf8f6  \uf8f8  i∈N  ������  E  ≥  ������  \uf8eb  \uf8edα1Tx(i) +  �  j≥2  αjTjx(i)  \uf8f6  \uf8f8  i∈N1  ������  E  = |α1|  ��(Tx(i))i∈N1  ��  E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Then  ∥Szi∥E ≥ lim  k |α(k)  1 |  ��(Tzi(i))i∈N1  ��  E  yields  ��(Szi)i∈Γ  ��  ℓt(E) ≥ lim  k |α(k)  1 | ·  ��(Tzi)i∈Γ  ��  ℓt(E) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  LARGE STRUCTURES WITHIN THE CLASS OF SUMMING OPERATORS  7  Let us turn to the case limk α(k)  1  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Since S ̸= 0, there are x ∈ E1 × · · · × Em and i0 ∈ N  such that Sx(i0) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' There are unique l, t ∈ N such that i0 = n(l)  t , thus  Sx(i0) = lim  k→∞ Ψkx(i0)  = lim  k→∞  \uf8eb  \uf8edα(k)  1 Tx(i0) +  �  j≥2  α(k)  j Tjx(i0)  \uf8f6  \uf8f8  = lim  k→∞  �  α(k)  1 Tx(n(l)  t ) + α(k)  l  Tlx(n(l)  t )  �  = lim  k→∞ α(k)  l  Tx(t),  and, hence limk→∞ α(k)  l  ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Moreover,  ∥Sx∥E =  ������  lim  k→∞  \uf8ee  \uf8f0α(k)  1 Tx +  �  j≥2  α(k)  j Tjx  \uf8f9  \uf8fb  ������  E  = lim  k→∞  ������  �  j≥2  α(k)  j Tjx  ������  E  ≥ lim  k→∞  ������  \uf8eb  \uf8ed�  j≥2  α(k)  j Tjx(i)  \uf8f6  \uf8f8  i∈Nl  ������  E  = lim  k→∞  ����  �  α(k)  l  Tlx(i)  �  i∈Nl  ����  E  = lim  k→∞  ���α(k)  l  ���  ���(Tlx(i))i∈Nl  ���  E  = lim  k→∞  ���α(k)  l  ���  ��(Tx(i))i∈N  ��  E  = lim  k→∞  ���α(k)  l  ��� ∥Tx∥E  for all x ∈ E1 × · · · × Em.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' In particular, for such zi as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='2),  ��(Szi)i∈Γ  ��  ℓt(E) ≥  �  lim  k→∞  ���α(k)  l  ���  � ��(Tzi)i∈Γ  ��  ℓt(E) = ∞,  this leads to S /∈ B and, therefore, the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  □  We obtain the following result proceeding in the same manner as in the proof of Theorem  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Furthermore, taking Λ = Diag (Nm) := {(n, · · · , n) ∈ Nm : n ∈ N} and Γ = Nm, we return  to the classical setting of absolutely and multiple summing multilinear operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Let Λ ⊂ Nm be a set of indexes and r, s ∈ [1, +∞)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Then  L (E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , Em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' E) \\ ΠΛ  (r,s) (E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , Em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' E)  is either empty or pointwise c-spaceable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Let q ∈ (0, ∞] and r, s, t, u ∈ [1, +∞)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Then each one of the sets  L (E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , Em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' ℓq) \\ Πms  (t,u) (E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , Em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' ℓq) ,  L (E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , Em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' ℓq) \\ Πas  (t,u) (E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , Em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' ℓq)  and  Πas  (r,s) (E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , Em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' ℓq) \\ Πms  (t,u) (E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , Em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' ℓq)  is either empty or pointwise c-spaceable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  8  ALBUQUERQUE AND COLETA  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Non-absolutely summing operators on classical sequence spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' We now turn our  attention to linear bounded operators on the classical quasi-Banach spaces ℓq for 0 < q < 1 that  fail to be absolutely summing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' More precisely, for 0 < q < 1, we investigate large topological  linear structures within L (ℓq, ℓq)\\�  1≤s≤r<∞ Π(r,s) (ℓq, ℓq) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' This matter was recently investigated  in [27], where the c-lineability of the set mentioned earlier was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' We take a step forward,  providing that the set is c-spaceable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' The following result generalizes [27, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Let 0 < q < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Then  L (ℓq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' ℓq) \\  �  1≤s≤r<∞  Π(r,s) (ℓq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' ℓq)  is c-spaceable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Moreover, the result is sharp, since it is not valid for q ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' The starting point is a key result due to Maddox, who proved that the identity map on  ℓp is never absolutely summing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' More precisely, if 0 < q < 1 and 1 ≤ s ≤ r < ∞, then the  identity map Id : ℓq → ℓq is not (r, s)-absolutely summing (for more details see [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Hence we  can take T := Id ∈ L (ℓq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' ℓq) \\ �  1≤s≤r<∞ Π(r,s) (ℓq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' ℓq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' The argument is similar to the proof of  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' We mention just the slight changes needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' In Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='2 we can take any  countably disjoint decomposition (Nk)k∈N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' and the operator Ψ in item (ii) must be defined as  Ψa := �∞  j=1 αjTj for a = (αj)j∈N ∈ ℓp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' The last statement was observed in [27] as for q ≥ 1 the  coincidence Πmax{2,q},1 (ℓq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' ℓq) = L (ℓq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' ℓq) is well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Absolutely summing and Dunford-Pettis operators  It is well known that the class of Dunford-Pettis operators plays an important role in general  Banach theory and Operator Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' An operator T : E → F is called Dunford-Pettis (or  completely continuous) if Txn converges to Tx (in F), whenever xn converges weakly to x in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  From now on the class of Dunford-Pettis operators from Banach spaces E into F will be denoted  by DP(E, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Observe that a compact operator must be a Dunford-Pettis operator and, clearly,  the two notions coincide when E is reflexive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' The interested reader can find more information  and a panorama on the subject in the work [14] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Bennet, in [7], showed a connection between absolutely (r, s)-summing and Dunford-Pettis  operators: for any Banach spaces E and F, there exists an absolutely (r, s)-summing operator  T : E → F, with 1 ≤ r < s < ∞, that does not satisfy the Dunford-Pettis property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' With this we  have a suitable environment to investigate exotic properties such as lineability and spaceability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  In fact, the next result shows that the set of absolutely (r, s)-summing operators with values on  ℓq that fails to be Dunford-Pettis is pointwise c-spaceable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Let 1 ≤ q ≤ ∞ and 1 ≤ r < s < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Then  Π(r,s)(E, ℓq) \\ DP(E, ℓq)  is either empty or pointwise c-spaceable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' To shorten notation we write A := �  (r,s)(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' ℓq) and DP := DP(E, ℓq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' We fix a non-  trivial operator T in A \\ DP and take a sequence (xn)n be such that it weakly converges to  x ∈ E but Txn does not converge to Tx in ℓq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Also let x0 ∈ E and j0 ∈ N such that Tx0(j0) ̸= 0,  that is, the j0-th coordinate of the sequence Tx0 ∈ ℓq is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Then we can choose ǫ0 > 0  and N1 ⊂ N with j0 ∈ N1, card N1 = card(N \\ N1) = ℵ0 and such that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='1)  \uf8eb  \uf8ed�  l∈N1  |Txn(l) − Tx(l)|q  \uf8f6  \uf8f8  1  q  ≥ ǫ0,  for some n large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  We proceed as in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='1, thus we omit some details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='2 is  used with M = Π(r,s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Fixed a countably disjoint decomposition N\\N1 = �  j≥2 Nj, the sequence  of operators (Tk)k∈N from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='2 (i) fulfills ∥Tk∥A = ∥T∥A and Tk ∈ A \\ DP for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Since A is Banach, in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='2 (ii) we have p = 1 and the map constructed, Ψ : ℓ1 → A, is  LARGE STRUCTURES WITHIN THE CLASS OF SUMMING OPERATORS  9  such that Ψ(ℓp) ⊂ A is a c-dimensional space containing T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' We just need to prove that Ψa /∈ DP  for all non-zero a = (αj)∞  j=1 ∈ ℓ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  First, suppose that α1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Since {Tkx}k∈N, are sequences in ℓq with supports pairwise  disjoint for all x,  ∞  �  l=1  |Ψaxn(l) − Ψax(l)|q =  ∞  �  l=1  ������  ∞  �  j=2  αjTjxn(l) − αjTjx(l)  ������  q  =  ∞  �  l=1  ������  ∞  �  j=2  αjTj(xn − x)(l)  ������  q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  For each l ∈ N, there are unique t, s ∈ N such that l = n(s)  t , hence  ∞  �  l=1  ������  ∞  �  j=1  αjTj(xn − x)(l)  ������  q  =  ∞  �  s,t=1  ������  ∞  �  j=1  αjTj(xn − x)  �  n(s)  t  �  ������  q  =  ∞  �  s,t=1  ���αsTs(xn − x)  �  n(s)  t  ����  q  =  ∞  �  s=1  |αs|q  ∞  �  t=1  |T(xn − x)(t)|q  ≥ ∥a∥q  ℓq · ǫq  0,  where the last inequality follows by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' On the other hand, α1 ̸= 0 implies  ∥Ψaxn − Ψax∥q  ℓq  ≥  �  l∈N1  ������  α1T(xn − x)(l) +  ∞  �  j=2  αjTj(xn − x)(l)  ������  q  = |α1|q ·  �  l∈N1  |T(xn − x)(l)|q  ≥ |α1|q · ǫq  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Again, the last inequality follows by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' This yields Ψa /∈ DP and, therefore, the pointwise  c-lineability of A \\ DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Now we prove that Ψ(ℓ1) ⊂ (A \\ DP) ∪ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Let S = limk Ψa(k) ∈ Ψ(ℓ1) \\ {0}, with a(k) =  �  α(k)  j  �  j ∈ ℓ1 for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' It remains for us to verify that S /∈ DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Let us consider ǫ0 > 0 and  (xn)n∈N, x ∈ E as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' By the same reasoning from the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='1, there exists  limk α(k)  1  and first we suppose that this limit is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Thus we have  ∥S(xn − x)∥q  ℓq =  ������  lim  k→∞  \uf8ee  \uf8f0α(k)  1 T(xn − x) +  ∞  �  j=2  α(k)  j Tj(xn − x)  \uf8f9  \uf8fb  ������  q  ℓq  =  �  l∈N  lim  k→∞  ������  α(k)  1 T(xn − x)(l) +  ∞  �  j=2  α(k)  j Tj(xn − x)(l)  ������  q  ≥  �  l∈N1  lim  k→∞  ������  α(k)  1 T(xn − x)(l) +  ∞  �  j=2  α(k)  j Tj(xn − x)(l)  ������  q  =  �  l∈N1  lim  k→∞  ���α(k)  1 T(xn − x)(l)  ���  q  = lim  k→∞ |α(k)  1 |q ·  �  l∈N1  |T(xn − x)(l)|q  10  ALBUQUERQUE AND COLETA  ≥ lim  k→∞ |α(k)  1 |q · ǫq  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  The argument of the case limk α(k)  1  = 0 is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Hence S fails the be a Dunford-Petis operator,  and this concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Summing operators on general summable scalar families  In this section we deal with operators taking values on ℓq(I), the space of q-summable scalar  family indexed over a fixed non-empty set I, and q ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Recall that ℓq(I) is the vector  space of all functions f : I → K such that �  i∈I |f(i)|q < ∞, with this sum defined by  �  i∈I  |f(i)|q := sup  ��  i∈F  |f(i)|q : F is a finite subset of I  �  ,  and also recall that ℓq(I) endowed with ∥f∥q =  ��  i∈I |f(i)|q�1/q is a quasi-Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' For  details about the space ℓq(I) and related results we refer to [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  In this setting we investigate the geometric notion of (α, β)-lineability, recently introduced  by F´avaro, Pellegrino and Tomaz in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' More precisely, let α, β, λ be cardinal numbers and  V be a vector space, with dim V = λ and α < β ≤ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' A set A ⊂ V is (α, β)-lineable if it is  α-lineable and for every subspace Wα ⊂ V with Wα ⊂ A ∪ {0} and dim Wα = α, there is a  subspace Wβ ⊂ V with dim Wβ = β and Wα ⊂ Wβ ⊂ A ∪ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' When V is a topological vector  space and the subspace Wβ can be chosen closed, A is said to be (α, β)-spaceable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Observe that  this encompasses the lineability notion (take α = 0) and it is clear that pointwise α-lineability  / spaceability implies (1, α)-lineability / spaceability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Recent papers provided results in this  fashion [15, 16, 17, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Next, we obtain a result of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Let q ∈ (0, ∞], I be a non-empty set, and Ei be infinite-dimensional Banach  spaces with dim Ei < card I for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Also consider r, s, t, u ∈ [1, +∞)m and Λ ⊂ Γ ⊂  Nm sets of indexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Then  ΠΛ  (r,s) (E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , Em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' ℓq (I)) \\ ΠΓ  (t,u) (E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , Em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' ℓq (I))  is either empty or (α, card I)-lineable for a cardinal α with α < card I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' We will write A = ΠΛ  (r,s) (E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , Em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' ℓq (I)) and B = ΠΓ  (t,u) (E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' , Em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' ℓq (I)) to sim-  plify the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Suppose there is T ∈ A\\B and write I = �  j∈I Ij as a pairwise disjoint union,  with card Ij = card I for all j ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' For each j ∈ I, we denote Ij :=  �  η(j)  i  : i ∈ I  �  , and thus  I =  �  η(j)  i  : i, j ∈ I  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Then for any ν ∈ I, there are unique i, j ∈ I such that ν = η(j)  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Similarly  to the construction in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='2 (i), for all ν ∈ I we define Tν : E1 × · · · × Em → ℓq(I) by  Tνx  �  η(j)  i  �  =  �  0,  if j ̸= ν  Tx(i),  if j = ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Thus ∥Tν∥A = ∥T∥A and this implies Tν ∈ A \\ B for all ν ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Moreover, {Tν : ν ∈ I} is linearly  independent and X := span {Tν : ν ∈ I} ⊂ (A \\ B) ∪ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Notice that, since dim X = card I,  this yields the card I-lineability of A \\ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  Now we turn to proving that A\\B is (α, card I)-lineable for all α < card I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Let V ⊂ A\\B∪{0}  be an arbitrary α-dimensional subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Consider  I := {V x : V ∈ V and x ∈ E1 × · · · × Em} ⊂ ℓq(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  and also denote by ∆ the set of indexes formed by all non-zero coordinates of each scalar family  V x ∈ I ⊂ ℓq(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Note that the number of such coordinates is not bigger than  β := card (E1 × · · · × Em) · ℵ0 · card V < card I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  LARGE STRUCTURES WITHIN THE CLASS OF SUMMING OPERATORS  11  Consider a new disjoint decomposition of I,  I =  \uf8eb  \uf8ed�  j∈I  �Ij  \uf8f6  \uf8f8 ∪ ∆,  with card �Ij = card I for all j ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  As before, we have to construct operators Fν : E1 ×  · · × Em → ℓq(I) such that the support of the scalar family Fνx ∈ ℓq(I) lies in �Iν, and also  ∥Fνx∥ℓp(I) = ∥Tνx∥ℓq(I), for all x ∈ E1 × · · · × Em.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Hence Fν belongs to A \\ B for all ν ∈ I and  the subspace  W := span {Fν, V : ν ∈ I and V ∈ V}  fulfills V ⊂ W ⊂ (A \\ B) ∪ {0} and dim W = card I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
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+page_content=' Pellegrino, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
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+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
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+page_content=' Puglisi and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
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+page_content=' 338 (2008), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' 1, 292–298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  [27] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Tom´az, Linear structure in certain subsets of quasi-Banach sequence spaces, Linear Multilinear Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  67 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' 8, 1561–1566.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='  (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content=' Albuquerque) Departamento de Matem´atica  Universidade Federal da Para´ıba  Jo˜ao Pessoa - PB  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='051-900 (Brazil)  Email address: nacib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='albuquerque@academico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
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+page_content=' Coleta) Departamento de Matem´atica  Universidade Federal da Para´ıba  Jo˜ao Pessoa - PB  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
+page_content='051-900 (Brazil)  Email address: lindinescoleta@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfUPcx/content/2301.00120v1.pdf'}
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+arXiv:2301.05153v1  [math.RT]  12 Jan 2023
+EQUIVALENCE OF DECOMPOSITION MATRICES FOR BLOCKS OF ARIKI-KOIKE
+ALGEBRAS
+ALICE DELL’ARCIPRETE
+School of Mathematics,
+University of East Anglia,
+Norwich NR4 7TJ, UK.
+Abstract. We consider the representation theory of the Ariki-Koike algebra, a q-deformation of the group
+algebra of the complex reflection group Cr ≀Sn. We examine blocks of the Ariki-Koike algebra. In particular,
+we prove a sufficient condition such that restriction of modules leads to a natural correspondence between
+the multipartitions of n whose Specht modules belong to a block B and those of n − δi(B) whose Specht
+modules belong to the block B′, obtained from B applying a Scopes’ equivalence. This bijection gives us an
+equivalence for the decomposition numbers of the Ariki-Koike algebras.
+1. Introduction
+Let n be a positive integer. Let Sn denote the symmetric group of degree n. This has the famous Coxeter
+presentation with generators T1, . . . , Tn−1 and relations
+T 2
+i = 1,
+for 1 ≤ i ≤ n − 1,
+TiTj = TjTi,
+for 1 ≤ i < j − 1 ≤ n − 2,
+TiTi+1Ti = Ti+1TiTi+1,
+for 1 ≤ i ≤ n − 2.
+If we view this as a presentation for a (unital associative) algebra over a field F, then the algebra we get is
+the group algebra FSn.
+Let q be a non-zero element of F. Now we can introduce a ‘deformation’, by replacing the relation T 2
+i = 1
+with
+(Ti + q)(Ti − 1) = 0
+for each i. The resulting algebra is the Iwahori-Hecke algebra Hn = HF,q(Sn) of the symmetric group Sn.
+This algebra (of which the group algebra FSn is a special case) arises naturally, and its representation theory
+has been extensively studied. An excellent introduction to this theory is provided by Mathas’s book [M99].
+As long as q is non-zero, the representation theory of Hn bears a remarkable resemblance to the representation
+theory of Sn. Indeed, there are many theorems concerning Hn which reduce representation-theoretic notions
+to statements about the combinatorics of partitions.
+In this paper, we consider the representation theory of the Ariki-Koike algebra. This algebra is a deforma-
+tion of the group algebra of the complex reflection group Cr ≀Sn, defined using parameters q, Q1, . . . , Qr ∈ F.
+The representation theory of these algebras is beginning to be well understood. For example, the simple
+modules of the Ariki-Koike algebras have been classified; the blocks are known; and, in principle, the de-
+composition matrices of the Ariki-Koike algebras can be computed in characteristic zero. A comprehensive
+review of the representation theory of the Ariki-Koike algebras can be found in Mathas’s paper [M04]. In
+many respects it seems that the Ariki-Koike algebra behaves in the same way as the Iwahori-Hecke algebra
+Hn; many of the combinatorial theorems concerning Hn have been generalised to the Ariki-Koike algebra,
+with the role of partitions being played by multipartitions. In fact, much of the difficulty of understanding
+the Ariki-Koike algebra seems to lie in finding the right generalisations of the combinatorics of partitions to
+E-mail address: A.Dell-Arciprete@uea.ac.uk.
+Key words and phrases. Ariki-Koike algebras, blocks, decomposition numbers.
+1
+
+2
+A. DELL’ARCIPRETE
+multipartitions - very simple combinatorial notions (such as the definition of an e-restricted partition) can
+have rather nebulous generalisations (such as ‘Kleshchev’ multipartitions).
+In [S91], under some conditions, Scopes establishes a natural correspondence between Specht modules
+and simple modules in the blocks B and φi(B) of the symmetric groups where φi is the map swapping the
+runners i − 1 and i of each partition in the block B. This leads to an equivalence of decomposition matrices
+for these two blocks, meaning that the blocks B and φi(B) have the same decomposition matrices. In the
+last part of this paper Scopes proves Donovan’s conjecture for blocks of the symmetric groups. In particular,
+given a partition λ of n, the bijection φi gives a Morita equivalence between the blocks B of λ and the
+block φi(B) of φi(λ) for the symmetric group algebra FSn. The generalisation of Donovan’s conjecture and
+hence of the Morita equivalence between the blocks B and Φi(B) of Ariki-Koike algebras (with Φi the map
+φi extended componentwise) can be found in [AS] where Amara-Omari and Schaps prove this combining
+some combinatorics with the Chuang-Rouquier categorification of integrable highest weight modules over
+Kac-Moody algebra of affine type A. We would like to underline the fact that Morita equivalence does not
+imply the equivalence of decomposition matrices. Indeed, if we consider n = 8 and p = 3, we have that
+the block of the partition (8) and the block of the partition (18) for the symmetric group algebra F3S8 are
+Morita equivalent, but they have different decomposition matrices.
+This paper is intended as the Ariki-Koike algebra version of Scopes’ paper about decomposition numbers.
+In particular, we prove a sufficient condition such that two blocks of the Ariki-Koike algebras have the same
+decomposition matrices. In Section 2, we define the Ariki-Koike algebras and the combinatorical objects that
+we will use. Thus, we introduce the r-multipartition of an integer n and we describe the construction of its
+abacus configuration with e runners. In Section 3, we summarise all the relevant results from [F06] and [F07]
+concerning weight of multipartitions and core blocks of Ariki-Koike algebras. In Section 4, we show that
+there is a sequence of multicores with non-increasing weights from a given multicore to a multicore in a core
+block. Then, we give a lower bound for the minimal difference between the positions of the lowest beads of
+two consecutive runners. In Section 5, we use this lower bound to get a condition on the weight of the block
+B of a multipartition so that no configuration i − 1
+i
+✉
+appears in the abacus display of any multipartition in
+B. In turn, we show that there is a natural correspondence between Specht modules and simple modules in
+the blocks B and Φi(B) of the Ariki-Koike algebras.
+2. Basic definitions
+2.1. The Ariki-Koike algebras. Let r ≥ 1 and n ≥ 0. Let Wr,n be the complex reflection group Cr ≀ Sn.
+We can define the Ariki-Koike algebra as a deformation of the group algebra FWr,n.
+Definition 2.1. Let F be a field and q, Q1, . . . , Qr be elements of F, with q non-zero. Let Q = (Q1, . . . , Qr).
+The Ariki-Koike algebra HF,q,Q(Wr,n) of Wr,n is defined to be the unital associative F-algebra with
+generators T0, . . . , Tn−1 and relations
+(T0 − Q1) · · · (T0 − Qr) = 0,
+T0T1T0T1 = T1T0T1T0,
+(Ti + 1)(Ti − q) = 0,
+for 1 ≤ i ≤ n − 1,
+TiTj = TjTi,
+for 0 ≤ i < j − 1 ≤ n − 2,
+TiTi+1Ti = Ti+1TiTi+1,
+for 1 ≤ i ≤ n − 2.
+For brevity, we may write Hr,n for HF,q,Q(Wr,n). Define e to be minimal such that 1 + q + . . . + qe−1 = 0,
+or set e = ∞ if no such value exists. Throughout this paper we shall assume that e is finite and we shall
+refer to e as the quantum characteristic. Set I = {0, 1, . . ., e − 1}: we will identify I with Z/eZ. We
+refer to any r-tuple of integers (a1, . . . , ar) ∈ Zr as a multicharge. We say Q is q-connected if, for each
+i ∈ {1, . . . , r}, Qi = qai for some ai ∈ Z. In [DM02], Dipper and Mathas prove that any Ariki-Koike algebra
+is Morita equivalent to a direct sum of tensor products of smaller Ariki-Koike algebras, each of which has
+q-connected parameters. Thus, we may assume that we are always working with a Ariki-Koike algebra with
+each Qi being an integral power of q. So we assume that we can find an r-tuple of integers a = (a1, . . . , ar)
+
+EQUIVALENCE OF DECOMPOSITION MATRICES FOR BLOCKS OF ARIKI-KOIKE ALGEBRAS
+3
+such that Qi = qai for each i where q ̸= 1, so that q is a primitive eth root of unity in F. We call such a a
+multicharge of Hr,n. Since e is finite then we may change any of the ai by adding a multiple of e, and we
+shall still have Qi = qai.
+For the rest of this section we will fix a multicharge a = (a1, . . . , ar) ∈ Zr. Notice that at this stage we
+are not requiring that a is a multicharge of the Ariki-Koike algebra.
+2.2. Multipartitions. A partition of n is defined to be a non-increasing sequence λ = (λ1, λ2, . . . ) of non-
+negative integers whose sum is n. The integers λb, for b ≥ 1, are called the parts of λ. We write |λ| = n.
+Since n < ∞, there is a k such that λb = 0 for b > k and we write λ = (λ1, . . . , λk). We write ∅ for the unique
+empty partition (0, 0, . . . , 0). If a partition has repeated parts, for convenience we group them together with
+an index. For example,
+(4, 4, 2, 1, 0, 0, . . .) = (4, 4, 2, 1) = (42, 2, 1).
+The Young diagram of a partition λ is the subset
+[λ] := {(b, c) ∈ N>0 × N>0 | c ≤ λb}.
+Definition 2.2. An r-multipartition of n is an ordered r-tuple λ = (λ(1), . . . , λ(r)) of partitions such that
+|λ| := |λ(1)| + . . . + |λ(r)| = n.
+If r is understood, we shall just call this a multipartition of n.
+As with partitions, we write the unique multipartition of 0 as ∅∅∅. The Young diagram of a multipartition
+λ is the subset
+[λ] := {(b, c, j) ∈ N>0 × N>0 × {1, . . . , r} | c ≤ λ(j)
+b }.
+We may abuse notation by not distinguishing a multipartition from its Young diagram. The elements of [λ]
+are called nodes of λ. We say that a node n ∈ [λ] is removable if [λ] \ {n} is also the Young diagram of
+a multipartion. We say that an element n ∈ N2 × {1, . . ., r} is an addable node if n /∈ [λ] and [λ] ∪ {n} is
+the Young diagram of a multipartition. Given a multicharge a = (a1, . . . , ar), to each node (b, c, j) ∈ [λ] we
+associate its residue resa(b, c, j) = aj + c − b (mod e). We draw the residue diagram of λ by replacing each
+node in the Young diagram by its residue.
+Example 2.3. Suppose r = 3 and a = (1, 0, 2). Let λ = ((12), (2), (2, 1)) and µ = ((1), (2, 1), (13)) be two
+multipartitions of 7. If e = 4, then the 4-residue diagram of [λ] and of [µ] are
+1
+0
+0
+1
+2
+3
+1
+and
+1
+0
+1
+3
+2
+1
+0
+.
+For an r-multipartition λ, define the residue set of λ to be the multiset Resa(λ) = {res(n) | n ∈ [λ]}.
+Notice that in the example above
+Resa(λ) = {0, 0, 1, 1, 1, 2, 3} = Resa(µ).
+2.3. Kleshchev multipartitions. Residues of nodes are useful in classifying the simple Hr,n-modules. In-
+deed, the notion of residue helps us to describe a certain subset K of the set of all multipartitions, which
+index the simple modules for Hr,n. The multipartions in the set K are called Kleshchev multipartitions.
+The recursive definition of Kleshchev multipartition can be found in [F07]. For the scope of this paper, it is
+enough to consider them as the multipartitions indexing the simple modules of Hr,n.
+2.4. Specht module and simple modules. The algebra Hr,n is a cellular algebra [DJM,GL] with the cell
+modules indexed by r-multipartitions of n. For each r-multipartition λ of n we define a Hr,n-module Sλ called
+a Specht module; these modules are the cell modules given by the cellular basis of Hr,n [DJM]. When
+Hr,n is semisimple, the Specht modules form a complete set of non-isomorphic irreducible Hr,n-modules.
+However, we are mainly interested in the case when Hr,n is not semisimple. In this case, for each Kleshchev
+r-multipartition λ, the Specht module Sλ has an irreducible cosocle Dλ, and the Dλ provide a complete set
+of irreducible modules for Hr,n as λ ranges over the set of Kleshchev multipartitions of n [A01, Theorem
+4.2]. If λ and µ are r-multipartition of n with µ Kleshchev, let dλµ = [Sλ : Dµ] denote the multiplicity of
+
+4
+A. DELL’ARCIPRETE
+the simple module Dµ as a composition factor of the Specht module Sλ. The matrix D = (dλµ) is called
+the decomposition matrix of Hr,n and determining its entries is one of the most important open problems
+in the representation theory of the Ariki-Koike algebras. It follows from the cellularity of Hr,n that the
+decomposition matrix is ‘triangular’; to state this we need to define the dominance order on multipartitions.
+Given two multipartitions λ and µ of n, we say that λ dominates µ, and write λ ⊵ µ, if
+j−1
+�
+a=1
+|λ(a)| +
+i
+�
+b=1
+λ(j)
+b
+≥
+j−1
+�
+a=1
+|µ(a)| +
+i
+�
+b=1
+µ(j)
+b
+for j = 1, 2, . . ., r and for all i ≥ 1. Then we have the following.
+Theorem 2.4.
+[DJM,GL] Let F be a field. Suppose λ and µ are r-multipartitions of n with µ Kleshchev.
+(1) If µ = λ, then [Sλ : Dµ] = 1.
+(2) If [Sλ : Dµ] > 0, then λ ⊵ µ.
+Note that the dominance order is a partial order on the set of multipartitions. The dominance order is
+certainly the ‘correct’ order to use for multipartitions, but it is sometimes useful to have a total order, >, on
+the set of multipartitions. The one we use is given as follows.
+Definition 2.5. Given two multipartitions λ and µ of n with λ ̸= µ, we write λ > µ if and only if the
+minimal j ∈ {1, . . . , r} for which λ(j) ̸= µ(j) and the minimal i ≥ 1 such that λ(j)
+i
+̸= µ(j)
+i
+satisfy λ(j)
+i
+> µ(j)
+i .
+This is called the lexicographic order on multipartitions.
+2.5. Blocks of Ariki-Koike algebras. It follows from the cellularity of Hr,n that each Specht module Sλ
+lies in one block of Hr,n, and we abuse notation by saying that a multipartition λ lies in a block B if Sλ lies
+in B. On the other hand, each block contains at least one Specht module, so in order to classify the blocks of
+Hr,n, it suffices to describe the corresponding partition of the set of multipartitions. We have the following
+classification of the blocks of Hr,n.
+Theorem 2.6.
+[LM07, Theorem 2.11] Let a ∈ Ir be a multicharge of Hr,n.
+Suppose λ and µ are r-
+multipartitions of n. Then, Sλ and Sµ lie in the same block of Hr,n if and only if Resa(λ) = Resa(µ).
+Example 2.7. Continuing the Example 2.3, we see that the residue sets of λ and µ are equal. Hence λ and
+µ lie in the same block of H3,7 with e = 4.
+2.6. Induction and restriction. If n > 1, then Hr,n−1 is naturally a subalgebra of Hr,n, and in fact Hr,n is
+free as an Hr,n−1-module. So there are well-behaved induction and restriction functors between the module
+categories of Hr,n−1 and Hr,n. Given modules M, N for Hr,n−1 and Hr,n, respectively, we write M ↑Hr,n and
+N ↓Hr,n−1 for the induced and restricted modules. If B and C are blocks of Hr,n−1 and Hr,n, respectively,
+then we may write M ↑C and N ↓B for the projections of the induced and restricted modules onto B and C.
+Theorem 2.8.
+[A96, Lemma 2.1]
+• Suppose λ is a multipartition of n − 1, and let n1, . . . , ns be the addable nodes of [λ].
+For each
+i = 1, . . . , s, let λ+i be the multipartition of n with [λ+i] = [λ] ∪ {ni}. Then Sλ ↑Hr,n has a filtration
+in which the factors are Sλ+1, . . . , Sλ+s.
+• Suppose λ is a multipartition of n, and let n1, . . . , nt be the removable nodes of [λ]. For each i =
+1, . . . , t, let λ−i be the multipartition of n−1 with [λ−i] = [λ]\{ni}. Then Sλ ↓Hr,n−1 has a filtration
+in which the factors are Sλ−1, . . . , Sλ−t.
+If a module M has a filtration M = M0 ⊇ M1 ⊇ . . . ⊇ Mt−1 ⊇ Mt = 0 with composition factors
+Mi/Mi+1 ∼= Ni for i = 0, . . . , t − 1, we will write
+M ∼
+t−1
+�
+i=0
+Ni.
+
+EQUIVALENCE OF DECOMPOSITION MATRICES FOR BLOCKS OF ARIKI-KOIKE ALGEBRAS
+5
+2.7. β-numbers and abacus. Given the assumption that the cyclotomic parameters of Hr,n are all powers
+of q, we may conveniently represent multipartitions on an abacus display. Fix a = (a1, . . . , ar) ∈ Zr to be a
+multicharge of Hr,n.
+Definition 2.9. Let λ = (λ(1), . . . , λ(r)) be a multipartition of n. For every i ≥ 1 and for every j ∈ {1, . . . , r},
+we define the β-number βj
+i to be
+βj
+i := λ(j)
+i
++ aj − i.
+The set Bj
+aj = {βj
+1, βj
+2, . . .} is the set of β-numbers (defined using the integer aj) of partition λ(j). It is
+easy to see that any set Bj
+aj = {βj
+1, βj
+2, . . .} is a set containing exactly aj + N integers greater than or equal
+to −N, for sufficiently large N.
+For each set Bj
+aj we can define an abacus display in the following way. We take an abacus with e infinite
+vertical runners, which we label 0, 1, . . . , e − 1 from left to right, and we mark positions on runner l and
+label them with the integers congruent to l modulo e, so that position (x + 1)e + l lies immediately below
+position xe + l, for each x. Now we place a bead at position βj
+i , for each i. The resulting configuration is
+called e-abacus display, or e-abacus configuration, for λ(j) with respect to aj. Moreover, we say that the bead
+corresponding to the β-number xe + l is at level x for x ∈ Z.
+Hence, we can now define the e-abacus display, or the e-abacus configuration, for a multipartition λ
+with respect to a to be the r-tuple of e-abacus displays associated to each component λ(j). If it is clear which
+e we are referring to, we simply say abacus configuration. When we draw abacus configurations we will draw
+only a finite part of the runners and we will assume that above this point the runners are full of beads and
+below this point there are no beads.
+Example 2.10. Suppose that r = 3, a = (−1, 0, 1) and λ = ((1), ∅, (12)). Then we have
+B1
+−1 = {−1, −3, −4, −5, . . .};
+B2
+0 = {−1, −2, −3, −4 . . .};
+B3
+1 = {1, 0, −2, −3, −4, . . .}.
+So, the abacus display with respect to the multicharge a for λ when e = 4 is
+0
+1
+2
+3
+qq
+q
+qq
+q
+qq
+q
+qq
+q
+④
+④
+④
+④
+④
+④
+④
+qq
+q
+qq
+q
+qq
+q
+qq
+q
+0
+1
+2
+3
+qq
+q
+qq
+q
+qq
+q
+qq
+q
+④
+④
+④
+④
+④
+④
+④
+④
+qq
+q
+qq
+q
+qq
+q
+qq
+q
+0
+1
+2
+3
+level
+qq
+q
+qq
+q
+qq
+q
+qq
+q
+④
+④
+④
+④
+−2
+④
+④
+④
+−1
+④
+④
+0
+qq
+q
+qq
+q
+qq
+q
+qq
+q
+.
+2.8. Rim e-hooks and e-core. We recall some other definitions about diagram of a partition since their
+generalization for multipartitions will be really useful in this paper.
+Definition 2.11. Suppose λ is a partition of n and (i, j) is a node of [λ].
+(1) The rim of [λ] is defined to be the set of nodes
+{(i, j) ∈ [λ] | (i + 1, j + 1) /∈ [λ]}.
+(2) Define an e-rim hook to be a connected subset R of the rim containing exactly e nodes such that
+[λ] \ R is the diagram of a partition.
+(3) If λ has no e-rim hooks then we say that λ is an e-core.
+(4) If we can remove w e-rim hooks from [λ] to produce an e-core, then we say that λ has e-weight w.
+In particular, an e-core has weight 0.
+
+6
+A. DELL’ARCIPRETE
+An abacus display for a partition is useful for visualising the removal of e-rim hooks. If we are given an
+abacus display for λ with β-numbers in a set B, then [λ] has a e-rim hook if and only if there is a β-number
+βi ∈ B such that βi − e /∈ B. Furthermore, removing a e-rim hook corresponds to reducing such a β-number
+by e. On the abacus, this corresponds to sliding a bead up one position on its runner. So, λ is an e-core if
+and only if every bead in the abacus display has a bead immediately above it. Using this, we can see that the
+definition of e-weight and e-core of λ are well defined. Finally, we can also notice that each bead corresponds
+to the end of a row of the diagram of λ (or to a row of length 0).
+For multipartitions, by the definition of the β-numbers the node at the end of the row (if it exists) has
+residue i if and only if the corresponding bead is on runner i of the abacus for i ∈ I. Thus, if we increase
+any β-number by one, this is equivalent to moving a bead from runner i to runner i + 1 which is equivalent
+to adding a node of residue i + 1 to the diagram of a component λ(j). Similarly, decreasing a β-number by
+one is equivalent to moving a bead from runner i to runner i − 1 which is equivalent to removing a node of
+residue i from the diagram of λ(j).
+3. Background results about weight of multipartitions and core block
+Here we summarise the main definitions and some results from [F06] and [F07], mostly concerning weight
+of multipartitions and core blocks of Ariki-Koike algebras. These are the generalisations to multipartitions
+of the notions of e-weight and e-core of partitions.
+3.1. Weight and hub of multipartitions. In [F06], Fayers generalises the notion of weight from partitions
+to multipartitions which will be of great utility in this paper. Fix a multicharge a = (a1, . . . , ar) of Hr,n.
+Definition 3.1. Let λ = (λ(1), . . . , λ(r)) be a multipartition of n. Let ci(λ) denote the number of nodes in
+[λ] of residue i ∈ I and aj denote aj (mod e). Define the weight w(λ) of λ to be the non-negative integer
+w(λ) =
+
+
+r
+�
+j=1
+caj(λ)
+
+ − 1
+2
+�
+i∈I
+(ci(λ) − ci+1(λ))2.
+It is also useful to define the hub of a multipartition λ. For each i ∈ I and j ∈ {1, . . . , r}, define
+δj
+i (λ) =(the number of removable nodes of [λ(j)] of residue i)
+− (the number of addable nodes of [λ(j)] of residue i),
+and put δi(λ) = �r
+j=1 δj
+i (λ). The collection (δi(λ) | i ∈ I) of integers is called the hub of λ.
+Moreover, we want to highlight that an important feature of the weight and hub of a multipartition: they
+are invariants of the block containing λ, and in fact determine the block.
+Proposition 3.2.
+[F06, Proposition 3.2 & Lemma 3.3] Suppose λ is a multipartition of n and µ is a
+multipartition of m. Then
+(1) if λ and µ have the same hub, then m ≡ n (mod e), and
+w(λ) − w(µ) = r(n − m)
+e
+;
+(2) if n = m, then λ and µ lie in the same block of Hr,n if and only if they have the same hub.
+In view of this result, we may define the hub of a block B to be the hub of any multipartition λ in B,
+and we write δi(B) = δi(λ).
+3.2. Comparing the weight from the abacus. Here we summarise some results from [F06] that shows
+how to compare weights of multipartitions efficiently from an abacus display.
+Proposition 3.3.
+[F06, Corollary 3.4] Suppose λ = (λ(1), . . . , λ(r)) is a multipartition, and that λ− is the
+multipartition obtained from λ by removing an e-rim hook from some λ(j). Then w(λ) = w(λ−) + r.
+Definition 3.4. If λ = (λ(1), . . . , λ(r)) is a multipartition and each λ(j) is an e-core, then we say that λ is a
+multicore.
+
+EQUIVALENCE OF DECOMPOSITION MATRICES FOR BLOCKS OF ARIKI-KOIKE ALGEBRAS
+7
+Suppose λ = (λ(1), . . . , λ(r)) is a multicore and a = (a1, . . . , ar) is a multicharge of Hr,n. We construct
+the corresponding abacus display for λ as in Section 2.7, and then for each i ∈ I and 1 ≤ j ≤ r we define
+ba
+ij(λ) to be the position of the lowest bead on runner i of the abacus for λ(j); that is, the largest element of
+Bj congruent to i modulo e.
+Now, fix e ∈ {2, 3, 4, . . .}. Let i ∈ I and j, k ∈ {1, . . . , r}, we define
+γjk
+i (λ) = 1
+e(ba
+ij(λ) − ba
+ik(λ)).
+γjk
+i (λ) may then be regarded as the difference in height between the lowest bead on runner i of the abacus
+display for λ(j) and the lowest bead on runner i of the abacus display for λ(k). Since e is finite, the integers
+γjk
+i (λ) depend on the choice of a if we change any aj by a multiple of e, but the differences
+γjk
+il (λ) := γjk
+i (λ) − γjk
+l (λ)
+do not.
+Suppose λ = (λ(1), . . . , λ(r)) is a multicore, i, l ∈ I and j, k ∈ {1, . . . , r}. We define sjk
+il (λ) to be the
+multicore whose abacus configuration is obtained from that of λ by moving a bead from runner i to runner
+l on the abacus for λ(j), and moving a bead from runner l to runner i on the abacus for λ(k). It is worth
+noting that sjk
+il (λ) = skj
+li (λ) for all i, j, k, l.
+Proposition 3.5.
+[F07, Proposition 1.6] Let λ be a multicore and let sjk
+il (λ) be defined as above. Then
+(1) sjk
+il (λ) has the same hub as λ, and,
+(2) w(sjk
+il (λ)) = w(λ) − r(γjk
+il (λ) − 2).
+Using Proposition 3.5, we may reduce the calculation of the weight of a multicore λ to the case where we
+have γjk
+il (λ) ≤ 2 because if γjk
+il (λ) ≥ 3, we can obtain w(λ) from w(sjk
+il (λ)) inductively.
+3.3. Scopes isometries. Here we introduce maps between blocks of Ariki-Koike algebras analogous to those
+defined by Scopes [S91] between blocks of symmetric groups. Suppose i ∈ I, and let φi : Z → Z be the map
+given by
+φi(x) =
+
+
+
+
+
+x + 1
+x ≡ i − 1
+(mod e)
+x − 1
+x ≡ i
+(mod e)
+x
+otherwise.
+φi descends to give a map from I to I; we abuse notation by referring to this map as φi also.
+Now suppose λ is a multipartition, and that we have chosen an abacus display for λ. For each j, we define
+a partition Φi(λ(j)) by replacing each beta-number β with φi(β). Equivalently, we simultaneously remove all
+removable nodes of residue i from [λ(j)] and add all addable nodes of [λ(j)] of residue i, or in terms of abacus
+configuration we swap the runners (i − 1) and i of each abacus in the abacus display of λ. If i = 0, we swap
+runners e − 1 and runners 0. We define Φi(λ) to be the multipartition (Φi(λ(1)), . . . , Φi(λ(r))).
+Example 3.6. Here we give an example of the definition of φi for a partition. For e = 3, consider the abacus
+configuration of the partition λ given below and apply to λ the map φ1. Then,
+
+8
+A. DELL’ARCIPRETE
+λ
+φ1(λ)
+0
+1
+2
+qq
+q
+qq
+q
+qq
+q
+④
+④
+④
+④
+④
+④
+④
+④
+④
+④
+qq
+q
+qq
+q
+qq
+q
+−→
+0
+1
+2
+qq
+q
+qq
+q
+qq
+q
+④
+④
+④
+④
+④
+④
+④
+④
+④
+④
+qq
+q
+qq
+q
+qq
+q
+.
+So, it can be seen that the removal and the addition of nodes for λ due to the application of φ1 is simultaneous.
+Proposition 3.7.
+[F06, Proposition 4.6] Suppose B is a block of Hr,n and i ∈ I. Then there is a block ¯B
+of Hr,n−δi(B) with the same weight as B. Moreover, Φi gives a bijection between the set of multipartitions in
+B and the set of multipartitions in ¯B.
+We write Φi(B) for the block ¯B described in Proposition 3.7.
+3.4. Core blocks of Ariki-Koike algebras. Here we introduce the notion of core blocks of Ariki-Koike
+algebras, giving several equivalent definitions following the work of Fayers in [F07]. This definition is the one
+generalising the e-core notion to multipartitions.
+Theorem 3.8. Suppose that e is finite, and that λ is a multipartition lying in a block B of Hr,n. Let κ be a
+multicharge of Hr,n. The following are equivalent.
+(1) λ is a multicore, and there exists a multicharge a = (a1, . . . , ar) such that aj ≡ κj (mod e) and
+integers α0, . . . , αe−1 such that for each i, j, ba
+ij(λ) equals either αi or αi + e.
+(2) There is no block of any Hr,m with the same hub as B and smaller weight than B.
+(3) Every multipartition in B is a multicore.
+Now we can make the definition of a core block for the Ariki-Koike algebra Hr,n.
+Definition 3.9. Suppose B is a block of Hr,n. Then we say that B is a core block if and only if e is finite
+and the equivalent conditions of Theorem 3.8 are satisfied for any multipartition λ in B,
+Theorem 3.8 gives us several equivalent conditions for a multipartition to lie in a core block. Moreover,
+condition (2) of Theorem 3.8 implies that, of the blocks with a given hub, only the one with the smallest
+weight is a core block. So, if λ is a multipartition with this hub, then we may speak of this core block as the
+core block of λ.
+Now, let λ be a multipartition and κ be a multicharge for Hr,n. Recalling the definition of level of a bead
+in an e-abacus display given in Section 2.7, define ℓκ
+ij(λ) to be the level of the lowest bead on runner i of the
+abacus display for λ(j) with respect to κ. Note that γjk
+i (λ) = ℓa
+ij(λ) − ℓa
+ik(λ). Using Theorem 3.8, we see
+that for λ corresponding to Sλ in a core block, there exists a multicharge a = (a1, . . . , ar) ∈ Zr such that
+ai ≡ κi mod e and integers b0, b1, . . . , be−1 such that for each i ∈ I and j ∈ {1, . . . , r}, ℓa
+ij(λ) equals either bi
+or bi + 1. We call such an e-tuple (b0, b1, . . . , be−1) a base tuple for λ. Adapting Theorem 3.8 we have the
+following result.
+Proposition 3.10. Suppose λ is a multicore and κ = (κ1, . . . , κr) is a multicharge for Hr,n. Then Sλ lies
+in a core block of Hr,n if and only if there is a = (a1, . . . , ar) ∈ Zr such that aj ≡ κj (mod e) and an abacus
+configuration for λ such that
+|γjk
+i (λ)| = |ℓa
+ij(λ) − ℓa
+ik(λ)| ≤ 1 for each i ∈ I and j, k ∈ {1, . . ., r}.
+
+EQUIVALENCE OF DECOMPOSITION MATRICES FOR BLOCKS OF ARIKI-KOIKE ALGEBRAS
+9
+Corollary 3.11. Assume the conditions of Proposition 3.10. If Sλ lies in a core block of Hr,n, then there
+exists a = (a1, . . . , ar) ∈ Zr such that aj ≡ κj (mod e) and an abacus configuration for λ such that
+|δj
+i (λ) − δk
+i (λ)| ≤ 2 for each i ∈ I and j, k ∈ {1, . . ., r}.
+Proof. By Proposition 3.10, since there exists a multicharge a such that |γjk
+i (λ)| ≤ 1 for each i ∈ I and
+j, k ∈ {1, . . ., r} we have
+|δj
+i (λ) − δk
+i (λ)| = |ℓa
+ij − ℓa
+i−1,j − (ℓa
+ik − ℓa
+i−1,k)|
+= |ℓa
+ij − ℓa
+ik + ℓa
+i−1,j − ℓa
+i−1,k|
+≤ |ℓa
+ij − ℓa
+ik| + |ℓa
+i−1,j − ℓa
+i−1,k|
+≤ 1 + 1 = 2.
+□
+4. Results about multicores
+In this section, we give some results concerning properties of multicores that play a fundamental role in
+the proof of the main result of this paper. We fix some notation.
+Let F be a field, and let q, Q1, . . . , Qr be non-zero elements of F. Assume that (Q1, . . . , Qr) are q-connected
+parameters. For i ∈ I and a multicore m, denote by:
+di(m) = min{δj
+i (m) | j ∈ {1, . . ., r}}.
+If the value of i is clear, we will write d(m) instead of di(m).
+Firstly, we notice an important and useful property of the abacus display of a multicore µ lying in a core
+block C of Hr,n and then we give some results about multicores not necessarily in a core block.
+If µ is a multipartition lying in a core block C of Hr,n then, by the definition of a base tuple, there exists a
+multicharge a = (a1, . . . , ar) of Hr,n and at least one base tuple (b0, . . . , be−1) such that µ has abacus display
+where for any i ∈ I and j ∈ {1, . . ., r}
+δj
+i (µ) ∈ {bi − bi−1 − 1, bi − bi−1, bi − bi−1 + 1}.
+(4.1)
+Notice that (4.1) holds for all the multicores in the core block C since the base tuple is an invariant of a core
+block.
+Fix ¯i ∈ I, ¯i ≥ 1 and a multicharge a = (a1, . . . , ar) such that (4.1) holds. Let (b0, . . . , be−1) be the
+corresponding base tuple such that b¯i is as big as possible and b¯i−1 is as small as possible between all the
+possible choices of base tuples corresponding to a. Then we can define
+K¯i := b¯i − b¯i−1 − 1
+(4.2)
+If it is clear what ¯i we are referring to, we will simply write K instead of K¯i. Note that
+K ≤ d(µ)
+(4.3)
+for all µ ∈ C.
+Example 4.1. Suppose e = 5, r = 3 and (Q1, Q2, Q3) = (1, q3, q). So, κ = (0, 3, 1). Let µ be the multicore
+((4, 3, 1), (4, 23), (3, 2)) which has abacus display with respect to the multicharge a = (0, −2, 1)
+0
+1
+2
+3
+4
+qq
+q
+qq
+q
+qq
+q
+qq
+q
+qq
+q
+④
+④
+④
+④
+④
+④
+④
+④
+④
+④
+④
+④
+④
+④
+④
+qq
+q
+qq
+q
+qq
+q
+qq
+q
+qq
+q
+0
+1
+2
+3
+4
+qq
+q
+qq
+q
+qq
+q
+qq
+q
+qq
+q
+④
+④
+④
+④
+④
+④
+④
+④
+④
+④
+④
+④
+④
+qq
+q
+qq
+q
+qq
+q
+qq
+q
+qq
+q
+0
+1
+2
+3
+4
+qq
+q
+qq
+q
+qq
+q
+qq
+q
+qq
+q
+④
+④
+④
+④
+④
+④
+④
+④
+④
+④
+④
+④
+④
+④
+④
+④
+qq
+q
+qq
+q
+qq
+q
+qq
+q
+qq
+q
+.
+
+10
+A. DELL’ARCIPRETE
+Since |γjk
+i (µ)| ≤ 1 for all j, k ∈ {1, 2, 3} and i ∈ {0, . . . , 4}, µ lies in a core block C by Proposition 3.10.
+Then we can define a base tuple for µ and thus for C. In particular, we can consider the following two base
+tuples:
+(1) (b0, b1, b2, b3, b4) = (2, 4, 2, 3, 1);
+(2) (b′
+0, b′
+1, b′
+2, b′
+3, b′
+4) = (2, 3, 2, 3, 1).
+Take ¯i = 1. In order to define K1 we need to choose (b0, b1, b2, b3, b4) = (2, 4, 2, 3, 1) because b1 > b′
+1 and
+b0 = b′
+0, so we get K1 = 1.
+Take ¯i = 3. In order to define K3 we can choose either (b0, b1, b2, b3, b4) or (b′
+0, b′
+1, b′
+2, b′
+3, b′
+4) because b3 = b′
+3
+and b2 = b′
+2, so we get K3 = 0.
+We now want to exhibit, for all multicores of Hr,n, a sequence of multicores of non-increasing weight
+ending in a multicore lying in a core block. In order to do this, we need a preliminary lemma adapted from
+the proof of Proposition 3.7 in [F07] to the case of multicores.
+Lemma 4.2. If λ is a multicore not lying in a core block of Hr,n, then there is a sequence λ = λ0, . . . , λu
+of multicores such that, for all t = 0, . . . , u − 1, we have λt+1 = sjk
+il (λt) for some i, l, j, k. Furthermore,
+w(λt+1) ≤ w(λt) for all t = 0, . . . , u − 1.
+Proposition 4.3. Let m be a multicore of Hr,n. Then there exist a multicore µ in the core block C of m
+and a sequence of multicores
+m = m0, m1, . . . , ms−1, ms = µ
+such that:
+1) the core block of mt is C for all t = 0, . . . , s;
+2) mt+1 = sjk
+il (mt) for some j, k ∈ {1, . . ., r}, i, l ∈ {1, . . ., e − 1} for all t = 0, . . . , s − 1;
+3) there exists 0 ≤ v ≤ s such that
+i) w(mt+1) < w(mt) for all t = 0, . . . , v − 1 and w(mt+1) ≤ w(mt) for all t = v, . . . , s − 1;
+ii) |γjk
+il (mv)| ≤ 2 for all i, l and j, k.
+Proof. Let m0 := m be a multicore of Hr,n. Then, apply the following procedure for t ≥ 0.
+1. Calculate γjk
+i (mt) for all i ∈ {0, . . ., e − 1} and j, k ∈ {1, . . . , r};
+2. If there is a choice of i, l and j, k such that γjk
+il (mt) ≥ 3, set mt+1 = sjk
+il (mt). By Proposition 3.5
+we have
+w(mt+1) < w(mt).
+3. Repeat this step until we have mt+1 with γjk
+il (mt+1) ≤ 2 for all i, l and j, k.
+Suppose that we stop for t + 1 = v. Notice that γjk
+il (mv) ≤ 2 for all i, l and j, k implies |γjk
+il (mv)| ≤ 2 for
+all i, l and j, k. Indeed,
+γjk
+il (mv) = −γjk
+li (mv) = −γkj
+il (mv).
+Now, if mv is not in the core block C, apply Lemma 4.2 until we get a multicore µ in the core block C.
+□
+Before stating the main result, we need some preliminary lemmas.
+Lemma 4.4. Suppose that m is a multicore such that |γjk
+il (m)| ≤ 2 for all i, l ∈ I and j, k ∈ {1, . . . r}. Fix
+¯i ∈ I, and let d be the integer such that d(m) = d − 1. Then
+δj
+¯i (m) ∈ {d − 1, d, d + 1} for all j ∈ {1, . . . , r}.
+Proof. Consider m a multicore such that |γjk
+il (m)| ≤ 2 for all i, l ∈ I and j, k ∈ {1, . . . r}. Fix ¯i ∈ I. Then,
+since |γjk
+il (m)| ≤ 2 for all i, l ∈ I and j, k ∈ {1, . . . r}, we have that
+|γjk
+(¯i−1)¯i(m)| = |δk
+¯i (m) − δj
+¯i (m)| ≤ 2.
+Hence, since d(m) = d − 1, we get δj
+¯i (m) ∈ {d − 1, d, d + 1} for all j ∈ {1, . . ., r}.
+□
+Lemma 4.5. Let m be a multicore with core block C and such that |γjk
+il (m)| ≤ 2 for all i, l ∈ I and
+j, k ∈ {1, . . .r}. Suppose that µ is a multicore in the core block C. Fix ¯i ∈ I, and let d(m) = d − 1 and
+d(µ) = d′ − 1 for some integer d and d′. Then d′ ≤ d + 1.
+
+EQUIVALENCE OF DECOMPOSITION MATRICES FOR BLOCKS OF ARIKI-KOIKE ALGEBRAS
+11
+Proof. By Remark 4.4, we know that δj
+¯i (m) ∈ {d − 1, d, d + 1} and δj
+¯i (µ) ∈ {d′ − 1, d′, d′ + 1} for all
+j ∈ {1, . . ., r}. Let a, b, and c be the number of δj
+¯i (m) equal respectively to d − 1, d, and d + 1. Let a′, b′,
+and c′ be the number of δj
+¯i (µ) equal respectively to d′ − 1, d′, and d′ + 1. Notice that a > 0 and a′ > 0 by
+definition of d − 1 and d′ − 1. By Proposition 4.3, we can go from the multicore m to the multicore µ in the
+core block C via a sequence of multicores mt such that mt+1 = sjk
+il (mt) for some i, l ∈ I and j, k ∈ {1, . . . , r}.
+By point (1) of Proposition 3.5, we know that each multicore mt occuring in this sequence has the same hub
+of m, then m and µ have the same hub. So,
+a(d − 1) + b(d) + c(d + 1) = a′(d′ − 1) + b′(d′) + c′(d′ + 1),
+(4.4)
+where a + b + c = a′ + b′ + c′ = r. Suppose by contradiction that d′ > d + 1. Then, looking at (4.4) we have
+LHS ≤ r(d + 1) with equality if and only if a = b = 0;
+RHS ≥ a′(d + 1) + b′(d + 2) + c′(d + 3) ≥ r(d + 1) with equality if and only if b′ = c′ = 0;
+We must have equality in both terms, but this is a contradiction since a > 0.
+□
+Lemma 4.6. Let m be a multicore of Hr,n and m′ = sjk
+il (m) for some i, l, j, k. Fix ¯i ∈ I. Then d(m′) ≥
+d(m) − 2. Moreover, if γjk
+il (m) = 1, then d(m′) ≥ d(m) − 1.
+Proof. The fact that d(m′) ≥ d(m) − 2 follows from the definition of sjk
+il and that |δj
+¯i (m) − δj
+¯i (m′)| = 2 if
+and only if {i, l} = {¯i − 1,¯i}.
+Suppose γjk
+il (m) = 1. Then,
+• if {i, l} ̸= {¯i − 1,¯i}, then d(m′) ≥ d(m) − 1 since the only case in which δj
+¯i (m′) decreases by 2 with
+respect to δj
+¯i (m) is when {i, l} = {¯i − 1,¯i};
+• if {i, l} = {¯i − 1,¯i}, then
+γjk
+¯i−1,¯i(m) = 1 ⇔ δk
+¯i (m) − δj
+¯i (m) = 1 ⇔ δk
+¯i (m) = δj
+¯i (m) + 1.
+Thus, δk
+¯i (m′) = δk
+¯i (m) + 2 and δk
+¯i (m′) = δk
+¯i (m) − 1. Hence, d(m′) ≥ d(m) − 1.
+□
+Lemma 4.7. Let m be a multicore of Hr,n. Suppose that m = m0, m1, . . . , mv is a sequence of multicores
+such that mt+1 = sjk
+il (mt) for some i, l, j, k and w(mt+1) < w(mt) for all t = 0, . . . , v − 1. Let w(m) =
+w(mv) + hr with h > 0. Then d(m) ≥ d(mv) − h.
+Proof. We proceed by induction on v.
+If v = 1, the sequence of multicores consists of m0 = m and
+m1 = sjk
+il (m0) for some i, l, j, k with w(m0) = w(m1) + hr for h > 0. Note that m0 = sjk
+li (m1). By
+Proposition 3.5(2), the weight of m0 is w(m0) = w(m1) − r(γjk
+li (m1) − 2), so h = −γjk
+li (m1) + 2. Moreover,
+h > 0 and so we have that γjk
+li (m1) ≤ 1. Hence, we just need to check the following two cases.
+• If γjk
+li (m1) ≤ 0, then d(m0) ≥ d(m1) − 2 by Remark 4.6 and h = −γjk
+li (m1) + 2 ≥ 2. Thus,
+d(m0) ≥ d(m1) − 2 ≥ d(m1) + γjk
+li (m0) − 2 = d(m1) − h.
+• If γjk
+il (m0) = 1 by Remark 4.6, then h = 1 and d(m0) ≥ d(m1) − 1. Thus,
+d(m0) ≥ d(m1) − 1 = d(m1) − h.
+Suppose v > 1. Let w(m) = w(mv−1) + h′r with 0 ≤ h′ < h and w(mv−1) = w(mv) + h′′r with h′′ > 0
+so that h = h′′ + h′. By induction hypothesis we know that d(m) ≥ d(mv−1) − h′. In order to get the result
+we want to show that d(m) ≥ d(mv) − h. We know from the base step that that d(mv−1) ≥ d(mv) − h′′.
+Thus,
+d(m) ≥ d(mv−1) − h′ ≥ d(mv) − h′′ − h′ = d(mv) − h.
+□
+
+12
+A. DELL’ARCIPRETE
+Proposition 4.8. Fix ¯i ∈ {1, . . . , e − 1}. Let K = K¯i be the integer defined in (4.2) for a core block C.
+Suppose that m is a multicore with core block C and weight
+w(m) = w(C) + hr
+with 0 ≤ h ≤ K. Then d(m) ≥ K − h.
+Proof. Let
+m = m0, . . . , mv, mv+1, . . . , ms = µ
+be the sequence defined in Proposition 4.3, where v is such that |γjk
+il (mv)| ≤ 2 for all i, l, j, k, and µ ∈ C.
+By Lemma 4.7, we have that
+d(m) ≥ d(mv) − h′,
+(4.5)
+where 0 ≤ h′ ≤ h is such that w(m) = w(mv) + h′r.
+If h′ = h, then mv = ms = µ ∈ C; therefore d(m) ≥ d(µ) − h ≥ K − h by (4.3).
+Otherwise h′ < h, and Lemma 4.5 can be applied to get that
+d(mv) ≥ d(µ) − 1 ≥ d(µ) − (h − h′).
+Combining this with (4.5), we have:
+d(m) ≥ d(mv) − h′ ≥ d(µ) − (h − h′) − h′ = d(µ) − h ≥ K − h.
+□
+5. Decomposition numbers for blocks of Hr,n
+Now, we want to generalise Lemma 2.1 of Scopes’ paper [S91] to the Ariki-Koike algebras Hr,n. Thus,
+we want to show that, for i ∈ I, restriction of modules leads to a natural correspondence between the
+multipartitions of n whose Specht modules belong to B and those of n − δi(B) whose Specht modules belong
+to Φi(B), provided that
+w(B) ≤ w(C) + Kir
+(5.1)
+where
+• C is the core block of B,
+• Ki is the integer defined in (4.2).
+Notice that this condition is a block condition, i.e., it is satisfied by all the multipartitions in the block.
+When i is understood, we write K instead of Ki.
+Lemma 5.1. Fix i ∈ I. Let B be a block of Hr,n such that (5.1) holds and δi(B) ≥ 0. Then, in each
+component of every r-multipartition λ of n such that Sλ belongs to the block B, there is no abacus configuration
+of the type ✉
+in runners i − 1 and i.
+Proof. First of all, notice that
+w(B) − w(C) = ℓr,
+for some ℓ ∈ {0, . . . , K}
+(5.2)
+since w(B) − w(C) can take as values only integral multiples of r.
+Now, consider the multicore m associated to λ, that is the multicore whose abacus display is obtained by
+the one of λ sliding all the beads up as high as possible. Then m has the same core block C of λ and weight
+w(m) = w(C) + sr with 0 ≤ s ≤ ℓ. Thus, by Proposition 4.8 we can say that
+d(m) ≥ K − s ≥ ℓ − s,
+since ℓ ≤ K by (5.2). Moreover, in order to get λ from m we just need to slide beads down of a total number
+of ℓ − s spaces. This implies that in each component m(j) of m we need to slide beads down of at most ℓ − s
+spaces. Since d ≥ ℓ − s, similarly to the proof of [S91, Lemma 2.1] we can conclude that each component λ(j)
+of λ has no configuration ✉
+in runners i − 1 and i.
+□
+
+EQUIVALENCE OF DECOMPOSITION MATRICES FOR BLOCKS OF ARIKI-KOIKE ALGEBRAS
+13
+Proposition 5.2. Fix i ∈ I. Let B be a block of Hr,n such that (5.1) holds and δi(B) ≥ 0. Suppose that λ
+is an r-multipartition of n such that Sλ belongs to the block B. Then the multipartition Φi(λ) of n − δi(B)
+is such that SΦi(λ) belongs to Φi(B) and
+Sλ ↓Φi(B) ∼ δi(B)!SΦi(λ),
+SΦi(λ) ↑B ∼ δi(B)!Sλ.
+Proof. Suppose that µ is a multipartition of n − δi(B) and Sµ is a factor of Sλ ↓Hr,n−δi(B) using the Specht
+filtration given in Theorem 2.8. The diagram [µ] can be obtained from [λ] by removing δi(B) nodes. The
+multiplicity of Sµ as a factor is the number of ways in which the node removal can be effected. The abacus
+of µ is obtained from that of λ by successively moving δi(B) beads one place to the left. The module Sµ
+belongs to Φi(B) if and only if the overall effect is moving δi(B) beads from each runner i to each runner
+i − 1 by point (2) of Proposition 3.2.
+By Lemma 5.1, in each component of the multipartition λ, we have no abacus configuration of the type
+✉
+in runners i−1 and i. This implies that λ has no addable i-nodes and so Φi consists only of removing
+i-nodes. Hence, the number of ways in which the node removal can be effected is δi(B)! because they have all
+the same residue i and so we can remove these nodes in any order. This gives the first result about restriction.
+This also shows that there is exacly one µ that can be found by removing δi(B) i-nodes.
+Similarly, the second result about induction can be proved by adding i-nodes instead of removing i-
+nodes.
+□
+Lemma 5.3. Let i ∈ I. If condition (5.1) holds, then Φi preserves the lexicographic order of multipartitions.
+Proof. Let λ and µ be multipartitions whose corresponding Specht modules belong to block B. Let Λ =
+(Λ1, . . . , Λr) and M = (M1, . . . , Mr) be the associated sets of β-numbers.
+Let ¯λ = Φi(λ) and ¯µ = Φi(µ), and let their corresponding sets of β-numbers be ¯Λ = (¯Λ1, . . . , ¯Λr) and
+¯
+M = ( ¯
+M1, . . . , ¯
+Mr).
+Now if λ > µ, then as sets of numbers using the lexicographic order Λ > M. Similarly Λ > M implies
+that λ > µ and we obtain
+λ > µ ⇔ Λ > M ⇔ Λ \ (Λ ∩ M) > M \ (Λ ∩ M).
+Assume λ > µ. Let j0 be the minimal j ∈ {1, . . ., r} such that
+Λj \ (Λj ∩ Mj) > Mj \ (Λj ∩ Mj).
+Let Λj0 \ (Λj0 ∩ Mj0) = {x1, . . . , xt}, with xl > xl+1 for l = 1, . . . , t − 1, and
+let Mj0 \ (Λj0 ∩ Mj0) = {y1, . . . , ys}, with yl > yl+1 for l = 1, . . . , s − 1.
+Then,
+¯Λj0 \ (¯Λj0 ∩ ¯
+Mj0) = {φi(x1), . . . , φi(xt)}
+and
+¯
+Mj0 \ (¯Λj0 ∩ ¯
+Mj0) = {φi(y1), . . . , φi(ys)}.
+Since λ > µ, it follows that x1 > y1. We have three cases to consider.
+Case 1: x1 belongs to column i − 1. In this case φi(x1) = x1 + 1 > yl + 1 ≥ φi(yl) for all l. Hence
+¯Λ \ (¯Λ ∩ ¯
+M) > ¯
+M \ (¯Λ ∩ ¯
+M), so ¯λ > ¯µ.
+Case 2: x1 belongs to column i. Clearly φi(x1) = x1 − 1 ≥ yl + l − 1 > yl + 1 ≥ φi(yl) for all l ≥ 3 as
+x1 > y1 > y2 > y3 > · · · .
+If y2 does not belong to column i − 1, then φi(y2) ≤ y2 and x1 − 1 > y2 as x1 > y1 > y2.
+So φi(x1) = x1 − 1 > y2 ≥ φi(y2).
+If y2 belongs to column i − 1 then φi(y2) = y2 + 1. Since
+x1 − 1 and y2 belongs to column i − 1 and y2 < x1 − 1 as above, so y2 ≤ x1 − 1 − e.
+Hence
+φi(x1) = x1 − 1 > y2 + 1 = φi(y2).
+If y1 lies in column i, then φi(x1) = x1 − 1 > y1 − 1 = φi(y1). If y1 lies in column k, where
+k ̸= i, i − 1, then y1 ≤ x1 − 2, so φi(x1) = x1 − 1 > x1 − 2 ≥ y1 = φi(y1). Suppose y1 lies in
+column i − 1. If y1 = x1 − 1, then the abacus of µ(j0) presents a configuration
+✉
+in runners
+i − 1 and i where the bead corresponds to y1. However, by Lemma 5.1 a multipartition in the block
+B cannot have this configuration in runners i − 1 and i. Hence y1 ̸= x1 − 1, so y1 + e < x1 and
+
+14
+A. DELL’ARCIPRETE
+φi(y1) = y1 + 1 < x1 − 1 = φi(x1). Thus φi(x1) > φi(yl) for all l, and ¯Λ \ (¯Λ ∩ ¯
+M) > ¯
+M \ (¯Λ ∩ ¯
+M),
+so ¯λ > ¯µ.
+Case 3: x1 does not belong to column i nor to column i−1. By similar arguments we see that φi(x1) =
+x1 > φ(yl) for all l ≥ 2.
+If y1 does not lie in column i−1 then φi(y1) = y1 or φi(y1) = y1 −1. So φi(x1) = x1 > y1 ≥ φi(yl).
+If y1 lies in column i − 1, then φi(y1) = y1 + 1. Since x1 is not in runner i − 1 or in runner i, then
+y1 < x1 −1. So again φi(x1) > φi(y1). Thus φi(x1) > φi(yl) for all l, and ¯Λ\(¯Λ∩ ¯
+M) > ¯
+M \(¯Λ∩ ¯
+M),
+so ¯λ > ¯µ.
+□
+Now we note that if condition (5.1) holds, Φi preserves the Kleshchev property. Indeed, we have the
+following result.
+Lemma 5.4.
+[F07, Lemma 1.9] Suppose λ is a multipartition, and that [λ] has no addable nodes of residue
+i for i ∈ I. Then λ is Kleshchev if and only if Φi(λ) is.
+Given that, we just need to notice that condition (5.1) implies that [λ] has no addable nodes of residue i.
+Hence, we can conclude that if condition (5.1) holds, then Kleshchev multipartitions are preserved by Φi.
+Similarly to Lemma 2.4 in [S91], we have the following.
+Proposition 5.5. Fix i ∈ {1, . . . , e − 1}. Let B be a block of Hr,n such that (5.1) holds and δi(B) ≥ 0.
+Suppose that λ is a r-multipartition of n such that Sλ belongs to the block B. Then
+(1) Dλ ↓Φi(B)∼ δi(B)!DΦi(λ).
+(2) DΦi(λ) ↑B∼ δi(B)!Dλ.
+(3) The blocks B and Φi(B) have the same decomposition matrix.
+(4) The blocks B and Φi(B) have the same Cartan matrix.
+Proof. Let λ1 > λ2 > . . . > λy be the Kleshchev multipartitions whose Specht modules belong to B; then
+Φi(λ1) > Φi(λ2) > . . . > Φi(λy) are the Kleshchev multipartitions whose Specht modules belong to Φi(B)
+by Lemmas 5.3 and 5.4. Suppose
+Sλ ∼
+y
+�
+j=1
+dλλjDλj,
+dλλj ∈ N
+(5.3)
+If λ ⊵ µ, then λ ≥ µ, that is equivalent to say if λ < µ, then λ ⋭ µ. Hence, by point (2) of Theorem 2.4
+we have that if λ < µ, then dλµ = 0.
+Hence, (5.3) becomes
+Sλ ∼
+y
+�
+j=1
+dλλjDλj,
+dλλj =
+�
+1
+if λ = λj
+0
+if λ < λj
+.
+(5.4)
+In particular, Sλ1 = Dλ1. We want to prove points (1) and (2) by induction on l ∈ {1, . . . , y}.
+If l = 1, then by Proposition 5.2 we have
+Sλ1 ↓Φi(B)∼ δi(B)!SΦi(λ1)
+and
+SΦi(λi) ↑B∼ δi(B)!Sλ1,
+so, since Sλ1 = Dλ1 we get the result:
+Dλ1 ↓Φi(B)∼ δi(B)!DΦi(λ1)
+and
+DΦi(λ1) ↑B∼ δi(B)!Dλ1.
+If l > 1, suppose that points (1) and (2) holds for λ1, . . . , λl−1, then we want to prove points (1) and (2)
+for λl. The inductive hypothesis is the following: for j < l we have
+Dλj ↓Φi(B)∼ δi(B)!DΦi(λj)
+and
+DΦi(λj) ↑B∼ δi(B)!Dλj .
+Then
+(Sλl) ↓Φi(B)↑B∼ (δi(B)!)2Sλl ∼ (δi(B)!)2(
+l−1
+�
+j=1
+δλlλjDλj + Dλl),
+
+EQUIVALENCE OF DECOMPOSITION MATRICES FOR BLOCKS OF ARIKI-KOIKE ALGEBRAS
+15
+and,
+(Sλl) ↓Φi(B)↑B∼
+l−1
+�
+j=1
+(δi(B)!)2δλlλjDλj + (Dλl) ↓Φi(B)↑B .
+So
+Dλl ↓Φi(B)↑B∼ (δi(B)!)2Dλl.
+(5.5)
+Now notice that for some αj, βj ∈ N
+Dλl ↓Φi(B) ∼
+l
+�
+j=1
+αjDΦi(λj),
+(5.6)
+DΦi(λl) ↑B ∼
+l
+�
+j=1
+βjDλj.
+(5.7)
+Then, by the inductive hypothesis and using (5.6), (5.7)
+(Dλl) ↓Φi(B)↑B∼
+l−1
+�
+j=1
+(δi(B)!αj + αlβj)Dλj + αlβlDλl.
+(5.8)
+Now combining (5.5) and (5.8), we get
+(δi(B)!)2Dλl ∼
+l−1
+�
+j=1
+(δi(B)!αj + αlβj)Dλj + αlβlDλl.
+and so by the uniqueness of the composition series of (Dλl) ↓Φi(B)↑B we have αlβl = (δi(B)!)2 and αj = 0 = βj
+for all 1 ≤ j ≤ l − 1. Thus, by (5.6) and (5.7) we obtain
+Dλl ↓Φi(B)∼ αlDΦi(λl)
+and
+DΦi(λl) ↑B∼ βlDλj with αlβl = (δi(B)!)2.
+Hence, using Proposition 5.2 and (5.4) we have that
+(Sλl) ↓Φi(B)∼
+l−1
+�
+j=1
+δi(B)!dλlλjDΦi(λj) + αlDΦi(λl)
+and,
+δi(B)!SΦi(λl) ∼ δi(B)!
+
+
+l−1
+�
+j=1
+dΦi(λl)Φi(λj)DΦi(λj) + DΦi(λl)
+
+
+and so we can conclude that dλlλj = dΦi(λl)Φi(λj) for all 1 ≤ j ≤ l − 1, αl = δi(B)! and so βl = δi(B)!.
+Therefore,
+Dλl ↓Φi(B)∼ δi(B)!DΦi(λl)
+and
+DΦi(λl) ↑B∼ δi(B)!Dλl
+and
+SΦi(λl) ∼
+y
+�
+j=1
+dλlλjDΦi(λj).
+(5.9)
+Thus, points 1. and 2. are proved by induction, and points 3. and 4. follow immediately from (5.9).
+□
+Acknowledgements
+This paper forms part of work towards a PhD degree under the supervision of Dr. Sin´ead Lyle at University
+of East Anglia, and the author wishes to thank her for her direction and support throughout.
+
+16
+A. DELL’ARCIPRETE
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+[M99]
+A. Mathas, Iwahori-Hecke algebras and Schur algebras of the symmetric group, University Lecture Series 15, American
+Mathematical Society, Providence, RI, 1999.
+[M04]
+A. Mathas, ‘The representation theory of the Ariki-Koike and cyclotomic q-Schur algebras’, Representation theory of
+algebraic groups and quantum groups, Adv. Stud. Pure Math, Math. Soc. Japan, Tokyo 40 (2004), 261–320.
+[S91]
+J. Scopes, ‘Cartan matrices and Morita equivalence for blocks of the symmetric groups’, J. Algebra 142 (1991), 441–55.
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf,len=1112
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='05153v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='RT]  12 Jan 2023  EQUIVALENCE OF DECOMPOSITION MATRICES FOR BLOCKS OF ARIKI-KOIKE  ALGEBRAS  ALICE DELL’ARCIPRETE  School of Mathematics,  University of East Anglia,  Norwich NR4 7TJ, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' We consider the representation theory of the Ariki-Koike algebra, a q-deformation of the group  algebra of the complex reflection group Cr ≀Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' We examine blocks of the Ariki-Koike algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' In particular,  we prove a sufficient condition such that restriction of modules leads to a natural correspondence between  the multipartitions of n whose Specht modules belong to a block B and those of n − δi(B) whose Specht  modules belong to the block B′, obtained from B applying a Scopes’ equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' This bijection gives us an  equivalence for the decomposition numbers of the Ariki-Koike algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Introduction  Let n be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Let Sn denote the symmetric group of degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' This has the famous Coxeter  presentation with generators T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , Tn−1 and relations  T 2  i = 1,  for 1 ≤ i ≤ n − 1,  TiTj = TjTi,  for 1 ≤ i < j − 1 ≤ n − 2,  TiTi+1Ti = Ti+1TiTi+1,  for 1 ≤ i ≤ n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  If we view this as a presentation for a (unital associative) algebra over a field F, then the algebra we get is  the group algebra FSn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Let q be a non-zero element of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Now we can introduce a ‘deformation’, by replacing the relation T 2  i = 1  with  (Ti + q)(Ti − 1) = 0  for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' The resulting algebra is the Iwahori-Hecke algebra Hn = HF,q(Sn) of the symmetric group Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  This algebra (of which the group algebra FSn is a special case) arises naturally, and its representation theory  has been extensively studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' An excellent introduction to this theory is provided by Mathas’s book [M99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  As long as q is non-zero, the representation theory of Hn bears a remarkable resemblance to the representation  theory of Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Indeed, there are many theorems concerning Hn which reduce representation-theoretic notions  to statements about the combinatorics of partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  In this paper, we consider the representation theory of the Ariki-Koike algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' This algebra is a deforma-  tion of the group algebra of the complex reflection group Cr ≀Sn, defined using parameters q, Q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , Qr ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  The representation theory of these algebras is beginning to be well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' For example, the simple  modules of the Ariki-Koike algebras have been classified;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' the blocks are known;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' and, in principle, the de-  composition matrices of the Ariki-Koike algebras can be computed in characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' A comprehensive  review of the representation theory of the Ariki-Koike algebras can be found in Mathas’s paper [M04].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' In  many respects it seems that the Ariki-Koike algebra behaves in the same way as the Iwahori-Hecke algebra  Hn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' many of the combinatorial theorems concerning Hn have been generalised to the Ariki-Koike algebra,  with the role of partitions being played by multipartitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' In fact, much of the difficulty of understanding  the Ariki-Koike algebra seems to lie in finding the right generalisations of the combinatorics of partitions to  E-mail address: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='Dell-Arciprete@uea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Ariki-Koike algebras, blocks, decomposition numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  1  2  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' DELL’ARCIPRETE  multipartitions - very simple combinatorial notions (such as the definition of an e-restricted partition) can  have rather nebulous generalisations (such as ‘Kleshchev’ multipartitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  In [S91], under some conditions, Scopes establishes a natural correspondence between Specht modules  and simple modules in the blocks B and φi(B) of the symmetric groups where φi is the map swapping the  runners i − 1 and i of each partition in the block B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' This leads to an equivalence of decomposition matrices  for these two blocks, meaning that the blocks B and φi(B) have the same decomposition matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' In the  last part of this paper Scopes proves Donovan’s conjecture for blocks of the symmetric groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' In particular,  given a partition λ of n, the bijection φi gives a Morita equivalence between the blocks B of λ and the  block φi(B) of φi(λ) for the symmetric group algebra FSn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' The generalisation of Donovan’s conjecture and  hence of the Morita equivalence between the blocks B and Φi(B) of Ariki-Koike algebras (with Φi the map  φi extended componentwise) can be found in [AS] where Amara-Omari and Schaps prove this combining  some combinatorics with the Chuang-Rouquier categorification of integrable highest weight modules over  Kac-Moody algebra of affine type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' We would like to underline the fact that Morita equivalence does not  imply the equivalence of decomposition matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Indeed, if we consider n = 8 and p = 3, we have that  the block of the partition (8) and the block of the partition (18) for the symmetric group algebra F3S8 are  Morita equivalent, but they have different decomposition matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  This paper is intended as the Ariki-Koike algebra version of Scopes’ paper about decomposition numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  In particular, we prove a sufficient condition such that two blocks of the Ariki-Koike algebras have the same  decomposition matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' In Section 2, we define the Ariki-Koike algebras and the combinatorical objects that  we will use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Thus, we introduce the r-multipartition of an integer n and we describe the construction of its  abacus configuration with e runners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' In Section 3, we summarise all the relevant results from [F06] and [F07]  concerning weight of multipartitions and core blocks of Ariki-Koike algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' In Section 4, we show that  there is a sequence of multicores with non-increasing weights from a given multicore to a multicore in a core  block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Then, we give a lower bound for the minimal difference between the positions of the lowest beads of  two consecutive runners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' In Section 5, we use this lower bound to get a condition on the weight of the block  B of a multipartition so that no configuration i − 1  i  ✉  appears in the abacus display of any multipartition in  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' In turn, we show that there is a natural correspondence between Specht modules and simple modules in  the blocks B and Φi(B) of the Ariki-Koike algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Basic definitions  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' The Ariki-Koike algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Let r ≥ 1 and n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Let Wr,n be the complex reflection group Cr ≀ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  We can define the Ariki-Koike algebra as a deformation of the group algebra FWr,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Let F be a field and q, Q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , Qr be elements of F, with q non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Let Q = (Q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , Qr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  The Ariki-Koike algebra HF,q,Q(Wr,n) of Wr,n is defined to be the unital associative F-algebra with  generators T0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , Tn−1 and relations  (T0 − Q1) · · · (T0 − Qr) = 0,  T0T1T0T1 = T1T0T1T0,  (Ti + 1)(Ti − q) = 0,  for 1 ≤ i ≤ n − 1,  TiTj = TjTi,  for 0 ≤ i < j − 1 ≤ n − 2,  TiTi+1Ti = Ti+1TiTi+1,  for 1 ≤ i ≤ n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  For brevity, we may write Hr,n for HF,q,Q(Wr,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Define e to be minimal such that 1 + q + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' + qe−1 = 0,  or set e = ∞ if no such value exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Throughout this paper we shall assume that e is finite and we shall  refer to e as the quantum characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Set I = {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=', e − 1}: we will identify I with Z/eZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' We  refer to any r-tuple of integers (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , ar) ∈ Zr as a multicharge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' We say Q is q-connected if, for each  i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , r}, Qi = qai for some ai ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' In [DM02], Dipper and Mathas prove that any Ariki-Koike algebra  is Morita equivalent to a direct sum of tensor products of smaller Ariki-Koike algebras, each of which has  q-connected parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Thus, we may assume that we are always working with a Ariki-Koike algebra with  each Qi being an integral power of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' So we assume that we can find an r-tuple of integers a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , ar)  EQUIVALENCE OF DECOMPOSITION MATRICES FOR BLOCKS OF ARIKI-KOIKE ALGEBRAS  3  such that Qi = qai for each i where q ̸= 1, so that q is a primitive eth root of unity in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' We call such a a  multicharge of Hr,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Since e is finite then we may change any of the ai by adding a multiple of e, and we  shall still have Qi = qai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  For the rest of this section we will fix a multicharge a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , ar) ∈ Zr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Notice that at this stage we  are not requiring that a is a multicharge of the Ariki-Koike algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Multipartitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' A partition of n is defined to be a non-increasing sequence λ = (λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' ) of non-  negative integers whose sum is n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' The integers λb, for b ≥ 1, are called the parts of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' We write |λ| = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Since n < ∞, there is a k such that λb = 0 for b > k and we write λ = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , λk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' We write ∅ for the unique  empty partition (0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' If a partition has repeated parts, for convenience we group them together with  an index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' For example,  (4, 4, 2, 1, 0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=') = (4, 4, 2, 1) = (42, 2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  The Young diagram of a partition λ is the subset  [λ] := {(b, c) ∈ N>0 × N>0 | c ≤ λb}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' An r-multipartition of n is an ordered r-tuple λ = (λ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , λ(r)) of partitions such that  |λ| := |λ(1)| + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' + |λ(r)| = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  If r is understood, we shall just call this a multipartition of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  As with partitions, we write the unique multipartition of 0 as ∅∅∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' The Young diagram of a multipartition  λ is the subset  [λ] := {(b, c, j) ∈ N>0 × N>0 × {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , r} | c ≤ λ(j)  b }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  We may abuse notation by not distinguishing a multipartition from its Young diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' The elements of [λ]  are called nodes of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' We say that a node n ∈ [λ] is removable if [λ] \\ {n} is also the Young diagram of  a multipartion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' We say that an element n ∈ N2 × {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=', r} is an addable node if n /∈ [λ] and [λ] ∪ {n} is  the Young diagram of a multipartition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Given a multicharge a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , ar), to each node (b, c, j) ∈ [λ] we  associate its residue resa(b, c, j) = aj + c − b (mod e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' We draw the residue diagram of λ by replacing each  node in the Young diagram by its residue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Suppose r = 3 and a = (1, 0, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Let λ = ((12), (2), (2, 1)) and µ = ((1), (2, 1), (13)) be two  multipartitions of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' If e = 4, then the 4-residue diagram of [λ] and of [µ] are  1  0  0  1  2  3  1  and  1  0  1  3  2  1  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  For an r-multipartition λ, define the residue set of λ to be the multiset Resa(λ) = {res(n) | n ∈ [λ]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Notice that in the example above  Resa(λ) = {0, 0, 1, 1, 1, 2, 3} = Resa(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Kleshchev multipartitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Residues of nodes are useful in classifying the simple Hr,n-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' In-  deed, the notion of residue helps us to describe a certain subset K of the set of all multipartitions, which  index the simple modules for Hr,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' The multipartions in the set K are called Kleshchev multipartitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  The recursive definition of Kleshchev multipartition can be found in [F07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' For the scope of this paper, it is  enough to consider them as the multipartitions indexing the simple modules of Hr,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Specht module and simple modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' The algebra Hr,n is a cellular algebra [DJM,GL] with the cell  modules indexed by r-multipartitions of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' For each r-multipartition λ of n we define a Hr,n-module Sλ called  a Specht module;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' these modules are the cell modules given by the cellular basis of Hr,n [DJM].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' When  Hr,n is semisimple, the Specht modules form a complete set of non-isomorphic irreducible Hr,n-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  However, we are mainly interested in the case when Hr,n is not semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' In this case, for each Kleshchev  r-multipartition λ, the Specht module Sλ has an irreducible cosocle Dλ, and the Dλ provide a complete set  of irreducible modules for Hr,n as λ ranges over the set of Kleshchev multipartitions of n [A01, Theorem  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' If λ and µ are r-multipartition of n with µ Kleshchev, let dλµ = [Sλ : Dµ] denote the multiplicity of  4  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' DELL’ARCIPRETE  the simple module Dµ as a composition factor of the Specht module Sλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' The matrix D = (dλµ) is called  the decomposition matrix of Hr,n and determining its entries is one of the most important open problems  in the representation theory of the Ariki-Koike algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' It follows from the cellularity of Hr,n that the  decomposition matrix is ‘triangular’;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' to state this we need to define the dominance order on multipartitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Given two multipartitions λ and µ of n, we say that λ dominates µ, and write λ ⊵ µ, if  j−1  �  a=1  |λ(a)| +  i  �  b=1  λ(j)  b  ≥  j−1  �  a=1  |µ(a)| +  i  �  b=1  µ(j)  b  for j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=', r and for all i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Then we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  [DJM,GL] Let F be a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Suppose λ and µ are r-multipartitions of n with µ Kleshchev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  (1) If µ = λ, then [Sλ : Dµ] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  (2) If [Sλ : Dµ] > 0, then λ ⊵ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Note that the dominance order is a partial order on the set of multipartitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' The dominance order is  certainly the ‘correct’ order to use for multipartitions, but it is sometimes useful to have a total order, >, on  the set of multipartitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' The one we use is given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Given two multipartitions λ and µ of n with λ ̸= µ, we write λ > µ if and only if the  minimal j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , r} for which λ(j) ̸= µ(j) and the minimal i ≥ 1 such that λ(j)  i  ̸= µ(j)  i  satisfy λ(j)  i  > µ(j)  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  This is called the lexicographic order on multipartitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Blocks of Ariki-Koike algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' It follows from the cellularity of Hr,n that each Specht module Sλ  lies in one block of Hr,n, and we abuse notation by saying that a multipartition λ lies in a block B if Sλ lies  in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' On the other hand, each block contains at least one Specht module, so in order to classify the blocks of  Hr,n, it suffices to describe the corresponding partition of the set of multipartitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' We have the following  classification of the blocks of Hr,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  [LM07, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='11] Let a ∈ Ir be a multicharge of Hr,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Suppose λ and µ are r-  multipartitions of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Then, Sλ and Sµ lie in the same block of Hr,n if and only if Resa(λ) = Resa(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Continuing the Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='3, we see that the residue sets of λ and µ are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Hence λ and  µ lie in the same block of H3,7 with e = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Induction and restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' If n > 1, then Hr,n−1 is naturally a subalgebra of Hr,n, and in fact Hr,n is  free as an Hr,n−1-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' So there are well-behaved induction and restriction functors between the module  categories of Hr,n−1 and Hr,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Given modules M, N for Hr,n−1 and Hr,n, respectively, we write M ↑Hr,n and  N ↓Hr,n−1 for the induced and restricted modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' If B and C are blocks of Hr,n−1 and Hr,n, respectively,  then we may write M ↑C and N ↓B for the projections of the induced and restricted modules onto B and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  [A96, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='1]  Suppose λ is a multipartition of n − 1, and let n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , ns be the addable nodes of [λ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  For each  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , s, let λ+i be the multipartition of n with [λ+i] = [λ] ∪ {ni}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Then Sλ ↑Hr,n has a filtration  in which the factors are Sλ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , Sλ+s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Suppose λ is a multipartition of n, and let n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , nt be the removable nodes of [λ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' For each i =  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , t, let λ−i be the multipartition of n−1 with [λ−i] = [λ]\\{ni}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Then Sλ ↓Hr,n−1 has a filtration  in which the factors are Sλ−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , Sλ−t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  If a module M has a filtration M = M0 ⊇ M1 ⊇ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' ⊇ Mt−1 ⊇ Mt = 0 with composition factors  Mi/Mi+1 ∼= Ni for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , t − 1, we will write  M ∼  t−1  �  i=0  Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  EQUIVALENCE OF DECOMPOSITION MATRICES FOR BLOCKS OF ARIKI-KOIKE ALGEBRAS  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' β-numbers and abacus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Given the assumption that the cyclotomic parameters of Hr,n are all powers  of q, we may conveniently represent multipartitions on an abacus display.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Fix a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , ar) ∈ Zr to be a  multicharge of Hr,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Let λ = (λ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , λ(r)) be a multipartition of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' For every i ≥ 1 and for every j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , r},  we define the β-number βj  i to be  βj  i := λ(j)  i  + aj − i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  The set Bj  aj = {βj  1, βj  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='} is the set of β-numbers (defined using the integer aj) of partition λ(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' It is  easy to see that any set Bj  aj = {βj  1, βj  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='} is a set containing exactly aj + N integers greater than or equal  to −N, for sufficiently large N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  For each set Bj  aj we can define an abacus display in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' We take an abacus with e infinite  vertical runners, which we label 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , e − 1 from left to right, and we mark positions on runner l and  label them with the integers congruent to l modulo e, so that position (x + 1)e + l lies immediately below  position xe + l, for each x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Now we place a bead at position βj  i , for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' The resulting configuration is  called e-abacus display, or e-abacus configuration, for λ(j) with respect to aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Moreover, we say that the bead  corresponding to the β-number xe + l is at level x for x ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Hence, we can now define the e-abacus display, or the e-abacus configuration, for a multipartition λ  with respect to a to be the r-tuple of e-abacus displays associated to each component λ(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' If it is clear which  e we are referring to, we simply say abacus configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' When we draw abacus configurations we will draw  only a finite part of the runners and we will assume that above this point the runners are full of beads and  below this point there are no beads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Suppose that r = 3, a = (−1, 0, 1) and λ = ((1), ∅, (12)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Then we have  B1  −1 = {−1, −3, −4, −5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  B2  0 = {−1, −2, −3, −4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  B3  1 = {1, 0, −2, −3, −4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  So, the abacus display with respect to the multicharge a for λ when e = 4 is  0  1  2  3  qq  q  qq  q  qq  q  qq  q  ④  ④  ④  ④  ④  ④  ④  qq  q  qq  q  qq  q  qq  q  0  1  2  3  qq  q  qq  q  qq  q  qq  q  ④  ④  ④  ④  ④  ④  ④  ④  qq  q  qq  q  qq  q  qq  q  0  1  2  3  level  qq  q  qq  q  qq  q  qq  q  ④  ④  ④  ④  −2  ④  ④  ④  −1  ④  ④  0  qq  q  qq  q  qq  q  qq  q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Rim e-hooks and e-core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' We recall some other definitions about diagram of a partition since their  generalization for multipartitions will be really useful in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Suppose λ is a partition of n and (i, j) is a node of [λ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  (1) The rim of [λ] is defined to be the set of nodes  {(i, j) ∈ [λ] | (i + 1, j + 1) /∈ [λ]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  (2) Define an e-rim hook to be a connected subset R of the rim containing exactly e nodes such that  [λ] \\ R is the diagram of a partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  (3) If λ has no e-rim hooks then we say that λ is an e-core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  (4) If we can remove w e-rim hooks from [λ] to produce an e-core, then we say that λ has e-weight w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  In particular, an e-core has weight 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  6  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' DELL’ARCIPRETE  An abacus display for a partition is useful for visualising the removal of e-rim hooks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' If we are given an  abacus display for λ with β-numbers in a set B, then [λ] has a e-rim hook if and only if there is a β-number  βi ∈ B such that βi − e /∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Furthermore, removing a e-rim hook corresponds to reducing such a β-number  by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' On the abacus, this corresponds to sliding a bead up one position on its runner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' So, λ is an e-core if  and only if every bead in the abacus display has a bead immediately above it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Using this, we can see that the  definition of e-weight and e-core of λ are well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Finally, we can also notice that each bead corresponds  to the end of a row of the diagram of λ (or to a row of length 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  For multipartitions, by the definition of the β-numbers the node at the end of the row (if it exists) has  residue i if and only if the corresponding bead is on runner i of the abacus for i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Thus, if we increase  any β-number by one, this is equivalent to moving a bead from runner i to runner i + 1 which is equivalent  to adding a node of residue i + 1 to the diagram of a component λ(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Similarly, decreasing a β-number by  one is equivalent to moving a bead from runner i to runner i − 1 which is equivalent to removing a node of  residue i from the diagram of λ(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Background results about weight of multipartitions and core block  Here we summarise the main definitions and some results from [F06] and [F07], mostly concerning weight  of multipartitions and core blocks of Ariki-Koike algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' These are the generalisations to multipartitions  of the notions of e-weight and e-core of partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Weight and hub of multipartitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' In [F06], Fayers generalises the notion of weight from partitions  to multipartitions which will be of great utility in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Fix a multicharge a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , ar) of Hr,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Let λ = (λ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , λ(r)) be a multipartition of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Let ci(λ) denote the number of nodes in  [λ] of residue i ∈ I and aj denote aj (mod e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Define the weight w(λ) of λ to be the non-negative integer  w(λ) =  \uf8eb  \uf8ed  r  �  j=1  caj(λ)  \uf8f6  \uf8f8 − 1  2  �  i∈I  (ci(λ) − ci+1(λ))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  It is also useful to define the hub of a multipartition λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' For each i ∈ I and j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , r}, define  δj  i (λ) =(the number of removable nodes of [λ(j)] of residue i)  − (the number of addable nodes of [λ(j)] of residue i),  and put δi(λ) = �r  j=1 δj  i (λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' The collection (δi(λ) | i ∈ I) of integers is called the hub of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Moreover, we want to highlight that an important feature of the weight and hub of a multipartition: they  are invariants of the block containing λ, and in fact determine the block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  [F06, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='2 & Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='3] Suppose λ is a multipartition of n and µ is a  multipartition of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Then  (1) if λ and µ have the same hub, then m ≡ n (mod e), and  w(λ) − w(µ) = r(n − m)  e  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  (2) if n = m, then λ and µ lie in the same block of Hr,n if and only if they have the same hub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  In view of this result, we may define the hub of a block B to be the hub of any multipartition λ in B,  and we write δi(B) = δi(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Comparing the weight from the abacus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Here we summarise some results from [F06] that shows  how to compare weights of multipartitions efficiently from an abacus display.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  [F06, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='4] Suppose λ = (λ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , λ(r)) is a multipartition, and that λ− is the  multipartition obtained from λ by removing an e-rim hook from some λ(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Then w(λ) = w(λ−) + r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' If λ = (λ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , λ(r)) is a multipartition and each λ(j) is an e-core, then we say that λ is a  multicore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  EQUIVALENCE OF DECOMPOSITION MATRICES FOR BLOCKS OF ARIKI-KOIKE ALGEBRAS  7  Suppose λ = (λ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , λ(r)) is a multicore and a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , ar) is a multicharge of Hr,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' We construct  the corresponding abacus display for λ as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='7, and then for each i ∈ I and 1 ≤ j ≤ r we define  ba  ij(λ) to be the position of the lowest bead on runner i of the abacus for λ(j);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' that is, the largest element of  Bj congruent to i modulo e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Now, fix e ∈ {2, 3, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Let i ∈ I and j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , r}, we define  γjk  i (λ) = 1  e(ba  ij(λ) − ba  ik(λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  γjk  i (λ) may then be regarded as the difference in height between the lowest bead on runner i of the abacus  display for λ(j) and the lowest bead on runner i of the abacus display for λ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Since e is finite, the integers  γjk  i (λ) depend on the choice of a if we change any aj by a multiple of e, but the differences  γjk  il (λ) := γjk  i (λ) − γjk  l (λ)  do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Suppose λ = (λ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , λ(r)) is a multicore, i, l ∈ I and j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' We define sjk  il (λ) to be the  multicore whose abacus configuration is obtained from that of λ by moving a bead from runner i to runner  l on the abacus for λ(j), and moving a bead from runner l to runner i on the abacus for λ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' It is worth  noting that sjk  il (λ) = skj  li (λ) for all i, j, k, l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  [F07, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='6] Let λ be a multicore and let sjk  il (λ) be defined as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Then  (1) sjk  il (λ) has the same hub as λ, and,  (2) w(sjk  il (λ)) = w(λ) − r(γjk  il (λ) − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='5, we may reduce the calculation of the weight of a multicore λ to the case where we  have γjk  il (λ) ≤ 2 because if γjk  il (λ) ≥ 3, we can obtain w(λ) from w(sjk  il (λ)) inductively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Scopes isometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Here we introduce maps between blocks of Ariki-Koike algebras analogous to those  defined by Scopes [S91] between blocks of symmetric groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Suppose i ∈ I, and let φi : Z → Z be the map  given by  φi(x) =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  x + 1  x ≡ i − 1  (mod e)  x − 1  x ≡ i  (mod e)  x  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  φi descends to give a map from I to I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' we abuse notation by referring to this map as φi also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Now suppose λ is a multipartition, and that we have chosen an abacus display for λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' For each j, we define  a partition Φi(λ(j)) by replacing each beta-number β with φi(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Equivalently, we simultaneously remove all  removable nodes of residue i from [λ(j)] and add all addable nodes of [λ(j)] of residue i, or in terms of abacus  configuration we swap the runners (i − 1) and i of each abacus in the abacus display of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' If i = 0, we swap  runners e − 1 and runners 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' We define Φi(λ) to be the multipartition (Φi(λ(1)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , Φi(λ(r))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Here we give an example of the definition of φi for a partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' For e = 3, consider the abacus  configuration of the partition λ given below and apply to λ the map φ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Then,  8  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' DELL’ARCIPRETE  λ  φ1(λ)  0  1  2  qq  q  qq  q  qq  q  ④  ④  ④  ④  ④  ④  ④  ④  ④  ④  qq  q  qq  q  qq  q  −→  0  1  2  qq  q  qq  q  qq  q  ④  ④  ④  ④  ④  ④  ④  ④  ④  ④  qq  q  qq  q  qq  q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  So, it can be seen that the removal and the addition of nodes for λ due to the application of φ1 is simultaneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  [F06, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='6] Suppose B is a block of Hr,n and i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Then there is a block ¯B  of Hr,n−δi(B) with the same weight as B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Moreover, Φi gives a bijection between the set of multipartitions in  B and the set of multipartitions in ¯B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  We write Φi(B) for the block ¯B described in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Core blocks of Ariki-Koike algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Here we introduce the notion of core blocks of Ariki-Koike  algebras, giving several equivalent definitions following the work of Fayers in [F07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' This definition is the one  generalising the e-core notion to multipartitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Suppose that e is finite, and that λ is a multipartition lying in a block B of Hr,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Let κ be a  multicharge of Hr,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' The following are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  (1) λ is a multicore, and there exists a multicharge a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , ar) such that aj ≡ κj (mod e) and  integers α0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , αe−1 such that for each i, j, ba  ij(λ) equals either αi or αi + e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  (2) There is no block of any Hr,m with the same hub as B and smaller weight than B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  (3) Every multipartition in B is a multicore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Now we can make the definition of a core block for the Ariki-Koike algebra Hr,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Suppose B is a block of Hr,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Then we say that B is a core block if and only if e is finite  and the equivalent conditions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='8 are satisfied for any multipartition λ in B,  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='8 gives us several equivalent conditions for a multipartition to lie in a core block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Moreover,  condition (2) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='8 implies that, of the blocks with a given hub, only the one with the smallest  weight is a core block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' So, if λ is a multipartition with this hub, then we may speak of this core block as the  core block of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Now, let λ be a multipartition and κ be a multicharge for Hr,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Recalling the definition of level of a bead  in an e-abacus display given in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='7, define ℓκ  ij(λ) to be the level of the lowest bead on runner i of the  abacus display for λ(j) with respect to κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Note that γjk  i (λ) = ℓa  ij(λ) − ℓa  ik(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='8, we see  that for λ corresponding to Sλ in a core block, there exists a multicharge a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , ar) ∈ Zr such that  ai ≡ κi mod e and integers b0, b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , be−1 such that for each i ∈ I and j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , r}, ℓa  ij(λ) equals either bi  or bi + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' We call such an e-tuple (b0, b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , be−1) a base tuple for λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Adapting Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='8 we have the  following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Suppose λ is a multicore and κ = (κ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , κr) is a multicharge for Hr,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Then Sλ lies  in a core block of Hr,n if and only if there is a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , ar) ∈ Zr such that aj ≡ κj (mod e) and an abacus  configuration for λ such that  |γjk  i (λ)| = |ℓa  ij(λ) − ℓa  ik(λ)| ≤ 1 for each i ∈ I and j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=', r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  EQUIVALENCE OF DECOMPOSITION MATRICES FOR BLOCKS OF ARIKI-KOIKE ALGEBRAS  9  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Assume the conditions of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' If Sλ lies in a core block of Hr,n, then there  exists a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , ar) ∈ Zr such that aj ≡ κj (mod e) and an abacus configuration for λ such that  |δj  i (λ) − δk  i (λ)| ≤ 2 for each i ∈ I and j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=', r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='10, since there exists a multicharge a such that |γjk  i (λ)| ≤ 1 for each i ∈ I and  j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=', r} we have  |δj  i (λ) − δk  i (λ)| = |ℓa  ij − ℓa  i−1,j − (ℓa  ik − ℓa  i−1,k)|  = |ℓa  ij − ℓa  ik + ℓa  i−1,j − ℓa  i−1,k|  ≤ |ℓa  ij − ℓa  ik| + |ℓa  i−1,j − ℓa  i−1,k|  ≤ 1 + 1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Results about multicores  In this section, we give some results concerning properties of multicores that play a fundamental role in  the proof of the main result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' We fix some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Let F be a field, and let q, Q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , Qr be non-zero elements of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Assume that (Q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , Qr) are q-connected  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' For i ∈ I and a multicore m, denote by:  di(m) = min{δj  i (m) | j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=', r}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  If the value of i is clear, we will write d(m) instead of di(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Firstly, we notice an important and useful property of the abacus display of a multicore µ lying in a core  block C of Hr,n and then we give some results about multicores not necessarily in a core block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  If µ is a multipartition lying in a core block C of Hr,n then, by the definition of a base tuple, there exists a  multicharge a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , ar) of Hr,n and at least one base tuple (b0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , be−1) such that µ has abacus display  where for any i ∈ I and j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=', r}  δj  i (µ) ∈ {bi − bi−1 − 1, bi − bi−1, bi − bi−1 + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='1)  Notice that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='1) holds for all the multicores in the core block C since the base tuple is an invariant of a core  block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Fix ¯i ∈ I, ¯i ≥ 1 and a multicharge a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , ar) such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Let (b0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , be−1) be the  corresponding base tuple such that b¯i is as big as possible and b¯i−1 is as small as possible between all the  possible choices of base tuples corresponding to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Then we can define  K¯i := b¯i − b¯i−1 − 1  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='2)  If it is clear what ¯i we are referring to, we will simply write K instead of K¯i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Note that  K ≤ d(µ)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='3)  for all µ ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Suppose e = 5, r = 3 and (Q1, Q2, Q3) = (1, q3, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' So, κ = (0, 3, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Let µ be the multicore  ((4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' 23),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' 2)) which has abacus display with respect to the multicharge a = (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' −2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' 1)  0  1  2  3  4  qq  q  qq  q  qq  q  qq  q  qq  q  ④  ④  ④  ④  ④  ④  ④  ④  ④  ④  ④  ④  ④  ④  ④  qq  q  qq  q  qq  q  qq  q  qq  q  0  1  2  3  4  qq  q  qq  q  qq  q  qq  q  qq  q  ④  ④  ④  ④  ④  ④  ④  ④  ④  ④  ④  ④  ④  qq  q  qq  q  qq  q  qq  q  qq  q  0  1  2  3  4  qq  q  qq  q  qq  q  qq  q  qq  q  ④  ④  ④  ④  ④  ④  ④  ④  ④  ④  ④  ④  ④  ④  ④  ④  qq  q  qq  q  qq  q  qq  q  qq  q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  10  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' DELL’ARCIPRETE  Since |γjk  i (µ)| ≤ 1 for all j, k ∈ {1, 2, 3} and i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , 4}, µ lies in a core block C by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Then we can define a base tuple for µ and thus for C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' In particular, we can consider the following two base  tuples:  (1) (b0, b1, b2, b3, b4) = (2, 4, 2, 3, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  (2) (b′  0, b′  1, b′  2, b′  3, b′  4) = (2, 3, 2, 3, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Take ¯i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' In order to define K1 we need to choose (b0, b1, b2, b3, b4) = (2, 4, 2, 3, 1) because b1 > b′  1 and  b0 = b′  0, so we get K1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Take ¯i = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' In order to define K3 we can choose either (b0, b1, b2, b3, b4) or (b′  0, b′  1, b′  2, b′  3, b′  4) because b3 = b′  3  and b2 = b′  2, so we get K3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  We now want to exhibit, for all multicores of Hr,n, a sequence of multicores of non-increasing weight  ending in a multicore lying in a core block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' In order to do this, we need a preliminary lemma adapted from  the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='7 in [F07] to the case of multicores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' If λ is a multicore not lying in a core block of Hr,n, then there is a sequence λ = λ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , λu  of multicores such that, for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , u − 1, we have λt+1 = sjk  il (λt) for some i, l, j, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Furthermore,  w(λt+1) ≤ w(λt) for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , u − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Let m be a multicore of Hr,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Then there exist a multicore µ in the core block C of m  and a sequence of multicores  m = m0, m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , ms−1, ms = µ  such that:  1) the core block of mt is C for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  2) mt+1 = sjk  il (mt) for some j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=', r}, i, l ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=', e − 1} for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , s − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  3) there exists 0 ≤ v ≤ s such that  i) w(mt+1) < w(mt) for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , v − 1 and w(mt+1) ≤ w(mt) for all t = v, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , s − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  ii) |γjk  il (mv)| ≤ 2 for all i, l and j, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Let m0 := m be a multicore of Hr,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Then, apply the following procedure for t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Calculate γjk  i (mt) for all i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=', e − 1} and j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , r};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' If there is a choice of i, l and j, k such that γjk  il (mt) ≥ 3, set mt+1 = sjk  il (mt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='5  we have  w(mt+1) < w(mt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Repeat this step until we have mt+1 with γjk  il (mt+1) ≤ 2 for all i, l and j, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Suppose that we stop for t + 1 = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Notice that γjk  il (mv) ≤ 2 for all i, l and j, k implies |γjk  il (mv)| ≤ 2 for  all i, l and j, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Indeed,  γjk  il (mv) = −γjk  li (mv) = −γkj  il (mv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Now, if mv is not in the core block C, apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='2 until we get a multicore µ in the core block C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  □  Before stating the main result, we need some preliminary lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Suppose that m is a multicore such that |γjk  il (m)| ≤ 2 for all i, l ∈ I and j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Fix  ¯i ∈ I, and let d be the integer such that d(m) = d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Then  δj  ¯i (m) ∈ {d − 1, d, d + 1} for all j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Consider m a multicore such that |γjk  il (m)| ≤ 2 for all i, l ∈ I and j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Fix ¯i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Then,  since |γjk  il (m)| ≤ 2 for all i, l ∈ I and j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' r}, we have that  |γjk  (¯i−1)¯i(m)| = |δk  ¯i (m) − δj  ¯i (m)| ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Hence, since d(m) = d − 1, we get δj  ¯i (m) ∈ {d − 1, d, d + 1} for all j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=', r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  □  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Let m be a multicore with core block C and such that |γjk  il (m)| ≤ 2 for all i, l ∈ I and  j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Suppose that µ is a multicore in the core block C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Fix ¯i ∈ I, and let d(m) = d − 1 and  d(µ) = d′ − 1 for some integer d and d′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Then d′ ≤ d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  EQUIVALENCE OF DECOMPOSITION MATRICES FOR BLOCKS OF ARIKI-KOIKE ALGEBRAS  11  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' By Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='4, we know that δj  ¯i (m) ∈ {d − 1, d, d + 1} and δj  ¯i (µ) ∈ {d′ − 1, d′, d′ + 1} for all  j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=', r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Let a, b, and c be the number of δj  ¯i (m) equal respectively to d − 1, d, and d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Let a′, b′,  and c′ be the number of δj  ¯i (µ) equal respectively to d′ − 1, d′, and d′ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Notice that a > 0 and a′ > 0 by  definition of d − 1 and d′ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='3, we can go from the multicore m to the multicore µ in the  core block C via a sequence of multicores mt such that mt+1 = sjk  il (mt) for some i, l ∈ I and j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  By point (1) of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='5, we know that each multicore mt occuring in this sequence has the same hub  of m, then m and µ have the same hub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' So,  a(d − 1) + b(d) + c(d + 1) = a′(d′ − 1) + b′(d′) + c′(d′ + 1),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='4)  where a + b + c = a′ + b′ + c′ = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Suppose by contradiction that d′ > d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Then, looking at (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='4) we have  LHS ≤ r(d + 1) with equality if and only if a = b = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  RHS ≥ a′(d + 1) + b′(d + 2) + c′(d + 3) ≥ r(d + 1) with equality if and only if b′ = c′ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  We must have equality in both terms, but this is a contradiction since a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  □  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Let m be a multicore of Hr,n and m′ = sjk  il (m) for some i, l, j, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Fix ¯i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Then d(m′) ≥  d(m) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Moreover, if γjk  il (m) = 1, then d(m′) ≥ d(m) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' The fact that d(m′) ≥ d(m) − 2 follows from the definition of sjk  il and that |δj  ¯i (m) − δj  ¯i (m′)| = 2 if  and only if {i, l} = {¯i − 1,¯i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Suppose γjk  il (m) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Then,  if {i, l} ̸= {¯i − 1,¯i}, then d(m′) ≥ d(m) − 1 since the only case in which δj  ¯i (m′) decreases by 2 with  respect to δj  ¯i (m) is when {i, l} = {¯i − 1,¯i};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  if {i, l} = {¯i − 1,¯i}, then  γjk  ¯i−1,¯i(m) = 1 ⇔ δk  ¯i (m) − δj  ¯i (m) = 1 ⇔ δk  ¯i (m) = δj  ¯i (m) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Thus, δk  ¯i (m′) = δk  ¯i (m) + 2 and δk  ¯i (m′) = δk  ¯i (m) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Hence, d(m′) ≥ d(m) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  □  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Let m be a multicore of Hr,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Suppose that m = m0, m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , mv is a sequence of multicores  such that mt+1 = sjk  il (mt) for some i, l, j, k and w(mt+1) < w(mt) for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , v − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Let w(m) =  w(mv) + hr with h > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Then d(m) ≥ d(mv) − h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' We proceed by induction on v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  If v = 1, the sequence of multicores consists of m0 = m and  m1 = sjk  il (m0) for some i, l, j, k with w(m0) = w(m1) + hr for h > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Note that m0 = sjk  li (m1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' By  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='5(2), the weight of m0 is w(m0) = w(m1) − r(γjk  li (m1) − 2), so h = −γjk  li (m1) + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Moreover,  h > 0 and so we have that γjk  li (m1) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Hence, we just need to check the following two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  If γjk  li (m1) ≤ 0, then d(m0) ≥ d(m1) − 2 by Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='6 and h = −γjk  li (m1) + 2 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Thus,  d(m0) ≥ d(m1) − 2 ≥ d(m1) + γjk  li (m0) − 2 = d(m1) − h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  If γjk  il (m0) = 1 by Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='6, then h = 1 and d(m0) ≥ d(m1) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Thus,  d(m0) ≥ d(m1) − 1 = d(m1) − h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Suppose v > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Let w(m) = w(mv−1) + h′r with 0 ≤ h′ < h and w(mv−1) = w(mv) + h′′r with h′′ > 0  so that h = h′′ + h′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' By induction hypothesis we know that d(m) ≥ d(mv−1) − h′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' In order to get the result  we want to show that d(m) ≥ d(mv) − h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' We know from the base step that that d(mv−1) ≥ d(mv) − h′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Thus,  d(m) ≥ d(mv−1) − h′ ≥ d(mv) − h′′ − h′ = d(mv) − h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  □  12  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' DELL’ARCIPRETE  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Fix ¯i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , e − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Let K = K¯i be the integer defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='2) for a core block C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Suppose that m is a multicore with core block C and weight  w(m) = w(C) + hr  with 0 ≤ h ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Then d(m) ≥ K − h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Let  m = m0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , mv, mv+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , ms = µ  be the sequence defined in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='3, where v is such that |γjk  il (mv)| ≤ 2 for all i, l, j, k, and µ ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='7, we have that  d(m) ≥ d(mv) − h′,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='5)  where 0 ≤ h′ ≤ h is such that w(m) = w(mv) + h′r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  If h′ = h, then mv = ms = µ ∈ C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' therefore d(m) ≥ d(µ) − h ≥ K − h by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Otherwise h′ < h, and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='5 can be applied to get that  d(mv) ≥ d(µ) − 1 ≥ d(µ) − (h − h′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Combining this with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='5), we have:  d(m) ≥ d(mv) − h′ ≥ d(µ) − (h − h′) − h′ = d(µ) − h ≥ K − h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Decomposition numbers for blocks of Hr,n  Now, we want to generalise Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='1 of Scopes’ paper [S91] to the Ariki-Koike algebras Hr,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Thus,  we want to show that, for i ∈ I, restriction of modules leads to a natural correspondence between the  multipartitions of n whose Specht modules belong to B and those of n − δi(B) whose Specht modules belong  to Φi(B), provided that  w(B) ≤ w(C) + Kir  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='1)  where  C is the core block of B,  Ki is the integer defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Notice that this condition is a block condition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=', it is satisfied by all the multipartitions in the block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  When i is understood, we write K instead of Ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Fix i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Let B be a block of Hr,n such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='1) holds and δi(B) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Then, in each  component of every r-multipartition λ of n such that Sλ belongs to the block B, there is no abacus configuration  of the type ✉  in runners i − 1 and i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' First of all, notice that  w(B) − w(C) = ℓr,  for some ℓ ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , K}  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='2)  since w(B) − w(C) can take as values only integral multiples of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Now, consider the multicore m associated to λ, that is the multicore whose abacus display is obtained by  the one of λ sliding all the beads up as high as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Then m has the same core block C of λ and weight  w(m) = w(C) + sr with 0 ≤ s ≤ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Thus, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='8 we can say that  d(m) ≥ K − s ≥ ℓ − s,  since ℓ ≤ K by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Moreover, in order to get λ from m we just need to slide beads down of a total number  of ℓ − s spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' This implies that in each component m(j) of m we need to slide beads down of at most ℓ − s  spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Since d ≥ ℓ − s, similarly to the proof of [S91, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='1] we can conclude that each component λ(j)  of λ has no configuration ✉  in runners i − 1 and i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  □  EQUIVALENCE OF DECOMPOSITION MATRICES FOR BLOCKS OF ARIKI-KOIKE ALGEBRAS  13  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Fix i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Let B be a block of Hr,n such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='1) holds and δi(B) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Suppose that λ  is an r-multipartition of n such that Sλ belongs to the block B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Then the multipartition Φi(λ) of n − δi(B)  is such that SΦi(λ) belongs to Φi(B) and  Sλ ↓Φi(B) ∼ δi(B)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='SΦi(λ),  SΦi(λ) ↑B ∼ δi(B)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='Sλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Suppose that µ is a multipartition of n − δi(B) and Sµ is a factor of Sλ ↓Hr,n−δi(B) using the Specht  filtration given in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' The diagram [µ] can be obtained from [λ] by removing δi(B) nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' The  multiplicity of Sµ as a factor is the number of ways in which the node removal can be effected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' The abacus  of µ is obtained from that of λ by successively moving δi(B) beads one place to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' The module Sµ  belongs to Φi(B) if and only if the overall effect is moving δi(B) beads from each runner i to each runner  i − 1 by point (2) of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='1, in each component of the multipartition λ, we have no abacus configuration of the type  ✉  in runners i−1 and i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' This implies that λ has no addable i-nodes and so Φi consists only of removing  i-nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Hence, the number of ways in which the node removal can be effected is δi(B)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' because they have all  the same residue i and so we can remove these nodes in any order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' This gives the first result about restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  This also shows that there is exacly one µ that can be found by removing δi(B) i-nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Similarly, the second result about induction can be proved by adding i-nodes instead of removing i-  nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  □  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Let i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' If condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='1) holds, then Φi preserves the lexicographic order of multipartitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Let λ and µ be multipartitions whose corresponding Specht modules belong to block B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Let Λ =  (Λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , Λr) and M = (M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , Mr) be the associated sets of β-numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Let ¯λ = Φi(λ) and ¯µ = Φi(µ), and let their corresponding sets of β-numbers be ¯Λ = (¯Λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , ¯Λr) and  ¯  M = ( ¯  M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , ¯  Mr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Now if λ > µ, then as sets of numbers using the lexicographic order Λ > M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Similarly Λ > M implies  that λ > µ and we obtain  λ > µ ⇔ Λ > M ⇔ Λ \\ (Λ ∩ M) > M \\ (Λ ∩ M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Assume λ > µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Let j0 be the minimal j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=', r} such that  Λj \\ (Λj ∩ Mj) > Mj \\ (Λj ∩ Mj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Let Λj0 \\ (Λj0 ∩ Mj0) = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , xt}, with xl > xl+1 for l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , t − 1, and  let Mj0 \\ (Λj0 ∩ Mj0) = {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , ys}, with yl > yl+1 for l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , s − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Then,  ¯Λj0 \\ (¯Λj0 ∩ ¯  Mj0) = {φi(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , φi(xt)}  and  ¯  Mj0 \\ (¯Λj0 ∩ ¯  Mj0) = {φi(y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , φi(ys)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Since λ > µ, it follows that x1 > y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' We have three cases to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Case 1: x1 belongs to column i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' In this case φi(x1) = x1 + 1 > yl + 1 ≥ φi(yl) for all l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Hence  ¯Λ \\ (¯Λ ∩ ¯  M) > ¯  M \\ (¯Λ ∩ ¯  M), so ¯λ > ¯µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Case 2: x1 belongs to column i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Clearly φi(x1) = x1 − 1 ≥ yl + l − 1 > yl + 1 ≥ φi(yl) for all l ≥ 3 as  x1 > y1 > y2 > y3 > · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  If y2 does not belong to column i − 1, then φi(y2) ≤ y2 and x1 − 1 > y2 as x1 > y1 > y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  So φi(x1) = x1 − 1 > y2 ≥ φi(y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  If y2 belongs to column i − 1 then φi(y2) = y2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Since  x1 − 1 and y2 belongs to column i − 1 and y2 < x1 − 1 as above, so y2 ≤ x1 − 1 − e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Hence  φi(x1) = x1 − 1 > y2 + 1 = φi(y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  If y1 lies in column i, then φi(x1) = x1 − 1 > y1 − 1 = φi(y1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' If y1 lies in column k, where  k ̸= i, i − 1, then y1 ≤ x1 − 2, so φi(x1) = x1 − 1 > x1 − 2 ≥ y1 = φi(y1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Suppose y1 lies in  column i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' If y1 = x1 − 1, then the abacus of µ(j0) presents a configuration  ✉  in runners  i − 1 and i where the bead corresponds to y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' However, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='1 a multipartition in the block  B cannot have this configuration in runners i − 1 and i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Hence y1 ̸= x1 − 1, so y1 + e < x1 and  14  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' DELL’ARCIPRETE  φi(y1) = y1 + 1 < x1 − 1 = φi(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Thus φi(x1) > φi(yl) for all l, and ¯Λ \\ (¯Λ ∩ ¯  M) > ¯  M \\ (¯Λ ∩ ¯  M),  so ¯λ > ¯µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Case 3: x1 does not belong to column i nor to column i−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' By similar arguments we see that φi(x1) =  x1 > φ(yl) for all l ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  If y1 does not lie in column i−1 then φi(y1) = y1 or φi(y1) = y1 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' So φi(x1) = x1 > y1 ≥ φi(yl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  If y1 lies in column i − 1, then φi(y1) = y1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Since x1 is not in runner i − 1 or in runner i, then  y1 < x1 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' So again φi(x1) > φi(y1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Thus φi(x1) > φi(yl) for all l, and ¯Λ\\(¯Λ∩ ¯  M) > ¯  M \\(¯Λ∩ ¯  M),  so ¯λ > ¯µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  □  Now we note that if condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='1) holds, Φi preserves the Kleshchev property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Indeed, we have the  following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  [F07, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='9] Suppose λ is a multipartition, and that [λ] has no addable nodes of residue  i for i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Then λ is Kleshchev if and only if Φi(λ) is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Given that, we just need to notice that condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='1) implies that [λ] has no addable nodes of residue i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Hence, we can conclude that if condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='1) holds, then Kleshchev multipartitions are preserved by Φi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Similarly to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='4 in [S91], we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Fix i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , e − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Let B be a block of Hr,n such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='1) holds and δi(B) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Suppose that λ is a r-multipartition of n such that Sλ belongs to the block B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Then  (1) Dλ ↓Φi(B)∼ δi(B)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='DΦi(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  (2) DΦi(λ) ↑B∼ δi(B)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='Dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  (3) The blocks B and Φi(B) have the same decomposition matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  (4) The blocks B and Φi(B) have the same Cartan matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Let λ1 > λ2 > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' > λy be the Kleshchev multipartitions whose Specht modules belong to B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' then  Φi(λ1) > Φi(λ2) > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' > Φi(λy) are the Kleshchev multipartitions whose Specht modules belong to Φi(B)  by Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Suppose  Sλ ∼  y  �  j=1  dλλjDλj,  dλλj ∈ N  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='3)  If λ ⊵ µ, then λ ≥ µ, that is equivalent to say if λ < µ, then λ ⋭ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Hence, by point (2) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='4  we have that if λ < µ, then dλµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Hence, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='3) becomes  Sλ ∼  y  �  j=1  dλλjDλj,  dλλj =  �  1  if λ = λj  0  if λ < λj  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='4)  In particular, Sλ1 = Dλ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' We want to prove points (1) and (2) by induction on l ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  If l = 1, then by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='2 we have  Sλ1 ↓Φi(B)∼ δi(B)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='SΦi(λ1)  and  SΦi(λi) ↑B∼ δi(B)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='Sλ1,  so, since Sλ1 = Dλ1 we get the result:  Dλ1 ↓Φi(B)∼ δi(B)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='DΦi(λ1)  and  DΦi(λ1) ↑B∼ δi(B)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='Dλ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  If l > 1, suppose that points (1) and (2) holds for λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' , λl−1, then we want to prove points (1) and (2)  for λl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' The inductive hypothesis is the following: for j < l we have  Dλj ↓Φi(B)∼ δi(B)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='DΦi(λj)  and  DΦi(λj) ↑B∼ δi(B)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='Dλj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Then  (Sλl) ↓Φi(B)↑B∼ (δi(B)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' )2Sλl ∼ (δi(B)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' )2(  l−1  �  j=1  δλlλjDλj + Dλl),  EQUIVALENCE OF DECOMPOSITION MATRICES FOR BLOCKS OF ARIKI-KOIKE ALGEBRAS  15  and,  (Sλl) ↓Φi(B)↑B∼  l−1  �  j=1  (δi(B)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' )2δλlλjDλj + (Dλl) ↓Φi(B)↑B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  So  Dλl ↓Φi(B)↑B∼ (δi(B)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' )2Dλl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='5)  Now notice that for some αj, βj ∈ N  Dλl ↓Φi(B) ∼  l  �  j=1  αjDΦi(λj),  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='6)  DΦi(λl) ↑B ∼  l  �  j=1  βjDλj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='7)  Then, by the inductive hypothesis and using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='6), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='7)  (Dλl) ↓Φi(B)↑B∼  l−1  �  j=1  (δi(B)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='αj + αlβj)Dλj + αlβlDλl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='8)  Now combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='5) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='8), we get  (δi(B)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' )2Dλl ∼  l−1  �  j=1  (δi(B)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='αj + αlβj)Dλj + αlβlDλl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  and so by the uniqueness of the composition series of (Dλl) ↓Φi(B)↑B we have αlβl = (δi(B)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' )2 and αj = 0 = βj  for all 1 ≤ j ≤ l − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Thus, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='6) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='7) we obtain  Dλl ↓Φi(B)∼ αlDΦi(λl)  and  DΦi(λl) ↑B∼ βlDλj with αlβl = (δi(B)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  Hence, using Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='2 and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='4) we have that  (Sλl) ↓Φi(B)∼  l−1  �  j=1  δi(B)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='dλlλjDΦi(λj) + αlDΦi(λl)  and,  δi(B)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='SΦi(λl) ∼ δi(B)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  \uf8eb  \uf8ed  l−1  �  j=1  dΦi(λl)Φi(λj)DΦi(λj) + DΦi(λl)  \uf8f6  \uf8f8  and so we can conclude that dλlλj = dΦi(λl)Φi(λj) for all 1 ≤ j ≤ l − 1, αl = δi(B)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' and so βl = δi(B)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='.  Therefore,  Dλl ↓Φi(B)∼ δi(B)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='DΦi(λl)  and  DΦi(λl) ↑B∼ δi(B)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='Dλl  and  SΦi(λl) ∼  y  �  j=1  dλlλjDΦi(λj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='9)  Thus, points 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' are proved by induction, and points 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' follow immediately from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  □  Acknowledgements  This paper forms part of work towards a PhD degree under the supervision of Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content=' Sin´ead Lyle at University  of East Anglia, and the author wishes to thank her for her direction and support throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
+page_content='  16  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQfkw0x/content/2301.05153v1.pdf'}
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+Draft version January 11, 2023
+Typeset using LATEX default style in AASTeX63
+Emergent Spectral Fluxes of Hot Jupiters:
+an Abrupt Rise in Day Side Brightness Temperature Under Strong Irradiation
+Drake Deming,1 Michael R. Line,2 Heather A. Knutson,3 Ian J. M. Crossfield,4 Eliza M.-R. Kempton,1
+Thaddeus D. Komacek,1 Nicole L. Wallack,3 and Guangwei Fu1, 5
+1Department of Astronomy, University of Maryland, College Park, MD 20742, USA
+2School of Earth & Space Exploration, Arizona State University, Tempe, AZ 85287, USA
+3Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125, USA
+4Department of Physics and Astronomy, University of Kansas, Lawrence, KS 66045, USA
+5Present address: Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218 USA
+ABSTRACT
+We study the emergent spectral fluxes of transiting hot Jupiters, using secondary eclipses from
+Spitzer. To achieve a large and uniform sample, we have re-analyzed all secondary eclipses for all
+hot Jupiters observed by Spitzer at 3.6- and/or 4.5 µm. Our sample comprises 457 eclipses of 122
+planets, including eclipses of 13 planets not previously published. We use these eclipse depths to
+calculate the spectral fluxes emergent from the exoplanetary atmospheres, and thereby infer temper-
+ature and spectral properties of hot Jupiters. We find that an abrupt rise in brightness temperature,
+similar to a phase change, occurs on the day side atmospheres of the population at an equilibrium
+temperature between 1714 K and 1818 K (99-percent confidence limits). The amplitude of the rise is
+291 ± 49 Kelvins, and two viable causes are the onset of magnetic drag that inhibits longitudinal heat
+redistribution, and/or the rapid dissipation of day side clouds. We also study hot Jupiter spectral
+properties with respect to metallicity and temperature inversions. Models exhibiting 4.5 µm emis-
+sion from temperature inversions reproduce our fluxes statistically for the hottest planets, but the
+transition to emission is gradual, not abrupt. The Spitzer fluxes are sensitive to metallicity for planets
+cooler than ∼ 1200 Kelvins, and most of the hot Jupiter population falls between model tracks having
+solar to 30X-solar metallicity.
+Keywords: Exoplanet astronomy- Exoplanet atmospheres - Hot Jupiters - Infrared astronomy
+1. INTRODUCTION
+The Spitzer Space Telescope leaves a rich legacy of transiting exoplanetary science. Spitzer investigators obtained
+photometry at secondary eclipse for many transiting planets (Deming & Knutson 2020). Using Spitzer’s photometry,
+atmospheric properties such as metallicity, C/O, and temperature structure, have been inferred for dozens of hot
+Jupiters (e.g., Harrington et al. 2007; Charbonneau et al. 2008; Knutson et al. 2008; Fressin et al. 2010; Madhusudhan
+et al. 2011; Todorov et al. 2013; Beatty et al. 2014; Mansfield et al. 2018), albeit not without some debate (Hansen
+et al. 2014; Ingalls et al. 2016) and some revisions (Knutson et al. 2012; Diamond-Lowe et al. 2014; Line et al. 2016).
+The physical properties of hot Jupiters were recently reviewed by Fortney et al. (2021).
+Most of Spitzer’s exoplanetary eclipse observations comprise photometry in broad spectral bands. Atmospheric
+inferences using photometry can be problematic because photometric spectral resolution is insufficient to identify
+molecular absorptions in emergent spectra via band shapes. Moreover, if hot Jupiters vary widely in their proper-
+ties, then studies of individual planets may struggle to accurately infer average properties of the entire hot Jupiter
+population. One solution to these problems is to focus on population studies of relatively large samples of plan-
+ets. Population studies of emergent fluxes have been pursued using eclipses in the Spitzer data (Wallack et al. 2019,
+Corresponding author: Drake Deming
+ldeming@umd.edu
+arXiv:2301.03639v1  [astro-ph.EP]  9 Jan 2023
+
+2
+Deming et al.
+Garhart et al. 2020 - hereafter G20, Baxter et al. 2020 - hereafter B20, Wallack et al. 2021 - hereafter W21, and Goyal
+et al. 2021), and also using eclipses observed with HST (Mansfield et al. 2021).
+1.1. Motivation for This Work
+Although analyses of Spitzer’s secondary eclipses have been extensive, there are 13 hot Jupiters (listed below) whose
+eclipses have not yet been published. In some cases, hot Jupiters with published eclipse analyses have additional
+unanalyzed eclipses in the archive. Moreover, there is arguably value to re-analyzing all of the eclipses using one
+uniform method. To that end, we have re-analyzed all of Spitzer’s secondary eclipse data in the 3.6- and 4.5 µm bands
+using one method. These two bandpasses encompass the vast majority of Spitzer’s secondary eclipse measurements
+of hot Jupiters.
+We seek to establish a population-level context for observations of hot Jupiters by JWST, and by high-resolution
+ground-based spectroscopy using the ELTs. The context that we seek is to establish the major properties of the hot
+Jupiter population that can be deduced from their emergent spectral fluxes as observed by Spitzer. Emergent fluxes
+are basic observable properties of the planets, and we aim for a simple and robust analysis.
+1.2. Organization of This Paper
+Section 2 describes the broad strategy of our archival analysis. The sample of planets that we analyze is summa-
+rized in Section 2.1, and our technique for re-analyzing Spitzer’s secondary eclipse data to produce emergent fluxes
+is described broadly in Section 2.2, and in detail in the Appendix. We interpret the emergent fluxes using a grid
+of models that are described in Section 3. The magnitudes and observed properties of the emergent fluxes are de-
+scribed in Section 4, and the spectral properties by comparing fluxes versus wavelength are explored in Section 5.
+Section 6 discusses the atmospheric implications of our results, beginning with an acknowledgement of previous
+work in Section 6.1. The potential causes of an abrupt increase in day side flux are explored in Sections 6.2 to Sec-
+tion 6.6. Section 6.7 describes the implications for atmospheric metallicities, and spectral emission from temperature
+inversions is discussed in Section 6.8. Section 7.1 summarizes our results, and Section 7.2 notes the need for future
+work.
+2. AN ARCHIVAL ANALYSIS
+2.1. The Sample of Planets
+Our sample comprises 122 hot Jupiters, and we have re-analyzed every eclipse observed for those planets using
+Spitzer at 3.6- and/or 4.5 µm. That includes eclipses that occur during phase curve observations. Planets whose
+Spitzer eclipses were not previously published include: WASP-11b, -21b, -28b, -32b, -35b, -37b, -38b, -50b, -95b,
+-107b, -122b, HAT-P-34b, and HD 202772b. In total, we derive eclipse depths for 439 individual eclipses, and we
+perform some averaging for 18 additional eclipses of four planets. Our derived eclipse depths are listed in Table 1.
+We use the eclipse depths to derive emergent day side fluxes, and brightness temperatures (Tb), listed in Table 2. The
+source of stellar and planetary parameters used in our analysis (TEPCat) is ever-evolving. Hence, we list the stellar
+and planetary radii and temperatures that we adopted in Table 3. In addition to our re-analyzed eclipses, our analysis
+in this paper makes use of Spitzer eclipses observed at 8 µm, but we have not re-analyzed those eclipses, nor have
+we listed their emergent fluxes in Table 2. Our analysis also considers eclipses from 2 µm ground-based observations
+reported in the literature (i.e., Croll et al. 2015, and many others).
+2.2. Derivation of Eclipse Depths, Emergent Fluxes, and Brightness Temperatures
+We derive eclipse depths using the method described by G20, in fact using the same code. This method separates
+systematic signals due to the IRAC instrument from the eclipse signal using pixel-level-decorrelation (PLD, Deming
+et al. 2015). Some minor differences from the G20 methodology are as follows. First, to increase the practicality of
+our analysis, we perform aperture photometry on the Spitzer images using only apertures of fixed radius (e.g., not
+variable radii as per Lewis et al. 2013), and centering the apertures using a 2-D Gaussian fit to the stellar image (i.e.,
+not using flux-weighted centering, Piskorz et al. 2018). This reduces the volume of photometry by a factor of four,
+and G20 found that fixed radii apertures with Gaussian centroiding was their best method, and was consistently
+good for all of the planets in their sample. G20 also derived two depth solutions for each eclipse: one depth that
+maximized the fit to the Allan deviation relation for the photometric residuals (see their Section 3.4), and another
+depth derived from the centroid of the posterior distribution of eclipse depth. They preferred the former method
+
+Spectral Fluxes of Hot Jupiters
+3
+in their analysis (but they commented that the population-level results were not sensitive to the choice). Using our
+larger sample, we find that the centroid of the posterior distribution is preferable, but we have also verified that
+the conclusions of this paper would not change if we used the alternate choice of eclipse depth. There are other
+(minor) technical issues that arise in our analysis, and we include a more comprehensive technical discussion in the
+Appendix.
+We believe that our results are an improvement over previous Spitzer population studies in 5 ways: 1) We have
+added 13 additional planets to the sample, 2) We analyze the maximum number of eclipses per planet (previous
+population studies have sometimes relied on literature results that do not include all of the eclipses per planet, e.g.,
+see the discussion of HD 189733b in Section C of the Appendix). 3) Our results are homogenous, using the same
+method to extract all of the eclipse depths (whereas the literature is inhomogenous, with the early Spitzer papers
+using much more simplistic data analysis methods), 4) a systematic comparison of seven data analysis techniques
+(Ingalls et al. 2016) concluded that our PLD method was among the three most accurate methods, and had the least
+bias of the seven. Finally, 5) we have consulted recent high spatial resolution imaging results to revisit dilution
+corrections needed for eclipses where those corrections were not previously applied (Appendix Section B).
+For the host stars, we adopt stellar effective temperatures, radii, and surface gravities from TEPCat1, and also
+planetary equilibrium temperatures, radii, and orbital distances from TEPCat (Southworth 2011). We interpolate in
+a grid of ATLAS9 models (Castelli & Kurucz 2003) to determine the fluxes of the host stars, integrating over the band-
+pass sensitivity functions of Spitzer’s IRAC instrument. We derive an emergent flux at the top of the exoplanetary
+atmosphere, using the measured eclipse depth and the stellar fluxes from the ATLAS models. To derive planetary
+brightness temperatures, we calculate eclipse depths for a grid of blackbody planets, integrating over the IRAC band-
+passes, and we interpolate the measured eclipse depth in that grid to find the brightness temperature. To calculate
+uncertainties, we propagate the uncertainty of the observed eclipse depths through the process to produce error bars
+on flux and brightness temperature. The planetary fluxes and brightness temperatures are given as averages per
+planet per waveband in Table 2.
+In some instances, especially for weak eclipses, the error bars on eclipse depth can allow negative depths. Negative
+eclipse depths would formally imply negative brightness temperatures (as per some ranges in Table 2). Although
+negative eclipse depths and brightness temperatures are not physical, it is necessary to allow them, so that population
+averages and trends are free of Lucy-Sweeney bias (Lucy & Sweeney 1971).
+Our error bars on Tb for the planets listed in G20 are significantly less than the G20 values. The G20 Tb error
+bars were calculated using a Rayleigh-Jeans approximation, which is insufficiently accurate. That was a mistake by
+one of us (D.D.), and we suggest that the values in Table 2 should be used in preference to the G20 error bars. Our
+current error bars in the Table were calculated using the same method as B20, i.e. propagating the eclipse depth
+errors through the temperature dependence of the Planck function integrated over Spitzer’s bandpass responses.
+3. ATMOSPHERIC MODELS
+We interpret our results using a set of 1D self-consistent radiative-convective-thermochemical equilibrium mod-
+els. The models were previously described and used in Piskorz et al. (2018), Arcangeli et al. (2018), Mansfield et al.
+(2018), Kreidberg et al. (2018), Gharib-Nezhad & Line (2019), Mansfield et al. (2021), and recently benchmarked
+against M-dwarfs in Iyer et al. (2022). Briefly, the models solve the 1D radiative-convective equilibrium problem
+via a Newton-Raphson iteration to achieve zero net flux divergence across each layer interface (see McKay et al.
+1989). Layer-by-layer radiative fluxes are computed using the two stream source function method from Toon et al.
+(1989) (without clouds/scattering) and convective flux is computed via mixing length theory. Equilibrium chem-
+istry with rainout (depletion of elemental species in overlying layers due to the onset of condensate formation) is
+computed using the NASA CEA2 routine (McBride & Gordon 1996). We include ∼ 40 line and continuum opacity
+sources accounting for species relevant in chemical equilibrium from 300 - 4000 Kelvins covering 0.1 - 100 µm (UV
+species included are neutral atomics/ions, H2, CO, and SiO), with molecular species sourced from the latest ExoMol
+(Tennyson et al. 2020) line lists and neutral/ion atomic species from the Kurucz line-list database. Line opacities are
+computed using the methods described in Gharib-Nezhad et al. (2021) and then are converted into R=250 correlated-
+K coefficients (using a 10 point double-Gauss quadrature sampling for the bin weights/ordinates, e.g., Molli`ere et al.
+2015) with mixed opacities computed on-the-fly using the RORR method described in Amundsen et al. (2017).
+1 https://www.astro.keele.ac.uk/jkt/tepcat/
+
+4
+Deming et al.
+Figure 1. Examples of modeled emergent flux for plan-
+ets with a range of temperature, and in the wavelength
+region encompassing Spitzer’s 3.6- and 4.5 µm band-
+passes (response curves at the bottom of the panels).
+The left panel shows models having solar metallicity,
+and the right panel is 30-times solar metallicity for the
+same planets. The planets and equilibrium tempera-
+tures are: WASP-121b (Teq = 2358 K), XO-4b (1641 K),
+WASP-39b (1166 K), and HAT-P-12b (955 K). The red
+tic marks at 4.5 µm call attention to carbon monoxide
+emission from a temperature inversion in WASP-121b.
+The models are in two groups: first, we calculate model tracks using a planet of 1-Jupiter mass and radius, ir-
+radiated by a solar-type star at a range of orbital distances. For these tracks, we adopt solar abundances in the
+exoplanetary atmosphere, and either complete re-distribution of stellar irradiance, or no re-distribution (these are
+same models as in Mansfield et al. 2021). We found the no-redistribution case to be especially useful in our analyses,
+and we henceforth refer to that case as the no redistribution (NR) track. Our second group of models is constructed
+for individual planets, using their measured masses and radii, and the temperatures and radii of their host stars;
+these are “population”-level models. In all cases, we use PHOENIX model atmospheres to specify the irradiating
+spectrum from the star. We checked that using PHOENIX stellar models to irradiate the planets, versus ATLAS9
+stellar models to calculate emergent fluxes (Section 2.2) did not produce significant inconsistencies - see Section D of
+the Appendix.
+The models for individual planets vary the heavy element abundance (metallicity), using both solar metallicity,
+and 30-times (30X) solar. Because both groups of models omit the presence of clouds, their atmospheres are dark,
+with albedos near zero. Emergent fluxes of four planets in this group of models are illustrated in Figure 1, that shows
+some important properties of the modeled fluxes. Spectral absorption features weaken as temperature increases, and
+carbon monoxide appears in emission due to an atmospheric temperature inversion in the hottest planets. Increas-
+ing metallicity changes the nature of the absorption in Spitzer’s bands. At solar metallicity, methane absorption is
+prominent in the 3.6 µm band for the coolest models. At high metallicity, carbon becomes preferentially bound in
+carbon monoxide and carbon dioxide (Moses et al. 2013). That effect is also apparent on Figure 1, where the coolest
+30X metallicity models exhibit absorption in the 4.5 µm bandpass, not as much at 3.6 µm.
+4. EMERGENT FLUXES AND DAY SIDE TEMPERATURE
+The emergent flux of a hot Jupiter in a given spectral band is determined by the vertical atmospheric temperature
+gradients and the specifics of the opacity sources that are significant in that band. The day side brightness temper-
+ature in a given spectral band is thereby diagnostic of the temperature structure of the atmosphere, and how it may
+change as a function of stellar irradiance, and how that heating is redistributed by longitudinal circulation.
+Longitudinal heat circulations for individual hot Jupiters are best studied using thermal infrared phase curves, and
+those studies have been extensive (e.g., Knutson et al. 2007, 2012; Wong et al. 2015; Stevenson et al. 2017; Kreidberg
+et al. 2018; Bell et al. 2021; May et al. 2021, 2022; Dang et al. 2022). Phase curves can delineate emergent flux over
+the full range of planetary longitude, whereas eclipses yield only the day side flux. Nevertheless, it has long been
+known that the day side fluxes can yield important statistical information on the nature of heat circulation for hot
+
+TTTTTT
+10
+WASP-121b
+XO-4b
+WASP-39b
+CO
+HAT-P-12b
+10
+10
+3
+4
+5
+3
+4
+5
+Wavelength (microns)Spectral Fluxes of Hot Jupiters
+5
+Figure 2. Observed and modeled fluxes for hot Jupiters
+at secondary eclipse, in both the Spitzer 3.6- and 4.5 µm
+bands, versus planetary Teq and Tirr.
+Fluxes from
+the no redistribution (NR) model are also plotted as
+a track, as well as the tracks for two blackbodies. The
+upper blackbody adopts zero albedo and no heat re-
+distribution, whereas the lower blackbody adopts zero
+albedo and heat redistribution over the entire planet.
+The vertical dashed lines mark the 99% confidence re-
+gions for the quasi-discontinuous increase in flux (see
+text for discussion).
+Jupiters (Cowan & Agol 2011). Although phase curves are numerous, eclipse data are even more numerous. Eclipse
+fluxes provide an opportunity to probe physical effects on the day side atmosphere with maximum resolution in
+stellar irradiance as we now demonstrate.
+Figure 2 shows the emergent spectral fluxes that we derive for hot Jupiters at both 3.6- and 4.5 µm, plotted as a
+function of the equilibrium and irradiation temperatures, Tirr. 2 Note that analyzing fluxes at each wavelength allows
+us to make maximum use of planets observed in only one Spitzer bandpass (e.g., 4.5 µm, program 14059).
+For equilibrium temperature (Teq), we adopt an albedo of zero and complete redistribution of heat. Equilibrium
+temperature is less than irradiation temperature by a factor of
+√
+2. Our values refer to the emergent flux density at
+the top of the exoplanetary atmosphere, so they are independent of the planetary radius. We derive uncertainties on
+the fluxes by propagating uncertainties in the eclipse depth, stellar temperature, and planetary radius. Uncertainty
+in the planetary radius enters only via the ratio of planetary-to-stellar radius. That ratio contributes negligibly to
+the flux uncertainty because transit depths are known much more precisely than eclipse depths. We similarly derive
+uncertainties in the equilibrium and irradiation temperatures based on uncertainties in the stellar temperature and
+planetary orbit.
+Fluxes from the no redistribution (NR) models, and two blackbodies, are overplotted on Figure 2. The NR model
+track adopts no redistribution of stellar irradiance, solar composition, and surface gravity equal to Jupiter’s. Temper-
+ature via absorbed irradiation is the overwhelmingly dominant factor that determines the flux in the model track. We
+2 irradiation temperature is Tirr = Ts
+√Rs/a, where Ts and Rs are the stellar temperature and radius, and a is the orbital distance of the planet in a
+circular orbit.
+
+Irradiation temperature
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+4.5 μm
+10~5
+10F6
+99% confidence limits
+10
+3.6 μm
+NR model
+Blackbodies
+10°
+500
+1000
+1500
+2000
+2500
+Equilibrium temperature6
+Deming et al.
+Figure 3. Observed and modeled fluxes for hot Jupiters
+at secondary eclipse, in both the Spitzer 8 µm bands,
+and ground-based eclipses in the K-band (2.3 µm), ver-
+sus planetary Teq and Tirr.
+No redistribution (NR)
+model and blackbody tracks are overplotted as per Fig-
+ure 2. The vertical dashed lines mark the 99% confi-
+dence regions for the quasi-discontinuous increase in
+flux (see text for discussion).
+experimented with plotting models tailored to the parameters inferred for individual planets (i.e., surface gravity,
+composition), and thereby verified that those parameters have a minor (albeit detectable) effect in this context. The
+blackbody tracks on Figure 2 correspond to zero albedo and no heat redistribution, and also heat redistribution over
+the entire planet.
+Figure 3 shows fluxes that we derive from literature values of eclipses observed using the next two most widely-
+used bands, i.e. Spitzer’s 8 µm band, and ground-based eclipses in the K-band (e.g., Kov´acs & Kov´acs 2019, and many
+others), and also the NR model track and blackbody tracks.
+Both Figures 2 and 3 exhibit similar behavior in the Spitzer bands. For Teq exceeding ∼ 1700 K, the planetary fluxes
+are largely confined between the two blackbody curves. Below Teq ∼ 1700 K, fluxes can fall below the lower blackbody
+curve. This general behavior has been seen previously, for example by G20 (their Figure 16), and Goyal et al. (2021).
+Fluxes below the lower blackbody curve require either molecular absorption, or a non-zero albedo (e.g., reflecting
+clouds) that lower the atmospheric temperature below the complete redistribution case, or cold clouds that emit little
+thermal radiation. The Figures suggest that the transition from fluxes that drop below the lower blackbody, to fluxes
+that remain above it, occurs rather abruptly near Teq ∼ 1700 K. It has long been known that the hottest of the hot
+Jupiters circulate heat with the least efficiency, and thereby have the hottest day side temperatures (Cowan & Agol
+2011; Schwartz & Cowan 2015; Showman et al. 2020). Below, we define this rise in emergent flux level quantitatively,
+using the largest possible collection of Spitzer’s secondary eclipses.
+4.1. An Abrupt Rise in Flux
+
+Irradiation temperature
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+10-5
+8 μm
+99% confidence limits
+10
+2 μm
+10'5
+NR model
+Blackbodies
+10
+500
+1000
+1500
+2000
+2500
+Eguilibrium temperatureSpectral Fluxes of Hot Jupiters
+7
+We here demonstrate that an abrupt rise (i.e., nearly discontinuous versus Teq) in day side emergent flux occurs at
+an equilibrium temperature between 1714 K to 1818 K. In this section, we establish the data-related aspects of the
+rise, such as amplitude and statistical significance. In Section 6 we explore the physical implications for the day side
+atmosphere.
+First, we note that the 2 µm fluxes behave differently than the Spitzer fluxes. They occur more often above the NR
+model, which is physically difficult to explain. Ground-based eclipse observations are difficult, and it seems possible
+that they may be especially affected by selection bias. Observers may be reluctant to publish eclipses that give weak
+detections (and consequently low fluxes), but they may readily publish strong eclipses. Hence, we hypothesize that
+the ground-based eclipses may suffer from selection bias. In order to further investigate the abrupt rise in emergent
+flux, we combine the three Spitzer bands, but for now we omit the 2 µm band. Figure 4 plots the ratio of the emergent
+flux in the Spitzer bands to the value from the NR model track, versus Teq, with each point being a single eclipse.
+The NR model track is smooth (Figure 2 and Figure 3), so this division removes the gradual increase in flux with
+increasing temperature, but it does not create the abrupt rise in flux: that abrupt rise is caused by the observed data,
+not by the models.
+4.1.1. Partial Eclipses and Eccentric Planets
+Figure 4 omits the planets WASP-34b (Smalley et al. 2011), WASP-67b (Hellier et al. 2012), and WASP-140b (Hellier
+et al. 2017), because their eclipses are partial, and therefore the eclipse depth cannot be converted reliably to an
+emergent flux at the top of their atmospheres. We do include TrES-3b, because O’Donovan et al. (2007) concluded
+that it is not partial, although it is close. Consequently, we include it but we increase its error bars by calculating
+the fraction of flux that would be missed when varying the orbital parameters within the error ranges given by
+O’Donovan et al. (2007).
+Planets with eccentric orbits may have day side temperatures at eclipse that are not representative of their equilib-
+rium temperature averaged over their orbits. This primarily affects HD 80606b and HAT-P-2b, due to their substantial
+orbital eccentricities (Moutou et al. 2009; Bakos et al. 2007). For these two planets, we replaced their TEPCat values
+of Teq with the day side temperatures calculated at eclipse in the time-dependent models of Mayorga et al. (2021)
+(see Table 3). Planets with less extreme eccentricities (e.g., WASP-14b, Joshi et al. 2009) are included using Teq values
+from TEPCat.
+4.1.2. Amplitude and Location of the Flux Rise
+In order to investigate the magnitude and location of the emergent flux rise, we apply the Bayesian methodology
+described by Lee & Heghinian (1977) (hereafter, LH) to the ratio data of Figure 4. The LH method is specifically
+tailored to find abrupt changes in the values of serial data. It adopts a uniform prior for the change location, and
+assumes that the data values and change amplitude have Gaussian distributions. It gives the posterior distributions
+for the change magnitude and location as analytic expressions involving sums over the data values. (The posterior
+distributions are not Gaussian, or even necessarily smooth.)
+The LH algorithm detects an abrupt rise in the ratio of the emergent flux to the NR model. The distribution for
+temperature where the rise in flux ratio occurs is sharply peaked at Teq = 1730 K, as shown on Figure 4. The rise in
+flux is abrupt, i.e. not clearly resolved in the temperature coordinate in spite of using 437 eclipses. The posterior
+distribution for the rise amplitude has a mean value of 0.12 ± 0.0157 in the ratio of flux to NR model (from 0.58
+to 0.70), so the rise is nominally detected at the 7.6σ level of significance, assuming that the data are normally
+distributed. As a test, we impose a strong (and physically motivated) prior on the flux ratio by using only the flux
+ratios per eclipse that are between 0 and 1. That range would encompass low fluxes (e.g., due to high, cold clouds),
+ranging to high fluxes from a hot day side with no redistribution, and assuming a blackbody spectrum. In that case,
+the location and amplitude of the flux rise are not significantly changed, but the statistical significance would rise to
+9σ, principally because ratios above unity are eliminated.
+Although we do not include the 2 µm eclipses in our adopted solution, we explored whether including them would
+affect the result. In this case, we reject only the eclipse of WASP-10b at 2 µm, that is an extreme outlier (Cruz et al.
+2015), at 2.65 times the NR model. Including the balance of the 2 µm eclipses in the LH analysis does not significantly
+change the location of the flux rise, and the statistical significance actually rises to 7.8σ. Although it is informative
+that the 2 µm eclipses support our result from the Spitzer eclipses, we continue to base our analysis exclusively on
+the Spitzer data.
+
+8
+Deming et al.
+4.1.3. Three Tests
+We did three additional tests to verify the existence of an abrupt rise in flux on Figure 4. To verify the statistical
+significance of the rise, we made 106 synthetic realizations of the flux ratio data. Those realizations preserved the Teq
+values and the relative differences among multiple eclipses for the same planet, but they varied the average flux ratios
+for each planet. We adopted a uniform (not Gaussian) distribution for the flux ratios, and distributed them between
+0 and 1 (see above), with no trends or rises in average value. Because we adopted a uniform distribution for these flux
+ratios, not a Gaussian distribution assumed by the LH method, these synthetic data represent a ”worst case” for the
+rise analysis, assuming that there is no physical reason to justify ratios above unity (i.e., no planets hotter than the
+no redistribution case). Only 0.028% of these simulations produced a detected rise at a significance equal to the real
+data and we conclude that the flux rise is significant at greater than a 99.97% level of confidence.
+Our second test explored whether the flux rise might be gradual, rather than abrupt. We again constructed 106
+realizations of the data, preserving the Teq values and relative differences among eclipses of the same planet. We
+scattered the average flux ratio for each planet around a gradual ramp with a scatter matching the real data. The
+ramp varied between the two flux ratio levels detected in the real data, and spanning the range between 1000 K and
+2500 K in Teq. The purpose of these realizations is to investigate whether the LH algorithm could be fooled into mis-
+identifying a gradual flux rise as being abrupt. We found that only 0.009% of the simulations were mis-identified
+as being abrupt. We also did a third test, where we again made 106 realizations of the data that scattered around a
+slope consistent with the real data, but with no restriction to vary between two flux ratio levels. This test also found
+only 0.009% of the simulations were mis-identified as being abrupt. Hence, we trust the abrupt nature of the flux
+rise inferred from the LH algorithm. We also conducted further tests of the LH algorithm, as described below.
+Figure 4 contains many points, and it may be difficult to see the abrupt flux rise when so many points are plotted.
+To make the flux rise visually clearer, Figure 5 bins the 3.6- and 4.5 µm points from Figure 4 in arbitrary intervals of
+100 Kelvins in equilibrium temperature. The formulation of the data on Figure 5 is not used directly in our analysis;
+that Figure is intended merely to make the abrupt rise in flux more clear to the eye.
+4.2. Location of the Abrupt Rise in Flux
+The LH algorithm has been widely used in terrestrial meteorology (e.g., to detect rapid changes in weather pat-
+terns), but it is not often used in astronomy. We further tested the methodology using more Monte Carlo simulations
+applied to both the real data as well as synthetic data. We verified that, as the transition in flux level is broadened in
+temperature, the LH posterior distribution also broadens. We explored the effect of dithering the data in both X- and
+Y-coordinates, using the observational errors in both observed flux ratio and temperature. We found that the obser-
+vational errors in temperature are not sufficient to significantly broaden the posterior distribution in rise location,
+and the effect of the errors in temperature is included on Figure 4.
+The observed planets are not distributed uniformly in temperature, and we hypothesized that the sharpness of
+the LH posterior distribution in flux-rise temperature could be an indirect artifact of the temperature distribution
+of the planet population. We therefore perturbed the real data, forcing the flux rise at Teq = 1730 K to occur over
+broader ranges. We broadened a range of the data near 1730 K over intervals of 50, 100, 150, 200, 250, and 300 K.
+Broadening the data means that we adjusted the flux ratios (but not the equilibrium temperatures) to scatter around
+an average linear ramp within those intervals. As we increased the width of the ramp, the 1730 K peak in the posterior
+distribution decreased in amplitude (as expected), and a second peak became evident near 1800 K.
+Based on all the numerical experiments described above, we conclude that the flux rise at Teq = 1730 K is statis-
+tically robust and abrupt, and not a random or gradual fluctuation. The sharp peak in the posterior distribution at
+1730 K accurately reflects the specific distribution of equilibrium temperature and emergent flux in our actual data.
+However, we also conclude that small and statistically plausible variations in our data are consistent with the flux rise
+occurring near 1800 K. We compiled the Teq values that mark the 1-percent and 99-percent limits in the combined
+area of both peaks of the posterior distribution. That conservative prescription gives us 99% confidence that the flux
+rise occurs between equilibrium temperatures of 1714 K and 1818 K. That 104 K range is sufficiently small to indicate
+a rapid change in exoplanetary atmospheres, similar to a phase change.
+4.3. Temperature Amplitude of the Rise
+The analysis above focused on the rise in the ratio of emergent flux to the NR model’s flux. That ratio rises because
+the emergent fluxes from the planets increase with increasing Teq, not because the fluxes from the models drop.
+
+Spectral Fluxes of Hot Jupiters
+9
+Figure 4. Ratio of emergent flux to the flux from the no redistribution (NR) model, versus equilibrium and irradiation temperature.
+Each of the 437 points is a single Spitzer eclipse, except for a few planets (WASP-29, HAT-P-12, and HAT-P-15) where we averaged
+several eclipses due to low signal-to-noise ratio. Several outlying points lie above the range of this plot, but they have been
+included in the analysis. The fluxes exhibit an abrupt rise at an equilibrium temperature exceeding 1730K, and the horizontal
+lines mark the inferred levels (ratios of 0.58 and 0.70). The amplitude of the rise corresponds to a brightness temperature increase
+of 210 ± 22 K, consistent among the different wavelength bands. The lower trace is the posterior distribution for the rise position,
+based on the analytic Bayesian methodology of Lee & Heghinian (1977). The horizontal scale of the posterior distribution for the
+solid line has been expanded graphically by a factor of 5, to make the shape more visible. For visual clarity, we do not plot error
+bars on the X-axis (temperature), but those uncertainties (median value ±19 Kelvins) were included when deriving the posterior
+distribution for the temperature of the flux rise.
+Figure 5. 3.6- and 4.5 µm points from Figure 4, binned over
+intervals of 100 Kelvins, with error bars based on the standard
+deviation (scatter) within each bin, divided by the square root
+of the number of points in that bin (standard error of the mean).
+The bin centers are nominally at multiples of 100K in equilib-
+rium temperature, but the points deviate slightly in X because
+we average both the X- and Y-values within each bin. We plot
+only the 3.6- and 4.5 µm points so as to maximize the clarity
+of the abrupt rise in flux. The vertical dashed lines are the 99%
+confidence limits that we infer for the temperature of the abrupt
+rise in flux. Note also that the data plotted here were not used
+directly in our analysis - this Figure is only to make the flux rise
+optimally visible to the eye.
+However, it is more physically meaningful to express the rise in flux as an increase in the planetary brightness
+temperature, Tb. We calculate the temperature rise using two methods. First, we convert the rise in the flux ratio
+average at each wavelength (Figure 4) to flux units using the NR model fluxes, and then calculating the rise in Tb by
+reference to the temperature dependence of the model flux. Using this method, the rise in Tb is ∆Tb = 192 ± 34 K at
+3.6 µm, ∆Tb = 216±30 K at 4.5 µm, and ∆Tb = 249±69 K at 8 µm. Within the errors, the rise in brightness temperature
+
+Irradiation temperature
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+1.0
+ model
+0.8
+Emergent flux / NR
+0.6
+0.4
+0.2
+3.6 μm
+< Posterior for rise position
+0.0
+4.5 μm
+8 μm
+500
+1000
+1500
+2000
+2500
+Eguilibrium temperatureIrradiation temperature
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+Emergent flux / NR model
+0.8
+0.6
+0.4
+4.5 μm
+3.6 μm
+500
+1000
+1500
+2000
+2500
+Eguilibrium temperature10
+Deming et al.
+Figure 6. Brightness temperature difference (Tb minus Teq) versus equilibrium temperature. Each point represents the average
+of all eclipses for a given planet in a given wavelength band (see color legend). The location of the abrupt temperature rise is
+consistent with Teq = 1730K (as on Figure 4), with amplitude ∆Tb = 291 ± 49 K (see text). The red and blue lines are least-squares
+fits in the regimes hotter and cooler than the location of the abrupt temperature rise. The fits consider uncertainties in both the X-
+and Y-coordinates, but for visual clarity the (small) error bars on X are not plotted. The upward slopes of the lines are consistent
+with brightness temperature exceeding the equilibrium temperature by increasing amounts with increasing irradiance.
+is the same at all three Spitzer wavelengths. Weighting the temperature rise at each wavelength by σ−2 yields an
+average temperature rise of ∆Tb = 210 ± 22 K.
+A second method to calculate the rise in Tb is illustrated in Figure 6. In this case, we calculate Tb for each planet at
+each wavelength using the eclipse depths and the stellar model fluxes. Tb is generally proportional to the equilibrium
+temperature, so we plot Tb−Teq in order to remove the overall increase in temperature with increasing irradiance, and
+concentrate on the fine structure in the proportionality. Because the most strongly irradiated planets circulate heat
+with the least efficiency, we expect that Tb − Teq will slope upward (albeit, slightly) as a function of Teq. We applied
+the LH algorithm to the Y-axis of Figure 6, and it finds a discontinuity near Teq = 1800 K, with 6.7σ significance.
+As a further test, we solved for a linear slope via least-squares for the entire range of data on Figure 6, and a
+second solution that uses two separate least-squares lines. One of those two lines is for Teq less than the discontinuity
+(blue line on Figure 6), and a second line at temperatures above the discontinuity (red line on Figure 6). Based
+on the χ2 values, we calculate Bayesian Information Criteria for the one line versus two line solution, and we find
+∆BIC = 14. That value of ∆BIC indicates ”very strong” evidence (Kass & Raftery 1995) in favor of the two line
+solution, consistent with the LH algorithm finding a discontinuity in both the emergent flux (Figure 4) and brightness
+temperature (Figure 6).
+Each point on Figure 6 is a single wavelength band for a single planet, which would over-weight planets that were
+observed in multiple bands. To derive the most accurate value of the temperature rise at the discontinuity, we weight
+each planet in the population equally. That yields a rise in brightness temperature of ∆Tb = 291 ± 49 K, consistent (at
+1.7σ) with the calculations based on the emergent fluxes (Section 4.3, above).
+To summarize our brightness temperature analysis, we find that Tb increases as a function of equilibrium temper-
+ature with a slope exceeding unity, but that the relation has a discontinuity in the range of Teq between 1714 and
+1818 Kelvins. The existence of the discontinuity in brightness temperature is detected by the LH algorithm (Lee &
+Heghinian 1977), and supported very strongly by a Bayesian Information Criterion applied to results from least-
+squares. The amplitude of the discontinuity in brightness temperature (∆Tb = 291±49 K) is (within the errors) equal
+at 3.6-, 4.5- and 8 µm.
+We discuss the interpretation and atmospheric implications of the temperature rise in Section 6.
+5. SPECTRAL PROPERTIES OF THE EMERGENT FLUXES
+
+Irradiation temperature
+1000
+2000
+3000
+4000
+1500
+8 μm
+4.5 μm
+3.6 μm
+1000
+ (Kelvins)
+500
+0
+-500
+-1000
+1000
+1500
+2000
+2500
+3000
+Eguilibrium temperatureSpectral Fluxes of Hot Jupiters
+11
+We now turn to more general spectral properties of the emergent fluxes and brightness temperatures that are
+given in Table 2. Previous investigations have focused on this topic. Because eclipses of hot Jupiters have been most
+abundantly observed in Spitzer’s 3.6- and 4.5 µm bands, the comparison between those bands has been a specific focus
+of population studies (e.g. G20, B20, W21, and Goyal et al. 2021, and also Dransfield & Triaud 2020). We here use
+our increased sample size to revisit the 3.6- and 4.5 µm flux comparison, and also to construct a color-flux diagram.
+5.1. 3.6- Versus 4.5 Micron Brightness Temperatures
+Figure 7 shows the ratio of 4.5- to 3.6 µm brightness temperature, versus equilibrium temperature. G20 found that
+this ratio increases with equilibrium temperature with a slope of 100±24 parts-per-million per Kelvin over the range
+of equilibrium temperature from approximately 800 to 2500 Kelvins. Work with larger samples (B20, and W21)
+found a similar trend, and our complete sample of Spitzer’s hot Jupiters confirms the effect. A least-squares fit of
+a straight line to the data on Figure 7, and accounting for uncertainties of each point in both X and Y coordinates,
+yields a positive slope of 67 ± 13 ppm per Kelvin. The slope is thereby significantly different from zero by 5.0σ,
+revealing a spectral property of the hot Jupiter population. Recall that G20 - using a smaller sample - found a slope
+of 100 ± 24 ppm per Kelvin, so our result is 1.4σ below G20’s estimate and thereby consistent with their work. B20
+found essentially the same result as G20, and W21 found a similar relation (albeit expressed differently), at 3.2σ
+significance.
+We also allow for the fact that the scatter on Figure 7 exceeds the error bars, for example due to intrinsic variation
+in the population. We scaled the error bars to greater values so that the error bars statistically match the total scatter
+in the points (observational errors plus intrinsic variations). In that case, the least-squares solution gives a slope of
+65 ± 17 ppm per Kelvin, still exceeding zero by 3.7σ. We did another test, following W21. We made 105 synthetic
+data sets whose per-point scatter was identical to the real data, but no systematic trend was present. We calculated
+least-squares slopes for those synthetic data realizations. Because the synthetic data sets contain no trend, least-
+squares slopes as large as in the real data can only occur via random chance. If the number of those cases is very
+small, then we can conclude that the real data contain a real slope to appropriately high confidence. We find that
+only 0.30% of the realizations produce a slope as large as we find in the real data. We conclude that the real data
+exhibit a significant slope at 99.7% confidence. The brightness temperature at 4.5 µm increases relative to the 3.6 µm
+brightness temperature as the planets get hotter. The planets do not behave as blackbodies, because blackbodies have
+the same brightness temperature at all wavelengths (ratio everywhere equal to unity on Figure 7).
+We calculated 3.6- and 4.5 µm brightness temperatures for solar and 30X metallicity model tracks, and those tracks
+are also plotted on Figure 7. For equilibrium temperatures exceeding ∼ 1200 Kelvins, the models slope upward
+similarly to the data, so the models do a good job of accounting for the hottest planets (the solar metallicity track
+is best for the hottest planets). However, for equilibrium temperatures below ∼ 1200 Kelvins, the data and models
+differ widely, and we discuss the physical implications of this difference in Section 6.7.
+5.2. A Color-Flux Diagram
+By analogy with stellar population studies, color-color and color-magnitude diagrams have been used to infer
+properties of the hot Jupiter population (Triaud 2014; Triaud et al. 2014; Dransfield & Triaud 2020). We explore that
+approach using our complete sample, but with one key difference. Rather than plotting a color-magnitude diagram,
+we plot a color-flux diagram wherein the radii of the planets is not a factor, only the spectral character of their
+emergent flux matters.
+Figure 8 shows the color-flux diagram for our complete sample, using the 4.5 µm flux on the Y-axis, and with the
+planets color-coded by equilibrium temperature. We also plot two model tracks, based on collections of models for
+individual planets (Section 3). Those tracks are for solar metallicity, and for 30X solar metallicity. The hottest planets
+tend to emit the most flux, and thereby lie highest on the Y-axis. For those hottest planets, the model tracks are very
+close together, indicating that the flux ratio (4.5 µm to 3.6 µm) is not very sensitive to metallicity, in agreement with
+Figure 7. The models track the hottest planets very well, unlike the blackbody line that slopes slightly to the upper
+left. Dransfield & Triaud (2020) found that planets tend to scatter to the right of the model track in a color-magnitude
+diagram (their Figure 4); we find good agreement with the models for the hottest planets. Also, Figure 8 is consistent
+with Figure 7, wherein the brightness temperature ratio (4.5- to 3.6 µm) increases with equilibrium temperature
+(blackbodies having a constant ratio), and the models follow the planets in Figure 7 for equilibrium temperatures
+exceeding ∼ 1200 K.
+
+12
+Deming et al.
+Figure 7. Brightness temperature ratio (Tb at 4.5 µm divided by Tb at 3.6 µm) versus equilibrium temperature. The tracks of
+points in slate blue and blue-violet colors are theoretical points from the models (solar metallicity, and 30-times solar metallicity).
+The black line is a least-squares fit that considers error bars in both the X- and Y-coordinates (error bars in X are not plotted, to
+keep the Figure clear). The slope of the fitted line differs significantly from zero (see text), and the model tracks come close to
+bracketing the observed points for equilibrium temperatures below ∼ 1200 Kelvins.
+One aspect of the hottest planets deserves mention, i.e. that there is no obvious and abrupt shift toward greater
+4.5 µm flux relative to 3.6 µm for the hottest planets (i.e. no abrupt shift to the right on Figure 8). Such a shift
+was found by B20, that they attributed to excess emission in the fundamental carbon monoxide band caused by
+atmospheric temperature inversions. We discuss this issue in more depth in Section 6.8.
+The coolest planets in the sample fall to the lower left on Figure 8, unlike directly imaged planets and brown
+dwarfs that fall to the lower right (Dransfield & Triaud 2020). The error bars are large for individual planets, but
+the shift of the hot Jupiter population to the lower left is a real effect. This effect is consistent with previous work
+(e.g., Kammer et al. 2015), and is also consistent with the behavior seen on Figure 7. Much of this effect can be due to
+high metallicity, because the 30X solar model track follows the planets on Figure 8 much better than does the solar
+metallicity track. We discuss this issue in greater depth in Section 6.7.
+6. DISCUSSION
+We here discuss that the flux rise has been noticed in previous work, we explore the physical implications of the
+abrupt flux rise, and we also discuss the spectral properties of the emergent fluxes.
+6.1. Noticed Previously
+The flux rise is noticeable on the right panel of Figure 2 in Wallack et al. (2021), and was also noted by Goyal
+et al. (2021). The latter authors did not elaborate, but Wallack et al. (2021) discussed decreasing efficiency of heat
+transport for temperatures exceeding ∼ 1500 K, albeit they did not emphasize the abruptness of the transition in their
+paper. We believe that this is the same phenomenon that we measure in this paper, in spite of the modest difference
+in the location of the transition. B20 found an abrupt increase in 4.5 µm flux near Teq = 1660 ± 100 K that we believe
+is related to the same phenomenon, as we discuss in Section 6.8.
+6.2. Weakening Molecular Absorption
+
+1.6
+1.4
+Ratio of Tb (4.5/3.6)
+1.2
+1.0
+0.8
+Solar metallicity track
+30X metallicity track
+0.6
+Observed data
+0.4
+1000
+1500
+2000
+2500
+Equilibrium Temperature (KSpectral Fluxes of Hot Jupiters
+13
+Figure 8. Color-flux diagram for hot Jupiters. The X-axis is log of the flux ratio (4.5 µm flux divided by 3.6 µm flux) and the Y-axis
+the log of the 4.5 µm flux. The black line is a blackbody track, and the tracks for two collections of model planets are also included:
+solar metallicity models, and 30X solar metallicity. The observed planets are color-coded by calculated equilibrium temperature.
+
+Planet temperature (Kelvins)
+400
+650
+900 1150 1400 1650 1900 2150 2400
+-5.0
+blackbody track
+solar metallicity models
+30x metallicity models
+-5.5
+-6.0
+close-in planets
+-6.5
+-7.0
+-1.0
+-0.5
+0.0
+0.5
+1.0
+Log(Flux 5/Flux, 
+614
+Deming et al.
+The first question concerning the abrupt flux rise is whether it reflects an increase in the temperature of the exo-
+planetary atmosphere, or whether it is merely an apparent increase due to a change in the depth of formation. For
+example, if molecular absorption weakens, then the emergent flux can originate from deeper and hotter layers of
+the atmosphere. That could cause an increase in emergent flux without any change in the run of temperature versus
+height in the atmosphere. The depth of flux formation could change due to either molecular dissociation or cloud
+dissipation.
+The models show that molecular absorption does indeed weaken as the equilibrium temperature increases, but the
+weakening is gradual, not abrupt. Moreover, the abrupt increase in brightness temperature being closely equal at
+3.6- and 4.5 µm is unlikely to be caused by weakening of molecular absorption. The molecular opacities are different
+in the two bands, and the 4.5 µm band includes emission by carbon monoxide, not present at 3.6 µm. We conclude
+that weakening molecular absorption is not a likely explanation for the abrupt rise in flux.
+6.3. Water Vapor Dissociation
+The abrupt flux rise occurs near an equilibrium temperature where Mansfield et al. (2021) find that water absorp-
+tion disappears in emergent spectra due to the dissociation of water vapor (Parmentier et al. 2018). Water vapor is a
+cooling agent: stellar irradiance is absorbed primarily at optical and near-IR wavelengths, and water vapor helps to
+balance that heating by re-emitting in the infrared. Therefore the absence of water vapor can produce a heating of
+the atmosphere. To the extent that water dissociation occurs abruptly with rising equilibrium temperature, it could
+in principle produce the abrupt flux rise that we detect. However, this explanation also fails. Figure 4 shows the flux
+rise as an increase in observed flux divided by the NR model’s flux. The NR model includes the effects of water vapor
+and its dissociation on the atmospheric temperature structure. Therefore the abrupt flux rise seen in Figure 4 cannot
+be ascribed to lack of cooling by water vapor.
+6.4. Cloud Clearing
+Cloud dissipation that abruptly reveals deeper, hotter layers, is a possible explanation. Current understanding of
+cloud formation and dissipation (Helling 2019) predicts that the dominant cloud types change with temperature.
+The day sides of the ultra-hot Jupiters are expected to be cloud-free (Parmentier et al. 2016; Wakeford et al. 2017;
+Helling et al. 2021). However, near the temperature of the flux rise, multiple types of clouds are expected to be
+present (Adams et al. 2022). Could cloud-clearing occur over a small range of temperature and thereby lead to the
+abrupt flux increase? Or, does cloud-clearing occur more gradually in temperature, as clouds of a single composition
+are replaced by other cloud types as temperature increases? Shifts in Kepler phase curves (Demory et al. 2013) are
+diagnostic of the presence of reflecting clouds (Parmentier et al. 2016). Those phase shifts change gradually with
+equilibrium temperature (e.g., Figure 3 of Parmentier et al. 2016), suggesting that cloud clearing is not sufficiently
+abrupt to account for the flux rise that we detect. The trend of geometric albedo versus equilibrium temperature for
+hot and ultra-hot Jupiters (Wong et al. 2021) also seems inconsistent with a single rapid cloud-clearing process.
+Nevertheless, some models indicate that rapid cloud dissipation could indeed produce the abrupt increase in
+brightness temperature that we observe. Roman & Rauscher (2019) and Roman et al. (2021) modeled cloud for-
+mation in GCM models of hot Jupiters. Figure 2 of Roman et al. (2021) shows an abrupt increase in day side thermal
+emission between 1750 K and 2000 K for their extended nucleation-limited case. Also, Figure 2 of Gao & Powell
+(2021) shows a distinct jog upward in modeled day side brightness temperature near 1700 K, having approximately
+the amplitude that we observe and caused by the dissipation of day side silicate clouds.
+We conclude that cloud dissipation is a tentatively viable explanation for the abrupt increase in brightness temper-
+ature that we observe. However, additional work is needed in order to clarify the effects of cloud dissipation on the
+day side brightness temperature, and to reconcile observations at multiple wavelengths.
+6.5. Efficiency of Heat Transport
+It is also important to consider a rise in day side temperature due to an abrupt decrease in the efficiency of spatially
+redistributing stellar irradiance. Parmentier et al. (2021) found that non-gray radiative transfer enables a decrease
+in heat redistribution efficiency that begins suddenly near 1600 K, and continues to increase monotonically with in-
+creasing irradiance. Our data indicate a similar behavior but require a more abrupt decrease in heat redistribution
+concentrated near 1730 K. We discuss whether that abrupt decrease in efficiency could occur either as a thermody-
+namic effect of night side clouds, or via inhibition of winds via magnetic drag.
+
+Spectral Fluxes of Hot Jupiters
+15
+6.5.1. Thermodynamic Effect of Night Side Clouds
+Hot Jupiters having tidally locked rotation can balance stellar irradiation by re-emitting that energy input, espe-
+cially from their night sides. However, cloud formation on the night side reduces the efficiency with which the night
+side can radiate, leading to increased day-night contrasts Parmentier et al. (2021). As equilibrium temperatures rise,
+night side clouds are forced to higher altitudes (Gao & Powell 2021), but the temperature of the clouds tends to be
+approximately constant, determined by the condensation temperature of the cloud species. That can further decrease
+the efficiency of re-radiation by the night side, forcing the day side to become hotter and re-radiate more strongly
+in order to balance stellar irradiation. That effect is intertwined with the consequences of cloud-clearing on the day
+sides, where the dissipation of day side clouds allows deeper and hotter layers of the day side atmosphere to become
+visible.
+6.5.2. Magnetic Drag
+Perna et al. (2010) and Menou (2012) calculated that rising ionization in strongly irradiated hot Jupiters would
+produce a conductive atmosphere and magnetic induction and drag that damps the zonal winds, leading to a hotter
+day side. For a reasonable atmospheric field strength of 10 Gauss, Menou’s Figure 1 indicates that the zonal winds
+fall in amplitude by half near an equilibrium temperature of ∼ 1650 Kelvins. That is close to where we observe the
+abrupt rise in day side flux, and Menou’s scaling calculations predict that the decrease in zonal wind speed becomes
+complete over ∼ 200 Kelvins in equilibrium temperature.
+Self-consistent 3D MHD simulations (Rogers & Komacek 2014) also show a similar decrease in zonal wind (their
+Figure 6) at Teq ∼ 1500 to 1800 K, depending (like Menou’s predictions) on the assumed magnetic field strength.
+Hindle et al. (2021) also find a transition in MHD behavior as ionization sets in and the magnetic Reynolds number
+becomes large.
+Thorngren & Fortney (2018) infer the efficiency of stellar irradiation at inflating the radii of 281 giant planets via
+Ohmic dissipation. They find a peak in efficiency near the equilibrium temperature of our flux rise, and Sarkis et al.
+(2021) confirmed their results with an independent study. Inflation of radii by Ohmic dissipation is intimately tied
+to the presence of magnetic drag (Menou 2012; Rauscher & Menou 2012, 2013), and Knierim et al. (2022) applied the
+results from Thorngren & Fortney (2018) to an Ohmic dissipation model. They found that interpreting hot Jupiter
+radius inflation as due to magnetism implies a strong decrease in zonal wind speed at ∼ 1700 to 1800 K, consistent
+with our abrupt increase in day side brightness temperature.
+We conclude that the onset of magnetic drag joins possible cloud dissipation as viable explanations for the abrupt
+rise that we detect in day side emergent flux and brightness temperature. Section 7.2 discusses some tests that could
+help to distinguish between these possibilities, (or demonstrate that they both play a role).
+6.6. Other Factors in the Flux Rise
+We have also considered other factors that can contribute to the abrupt flux rise. The hottest planets tend to orbit
+the hottest stars. We applied the LH algorithm to the trend of stellar host temperature as a function of planetary equi-
+librium temperature, but we find no abrupt change in stellar temperature corresponding to the flux rise. However,
+indirect factors related to the stellar temperature are possible contributors. For example, the spectral distribution
+of stellar energy can enhance temperature inversions (Lothringer & Barman 2019), and potentially play a role in the
+flux rise. Other factors such as chemical disequilibrium, metallicity, and variations in C/O ratio (Section 6.7) may
+contribute via their effect on radiative opacities. Clouds can also scatter in the infrared, and make planets on the cool
+side of the flux rise appear colder than they actually are (Taylor et al. 2021).
+6.7. Metallicities and C/O
+Figures 7 and 8 provide some metallicity information concerning hot Jupiters (because opacities differ between
+the two Spitzer bands). For temperatures above ∼ 1200 Kelvins, the models track the brightness temperature ratio
+(Figure 7) and color (Figure 8) of the hot Jupiter population very well. Both of those Figures demonstrate that the
+planets are not blackbodies (Goyal et al. 2021), but their departures from blackbodies are due to a combination of
+continuous opacity and weak molecular absorption/emission in the 3.6- and 4.5 µm Spitzer bands, not strongly sensi-
+tive to metallicity. Spectroscopy of individual hot Jupiters has revealed both high metallicities (Sheppard et al. 2021;
+Wong et al. 2022) and sub-solar metallicites (Line et al. 2021). The Spitzer data prefer solar metallicites for equi-
+librium temperatures above ∼ 1200 Kelvins, because the solar metallicity track on Figure 7 matches the brightness
+
+16
+Deming et al.
+temperature ratio better than does the 30X metallicity track. However, Figure 8 shows that many of the coolest hot
+Jupiters are best matched by high metallicity models, because the 30X solar track falls well to the left of the solar
+metallicity track. Many of the coolest planets on Figure 7 fall between the solar and 30X metallicity tracks. (We do
+not expect that metallicity is a function of equilibrium temperature per se, but the cooler planets more easily reveal
+an enhanced metallicity).
+In principle, atmospheric metallicity can be a function of planetary mass and/or radius (Welbanks et al. 2019),
+and that relation might be reflected in the brightness temperature ratio. We searched for a correlation between the
+Tb ratio and both planetary mass and radius in the region where the model tracks for solar and 30X solar diverge
+(Teq < 1200 K, see Figure 7). We found no statistically significant correlations in either case, but only 21 planets lie in
+that region. Our results remain broadly consistent with modest enhancements in the atmospheric metallicity of hot
+Jupiters, those enhancements being below the upper limits inferred by Thorngren & Fortney (2019).
+The C/O ratio should also be considered when comparing the relative fluxes to models (Wallack et al. 2019). We
+compared the brightness temperature ratio to a model track with C/O=0.85. That comparison (not plotted) provides
+only worse agreement with the data than does the solar abundance C/O track. We conclude that enhanced C/O is
+not a significant help when interpreting the Spitzer eclipses.
+6.8. Temperature Inversions and CO Emission
+There is excellent (and growing) evidence that the most strongly irradiated planets host temperature inversions in
+their upper atmospheres (Mikal-Evans et al. 2020; Cont et al. 2021; Fu et al. 2022; Yan et al. 2022a,b). Our models
+for the hottest planets include temperature inversions, and emission in the v=1 to v=0 fundamental band of carbon
+monoxide that lies within the 4.5 µm bandpass. That spectral emission is a significant reason why the planets and
+the track of models slope to the upper right of the blackbody track on Figure 8. We see no evidence for a distinct
+transition in the population due to the emission phenomenon, as claimed by B20, but we did not look for such an
+effect statistically. Rather, we have focused on the abrupt increase in flux that we find to occur in both the 3.6- and
+4.5 µm spectral bands.
+B20 concluded for an abrupt transition to emission at an equilibrium temperature of 1660±100 Kelvins. Within the
+errors, that location agrees with the abrupt day side brightness temperature rise that we observe. The temperature
+rise is easier to detect at 4.5- than at 3.6 µm, but in our extensive and improved data we detect it in both bands. B20
+matched their 3.6 µm data with blackbodies, and then extended and removed those blackbodies from 4.5 µm data to
+detect excess emission. We suggest that B20 thereby detected the day side temperature rise only at 4.5 µm, whereas
+we find that the abrupt flux increase occurs in both bands.
+7. SUMMARY AND FUTURE WORK
+7.1. Summary
+We have re-analyzed Spitzer’s 3.6- and 4.5 µm secondary eclipses of hot Jupiters using one uniform method. Our
+sample comprises 457 eclipses of 122 planets. This large sample allows us to make general statements, with good
+statistical justification, concerning the emergent spectral properties of the hot Jupiter population. We find that the
+day side brightness temperatures in the population increase abruptly, similar to a phase change, at an equilibrium
+temperature between 1714 and 1818 Kelvins (99% confidence interval). The amplitude of the temperature rise is
+291 ± 49 Kelvins, and we conclude that it is caused either by rapid clearing of day side clouds, or by decreased
+redistribution of stellar irradiance due to magnetic drag, as predicted a decade ago by Menou (2012).
+We compare the fluxes and brightness temperatures in the 3.6- and 4.5 µm bands, and we confirm previous work
+showing that the brightness temperature ratio (4.5- to 3.6) increases with equilibrium temperature. Models having
+solar metallicity reproduce that brightness temperature ratio for planets with equilibrium temperatures exceeding ∼
+1200 Kelvins, but the Spitzer photometry is not very sensitive to metallicity for those hottest planets. For equilibrium
+temperatures below ∼ 1200 Kelvins, the brightness temperature ratio falls consistently below the models having solar
+metallicity. Most of those cooler planets are bracketed by models between solar and 30X solar metallicity. Our results
+are broadly consistent with modest enhancements in the atmospheric metallicity of hot Jupiters, those enhancements
+being below the upper limits inferred by Thorngren & Fortney (2019).
+The ultra-hot portion of the population exhibit temperature inversions in our models, with emission due to car-
+bon monoxide in the 4.5 µm bandpass. That emission is a significant factor that allows the models to statistically
+reproduce the observations for the hottest planets. We do not see a distinct transition due to temperature inversions
+
+Spectral Fluxes of Hot Jupiters
+17
+in the hot Jupiter population as deduced by Baxter et al. (2020), but we did not look for such an effect statistically.
+Moreover, the random errors per-planet are large, and it is generally not possible to point to individual planets and
+conclude that they host a temperature inversion based only on the Spitzer photometry. Additional data (Mansfield
+et al. 2021; Fu et al. 2022) are often needed to establish temperature inversions in individual planets.
+7.2. Future Work
+The cause of the abrupt rise in brightness temperature can be tested by future work, both theoretical and observa-
+tional. Theoretical investigations that specifically address the abruptness of the transition in the population would
+be illuminating. Observationally, it is possible that the longitudinal offset in the population of phase curves could
+exhibit a corresponding abrupt transition to near-zero at equilibrium temperatures exceeding our range of 1714 to
+1818 Kelvins. Some results in that regard are encouraging (i.e., Figure 12 of May et al. 2022), but not all (Bell et al.
+2021). It should also be possible to derive temperature-pressure profiles for hot Jupiters on both sides of the tran-
+sition, using eclipse spectroscopy from JWST. Those temperature profiles may encode the effects of magnetic drag,
+when compared to MHD general circulation models (Rogers & Komacek 2014; Beltz et al. 2022). The temperature
+structure should also encode the presence of clouds, and help to determine the role of cloud dissipation on the
+brightness temperatures of hot Jupiters. Finally, abrupt changes in Doppler signatures might be detectable using
+ground-based high resolution spectroscopy, and any such effects would be diagnostic of changes in the zonal winds
+and in the efficiency of heat transport.
+An aspect of secondary eclipses that we have omitted from the scope of this paper concerns the orbital phase of the
+eclipses, and the implications for orbital eccentricity and dynamics. Recent improvements in orbital ephemerides
+(Ivshina & Winn 2022) imply that the secondary eclipse phases should have significant utility for studies of orbital
+dynamics, and we will address that topic in a future paper.
+ACKNOWLEDGMENTS
+We thank Peter Gao for clarifying the signatures of cloud dissipation and for comments on this paper. We also
+thank the anonymous referee for comments that allowed us to improve our original version. We acknowledge an un-
+published email exchange among Heather Knutson, Jonathan Fortney, and Adam Showman, wherein they discussed
+a ”sharp transition” in the efficiency of heat transport in hot Jupiters. They concluded that a purely radiative time
+scale effect would be gradual, not abrupt, and also that the onset of magnetic drag would be more likely to be gradual
+than abrupt. Cloud clearing was mentioned by Adam Showman as possibly causing a sharp transition, due to rapid
+cloud dissipation with increasing temperature.
+
+18
+Deming et al.
+APPENDIX
+A. DERIVATION OF ECLIPSE DEPTHS
+A.1. Photometry
+Extraction of a secondary eclipse depth begins with photometry of the Spitzer images, either full frame or subarray
+images. For full frame images, we extract a 32x32 pixel subframe centered on the host star, and thereafter we process
+those data in a similar manner to the subarray data. We first repair hot pixels with a 4σ filter applied to each pixel
+in time, and we replace discrepant pixels by their median value over time. (Strictly speaking, discrepant pixels
+should be zero-weighted and not repaired, but Tamburo et al. (2018) found that it makes negligible difference in
+actual practice with Spitzer photometry.) Subarray data cubes are 32x32-pixels x 64 subframes. After the hot pixel
+correction, we process each of the 64 subframes in the data cubes independently. For each image we find the centroid
+of the star by fitting a 2-D Gaussian, after median filtering the image to assure that the Gaussian fit is not fooled by
+any discrepant pixels not repaired by the temporal filter. We calculate the flux within 11 circular apertures centered
+on the star, the aperture radii varying from 1.6 to 3.5 pixels. We determine the background flux by first masking the
+star, and then fitting to a histogram of background pixel values to determine the median. Scaling that value to the
+sizes of the apertures, we subtract the background from each image independently and thereby produce a time series
+of photometry for each eclipse of each planet.
+In the case of eclipses that occur within long time series such as phase curves, we use only a limited range of data
+centered on the eclipse, typically 0.4 to 0.6 in phase (when the eclipse is at phase 0.5). Since we process 457 eclipses,
+we find that no rigid rules are practical, because within that large collection of eclipses are multiple special cases that
+frustrate any rigid procedures. We accordingly sometimes vary the range of data that we extract from phase curves
+depending on factors such as the amplitude of the phase variation and any instrumental temporal ramps that may
+be present. In several instances, we checked to make sure that the derived eclipse depths do not vary systematically
+with the range of the data that are used. For eclipses observed independently of phase curves, we omit the first 30
+minutes of the data, to avoid temporal ramps that are often steepest at the start of the time series. That 30 minute
+trim does not include ”peak up” data that are often acquired before starting the primary time series observations. In
+some cases, when a temporal ramp is especially prominent, we omit 60-, or (in very few cases) 90 minutes of data at
+the start of the time series.
+A.2. Pixel-Level Decorrelation
+Spitzer data are affected by a well known intra-pixel sensitivity variation that must be removed from the data in
+order to derive accurate eclipse depths. We do that using pixel-level decorrelation (PLD, e.g., G20, Deming et al.
+2015; Wong et al. 2016; Kilpatrick et al. 2017; Benneke et al. 2017), based on 12 pixels centered on the star (a 4x4
+square, minus the corners). The PLD fitting process solves simultaneously for the eclipse depth, central phase, and
+temporal ramp coefficients. The analysis is formally Bayesian, but the nature of secondary eclipses give us either
+very weak prior constraints, or very strong ones, so in practice the solution is dominated by minimizing chi-squared.
+For example, the central phase of the eclipse is only weakly constrained by transit and radial velocity data, so we
+specify a uniform interval for the prior on central phase. Conversely, the shape of the eclipse curve is very strongly
+constrained by transit and RV data. Hence we use priors with zero variance for orbital parameters other than central
+phase (i.e., we freeze their values). This is consistent with previous work (G20, and O’Rourke et al. 2014; Evans et al.
+2015; Mansfield et al. 2018; Deming et al. 2019).
+The eclipse fit begins with a search over phase to find an initial starting value for center of eclipse. With the best
+central phase found, the code then explores many combinations of photometric aperture and bin size, choosing the
+combination that produces the best slope in the Allan deviation relation at that trial value of the central phase. The
+Allan deviation relation (Allan 1966) expresses that the standard deviation of the residuals (data minus best fit)
+should decrease as the square root of the bin size. The fitting process at this stage is a multi-variate linear regression.
+We bin both the PLD pixel coefficients and the photometry for the fit. The reasons for binning the data are explained
+below. We limit the bin size so that the number of data points after fitting exceeds the number of fitted parameters
+by at least a factor of 10. In our experience, this avoids overfitting. A histogram of the Allan variance slope is shown
+as Figure 9 for all eclipses we analyze (both 3.6- and 4.5 µm combined). The distribution of slopes is peaked at -0.48,
+
+Spectral Fluxes of Hot Jupiters
+19
+Figure 9. Histogram of Allan deviation slopes for the
+residuals (data minus fit) of 439 eclipses (3.6- and
+4.5 µm both included).
+The blue curve is the best-
+fitting Gaussian that represents the distribution of
+slopes, and it peaks at -0.48, with standard deviation
+0.047.
+and the distribution closely approximates a Gaussian. A Gaussian fit to the histogram (illustrated on Figure 9) has
+a standard deviation of 0.047, which reflects both the random uncertainty of each slope measurement as well as
+intrinsic variation in the quality of the data. Only one eclipse produces a fitted slope that is 3σ less than -0.5 (AOR
+39446016, for WASP-35). We verified that the seemingly unphysical slope in that instance is not due to overfitting,
+but occurs because of autocorrelation in the data.
+When the fit has selected the best combination of bin size and photometric aperture, it then refines the eclipse
+depth, central phase, and ramp parameters by exploring parameter space using a Markov Chain Monte Carlo
+(MCMC) sampling process. The MCMC is a classic Metropolis-Hastings algorithm with Gibbs sampling. During
+a 10,000 step burn-in process, we dynamically adjust the step size to achieve an acceptance ratio of 0.35. Then each
+MCMC chain runs for 800,000 steps, and we find that the sampling of parameter space is excellent in both eclipse
+depth and central phase.
+A.3. Binned Data
+Fitting to binned data is (in theory) non-optimal because the binning process destroys information (Kipping 2010).
+Recently, May et al. (2021) concluded against using binned data in fitting to phase curves (but not secondary eclipses
+per se). We therefore re-visited our rationale for using binned fits, and we conclude that the binning is desirable,
+at least in our method. We here explain our reasons. First, we note that the Spitzer IRAC instrument already bins
+the incident photons in time before transmitting the data to Earth. The observing cadence available in IRAC was
+set before transiting planets were discovered, so the observing cadences are not special for the purpose of secondary
+eclipses. We find that further binning of the data eliminates instrumental effects on short time scales (Challener et al.
+2021) that would otherwise be very difficult to adequately reproduce in the fitting process. In other words, although
+the binning destroys information, it destroys information that we want to destroy because we cannot reproduce those
+short-term instrument effects, and they occur on a faster time scale than the eclipse. We find that the binning allows
+the fitting process to concentrate on the time scales that are most relevant to the eclipse, and thereby it reduces red
+noise. We limit the size of the bins so that we do not distort the shape of the eclipse curve (e.g, by broadening the
+ingress/egress), and (as noted above) we limit the bin size so as not to overfit the data. For planets with multiple
+observed eclipses, our code usually selects different bin sizes from eclipse to eclipse, because they are independent
+events. In those cases, we checked to verify that eclipse depths do not correlate with the adopted bin size.
+A.4. Temporal Baselines
+We initially fit each eclipse with two different temporal baselines (ramps), a linear and quadratic ramp. We consult
+a Bayesian Information Criterion (BIC) analysis applied to the fitting residuals in order to decide between the linear
+and quadratic ramp. Although the BIC is very useful, we do not use it rigidly, and we also consider the totality of
+fit properties when deciding between temporal ramps. For example, the BIC may prefer a quadratic ramp, but that
+might produce a two-peaked posterior distribution for the central phase, that is not physically realistic. So in that
+case, we would adopt the linear ramp. For planets whose equilibrium temperatures exceed 2000 K, we use only a
+quadratic temporal ramp. The reason for that choice is to allow the ramp to account for the small portion of the
+
+1.0
+Relative ocurrence
+0.8
+0.6
+0.4
+0.2
+0.0
+-0.7
+-0.6
+-0.5
+-0.4
+-0.3
+Slope20
+Deming et al.
+phase curve that occurs within our data window, and thereby avoid bias in the fitted eclipse depth. We verified that
+a quadratic is a close approximation to the portion of the sinusoidal phase curve that occurs in our window.
+A.5. Chain Convergence
+After deciding on the ramp, we run an additional MCMC chain with the adopted ramp, and we inspect the posterior
+distributions for eclipse depth and central phase. The distributions for two independent chains should be, and are
+in almost all cases, closely coincident. We also calculate a Gelman-Rubin (GR) statistic (Gelman & Rubin 1992)
+for eclipse depths in the two adopted MCMC chains. Gelman & Rubin (1992) suggested that a value less than 1.1
+denotes good convergence, but more recent work has tended to emphasize GR values closer to unity. The GR statistic
+is derived from the shape and mean value of the posterior distributions, so closely equal distributions will always
+produce GR values near unity. Considering all planets and all eclipses, our median GR value is 1.002, our average
+value is 1.003, and and our maximum value is 1.05.
+Given the large number of eclipses we analyzed, we did encounter some where convergence was initially prob-
+lematic. Those cases usually arise when the temporal ramp coefficients become partially degenerate with the eclipse
+depth. That situation is more common when there is minimal out-of-eclipse baseline before and/or after the eclipse.
+In those cases, we achieve good convergence by freezing the temporal ramp coefficients during the MCMC. Because
+that freeze can cause the uncertainty in eclipse depth to be underestimated, in those cases we compare the width of
+the posterior distributions for eclipse depth (that determine the random error) with and without the freeze, and we
+thereby apply a correction to the error bars.
+A.6. Choice of Eclipse Depth
+Our analysis yields two eclipse depths for each planet; each of those depths is based on average from the same two
+(or more) converged MCMC chains. The first depth is the best fit using the criterion adopted by G20, who adopted
+the solution yielding the best Allan deviation relation. With our larger sample, we have come to prefer the (more
+conventional) practice of using the mean of the posterior distribution for depth. We verified that the conclusions of
+this paper do not change significantly if we adopt the choice used by G20.
+Our derived eclipse depths and uncertainties are given in Table 1. Eclipses for planets not previously published
+are illustrated in Figure 10.
+A.7. Special Cases
+There are several planets that represent special cases. WASP-33b orbits a pulsating star, and Deming et al. (2012)
+used a wavelet technique to account for the stellar pulsations while simultaneously solving for the eclipses. The
+wavelet technique is not easily included in our PLD code, so we have not re-analyzed the eclipses of WASP-33b, but we
+include the depths from Deming et al. (2012) and Zhang et al. (2018) in Table 1 (so that all of Spitzer’s eclipse depths
+at these wavelengths are available in a single Table). Also, some planets have very weak eclipses observed multiple
+times, and we treated those slightly differently. In those cases, we used the PLD code to correct for, and remove,
+Spitzer’s instrumental effect. We then combined the photometry for the multiple instrument-corrected eclipses and
+fitted an eclipse curve. Those fits obtained the eclipse depth and uncertainty based on the scatter, while constraining
+the central phase to equal 0.5 exactly. Those cases are for planets WASP-29b and HAT-P-15b in both bands, and
+WASP-67b and HAT-P-15b in the 3.6 µm band. The depths are listed in Table 1 associated with the first listed AOR,
+but all of the listed AORs were averaged to derive the quoted depth.
+B. DILUTION CORRECTIONS
+When a second star appears in the photometry aperture, the light from that companion star will dilute the eclipse,
+regardless of whether the companion star is physically bound to the planet-hosting star. We corrected the emergent
+fluxes for dilution using the multiplicative factors in Table 4. The eclipse depths in Table 1 are ”as observed”, not
+corrected for dilution. The dilution factors were applied to the observed eclipse depths to derive the emergent fluxes
+in Table 2.
+We adopted dilution factors from Shporer et al. (2014) and G20, and we calculated factors for planets not previ-
+ously corrected for dilution. In many cases, published Spitzer eclipse depths have not been previously corrected for
+dilution because the diluting companion star was not detected when the Spitzer eclipses were first analyzed. We con-
+sulted the literature of high resolution imaging to find stellar companions (Ginski et al. 2013; Ngo et al. 2015, 2016;
+
+Spectral Fluxes of Hot Jupiters
+21
+Figure 10. Eclipses of planets not previously published, and including WASP-29b that was only recently published by Wong et al.
+(2022). In these plots, the data have been binned, and binning scales are adopted so that the data for each eclipse plot are spanned
+by 50 plotted points. The 50-point binning is only for uniformity of illustration, the actual binning used in fitting for the eclipse
+depths is different in all cases. The X-axis of the plots is orbital phase, but values are not listed since the central phases of the
+eclipses are not analyzed in this paper.
+
+1.003 
+1.003
+1.0010
+1.0010 E
+1.0008 E
+1.002
+1.002
+WASP-107
+2
+2
+1.0005
+1.0006 F
+1.001
+[D2027
+1.001
+1.0004 
+1.000
+1.000
+1.0000
+1.0002 F
+W
+0.999
+H
+1.0000 E
+0.999
+0.998
+0.9995
+0.9998 E
+0.998
+L66:0
+1.004
+1.004
+1.003
+T
+1.003
+1.0010 
+WASP-122
+1.002
+1.002
+3
+1.002 E
+HD2027
+WASP-
+ASP.
+1.001
+1.0005
+1.000
+1.001 E
+1.000
+W
+0.999
+1.0000
+1.000 E
+0.998
+0.998
+i
++
+1.0015
+1.004 
+1.0015
+1.004 
+1.003 E
+1.0010 E
+1.0010
+WASP-122
+38
+2
+4
+1.002
+1.0005 
+WASP.
+1.002 
+P
+HAT-
+Tl
+1.0005
+WAs
+1.0000 
+1.000 
+1.001
+0.9995
+1.0000
+1.000 
+0.998
+0.9990 E
+0.9995
+0.999
+1.003
+1.0015
+1.004
+1.003
+1.0010
+1.002 
+OS-
+2
+1.002
+3.6 micron
+WASP-
+1.0005 E
+1.001
+P
+WASH
+1.001
+1.0000 
+1.000 
+1.000
+4.5 micron
+0.9995 
+0.999 
+0.999
+0.998
+0.9990
+0.998 
+1.004
+1.006
+1.002
+1.004
+WASP-50
+3
+1.002
+1.001
+WASP-
+1.002
+1.000
+1.000 
+1.000
+W
+0.998
+0.999 E
+0.998
+0.996
+1.003
+1.0020
+1.0015
+1.002 
+1.0015
+1.0010
+WASP-95
+5
+2
+1.0010
+1.001
+1.0005
+P
+ASI
+1.0005
+1.0000
+1.000 
+W
+W
+1.0000
+0.9995
+0.999 
+0.9995
+0.9990
+866'0
+1.0010
+1.002 
+1.002 
+ITTI
+WASP-107
+3
+1.001 
+1.0005
+WASP-
+1.001 E
+WASP.
+1.000 
+1.000
+1.0000
+0.999 E
+0.999 E
+0.9995
+0.998 E22
+Deming et al.
+Figure 11. Average eclipse depth per planet derived in this paper, versus literature values compiled by Baxter et al. (2020). The
+black lines are least-squares straight lines, and their slopes are within 2σ of unity, being 0.961±0.020 and 1.008±0.025 at 3.6- and
+4.5 µm, respectively. The red mark on the 3.6 µm panel marks the outlying point for HD 189733b - see text in Appendix Section
+C for discussion.
+Evans et al. 2018; Evans 2018; W¨ollert & Brandner 2015; W¨ollert et al. 2015). We also found companions that were
+sufficiently prominent, and spatially resolved from the primary star, using astronomical catalogs such as 2MASS.
+Upon identifying a companion, we used the spectral types and K-band magnitudes of both stars to calculate their
+fluxes in the 3.6- and 4.5 µm Spitzer bands, based on the STAR-PET tool3. Companions closer than ∼ 2arc − sec were
+not resolved from the primary star, and their fluxes simply add to produce the dilution. When the companion star
+was spatially resolved in the Spitzer data, we used the method described by G20 (their Section 3.5) to calculate the
+fraction of its flux that was scattered into the photometric aperture centered on the primary star. Using the calculated
+fluxes from both stars in the photometric aperture, we calculated the dilution correction factors at 3.6- and 4.5 µm
+listed in Table 4.
+C. COMPARISON TO LITERATURE VALUES
+Although our eclipse depths are more extensive than any extant eclipse data set, it is nevertheless interesting
+to compare subsets of our values to previous investigations. Previous compilations of eclipse depths have been
+presented by G20, B20, W20, and Dransfield & Triaud (2020). We choose to compare to the values compiled by B20,
+especially because those authors found an abrupt increase in the ratio of 4.5- to 3.6 µm eclipse depth - a phenomenon
+similar to the abrupt flux increase that we infer in this paper. Figure 11 shows our 3.6- and 4.5 µm average eclipse
+depths per planet, as a function of the depth from B20, for planets in common.
+Least-squares fits of straight lines to Figure 11, considering the uncertainties in both X and Y, yield slopes within 2σ
+of unity. We conclude that there is no systematic scale factor difference between our new eclipse depths and literature
+values. We also investigated outlying points on Figure 11. The most prominent outlier is HD 189733b at 3.6 µm
+(marked on Figure 11), where our average eclipse depth for nine eclipses is 1432±34 ppm (median value 1467 ppm).
+B20 list 2560 ± 140, based on one eclipse from Charbonneau et al. (2008), so this planet lies distinctly to the right
+of the fitted line in the left panel of Figure 11. Consequently, the 3.6 µm brightness temperature calculated by B20
+(1604 ± 32 Kelvins), is well above the equilibrium temperature of 1191 Kelvins. Our 3.6 µm brightness temperature
+is 1337 ± 30 Kelvins. Our values are consistent with a re-analysis by Knutson et al. (2012), who found eclipse depths
+consistent with our current results. Given that most planets at this level of irradiation have Tb not greatly exceeding
+their equilibrium temperatures, and given that our nine eclipses are mutually consistent, we confirm the result from
+3 https://irsa.ipac.caltech.edu/data/Spitzer/docs/dataanalysistools/tools/pet/starpet/
+
+3.6 μm
+4.5 μm
+Our eclipse depth (ppm)
+5000
+5000
+F
+4000
+ 4000
+3000
+3000
+2000
+2000
+1000
+1000
+0
+1000
+2000
+3000
+4000
+5000
+0
+1000
+2000
+3000
+4000
+5000
+B20 eclipse depth (ppm)Spectral Fluxes of Hot Jupiters
+23
+Figure 12. Ratio of the average eclipse depth per planet (ours divided by Baxter et al. 2020), versus planet equilibrium tempera-
+ture. We find no systematic variations at either wavelength as a function of equilibrium temperature. The red mark indicates the
+outlier Kepler-13b, discussed in Section C of the Appendix.
+Knutson et al. (2012) that the original 3.6 µm eclipse value from Charbonneau et al. (2008) is in need of revision,
+mostly likely due to improvements in observing and data analysis techniques. We point out that a similar revision
+for HD 209458b was found by Diamond-Lowe et al. (2014), and we note that our eclipse depths for HD 209458b are
+also in good agreement with that revision.
+We also compared the ratio of our eclipse depth to the value from B20, as a function of equilibrium temperature.
+That comparison is shown in Figure 12, and in this case also we have investigated a prominent outlier at 3.6 µm. The
+outlier (red mark on Figure 12) is Kepler-13b, where B20 use an eclipse depth of 1560±330 ppm, from Shporer et al.
+(2014). Our eclipse depth (3035 ± 257 ppm) is almost twice as large. This difficult system has a bright companion
+star separated by approximately 1 pixel, that produces a large dilution correction and adds noise to the photometry
+via pointing jitter. We use the same dilution correction as do Shporer et al. (2014), and we think that our PLD
+analysis method is well matched to correct for noise produced by the companion star. The relatively ”small” eclipse
+depths derived by Shporer et al. (2014) lead them to infer an ”efficient heat redistribution process” that is unusual
+for strongly irradiated hot Jupiters (Cowan & Agol 2011), and they also derive a geometric albedo (0.33) at the high
+end of the range for hot Jupiters. We have rechecked our eclipse data analyses for this planet: we find no issues that
+need correction, and we believe that our eclipse depths are correct, albeit with large random errors. Pending further
+analysis of other data on this system, we tentatively suggest that our eclipse depths are more realistic than are the
+values derived by Shporer et al. (2014), and used by B20. However, we also note that the population-level results in
+this paper are insensitive to the Kepler-13 results.
+D. PHOENIX VERSUS ATLAS MODELS
+PHOENIX stellar atmospheres (Hauschildt et al. 1997) are used in our planetary modeling code (Section 3),
+whereas we use ATLAS9 stellar models to calculate emergent planetary fluxes from eclipse depths (Section 2.2).
+These models are respectively deeply rooted in our procedures, and it would be impractical to alter those choices.
+Hence, we investigated to what degree different stellar models can affect our results. The principal concern was to
+what degree the choice of model type affected the emergent day side flux that we calculated for each planet (Sec-
+tion 2.2). B20 investigated PHOENIX versus ATLAS models and noted that planetary brightness temperatures were
+consistent between PHOENIX and ATLAS. Husser et al. (2013) compared stellar fluxes from PHOENIX and ATLAS
+and found excellent agreement. We investigated further by calculating stellar fluxes in the 3.6- and 4.5 µm Spitzer
+bands, from both PHOENIX (BT-NextGen) and ATLAS9. We calculated the ratio of ATLAS flux to PHOENIX flux
+
+2.5
+2.5
+3.6 μm
+4.5 μm
+2.0
+2.0
+Eclipse depth ratio
+1.5
+1.0
+0.5
+10.5
+0.0
+1000
+1500
+2000
+2500
+1000
+2000
+3000
+4000
+Eguilibrium temperature24
+Deming et al.
+at 9 stellar temperatures from 4000K to 9000K, at logg = 4.5. The ratio differed from unity with an average abso-
+lute difference of 0.68% and 1.02% at 3.6- and 4.5 µm, respectively. Those differences are sufficiently small to have
+no significant impact on our results. Only at 9000 Kelvins did we find a significant difference, where the ATLAS9
+stellar flux is 4% less than from PHOENIX. That would affect primarily KELT-9b, and to a lesser extent KELT-20b.
+However, Witzke et al. (2021) found that PHOENIX models underestimate ultraviolet flux for hot stars compared to
+observations (their Figure 17). Because the models constrain the emergent stellar bolometric flux to be closely equal
+to σT 4
+ef f , underestimating UV flux will require overestimating emergent flux at other wavelengths (e.g., the Spitzer
+bands). Consequently, we prefer the ATLAS9 models over PHOENIX especially for the hottest stars, to avoid that
+issue.
+
+Spectral Fluxes of Hot Jupiters
+25
+Table 1. Eclipse depths and uncertainties for 3.6- and 4.5 µm Spitzer observations, in units of the stellar flux, in parts-per-million.
+All planets are the ’b’ planet in the system unless otherwise noted. These are ”as observed” eclipse depths, not corrected for
+dilution by companion stars. AOR = Astronomical Observing Request number. In the rare instances where more than one eclipse
+occurs in a single AOR, the depths are listed in chronological order. For four planets (HAT-P-12 and -15, WASP-29 and -67),
+several AORs were averaged to yield a single eclipse depth. These eclipse depths are converted to emergent fluxes and brightness
+temperatures in Table 2. (This Table is published in its entirety in machine-readable format. A portion is shown here for guidance
+regarding its form and content.)
+Planet
+λ (µm)
+AOR
+Depth
+1σ
+CoRoT-01
+3.6
+36786176
+4328
+591
+CoRoT-01
+4.5
+36785920
+4285
+440
+CoRoT-02
+3.6
+31774976
+3134
+198
+CoRoT-02
+4.5
+28640000
+4788
+272
+57958144
+4660
+308
+57958656
+4310
+369
+HAT-P-01
+3.6
+2474547
+1339
+129
+HAT-P-01
+4.5
+21060608
+520
+351
+Table 2. Emergent fluxes in the 3.6- and 4.5 µm bands, and brightness
+temperatures in Kelvins. Planets are the b planet in each system. Units of
+fluxes are ergs cm−2 sec−1 Hz−1, or (equivalently) milli-Watts m−2 Hz−1.
+(This Table is published in its entirety in machine-readable format. A
+portion is shown here for guidance regarding its form and content.)
+Planet
+Fν 3.6 µm
+1σ
+Fν 4.5 µm
+1σ
+Tb 3.6 µm
+Tb 4.5 µm
+CoRoT-01
+6.42e-06
+8.76e-07
+4.01e-06
+4.12e-07
+2413+156
+−161
+2145+112
+−115
+CoRoT-02
+3.20e-06
+1.97e-07
+2.95e-06
+1.98e-07
+1785+43
+−44
+1844+58
+−59
+HAT-P-01
+2.97e-06
+2.87e-07
+7.26e-07
+4.90e-07
+1733+64
+−65
+1073+206
+−298
+HAT-P-02
+4.58e-06
+9.66e-07
+3.68e-06
+4.56e-07
+2067+185
+−196
+2052+127
+−130
+HAT-P-03
+1.87e-06
+2.83e-07
+1.55e-06
+2.20e-07
+1467+73
+−77
+1401+75
+−80
+HAT-P-04
+5.51e-06
+8.30e-07
+2.81e-06
+6.25e-07
+2247+153
+−159
+1803+182
+−192
+Table 3. Stellar and planetary temperatures and radii that we adopted
+from TEPCat. Planets are the ’b’ planet in each system. The stellar radii
+are in solar units, and the planetary radii in Jupiter units. Stellar tem-
+peratures are effective temperatures. The planetary temperatures are the
+calculated equilibrium temperatures based on the stellar irradiance, and
+adopting an albedo of zero and uniform redistribution over the planetary
+sphere. The values listed for HAT-P-2 and HD 80606 are not from TEP-
+Cat, they are the day side temperatures from the time-dependent calcu-
+lations of Mayorga et al. (2021). (This Table is published in its entirety in
+machine-readable format. A portion is shown here for guidance regard-
+ing its form and content.)
+Planet
+Rplanet
+Teq
+Rstar
+Tstar
+CoRoT-01
+1.551
+1915
+1.131
+5950
+CoRoT-02
+1.460
+1521
+0.901
+5598
+HAT-P-01
+1.319
+1322
+1.174
+5975
+
+26
+Deming et al.
+Table 4. Dilution corrections for secondary eclipse depth at both Spitzer wavelengths. Sources: G20 = Garhart et al.(2020), TH =
+this paper, SH = Shporer et al.(2014).
+Planet
+3.6 µm factor
+4.5 µm factor
+Source
+CoRoT-2b
+1.0276
+1.0255
+TH
+HAT-P-8b
+1.0109
+1.0102
+TH
+HAT-P-16b
+1.0006
+1.0006
+TH
+HAT-P-20b
+1.0070
+1.0074
+TH
+HAT-P-30b
+1.0121
+1.0117
+G20
+HAT-P-33b
+1.0377
+1.0332
+G20
+HAT-P-41b
+1.0069
+1.0111
+G20
+HD 202772b
+1.207
+1.207
+TH
+KELT-1b
+1.0065
+1.0059
+TH
+KELT-2b
+1.137
+1.123
+G20
+KELT-3b
+1.0125
+1.0127
+G20
+KELT-16b
+1.0189
+1.0168
+TH
+Kepler-5b
+1.0434
+1.0384
+TH
+Kepler-6b
+1.0041
+1.0049
+TH
+Kepler-13b
+1.9940
+1.9942
+SH
+TrES-2b
+1.0812
+1.0741
+TH
+TrES-4b
+1.0258
+1.0242
+TH
+WASP-1b
+1.0015
+1.0020
+TH
+WASP-2b
+1.0894
+1.0870
+TH
+WASP-8b
+1.1058
+1.0977
+TH
+WASP-11b
+1.0630
+1.0720
+TH
+WASP-12b
+1.1101
+1.0983
+G20
+WASP-14b
+1.0126
+1.0115
+TH
+WASP-26b
+1.0601
+1.0690
+TH
+WASP-33b
+1.0038
+1.0035
+TH
+WASP-36b
+1.0015
+1.0026
+G20
+WASP-49b
+1.0130
+1.0124
+G20
+WASP-72b
+1.0484
+1.0411
+TH
+WASP-76b
+1.1470
+1.1250
+G20
+WASP-77b
+1.0929
+1.0530
+G20
+WASP-87b
+1.0014
+1.0011
+G20
+WASP-103b
+1.1700
+1.1490
+G20
+WASP-131b
+1.0781
+1.0663
+TH
+
+Spectral Fluxes of Hot Jupiters
+27
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+page_content=' Line,2 Heather A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Knutson,3 Ian J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Crossfield,4 Eliza M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
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+page_content=' Komacek,1 Nicole L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
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+page_content=' 5  1Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
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+page_content=' College Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
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+page_content=' Arizona State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Tempe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
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+page_content=' California Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Pasadena,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' CA 91125,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' USA  4Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' University of Kansas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Lawrence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' KS 66045,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' USA  5Present address: Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Johns Hopkins University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Baltimore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' MD 21218 USA  ABSTRACT  We study the emergent spectral fluxes of transiting hot Jupiters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' using secondary eclipses from  Spitzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' To achieve a large and uniform sample, we have re-analyzed all secondary eclipses for all  hot Jupiters observed by Spitzer at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6- and/or 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Our sample comprises 457 eclipses of 122  planets, including eclipses of 13 planets not previously published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We use these eclipse depths to  calculate the spectral fluxes emergent from the exoplanetary atmospheres, and thereby infer temper-  ature and spectral properties of hot Jupiters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We find that an abrupt rise in brightness temperature,  similar to a phase change, occurs on the day side atmospheres of the population at an equilibrium  temperature between 1714 K and 1818 K (99-percent confidence limits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The amplitude of the rise is  291 ± 49 Kelvins, and two viable causes are the onset of magnetic drag that inhibits longitudinal heat  redistribution, and/or the rapid dissipation of day side clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We also study hot Jupiter spectral  properties with respect to metallicity and temperature inversions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Models exhibiting 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm emis-  sion from temperature inversions reproduce our fluxes statistically for the hottest planets, but the  transition to emission is gradual, not abrupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The Spitzer fluxes are sensitive to metallicity for planets  cooler than ∼ 1200 Kelvins, and most of the hot Jupiter population falls between model tracks having  solar to 30X-solar metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Keywords: Exoplanet astronomy- Exoplanet atmospheres - Hot Jupiters - Infrared astronomy  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' INTRODUCTION  The Spitzer Space Telescope leaves a rich legacy of transiting exoplanetary science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Spitzer investigators obtained  photometry at secondary eclipse for many transiting planets (Deming & Knutson 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Using Spitzer’s photometry,  atmospheric properties such as metallicity, C/O, and temperature structure, have been inferred for dozens of hot  Jupiters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=', Harrington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Charbonneau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Knutson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Fressin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Madhusudhan  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Todorov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Beatty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Mansfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2018), albeit not without some debate (Hansen  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Ingalls et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2016) and some revisions (Knutson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Diamond-Lowe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Line et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  The physical properties of hot Jupiters were recently reviewed by Fortney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Most of Spitzer’s exoplanetary eclipse observations comprise photometry in broad spectral bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Atmospheric  inferences using photometry can be problematic because photometric spectral resolution is insufficient to identify  molecular absorptions in emergent spectra via band shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Moreover, if hot Jupiters vary widely in their proper-  ties, then studies of individual planets may struggle to accurately infer average properties of the entire hot Jupiter  population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' One solution to these problems is to focus on population studies of relatively large samples of plan-  ets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Population studies of emergent fluxes have been pursued using eclipses in the Spitzer data (Wallack et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2019,  Corresponding author: Drake Deming  ldeming@umd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='edu  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='03639v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='EP]  9 Jan 2023  2  Deming et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Garhart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2020 - hereafter G20, Baxter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2020 - hereafter B20, Wallack et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2021 - hereafter W21, and Goyal  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2021), and also using eclipses observed with HST (Mansfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Motivation for This Work  Although analyses of Spitzer’s secondary eclipses have been extensive, there are 13 hot Jupiters (listed below) whose  eclipses have not yet been published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' In some cases, hot Jupiters with published eclipse analyses have additional  unanalyzed eclipses in the archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Moreover, there is arguably value to re-analyzing all of the eclipses using one  uniform method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' To that end, we have re-analyzed all of Spitzer’s secondary eclipse data in the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6- and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm bands  using one method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' These two bandpasses encompass the vast majority of Spitzer’s secondary eclipse measurements  of hot Jupiters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  We seek to establish a population-level context for observations of hot Jupiters by JWST, and by high-resolution  ground-based spectroscopy using the ELTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The context that we seek is to establish the major properties of the hot  Jupiter population that can be deduced from their emergent spectral fluxes as observed by Spitzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Emergent fluxes  are basic observable properties of the planets, and we aim for a simple and robust analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Organization of This Paper  Section 2 describes the broad strategy of our archival analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The sample of planets that we analyze is summa-  rized in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='1, and our technique for re-analyzing Spitzer’s secondary eclipse data to produce emergent fluxes  is described broadly in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='2, and in detail in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We interpret the emergent fluxes using a grid  of models that are described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The magnitudes and observed properties of the emergent fluxes are de-  scribed in Section 4, and the spectral properties by comparing fluxes versus wavelength are explored in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Section 6 discusses the atmospheric implications of our results, beginning with an acknowledgement of previous  work in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The potential causes of an abrupt increase in day side flux are explored in Sections 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='2 to Sec-  tion 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='7 describes the implications for atmospheric metallicities, and spectral emission from temperature  inversions is discussed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='1 summarizes our results, and Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='2 notes the need for future  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' AN ARCHIVAL ANALYSIS  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The Sample of Planets  Our sample comprises 122 hot Jupiters, and we have re-analyzed every eclipse observed for those planets using  Spitzer at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6- and/or 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' That includes eclipses that occur during phase curve observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Planets whose  Spitzer eclipses were not previously published include: WASP-11b, -21b, -28b, -32b, -35b, -37b, -38b, -50b, -95b,  107b, -122b, HAT-P-34b, and HD 202772b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' In total, we derive eclipse depths for 439 individual eclipses, and we  perform some averaging for 18 additional eclipses of four planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Our derived eclipse depths are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  We use the eclipse depths to derive emergent day side fluxes, and brightness temperatures (Tb), listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The  source of stellar and planetary parameters used in our analysis (TEPCat) is ever-evolving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Hence, we list the stellar  and planetary radii and temperatures that we adopted in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' In addition to our re-analyzed eclipses, our analysis  in this paper makes use of Spitzer eclipses observed at 8 µm, but we have not re-analyzed those eclipses, nor have  we listed their emergent fluxes in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Our analysis also considers eclipses from 2 µm ground-based observations  reported in the literature (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=', Croll et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2015, and many others).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Derivation of Eclipse Depths, Emergent Fluxes, and Brightness Temperatures  We derive eclipse depths using the method described by G20, in fact using the same code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' This method separates  systematic signals due to the IRAC instrument from the eclipse signal using pixel-level-decorrelation (PLD, Deming  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Some minor differences from the G20 methodology are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' First, to increase the practicality of  our analysis, we perform aperture photometry on the Spitzer images using only apertures of fixed radius (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=', not  variable radii as per Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2013), and centering the apertures using a 2-D Gaussian fit to the stellar image (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=',  not using flux-weighted centering, Piskorz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' This reduces the volume of photometry by a factor of four,  and G20 found that fixed radii apertures with Gaussian centroiding was their best method, and was consistently  good for all of the planets in their sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' G20 also derived two depth solutions for each eclipse: one depth that  maximized the fit to the Allan deviation relation for the photometric residuals (see their Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='4), and another  depth derived from the centroid of the posterior distribution of eclipse depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' They preferred the former method  Spectral Fluxes of Hot Jupiters  3  in their analysis (but they commented that the population-level results were not sensitive to the choice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Using our  larger sample, we find that the centroid of the posterior distribution is preferable, but we have also verified that  the conclusions of this paper would not change if we used the alternate choice of eclipse depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' There are other  (minor) technical issues that arise in our analysis, and we include a more comprehensive technical discussion in the  Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  We believe that our results are an improvement over previous Spitzer population studies in 5 ways: 1) We have  added 13 additional planets to the sample, 2) We analyze the maximum number of eclipses per planet (previous  population studies have sometimes relied on literature results that do not include all of the eclipses per planet, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=',  see the discussion of HD 189733b in Section C of the Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 3) Our results are homogenous, using the same  method to extract all of the eclipse depths (whereas the literature is inhomogenous, with the early Spitzer papers  using much more simplistic data analysis methods), 4) a systematic comparison of seven data analysis techniques  (Ingalls et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2016) concluded that our PLD method was among the three most accurate methods, and had the least  bias of the seven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Finally, 5) we have consulted recent high spatial resolution imaging results to revisit dilution  corrections needed for eclipses where those corrections were not previously applied (Appendix Section B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  For the host stars, we adopt stellar effective temperatures, radii, and surface gravities from TEPCat1, and also  planetary equilibrium temperatures, radii, and orbital distances from TEPCat (Southworth 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We interpolate in  a grid of ATLAS9 models (Castelli & Kurucz 2003) to determine the fluxes of the host stars, integrating over the band-  pass sensitivity functions of Spitzer’s IRAC instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We derive an emergent flux at the top of the exoplanetary  atmosphere, using the measured eclipse depth and the stellar fluxes from the ATLAS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' To derive planetary  brightness temperatures, we calculate eclipse depths for a grid of blackbody planets, integrating over the IRAC band-  passes, and we interpolate the measured eclipse depth in that grid to find the brightness temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' To calculate  uncertainties, we propagate the uncertainty of the observed eclipse depths through the process to produce error bars  on flux and brightness temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The planetary fluxes and brightness temperatures are given as averages per  planet per waveband in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  In some instances, especially for weak eclipses, the error bars on eclipse depth can allow negative depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Negative  eclipse depths would formally imply negative brightness temperatures (as per some ranges in Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Although  negative eclipse depths and brightness temperatures are not physical, it is necessary to allow them, so that population  averages and trends are free of Lucy-Sweeney bias (Lucy & Sweeney 1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Our error bars on Tb for the planets listed in G20 are significantly less than the G20 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The G20 Tb error  bars were calculated using a Rayleigh-Jeans approximation, which is insufficiently accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' That was a mistake by  one of us (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' ), and we suggest that the values in Table 2 should be used in preference to the G20 error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Our  current error bars in the Table were calculated using the same method as B20, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' propagating the eclipse depth  errors through the temperature dependence of the Planck function integrated over Spitzer’s bandpass responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' ATMOSPHERIC MODELS  We interpret our results using a set of 1D self-consistent radiative-convective-thermochemical equilibrium mod-  els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The models were previously described and used in Piskorz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2018), Arcangeli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2018), Mansfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  (2018), Kreidberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2018), Gharib-Nezhad & Line (2019), Mansfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2021), and recently benchmarked  against M-dwarfs in Iyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Briefly, the models solve the 1D radiative-convective equilibrium problem  via a Newton-Raphson iteration to achieve zero net flux divergence across each layer interface (see McKay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Layer-by-layer radiative fluxes are computed using the two stream source function method from Toon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  (1989) (without clouds/scattering) and convective flux is computed via mixing length theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Equilibrium chem-  istry with rainout (depletion of elemental species in overlying layers due to the onset of condensate formation) is  computed using the NASA CEA2 routine (McBride & Gordon 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We include ∼ 40 line and continuum opacity  sources accounting for species relevant in chemical equilibrium from 300 - 4000 Kelvins covering 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='1 - 100 µm (UV  species included are neutral atomics/ions, H2, CO, and SiO), with molecular species sourced from the latest ExoMol  (Tennyson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2020) line lists and neutral/ion atomic species from the Kurucz line-list database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Line opacities are  computed using the methods described in Gharib-Nezhad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2021) and then are converted into R=250 correlated-  K coefficients (using a 10 point double-Gauss quadrature sampling for the bin weights/ordinates, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=', Molli`ere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  2015) with mixed opacities computed on-the-fly using the RORR method described in Amundsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  1 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='keele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='uk/jkt/tepcat/  4  Deming et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Examples of modeled emergent flux for plan-  ets with a range of temperature, and in the wavelength  region encompassing Spitzer’s 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6- and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm band-  passes (response curves at the bottom of the panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  The left panel shows models having solar metallicity,  and the right panel is 30-times solar metallicity for the  same planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The planets and equilibrium tempera-  tures are: WASP-121b (Teq = 2358 K), XO-4b (1641 K),  WASP-39b (1166 K), and HAT-P-12b (955 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The red  tic marks at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm call attention to carbon monoxide  emission from a temperature inversion in WASP-121b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  The models are in two groups: first, we calculate model tracks using a planet of 1-Jupiter mass and radius, ir-  radiated by a solar-type star at a range of orbital distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' For these tracks, we adopt solar abundances in the  exoplanetary atmosphere, and either complete re-distribution of stellar irradiance, or no re-distribution (these are  same models as in Mansfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We found the no-redistribution case to be especially useful in our analyses,  and we henceforth refer to that case as the no redistribution (NR) track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Our second group of models is constructed  for individual planets, using their measured masses and radii, and the temperatures and radii of their host stars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  these are “population”-level models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' In all cases, we use PHOENIX model atmospheres to specify the irradiating  spectrum from the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We checked that using PHOENIX stellar models to irradiate the planets, versus ATLAS9  stellar models to calculate emergent fluxes (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='2) did not produce significant inconsistencies - see Section D of  the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  The models for individual planets vary the heavy element abundance (metallicity), using both solar metallicity,  and 30-times (30X) solar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Because both groups of models omit the presence of clouds, their atmospheres are dark,  with albedos near zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Emergent fluxes of four planets in this group of models are illustrated in Figure 1, that shows  some important properties of the modeled fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Spectral absorption features weaken as temperature increases, and  carbon monoxide appears in emission due to an atmospheric temperature inversion in the hottest planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Increas-  ing metallicity changes the nature of the absorption in Spitzer’s bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' At solar metallicity, methane absorption is  prominent in the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6 µm band for the coolest models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' At high metallicity, carbon becomes preferentially bound in  carbon monoxide and carbon dioxide (Moses et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' That effect is also apparent on Figure 1, where the coolest  30X metallicity models exhibit absorption in the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm bandpass, not as much at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' EMERGENT FLUXES AND DAY SIDE TEMPERATURE  The emergent flux of a hot Jupiter in a given spectral band is determined by the vertical atmospheric temperature  gradients and the specifics of the opacity sources that are significant in that band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The day side brightness temper-  ature in a given spectral band is thereby diagnostic of the temperature structure of the atmosphere, and how it may  change as a function of stellar irradiance, and how that heating is redistributed by longitudinal circulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Longitudinal heat circulations for individual hot Jupiters are best studied using thermal infrared phase curves, and  those studies have been extensive (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=', Knutson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2007, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Wong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Stevenson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Kreidberg  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Bell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' May et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2021, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Dang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Phase curves can delineate emergent flux over  the full range of planetary longitude, whereas eclipses yield only the day side flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Nevertheless, it has long been  known that the day side fluxes can yield important statistical information on the nature of heat circulation for hot  TTTTTT  10  WASP-121b  XO-4b  WASP-39b  CO  HAT-P-12b  10  10  3  4  5  3  4  5  Wavelength (microns)Spectral Fluxes of Hot Jupiters  5  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Observed and modeled fluxes for hot Jupiters  at secondary eclipse, in both the Spitzer 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6- and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm  bands, versus planetary Teq and Tirr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Fluxes from  the no redistribution (NR) model are also plotted as  a track, as well as the tracks for two blackbodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The  upper blackbody adopts zero albedo and no heat re-  distribution, whereas the lower blackbody adopts zero  albedo and heat redistribution over the entire planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  The vertical dashed lines mark the 99% confidence re-  gions for the quasi-discontinuous increase in flux (see  text for discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Jupiters (Cowan & Agol 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Although phase curves are numerous, eclipse data are even more numerous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Eclipse  fluxes provide an opportunity to probe physical effects on the day side atmosphere with maximum resolution in  stellar irradiance as we now demonstrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Figure 2 shows the emergent spectral fluxes that we derive for hot Jupiters at both 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6- and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm, plotted as a  function of the equilibrium and irradiation temperatures, Tirr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2 Note that analyzing fluxes at each wavelength allows  us to make maximum use of planets observed in only one Spitzer bandpass (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=', 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm, program 14059).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  For equilibrium temperature (Teq), we adopt an albedo of zero and complete redistribution of heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Equilibrium  temperature is less than irradiation temperature by a factor of  √  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Our values refer to the emergent flux density at  the top of the exoplanetary atmosphere, so they are independent of the planetary radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We derive uncertainties on  the fluxes by propagating uncertainties in the eclipse depth, stellar temperature, and planetary radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Uncertainty  in the planetary radius enters only via the ratio of planetary-to-stellar radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' That ratio contributes negligibly to  the flux uncertainty because transit depths are known much more precisely than eclipse depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We similarly derive  uncertainties in the equilibrium and irradiation temperatures based on uncertainties in the stellar temperature and  planetary orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Fluxes from the no redistribution (NR) models, and two blackbodies, are overplotted on Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The NR model  track adopts no redistribution of stellar irradiance, solar composition, and surface gravity equal to Jupiter’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Temper-  ature via absorbed irradiation is the overwhelmingly dominant factor that determines the flux in the model track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We  2 irradiation temperature is Tirr = Ts  √Rs/a, where Ts and Rs are the stellar temperature and radius, and a is the orbital distance of the planet in a  circular orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Irradiation temperature  1000  1500  2000  2500  3000  3500  4000  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 μm  10~5  10F6  99% confidence limits  10  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6 μm  NR model  Blackbodies  10°  500  1000  1500  2000  2500  Equilibrium temperature6  Deming et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Observed and modeled fluxes for hot Jupiters  at secondary eclipse, in both the Spitzer 8 µm bands,  and ground-based eclipses in the K-band (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='3 µm), ver-  sus planetary Teq and Tirr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  No redistribution (NR)  model and blackbody tracks are overplotted as per Fig-  ure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The vertical dashed lines mark the 99% confi-  dence regions for the quasi-discontinuous increase in  flux (see text for discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  experimented with plotting models tailored to the parameters inferred for individual planets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=', surface gravity,  composition), and thereby verified that those parameters have a minor (albeit detectable) effect in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The  blackbody tracks on Figure 2 correspond to zero albedo and no heat redistribution, and also heat redistribution over  the entire planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Figure 3 shows fluxes that we derive from literature values of eclipses observed using the next two most widely-  used bands, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Spitzer’s 8 µm band, and ground-based eclipses in the K-band (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=', Kov´acs & Kov´acs 2019, and many  others), and also the NR model track and blackbody tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Both Figures 2 and 3 exhibit similar behavior in the Spitzer bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' For Teq exceeding ∼ 1700 K, the planetary fluxes  are largely confined between the two blackbody curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Below Teq ∼ 1700 K, fluxes can fall below the lower blackbody  curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' This general behavior has been seen previously, for example by G20 (their Figure 16), and Goyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Fluxes below the lower blackbody curve require either molecular absorption, or a non-zero albedo (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=', reflecting  clouds) that lower the atmospheric temperature below the complete redistribution case, or cold clouds that emit little  thermal radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The Figures suggest that the transition from fluxes that drop below the lower blackbody, to fluxes  that remain above it, occurs rather abruptly near Teq ∼ 1700 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' It has long been known that the hottest of the hot  Jupiters circulate heat with the least efficiency, and thereby have the hottest day side temperatures (Cowan & Agol  2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Schwartz & Cowan 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Showman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Below, we define this rise in emergent flux level quantitatively,  using the largest possible collection of Spitzer’s secondary eclipses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=" An Abrupt Rise in Flux  Irradiation temperature  1000  1500  2000  2500  3000  3500  4000  10-5  8 μm  99% confidence limits  10  2 μm  10'5  NR model  Blackbodies  10  500  1000  1500  2000  2500  Eguilibrium temperatureSpectral Fluxes of Hot Jupiters  7  We here demonstrate that an abrupt rise (i." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=', nearly discontinuous versus Teq) in day side emergent flux occurs at  an equilibrium temperature between 1714 K to 1818 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' In this section, we establish the data-related aspects of the  rise, such as amplitude and statistical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' In Section 6 we explore the physical implications for the day side  atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  First, we note that the 2 µm fluxes behave differently than the Spitzer fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' They occur more often above the NR  model, which is physically difficult to explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Ground-based eclipse observations are difficult, and it seems possible  that they may be especially affected by selection bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Observers may be reluctant to publish eclipses that give weak  detections (and consequently low fluxes), but they may readily publish strong eclipses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Hence, we hypothesize that  the ground-based eclipses may suffer from selection bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' In order to further investigate the abrupt rise in emergent  flux, we combine the three Spitzer bands, but for now we omit the 2 µm band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Figure 4 plots the ratio of the emergent  flux in the Spitzer bands to the value from the NR model track, versus Teq, with each point being a single eclipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  The NR model track is smooth (Figure 2 and Figure 3), so this division removes the gradual increase in flux with  increasing temperature, but it does not create the abrupt rise in flux: that abrupt rise is caused by the observed data,  not by the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Partial Eclipses and Eccentric Planets  Figure 4 omits the planets WASP-34b (Smalley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2011), WASP-67b (Hellier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2012), and WASP-140b (Hellier  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2017), because their eclipses are partial, and therefore the eclipse depth cannot be converted reliably to an  emergent flux at the top of their atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We do include TrES-3b, because O’Donovan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2007) concluded  that it is not partial, although it is close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Consequently, we include it but we increase its error bars by calculating  the fraction of flux that would be missed when varying the orbital parameters within the error ranges given by  O’Donovan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Planets with eccentric orbits may have day side temperatures at eclipse that are not representative of their equilib-  rium temperature averaged over their orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' This primarily affects HD 80606b and HAT-P-2b, due to their substantial  orbital eccentricities (Moutou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Bakos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' For these two planets, we replaced their TEPCat values  of Teq with the day side temperatures calculated at eclipse in the time-dependent models of Mayorga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2021)  (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Planets with less extreme eccentricities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=', WASP-14b, Joshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2009) are included using Teq values  from TEPCat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Amplitude and Location of the Flux Rise  In order to investigate the magnitude and location of the emergent flux rise, we apply the Bayesian methodology  described by Lee & Heghinian (1977) (hereafter, LH) to the ratio data of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The LH method is specifically  tailored to find abrupt changes in the values of serial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' It adopts a uniform prior for the change location, and  assumes that the data values and change amplitude have Gaussian distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' It gives the posterior distributions  for the change magnitude and location as analytic expressions involving sums over the data values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (The posterior  distributions are not Gaussian, or even necessarily smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=')  The LH algorithm detects an abrupt rise in the ratio of the emergent flux to the NR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The distribution for  temperature where the rise in flux ratio occurs is sharply peaked at Teq = 1730 K, as shown on Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The rise in  flux is abrupt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' not clearly resolved in the temperature coordinate in spite of using 437 eclipses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The posterior  distribution for the rise amplitude has a mean value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0157 in the ratio of flux to NR model (from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='58  to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='70), so the rise is nominally detected at the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6σ level of significance, assuming that the data are normally  distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' As a test, we impose a strong (and physically motivated) prior on the flux ratio by using only the flux  ratios per eclipse that are between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' That range would encompass low fluxes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=', due to high, cold clouds),  ranging to high fluxes from a hot day side with no redistribution, and assuming a blackbody spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' In that case,  the location and amplitude of the flux rise are not significantly changed, but the statistical significance would rise to  9σ, principally because ratios above unity are eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Although we do not include the 2 µm eclipses in our adopted solution, we explored whether including them would  affect the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' In this case, we reject only the eclipse of WASP-10b at 2 µm, that is an extreme outlier (Cruz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  2015), at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='65 times the NR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Including the balance of the 2 µm eclipses in the LH analysis does not significantly  change the location of the flux rise, and the statistical significance actually rises to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='8σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Although it is informative  that the 2 µm eclipses support our result from the Spitzer eclipses, we continue to base our analysis exclusively on  the Spitzer data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  8  Deming et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Three Tests  We did three additional tests to verify the existence of an abrupt rise in flux on Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' To verify the statistical  significance of the rise, we made 106 synthetic realizations of the flux ratio data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Those realizations preserved the Teq  values and the relative differences among multiple eclipses for the same planet, but they varied the average flux ratios  for each planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We adopted a uniform (not Gaussian) distribution for the flux ratios, and distributed them between  0 and 1 (see above), with no trends or rises in average value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Because we adopted a uniform distribution for these flux  ratios, not a Gaussian distribution assumed by the LH method, these synthetic data represent a ”worst case” for the  rise analysis, assuming that there is no physical reason to justify ratios above unity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=', no planets hotter than the  no redistribution case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='028% of these simulations produced a detected rise at a significance equal to the real  data and we conclude that the flux rise is significant at greater than a 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='97% level of confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Our second test explored whether the flux rise might be gradual, rather than abrupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We again constructed 106  realizations of the data, preserving the Teq values and relative differences among eclipses of the same planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We  scattered the average flux ratio for each planet around a gradual ramp with a scatter matching the real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The  ramp varied between the two flux ratio levels detected in the real data, and spanning the range between 1000 K and  2500 K in Teq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The purpose of these realizations is to investigate whether the LH algorithm could be fooled into mis-  identifying a gradual flux rise as being abrupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We found that only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='009% of the simulations were mis-identified  as being abrupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We also did a third test, where we again made 106 realizations of the data that scattered around a  slope consistent with the real data, but with no restriction to vary between two flux ratio levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' This test also found  only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='009% of the simulations were mis-identified as being abrupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Hence, we trust the abrupt nature of the flux  rise inferred from the LH algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We also conducted further tests of the LH algorithm, as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Figure 4 contains many points, and it may be difficult to see the abrupt flux rise when so many points are plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  To make the flux rise visually clearer, Figure 5 bins the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6- and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm points from Figure 4 in arbitrary intervals of  100 Kelvins in equilibrium temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The formulation of the data on Figure 5 is not used directly in our analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  that Figure is intended merely to make the abrupt rise in flux more clear to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Location of the Abrupt Rise in Flux  The LH algorithm has been widely used in terrestrial meteorology (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=', to detect rapid changes in weather pat-  terns), but it is not often used in astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We further tested the methodology using more Monte Carlo simulations  applied to both the real data as well as synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We verified that, as the transition in flux level is broadened in  temperature, the LH posterior distribution also broadens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We explored the effect of dithering the data in both X- and  Y-coordinates, using the observational errors in both observed flux ratio and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We found that the obser-  vational errors in temperature are not sufficient to significantly broaden the posterior distribution in rise location,  and the effect of the errors in temperature is included on Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  The observed planets are not distributed uniformly in temperature, and we hypothesized that the sharpness of  the LH posterior distribution in flux-rise temperature could be an indirect artifact of the temperature distribution  of the planet population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We therefore perturbed the real data, forcing the flux rise at Teq = 1730 K to occur over  broader ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We broadened a range of the data near 1730 K over intervals of 50, 100, 150, 200, 250, and 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Broadening the data means that we adjusted the flux ratios (but not the equilibrium temperatures) to scatter around  an average linear ramp within those intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' As we increased the width of the ramp, the 1730 K peak in the posterior  distribution decreased in amplitude (as expected), and a second peak became evident near 1800 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Based on all the numerical experiments described above, we conclude that the flux rise at Teq = 1730 K is statis-  tically robust and abrupt, and not a random or gradual fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The sharp peak in the posterior distribution at  1730 K accurately reflects the specific distribution of equilibrium temperature and emergent flux in our actual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  However, we also conclude that small and statistically plausible variations in our data are consistent with the flux rise  occurring near 1800 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We compiled the Teq values that mark the 1-percent and 99-percent limits in the combined  area of both peaks of the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' That conservative prescription gives us 99% confidence that the flux  rise occurs between equilibrium temperatures of 1714 K and 1818 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' That 104 K range is sufficiently small to indicate  a rapid change in exoplanetary atmospheres, similar to a phase change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Temperature Amplitude of the Rise  The analysis above focused on the rise in the ratio of emergent flux to the NR model’s flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' That ratio rises because  the emergent fluxes from the planets increase with increasing Teq, not because the fluxes from the models drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Spectral Fluxes of Hot Jupiters  9  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Ratio of emergent flux to the flux from the no redistribution (NR) model, versus equilibrium and irradiation temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Each of the 437 points is a single Spitzer eclipse, except for a few planets (WASP-29, HAT-P-12, and HAT-P-15) where we averaged  several eclipses due to low signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Several outlying points lie above the range of this plot, but they have been  included in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The fluxes exhibit an abrupt rise at an equilibrium temperature exceeding 1730K, and the horizontal  lines mark the inferred levels (ratios of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='58 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The amplitude of the rise corresponds to a brightness temperature increase  of 210 ± 22 K, consistent among the different wavelength bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The lower trace is the posterior distribution for the rise position,  based on the analytic Bayesian methodology of Lee & Heghinian (1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The horizontal scale of the posterior distribution for the  solid line has been expanded graphically by a factor of 5, to make the shape more visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' For visual clarity, we do not plot error  bars on the X-axis (temperature), but those uncertainties (median value ±19 Kelvins) were included when deriving the posterior  distribution for the temperature of the flux rise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6- and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm points from Figure 4, binned over  intervals of 100 Kelvins, with error bars based on the standard  deviation (scatter) within each bin, divided by the square root  of the number of points in that bin (standard error of the mean).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  The bin centers are nominally at multiples of 100K in equilib-  rium temperature, but the points deviate slightly in X because  we average both the X- and Y-values within each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We plot  only the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6- and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm points so as to maximize the clarity  of the abrupt rise in flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The vertical dashed lines are the 99%  confidence limits that we infer for the temperature of the abrupt  rise in flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Note also that the data plotted here were not used  directly in our analysis - this Figure is only to make the flux rise  optimally visible to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  However, it is more physically meaningful to express the rise in flux as an increase in the planetary brightness  temperature, Tb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We calculate the temperature rise using two methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' First, we convert the rise in the flux ratio  average at each wavelength (Figure 4) to flux units using the NR model fluxes, and then calculating the rise in Tb by  reference to the temperature dependence of the model flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Using this method, the rise in Tb is ∆Tb = 192 ± 34 K at  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6 µm, ∆Tb = 216±30 K at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm, and ∆Tb = 249±69 K at 8 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Within the errors, the rise in brightness temperature  Irradiation temperature  1000  1500  2000  2500  3000  3500  4000  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0  model  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='8  Emergent flux / NR  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6 μm  < Posterior for rise position  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 μm  8 μm  500  1000  1500  2000  2500  Eguilibrium temperatureIrradiation temperature  1000  1500  2000  2500  3000  3500  4000  Emergent flux / NR model  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 μm  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6 μm  500  1000  1500  2000  2500  Eguilibrium temperature10  Deming et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Brightness temperature difference (Tb minus Teq) versus equilibrium temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Each point represents the average  of all eclipses for a given planet in a given wavelength band (see color legend).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The location of the abrupt temperature rise is  consistent with Teq = 1730K (as on Figure 4), with amplitude ∆Tb = 291 ± 49 K (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The red and blue lines are least-squares  fits in the regimes hotter and cooler than the location of the abrupt temperature rise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The fits consider uncertainties in both the X-  and Y-coordinates, but for visual clarity the (small) error bars on X are not plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The upward slopes of the lines are consistent  with brightness temperature exceeding the equilibrium temperature by increasing amounts with increasing irradiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  is the same at all three Spitzer wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Weighting the temperature rise at each wavelength by σ−2 yields an  average temperature rise of ∆Tb = 210 ± 22 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  A second method to calculate the rise in Tb is illustrated in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' In this case, we calculate Tb for each planet at  each wavelength using the eclipse depths and the stellar model fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Tb is generally proportional to the equilibrium  temperature, so we plot Tb−Teq in order to remove the overall increase in temperature with increasing irradiance, and  concentrate on the fine structure in the proportionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Because the most strongly irradiated planets circulate heat  with the least efficiency, we expect that Tb − Teq will slope upward (albeit, slightly) as a function of Teq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We applied  the LH algorithm to the Y-axis of Figure 6, and it finds a discontinuity near Teq = 1800 K, with 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='7σ significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  As a further test, we solved for a linear slope via least-squares for the entire range of data on Figure 6, and a  second solution that uses two separate least-squares lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' One of those two lines is for Teq less than the discontinuity  (blue line on Figure 6), and a second line at temperatures above the discontinuity (red line on Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Based  on the χ2 values, we calculate Bayesian Information Criteria for the one line versus two line solution, and we find  ∆BIC = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' That value of ∆BIC indicates ”very strong” evidence (Kass & Raftery 1995) in favor of the two line  solution, consistent with the LH algorithm finding a discontinuity in both the emergent flux (Figure 4) and brightness  temperature (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Each point on Figure 6 is a single wavelength band for a single planet, which would over-weight planets that were  observed in multiple bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' To derive the most accurate value of the temperature rise at the discontinuity, we weight  each planet in the population equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' That yields a rise in brightness temperature of ∆Tb = 291 ± 49 K, consistent (at  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='7σ) with the calculations based on the emergent fluxes (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='3, above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  To summarize our brightness temperature analysis, we find that Tb increases as a function of equilibrium temper-  ature with a slope exceeding unity, but that the relation has a discontinuity in the range of Teq between 1714 and  1818 Kelvins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The existence of the discontinuity in brightness temperature is detected by the LH algorithm (Lee &  Heghinian 1977), and supported very strongly by a Bayesian Information Criterion applied to results from least-  squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The amplitude of the discontinuity in brightness temperature (∆Tb = 291±49 K) is (within the errors) equal  at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6-, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5- and 8 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  We discuss the interpretation and atmospheric implications of the temperature rise in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' SPECTRAL PROPERTIES OF THE EMERGENT FLUXES  Irradiation temperature  1000  2000  3000  4000  1500  8 μm  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 μm  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6 μm  1000  (Kelvins)  500  0  500  1000  1000  1500  2000  2500  3000  Eguilibrium temperatureSpectral Fluxes of Hot Jupiters  11  We now turn to more general spectral properties of the emergent fluxes and brightness temperatures that are  given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Previous investigations have focused on this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Because eclipses of hot Jupiters have been most  abundantly observed in Spitzer’s 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6- and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm bands, the comparison between those bands has been a specific focus  of population studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' G20, B20, W21, and Goyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2021, and also Dransfield & Triaud 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We here use  our increased sample size to revisit the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6- and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm flux comparison, and also to construct a color-flux diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6- Versus 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 Micron Brightness Temperatures  Figure 7 shows the ratio of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5- to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6 µm brightness temperature, versus equilibrium temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' G20 found that  this ratio increases with equilibrium temperature with a slope of 100±24 parts-per-million per Kelvin over the range  of equilibrium temperature from approximately 800 to 2500 Kelvins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Work with larger samples (B20, and W21)  found a similar trend, and our complete sample of Spitzer’s hot Jupiters confirms the effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' A least-squares fit of  a straight line to the data on Figure 7, and accounting for uncertainties of each point in both X and Y coordinates,  yields a positive slope of 67 ± 13 ppm per Kelvin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The slope is thereby significantly different from zero by 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0σ,  revealing a spectral property of the hot Jupiter population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Recall that G20 - using a smaller sample - found a slope  of 100 ± 24 ppm per Kelvin, so our result is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='4σ below G20’s estimate and thereby consistent with their work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' B20  found essentially the same result as G20, and W21 found a similar relation (albeit expressed differently), at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='2σ  significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  We also allow for the fact that the scatter on Figure 7 exceeds the error bars, for example due to intrinsic variation  in the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We scaled the error bars to greater values so that the error bars statistically match the total scatter  in the points (observational errors plus intrinsic variations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' In that case, the least-squares solution gives a slope of  65 ± 17 ppm per Kelvin, still exceeding zero by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='7σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We did another test, following W21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We made 105 synthetic  data sets whose per-point scatter was identical to the real data, but no systematic trend was present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We calculated  least-squares slopes for those synthetic data realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Because the synthetic data sets contain no trend, least-  squares slopes as large as in the real data can only occur via random chance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' If the number of those cases is very  small, then we can conclude that the real data contain a real slope to appropriately high confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We find that  only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='30% of the realizations produce a slope as large as we find in the real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We conclude that the real data  exhibit a significant slope at 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='7% confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The brightness temperature at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm increases relative to the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6 µm  brightness temperature as the planets get hotter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The planets do not behave as blackbodies, because blackbodies have  the same brightness temperature at all wavelengths (ratio everywhere equal to unity on Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  We calculated 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6- and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm brightness temperatures for solar and 30X metallicity model tracks, and those tracks  are also plotted on Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' For equilibrium temperatures exceeding ∼ 1200 Kelvins, the models slope upward  similarly to the data, so the models do a good job of accounting for the hottest planets (the solar metallicity track  is best for the hottest planets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' However, for equilibrium temperatures below ∼ 1200 Kelvins, the data and models  differ widely, and we discuss the physical implications of this difference in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' A Color-Flux Diagram  By analogy with stellar population studies, color-color and color-magnitude diagrams have been used to infer  properties of the hot Jupiter population (Triaud 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Triaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Dransfield & Triaud 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We explore that  approach using our complete sample, but with one key difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Rather than plotting a color-magnitude diagram,  we plot a color-flux diagram wherein the radii of the planets is not a factor, only the spectral character of their  emergent flux matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Figure 8 shows the color-flux diagram for our complete sample, using the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm flux on the Y-axis, and with the  planets color-coded by equilibrium temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We also plot two model tracks, based on collections of models for  individual planets (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Those tracks are for solar metallicity, and for 30X solar metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The hottest planets  tend to emit the most flux, and thereby lie highest on the Y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' For those hottest planets, the model tracks are very  close together, indicating that the flux ratio (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6 µm) is not very sensitive to metallicity, in agreement with  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The models track the hottest planets very well, unlike the blackbody line that slopes slightly to the upper  left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Dransfield & Triaud (2020) found that planets tend to scatter to the right of the model track in a color-magnitude  diagram (their Figure 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' we find good agreement with the models for the hottest planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Also, Figure 8 is consistent  with Figure 7, wherein the brightness temperature ratio (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5- to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6 µm) increases with equilibrium temperature  (blackbodies having a constant ratio), and the models follow the planets in Figure 7 for equilibrium temperatures  exceeding ∼ 1200 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  12  Deming et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Brightness temperature ratio (Tb at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm divided by Tb at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6 µm) versus equilibrium temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The tracks of  points in slate blue and blue-violet colors are theoretical points from the models (solar metallicity, and 30-times solar metallicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  The black line is a least-squares fit that considers error bars in both the X- and Y-coordinates (error bars in X are not plotted, to  keep the Figure clear).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The slope of the fitted line differs significantly from zero (see text), and the model tracks come close to  bracketing the observed points for equilibrium temperatures below ∼ 1200 Kelvins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  One aspect of the hottest planets deserves mention, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' that there is no obvious and abrupt shift toward greater  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm flux relative to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6 µm for the hottest planets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' no abrupt shift to the right on Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Such a shift  was found by B20, that they attributed to excess emission in the fundamental carbon monoxide band caused by  atmospheric temperature inversions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We discuss this issue in more depth in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  The coolest planets in the sample fall to the lower left on Figure 8, unlike directly imaged planets and brown  dwarfs that fall to the lower right (Dransfield & Triaud 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The error bars are large for individual planets, but  the shift of the hot Jupiter population to the lower left is a real effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' This effect is consistent with previous work  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=', Kammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2015), and is also consistent with the behavior seen on Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Much of this effect can be due to  high metallicity, because the 30X solar model track follows the planets on Figure 8 much better than does the solar  metallicity track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We discuss this issue in greater depth in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' DISCUSSION  We here discuss that the flux rise has been noticed in previous work, we explore the physical implications of the  abrupt flux rise, and we also discuss the spectral properties of the emergent fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Noticed Previously  The flux rise is noticeable on the right panel of Figure 2 in Wallack et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2021), and was also noted by Goyal  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The latter authors did not elaborate, but Wallack et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2021) discussed decreasing efficiency of heat  transport for temperatures exceeding ∼ 1500 K, albeit they did not emphasize the abruptness of the transition in their  paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We believe that this is the same phenomenon that we measure in this paper, in spite of the modest difference  in the location of the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' B20 found an abrupt increase in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm flux near Teq = 1660 ± 100 K that we believe  is related to the same phenomenon, as we discuss in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Weakening Molecular Absorption  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='4  Ratio of Tb (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='8  Solar metallicity track  30X metallicity track  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6  Observed data  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='4  1000  1500  2000  2500  Equilibrium Temperature (KSpectral Fluxes of Hot Jupiters  13  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Color-flux diagram for hot Jupiters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The X-axis is log of the flux ratio (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm flux divided by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6 µm flux) and the Y-axis  the log of the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The black line is a blackbody track, and the tracks for two collections of model planets are also included:  solar metallicity models, and 30X solar metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The observed planets are color-coded by calculated equilibrium temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Planet temperature (Kelvins)  400  650  900 1150 1400 1650 1900 2150 2400  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0  blackbody track  solar metallicity models  30x metallicity models  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0  close-in planets  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0  Log(Flux 5/Flux,  614  Deming et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  The first question concerning the abrupt flux rise is whether it reflects an increase in the temperature of the exo-  planetary atmosphere, or whether it is merely an apparent increase due to a change in the depth of formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' For  example, if molecular absorption weakens, then the emergent flux can originate from deeper and hotter layers of  the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' That could cause an increase in emergent flux without any change in the run of temperature versus  height in the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The depth of flux formation could change due to either molecular dissociation or cloud  dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  The models show that molecular absorption does indeed weaken as the equilibrium temperature increases, but the  weakening is gradual, not abrupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Moreover, the abrupt increase in brightness temperature being closely equal at  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6- and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm is unlikely to be caused by weakening of molecular absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The molecular opacities are different  in the two bands, and the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm band includes emission by carbon monoxide, not present at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We conclude  that weakening molecular absorption is not a likely explanation for the abrupt rise in flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Water Vapor Dissociation  The abrupt flux rise occurs near an equilibrium temperature where Mansfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2021) find that water absorp-  tion disappears in emergent spectra due to the dissociation of water vapor (Parmentier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Water vapor is a  cooling agent: stellar irradiance is absorbed primarily at optical and near-IR wavelengths, and water vapor helps to  balance that heating by re-emitting in the infrared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Therefore the absence of water vapor can produce a heating of  the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' To the extent that water dissociation occurs abruptly with rising equilibrium temperature, it could  in principle produce the abrupt flux rise that we detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' However, this explanation also fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Figure 4 shows the flux  rise as an increase in observed flux divided by the NR model’s flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The NR model includes the effects of water vapor  and its dissociation on the atmospheric temperature structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Therefore the abrupt flux rise seen in Figure 4 cannot  be ascribed to lack of cooling by water vapor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Cloud Clearing  Cloud dissipation that abruptly reveals deeper, hotter layers, is a possible explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Current understanding of  cloud formation and dissipation (Helling 2019) predicts that the dominant cloud types change with temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  The day sides of the ultra-hot Jupiters are expected to be cloud-free (Parmentier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Wakeford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Helling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' However, near the temperature of the flux rise, multiple types of clouds are expected to be  present (Adams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Could cloud-clearing occur over a small range of temperature and thereby lead to the  abrupt flux increase?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Or, does cloud-clearing occur more gradually in temperature, as clouds of a single composition  are replaced by other cloud types as temperature increases?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Shifts in Kepler phase curves (Demory et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2013) are  diagnostic of the presence of reflecting clouds (Parmentier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Those phase shifts change gradually with  equilibrium temperature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=', Figure 3 of Parmentier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2016), suggesting that cloud clearing is not sufficiently  abrupt to account for the flux rise that we detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The trend of geometric albedo versus equilibrium temperature for  hot and ultra-hot Jupiters (Wong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2021) also seems inconsistent with a single rapid cloud-clearing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Nevertheless, some models indicate that rapid cloud dissipation could indeed produce the abrupt increase in  brightness temperature that we observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Roman & Rauscher (2019) and Roman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2021) modeled cloud for-  mation in GCM models of hot Jupiters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Figure 2 of Roman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2021) shows an abrupt increase in day side thermal  emission between 1750 K and 2000 K for their extended nucleation-limited case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Also, Figure 2 of Gao & Powell  (2021) shows a distinct jog upward in modeled day side brightness temperature near 1700 K, having approximately  the amplitude that we observe and caused by the dissipation of day side silicate clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  We conclude that cloud dissipation is a tentatively viable explanation for the abrupt increase in brightness temper-  ature that we observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' However, additional work is needed in order to clarify the effects of cloud dissipation on the  day side brightness temperature, and to reconcile observations at multiple wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Efficiency of Heat Transport  It is also important to consider a rise in day side temperature due to an abrupt decrease in the efficiency of spatially  redistributing stellar irradiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Parmentier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2021) found that non-gray radiative transfer enables a decrease  in heat redistribution efficiency that begins suddenly near 1600 K, and continues to increase monotonically with in-  creasing irradiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Our data indicate a similar behavior but require a more abrupt decrease in heat redistribution  concentrated near 1730 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We discuss whether that abrupt decrease in efficiency could occur either as a thermody-  namic effect of night side clouds, or via inhibition of winds via magnetic drag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Spectral Fluxes of Hot Jupiters  15  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Thermodynamic Effect of Night Side Clouds  Hot Jupiters having tidally locked rotation can balance stellar irradiation by re-emitting that energy input, espe-  cially from their night sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' However, cloud formation on the night side reduces the efficiency with which the night  side can radiate, leading to increased day-night contrasts Parmentier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' As equilibrium temperatures rise,  night side clouds are forced to higher altitudes (Gao & Powell 2021), but the temperature of the clouds tends to be  approximately constant, determined by the condensation temperature of the cloud species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' That can further decrease  the efficiency of re-radiation by the night side, forcing the day side to become hotter and re-radiate more strongly  in order to balance stellar irradiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' That effect is intertwined with the consequences of cloud-clearing on the day  sides, where the dissipation of day side clouds allows deeper and hotter layers of the day side atmosphere to become  visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Magnetic Drag  Perna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2010) and Menou (2012) calculated that rising ionization in strongly irradiated hot Jupiters would  produce a conductive atmosphere and magnetic induction and drag that damps the zonal winds, leading to a hotter  day side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' For a reasonable atmospheric field strength of 10 Gauss, Menou’s Figure 1 indicates that the zonal winds  fall in amplitude by half near an equilibrium temperature of ∼ 1650 Kelvins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' That is close to where we observe the  abrupt rise in day side flux, and Menou’s scaling calculations predict that the decrease in zonal wind speed becomes  complete over ∼ 200 Kelvins in equilibrium temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Self-consistent 3D MHD simulations (Rogers & Komacek 2014) also show a similar decrease in zonal wind (their  Figure 6) at Teq ∼ 1500 to 1800 K, depending (like Menou’s predictions) on the assumed magnetic field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Hindle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2021) also find a transition in MHD behavior as ionization sets in and the magnetic Reynolds number  becomes large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Thorngren & Fortney (2018) infer the efficiency of stellar irradiation at inflating the radii of 281 giant planets via  Ohmic dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' They find a peak in efficiency near the equilibrium temperature of our flux rise, and Sarkis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  (2021) confirmed their results with an independent study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Inflation of radii by Ohmic dissipation is intimately tied  to the presence of magnetic drag (Menou 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Rauscher & Menou 2012, 2013), and Knierim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2022) applied the  results from Thorngren & Fortney (2018) to an Ohmic dissipation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' They found that interpreting hot Jupiter  radius inflation as due to magnetism implies a strong decrease in zonal wind speed at ∼ 1700 to 1800 K, consistent  with our abrupt increase in day side brightness temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  We conclude that the onset of magnetic drag joins possible cloud dissipation as viable explanations for the abrupt  rise that we detect in day side emergent flux and brightness temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='2 discusses some tests that could  help to distinguish between these possibilities, (or demonstrate that they both play a role).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Other Factors in the Flux Rise  We have also considered other factors that can contribute to the abrupt flux rise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The hottest planets tend to orbit  the hottest stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We applied the LH algorithm to the trend of stellar host temperature as a function of planetary equi-  librium temperature, but we find no abrupt change in stellar temperature corresponding to the flux rise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' However,  indirect factors related to the stellar temperature are possible contributors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' For example, the spectral distribution  of stellar energy can enhance temperature inversions (Lothringer & Barman 2019), and potentially play a role in the  flux rise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Other factors such as chemical disequilibrium, metallicity, and variations in C/O ratio (Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='7) may  contribute via their effect on radiative opacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Clouds can also scatter in the infrared, and make planets on the cool  side of the flux rise appear colder than they actually are (Taylor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Metallicities and C/O  Figures 7 and 8 provide some metallicity information concerning hot Jupiters (because opacities differ between  the two Spitzer bands).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' For temperatures above ∼ 1200 Kelvins, the models track the brightness temperature ratio  (Figure 7) and color (Figure 8) of the hot Jupiter population very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Both of those Figures demonstrate that the  planets are not blackbodies (Goyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2021), but their departures from blackbodies are due to a combination of  continuous opacity and weak molecular absorption/emission in the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6- and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm Spitzer bands, not strongly sensi-  tive to metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Spectroscopy of individual hot Jupiters has revealed both high metallicities (Sheppard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Wong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2022) and sub-solar metallicites (Line et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The Spitzer data prefer solar metallicites for equi-  librium temperatures above ∼ 1200 Kelvins, because the solar metallicity track on Figure 7 matches the brightness  16  Deming et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  temperature ratio better than does the 30X metallicity track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' However, Figure 8 shows that many of the coolest hot  Jupiters are best matched by high metallicity models, because the 30X solar track falls well to the left of the solar  metallicity track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Many of the coolest planets on Figure 7 fall between the solar and 30X metallicity tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (We do  not expect that metallicity is a function of equilibrium temperature per se, but the cooler planets more easily reveal  an enhanced metallicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  In principle, atmospheric metallicity can be a function of planetary mass and/or radius (Welbanks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2019),  and that relation might be reflected in the brightness temperature ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We searched for a correlation between the  Tb ratio and both planetary mass and radius in the region where the model tracks for solar and 30X solar diverge  (Teq < 1200 K, see Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We found no statistically significant correlations in either case, but only 21 planets lie in  that region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Our results remain broadly consistent with modest enhancements in the atmospheric metallicity of hot  Jupiters, those enhancements being below the upper limits inferred by Thorngren & Fortney (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  The C/O ratio should also be considered when comparing the relative fluxes to models (Wallack et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We  compared the brightness temperature ratio to a model track with C/O=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' That comparison (not plotted) provides  only worse agreement with the data than does the solar abundance C/O track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We conclude that enhanced C/O is  not a significant help when interpreting the Spitzer eclipses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Temperature Inversions and CO Emission  There is excellent (and growing) evidence that the most strongly irradiated planets host temperature inversions in  their upper atmospheres (Mikal-Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Cont et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2022a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Our models  for the hottest planets include temperature inversions, and emission in the v=1 to v=0 fundamental band of carbon  monoxide that lies within the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm bandpass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' That spectral emission is a significant reason why the planets and  the track of models slope to the upper right of the blackbody track on Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We see no evidence for a distinct  transition in the population due to the emission phenomenon, as claimed by B20, but we did not look for such an  effect statistically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Rather, we have focused on the abrupt increase in flux that we find to occur in both the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6- and  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm spectral bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  B20 concluded for an abrupt transition to emission at an equilibrium temperature of 1660±100 Kelvins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Within the  errors, that location agrees with the abrupt day side brightness temperature rise that we observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The temperature  rise is easier to detect at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5- than at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6 µm, but in our extensive and improved data we detect it in both bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' B20  matched their 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6 µm data with blackbodies, and then extended and removed those blackbodies from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm data to  detect excess emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We suggest that B20 thereby detected the day side temperature rise only at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm, whereas  we find that the abrupt flux increase occurs in both bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' SUMMARY AND FUTURE WORK  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Summary  We have re-analyzed Spitzer’s 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6- and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm secondary eclipses of hot Jupiters using one uniform method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Our  sample comprises 457 eclipses of 122 planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' This large sample allows us to make general statements, with good  statistical justification, concerning the emergent spectral properties of the hot Jupiter population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We find that the  day side brightness temperatures in the population increase abruptly, similar to a phase change, at an equilibrium  temperature between 1714 and 1818 Kelvins (99% confidence interval).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The amplitude of the temperature rise is  291 ± 49 Kelvins, and we conclude that it is caused either by rapid clearing of day side clouds, or by decreased  redistribution of stellar irradiance due to magnetic drag, as predicted a decade ago by Menou (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  We compare the fluxes and brightness temperatures in the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6- and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm bands, and we confirm previous work  showing that the brightness temperature ratio (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5- to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6) increases with equilibrium temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Models having  solar metallicity reproduce that brightness temperature ratio for planets with equilibrium temperatures exceeding ∼  1200 Kelvins, but the Spitzer photometry is not very sensitive to metallicity for those hottest planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' For equilibrium  temperatures below ∼ 1200 Kelvins, the brightness temperature ratio falls consistently below the models having solar  metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Most of those cooler planets are bracketed by models between solar and 30X solar metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Our results  are broadly consistent with modest enhancements in the atmospheric metallicity of hot Jupiters, those enhancements  being below the upper limits inferred by Thorngren & Fortney (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  The ultra-hot portion of the population exhibit temperature inversions in our models, with emission due to car-  bon monoxide in the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm bandpass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' That emission is a significant factor that allows the models to statistically  reproduce the observations for the hottest planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We do not see a distinct transition due to temperature inversions  Spectral Fluxes of Hot Jupiters  17  in the hot Jupiter population as deduced by Baxter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2020), but we did not look for such an effect statistically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Moreover, the random errors per-planet are large, and it is generally not possible to point to individual planets and  conclude that they host a temperature inversion based only on the Spitzer photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Additional data (Mansfield  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2022) are often needed to establish temperature inversions in individual planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Future Work  The cause of the abrupt rise in brightness temperature can be tested by future work, both theoretical and observa-  tional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Theoretical investigations that specifically address the abruptness of the transition in the population would  be illuminating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Observationally, it is possible that the longitudinal offset in the population of phase curves could  exhibit a corresponding abrupt transition to near-zero at equilibrium temperatures exceeding our range of 1714 to  1818 Kelvins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Some results in that regard are encouraging (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=', Figure 12 of May et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2022), but not all (Bell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' It should also be possible to derive temperature-pressure profiles for hot Jupiters on both sides of the tran-  sition, using eclipse spectroscopy from JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Those temperature profiles may encode the effects of magnetic drag,  when compared to MHD general circulation models (Rogers & Komacek 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Beltz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The temperature  structure should also encode the presence of clouds, and help to determine the role of cloud dissipation on the  brightness temperatures of hot Jupiters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Finally, abrupt changes in Doppler signatures might be detectable using  ground-based high resolution spectroscopy, and any such effects would be diagnostic of changes in the zonal winds  and in the efficiency of heat transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  An aspect of secondary eclipses that we have omitted from the scope of this paper concerns the orbital phase of the  eclipses, and the implications for orbital eccentricity and dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Recent improvements in orbital ephemerides  (Ivshina & Winn 2022) imply that the secondary eclipse phases should have significant utility for studies of orbital  dynamics, and we will address that topic in a future paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  We thank Peter Gao for clarifying the signatures of cloud dissipation and for comments on this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We also  thank the anonymous referee for comments that allowed us to improve our original version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We acknowledge an un-  published email exchange among Heather Knutson, Jonathan Fortney, and Adam Showman, wherein they discussed  a ”sharp transition” in the efficiency of heat transport in hot Jupiters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' They concluded that a purely radiative time  scale effect would be gradual, not abrupt, and also that the onset of magnetic drag would be more likely to be gradual  than abrupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Cloud clearing was mentioned by Adam Showman as possibly causing a sharp transition, due to rapid  cloud dissipation with increasing temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  18  Deming et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  APPENDIX  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' DERIVATION OF ECLIPSE DEPTHS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Photometry  Extraction of a secondary eclipse depth begins with photometry of the Spitzer images, either full frame or subarray  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' For full frame images, we extract a 32x32 pixel subframe centered on the host star, and thereafter we process  those data in a similar manner to the subarray data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We first repair hot pixels with a 4σ filter applied to each pixel  in time, and we replace discrepant pixels by their median value over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (Strictly speaking, discrepant pixels  should be zero-weighted and not repaired, but Tamburo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2018) found that it makes negligible difference in  actual practice with Spitzer photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=') Subarray data cubes are 32x32-pixels x 64 subframes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' After the hot pixel  correction, we process each of the 64 subframes in the data cubes independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' For each image we find the centroid  of the star by fitting a 2-D Gaussian, after median filtering the image to assure that the Gaussian fit is not fooled by  any discrepant pixels not repaired by the temporal filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We calculate the flux within 11 circular apertures centered  on the star, the aperture radii varying from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We determine the background flux by first masking the  star, and then fitting to a histogram of background pixel values to determine the median.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Scaling that value to the  sizes of the apertures, we subtract the background from each image independently and thereby produce a time series  of photometry for each eclipse of each planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  In the case of eclipses that occur within long time series such as phase curves, we use only a limited range of data  centered on the eclipse, typically 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='4 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6 in phase (when the eclipse is at phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Since we process 457 eclipses,  we find that no rigid rules are practical, because within that large collection of eclipses are multiple special cases that  frustrate any rigid procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We accordingly sometimes vary the range of data that we extract from phase curves  depending on factors such as the amplitude of the phase variation and any instrumental temporal ramps that may  be present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' In several instances, we checked to make sure that the derived eclipse depths do not vary systematically  with the range of the data that are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' For eclipses observed independently of phase curves, we omit the first 30  minutes of the data, to avoid temporal ramps that are often steepest at the start of the time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' That 30 minute  trim does not include ”peak up” data that are often acquired before starting the primary time series observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' In  some cases, when a temporal ramp is especially prominent, we omit 60-, or (in very few cases) 90 minutes of data at  the start of the time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Pixel-Level Decorrelation  Spitzer data are affected by a well known intra-pixel sensitivity variation that must be removed from the data in  order to derive accurate eclipse depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We do that using pixel-level decorrelation (PLD, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=', G20, Deming et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Wong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Kilpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Benneke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2017), based on 12 pixels centered on the star (a 4x4  square, minus the corners).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The PLD fitting process solves simultaneously for the eclipse depth, central phase, and  temporal ramp coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The analysis is formally Bayesian, but the nature of secondary eclipses give us either  very weak prior constraints, or very strong ones, so in practice the solution is dominated by minimizing chi-squared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  For example, the central phase of the eclipse is only weakly constrained by transit and radial velocity data, so we  specify a uniform interval for the prior on central phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Conversely, the shape of the eclipse curve is very strongly  constrained by transit and RV data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Hence we use priors with zero variance for orbital parameters other than central  phase (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=', we freeze their values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' This is consistent with previous work (G20, and O’Rourke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Mansfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Deming et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  The eclipse fit begins with a search over phase to find an initial starting value for center of eclipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' With the best  central phase found, the code then explores many combinations of photometric aperture and bin size, choosing the  combination that produces the best slope in the Allan deviation relation at that trial value of the central phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The  Allan deviation relation (Allan 1966) expresses that the standard deviation of the residuals (data minus best fit)  should decrease as the square root of the bin size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The fitting process at this stage is a multi-variate linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  We bin both the PLD pixel coefficients and the photometry for the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The reasons for binning the data are explained  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We limit the bin size so that the number of data points after fitting exceeds the number of fitted parameters  by at least a factor of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' In our experience, this avoids overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' A histogram of the Allan variance slope is shown  as Figure 9 for all eclipses we analyze (both 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6- and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm combined).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The distribution of slopes is peaked at -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='48,  Spectral Fluxes of Hot Jupiters  19  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Histogram of Allan deviation slopes for the  residuals (data minus fit) of 439 eclipses (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6- and  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm both included).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  The blue curve is the best-  fitting Gaussian that represents the distribution of  slopes, and it peaks at -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='48, with standard deviation  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='047.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  and the distribution closely approximates a Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' A Gaussian fit to the histogram (illustrated on Figure 9) has  a standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='047, which reflects both the random uncertainty of each slope measurement as well as  intrinsic variation in the quality of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Only one eclipse produces a fitted slope that is 3σ less than -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 (AOR  39446016, for WASP-35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We verified that the seemingly unphysical slope in that instance is not due to overfitting,  but occurs because of autocorrelation in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  When the fit has selected the best combination of bin size and photometric aperture, it then refines the eclipse  depth, central phase, and ramp parameters by exploring parameter space using a Markov Chain Monte Carlo  (MCMC) sampling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The MCMC is a classic Metropolis-Hastings algorithm with Gibbs sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' During  a 10,000 step burn-in process, we dynamically adjust the step size to achieve an acceptance ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Then each  MCMC chain runs for 800,000 steps, and we find that the sampling of parameter space is excellent in both eclipse  depth and central phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Binned Data  Fitting to binned data is (in theory) non-optimal because the binning process destroys information (Kipping 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Recently, May et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2021) concluded against using binned data in fitting to phase curves (but not secondary eclipses  per se).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We therefore re-visited our rationale for using binned fits, and we conclude that the binning is desirable,  at least in our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We here explain our reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' First, we note that the Spitzer IRAC instrument already bins  the incident photons in time before transmitting the data to Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The observing cadence available in IRAC was  set before transiting planets were discovered, so the observing cadences are not special for the purpose of secondary  eclipses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We find that further binning of the data eliminates instrumental effects on short time scales (Challener et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  2021) that would otherwise be very difficult to adequately reproduce in the fitting process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' In other words, although  the binning destroys information, it destroys information that we want to destroy because we cannot reproduce those  short-term instrument effects, and they occur on a faster time scale than the eclipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We find that the binning allows  the fitting process to concentrate on the time scales that are most relevant to the eclipse, and thereby it reduces red  noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We limit the size of the bins so that we do not distort the shape of the eclipse curve (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='g, by broadening the  ingress/egress), and (as noted above) we limit the bin size so as not to overfit the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' For planets with multiple  observed eclipses, our code usually selects different bin sizes from eclipse to eclipse, because they are independent  events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' In those cases, we checked to verify that eclipse depths do not correlate with the adopted bin size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Temporal Baselines  We initially fit each eclipse with two different temporal baselines (ramps), a linear and quadratic ramp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We consult  a Bayesian Information Criterion (BIC) analysis applied to the fitting residuals in order to decide between the linear  and quadratic ramp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Although the BIC is very useful, we do not use it rigidly, and we also consider the totality of  fit properties when deciding between temporal ramps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' For example, the BIC may prefer a quadratic ramp, but that  might produce a two-peaked posterior distribution for the central phase, that is not physically realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' So in that  case, we would adopt the linear ramp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' For planets whose equilibrium temperatures exceed 2000 K, we use only a  quadratic temporal ramp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The reason for that choice is to allow the ramp to account for the small portion of the  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0  Relative ocurrence  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='3  Slope20  Deming et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  phase curve that occurs within our data window, and thereby avoid bias in the fitted eclipse depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We verified that  a quadratic is a close approximation to the portion of the sinusoidal phase curve that occurs in our window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Chain Convergence  After deciding on the ramp, we run an additional MCMC chain with the adopted ramp, and we inspect the posterior  distributions for eclipse depth and central phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The distributions for two independent chains should be, and are  in almost all cases, closely coincident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We also calculate a Gelman-Rubin (GR) statistic (Gelman & Rubin 1992)  for eclipse depths in the two adopted MCMC chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Gelman & Rubin (1992) suggested that a value less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='1  denotes good convergence, but more recent work has tended to emphasize GR values closer to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The GR statistic  is derived from the shape and mean value of the posterior distributions, so closely equal distributions will always  produce GR values near unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Considering all planets and all eclipses, our median GR value is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='002, our average  value is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='003, and and our maximum value is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Given the large number of eclipses we analyzed, we did encounter some where convergence was initially prob-  lematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Those cases usually arise when the temporal ramp coefficients become partially degenerate with the eclipse  depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' That situation is more common when there is minimal out-of-eclipse baseline before and/or after the eclipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  In those cases, we achieve good convergence by freezing the temporal ramp coefficients during the MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Because  that freeze can cause the uncertainty in eclipse depth to be underestimated, in those cases we compare the width of  the posterior distributions for eclipse depth (that determine the random error) with and without the freeze, and we  thereby apply a correction to the error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Choice of Eclipse Depth  Our analysis yields two eclipse depths for each planet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' each of those depths is based on average from the same two  (or more) converged MCMC chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The first depth is the best fit using the criterion adopted by G20, who adopted  the solution yielding the best Allan deviation relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' With our larger sample, we have come to prefer the (more  conventional) practice of using the mean of the posterior distribution for depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We verified that the conclusions of  this paper do not change significantly if we adopt the choice used by G20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Our derived eclipse depths and uncertainties are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Eclipses for planets not previously published  are illustrated in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Special Cases  There are several planets that represent special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' WASP-33b orbits a pulsating star, and Deming et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2012)  used a wavelet technique to account for the stellar pulsations while simultaneously solving for the eclipses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The  wavelet technique is not easily included in our PLD code, so we have not re-analyzed the eclipses of WASP-33b, but we  include the depths from Deming et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2012) and Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2018) in Table 1 (so that all of Spitzer’s eclipse depths  at these wavelengths are available in a single Table).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Also, some planets have very weak eclipses observed multiple  times, and we treated those slightly differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' In those cases, we used the PLD code to correct for, and remove,  Spitzer’s instrumental effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We then combined the photometry for the multiple instrument-corrected eclipses and  fitted an eclipse curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Those fits obtained the eclipse depth and uncertainty based on the scatter, while constraining  the central phase to equal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Those cases are for planets WASP-29b and HAT-P-15b in both bands, and  WASP-67b and HAT-P-15b in the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6 µm band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The depths are listed in Table 1 associated with the first listed AOR,  but all of the listed AORs were averaged to derive the quoted depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' DILUTION CORRECTIONS  When a second star appears in the photometry aperture, the light from that companion star will dilute the eclipse,  regardless of whether the companion star is physically bound to the planet-hosting star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We corrected the emergent  fluxes for dilution using the multiplicative factors in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The eclipse depths in Table 1 are ”as observed”, not  corrected for dilution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The dilution factors were applied to the observed eclipse depths to derive the emergent fluxes  in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  We adopted dilution factors from Shporer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2014) and G20, and we calculated factors for planets not previ-  ously corrected for dilution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' In many cases, published Spitzer eclipse depths have not been previously corrected for  dilution because the diluting companion star was not detected when the Spitzer eclipses were first analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We con-  sulted the literature of high resolution imaging to find stellar companions (Ginski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Ngo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2015, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Spectral Fluxes of Hot Jupiters  21  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Eclipses of planets not previously published, and including WASP-29b that was only recently published by Wong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' In these plots, the data have been binned, and binning scales are adopted so that the data for each eclipse plot are spanned  by 50 plotted points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The 50-point binning is only for uniformity of illustration, the actual binning used in fitting for the eclipse  depths is different in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The X-axis of the plots is orbital phase, but values are not listed since the central phases of the  eclipses are not analyzed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='003  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='003  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0010  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0010 E  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0008 E  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
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+page_content='002  WASP-107  2  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
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+page_content='0006 F  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='001  [D2027  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
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+page_content='0002 F  W  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='999  H  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
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+page_content='998  L66:0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
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+page_content='003  T  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
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+page_content='0010  WASP-122  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='002  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='002  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
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+page_content=' We also found companions that were  sufficiently prominent, and spatially resolved from the primary star, using astronomical catalogs such as 2MASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Upon identifying a companion, we used the spectral types and K-band magnitudes of both stars to calculate their  fluxes in the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6- and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm Spitzer bands, based on the STAR-PET tool3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Companions closer than ∼ 2arc − sec were  not resolved from the primary star, and their fluxes simply add to produce the dilution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' When the companion star  was spatially resolved in the Spitzer data, we used the method described by G20 (their Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5) to calculate the  fraction of its flux that was scattered into the photometric aperture centered on the primary star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Using the calculated  fluxes from both stars in the photometric aperture, we calculated the dilution correction factors at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6- and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm  listed in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' COMPARISON TO LITERATURE VALUES  Although our eclipse depths are more extensive than any extant eclipse data set, it is nevertheless interesting  to compare subsets of our values to previous investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Previous compilations of eclipse depths have been  presented by G20, B20, W20, and Dransfield & Triaud (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We choose to compare to the values compiled by B20,  especially because those authors found an abrupt increase in the ratio of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5- to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6 µm eclipse depth - a phenomenon  similar to the abrupt flux increase that we infer in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Figure 11 shows our 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6- and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm average eclipse  depths per planet, as a function of the depth from B20, for planets in common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Least-squares fits of straight lines to Figure 11, considering the uncertainties in both X and Y, yield slopes within 2σ  of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We conclude that there is no systematic scale factor difference between our new eclipse depths and literature  values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We also investigated outlying points on Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The most prominent outlier is HD 189733b at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6 µm  (marked on Figure 11), where our average eclipse depth for nine eclipses is 1432±34 ppm (median value 1467 ppm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  B20 list 2560 ± 140, based on one eclipse from Charbonneau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2008), so this planet lies distinctly to the right  of the fitted line in the left panel of Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Consequently, the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6 µm brightness temperature calculated by B20  (1604 ± 32 Kelvins), is well above the equilibrium temperature of 1191 Kelvins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Our 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6 µm brightness temperature  is 1337 ± 30 Kelvins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Our values are consistent with a re-analysis by Knutson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2012), who found eclipse depths  consistent with our current results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Given that most planets at this level of irradiation have Tb not greatly exceeding  their equilibrium temperatures, and given that our nine eclipses are mutually consistent, we confirm the result from  3 https://irsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='ipac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='edu/data/Spitzer/docs/dataanalysistools/tools/pet/starpet/  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6 μm  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 μm  Our eclipse depth (ppm)  5000  5000  F  4000  4000  3000  3000  2000  2000  1000  1000  0  1000  2000  3000  4000  5000  0  1000  2000  3000  4000  5000  B20 eclipse depth (ppm)Spectral Fluxes of Hot Jupiters  23  Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Ratio of the average eclipse depth per planet (ours divided by Baxter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 2020), versus planet equilibrium tempera-  ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We find no systematic variations at either wavelength as a function of equilibrium temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The red mark indicates the  outlier Kepler-13b, discussed in Section C of the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Knutson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2012) that the original 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6 µm eclipse value from Charbonneau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2008) is in need of revision,  mostly likely due to improvements in observing and data analysis techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We point out that a similar revision  for HD 209458b was found by Diamond-Lowe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2014), and we note that our eclipse depths for HD 209458b are  also in good agreement with that revision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  We also compared the ratio of our eclipse depth to the value from B20, as a function of equilibrium temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  That comparison is shown in Figure 12, and in this case also we have investigated a prominent outlier at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The  outlier (red mark on Figure 12) is Kepler-13b, where B20 use an eclipse depth of 1560±330 ppm, from Shporer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Our eclipse depth (3035 ± 257 ppm) is almost twice as large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' This difficult system has a bright companion  star separated by approximately 1 pixel, that produces a large dilution correction and adds noise to the photometry  via pointing jitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We use the same dilution correction as do Shporer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2014), and we think that our PLD  analysis method is well matched to correct for noise produced by the companion star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The relatively ”small” eclipse  depths derived by Shporer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2014) lead them to infer an ”efficient heat redistribution process” that is unusual  for strongly irradiated hot Jupiters (Cowan & Agol 2011), and they also derive a geometric albedo (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='33) at the high  end of the range for hot Jupiters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We have rechecked our eclipse data analyses for this planet: we find no issues that  need correction, and we believe that our eclipse depths are correct, albeit with large random errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Pending further  analysis of other data on this system, we tentatively suggest that our eclipse depths are more realistic than are the  values derived by Shporer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2014), and used by B20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' However, we also note that the population-level results in  this paper are insensitive to the Kepler-13 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' PHOENIX VERSUS ATLAS MODELS  PHOENIX stellar atmospheres (Hauschildt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' 1997) are used in our planetary modeling code (Section 3),  whereas we use ATLAS9 stellar models to calculate emergent planetary fluxes from eclipse depths (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  These models are respectively deeply rooted in our procedures, and it would be impractical to alter those choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Hence, we investigated to what degree different stellar models can affect our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The principal concern was to  what degree the choice of model type affected the emergent day side flux that we calculated for each planet (Sec-  tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' B20 investigated PHOENIX versus ATLAS models and noted that planetary brightness temperatures were  consistent between PHOENIX and ATLAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Husser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2013) compared stellar fluxes from PHOENIX and ATLAS  and found excellent agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We investigated further by calculating stellar fluxes in the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6- and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm Spitzer  bands, from both PHOENIX (BT-NextGen) and ATLAS9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' We calculated the ratio of ATLAS flux to PHOENIX flux  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6 μm  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 μm  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0  Eclipse depth ratio  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0  1000  1500  2000  2500  1000  2000  3000  4000  Eguilibrium temperature24  Deming et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  at 9 stellar temperatures from 4000K to 9000K, at logg = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The ratio differed from unity with an average abso-  lute difference of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='68% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='02% at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6- and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Those differences are sufficiently small to have  no significant impact on our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Only at 9000 Kelvins did we find a significant difference, where the ATLAS9  stellar flux is 4% less than from PHOENIX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' That would affect primarily KELT-9b, and to a lesser extent KELT-20b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  However, Witzke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2021) found that PHOENIX models underestimate ultraviolet flux for hot stars compared to  observations (their Figure 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Because the models constrain the emergent stellar bolometric flux to be closely equal  to σT 4  ef f , underestimating UV flux will require overestimating emergent flux at other wavelengths (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=', the Spitzer  bands).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Consequently, we prefer the ATLAS9 models over PHOENIX especially for the hottest stars, to avoid that  issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Spectral Fluxes of Hot Jupiters  25  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Eclipse depths and uncertainties for 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6- and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm Spitzer observations, in units of the stellar flux, in parts-per-million.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  All planets are the ’b’ planet in the system unless otherwise noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' These are ”as observed” eclipse depths, not corrected for  dilution by companion stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' AOR = Astronomical Observing Request number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' In the rare instances where more than one eclipse  occurs in a single AOR, the depths are listed in chronological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' For four planets (HAT-P-12 and -15, WASP-29 and -67),  several AORs were averaged to yield a single eclipse depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' These eclipse depths are converted to emergent fluxes and brightness  temperatures in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (This Table is published in its entirety in machine-readable format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' A portion is shown here for guidance  regarding its form and content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=')  Planet  λ (µm)  AOR  Depth  1σ  CoRoT-01  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6  36786176  4328  591  CoRoT-01  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5  36785920  4285  440  CoRoT-02  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6  31774976  3134  198  CoRoT-02  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5  28640000  4788  272  57958144  4660  308  57958656  4310  369  HAT-P-01  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6  2474547  1339  129  HAT-P-01  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5  21060608  520  351  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Emergent fluxes in the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6- and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm bands, and brightness  temperatures in Kelvins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Planets are the b planet in each system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Units of  fluxes are ergs cm−2 sec−1 Hz−1, or (equivalently) milli-Watts m−2 Hz−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  (This Table is published in its entirety in machine-readable format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' A  portion is shown here for guidance regarding its form and content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=')  Planet  Fν 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6 µm  1σ  Fν 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm  1σ  Tb 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6 µm  Tb 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm  CoRoT-01  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='42e-06  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='76e-07  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='01e-06  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='12e-07  2413+156  −161  2145+112  −115  CoRoT-02  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='20e-06  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='97e-07  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='95e-06  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='98e-07  1785+43  −44  1844+58  −59  HAT-P-01  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='97e-06  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='87e-07  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='26e-07  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='90e-07  1733+64  −65  1073+206  −298  HAT-P-02  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='58e-06  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='66e-07  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='68e-06  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='56e-07  2067+185  −196  2052+127  −130  HAT-P-03  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='87e-06  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='83e-07  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='55e-06  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='20e-07  1467+73  −77  1401+75  −80  HAT-P-04  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='51e-06  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='30e-07  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='81e-06  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='25e-07  2247+153  −159  1803+182  −192  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Stellar and planetary temperatures and radii that we adopted  from TEPCat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Planets are the ’b’ planet in each system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The stellar radii  are in solar units, and the planetary radii in Jupiter units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Stellar tem-  peratures are effective temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The planetary temperatures are the  calculated equilibrium temperatures based on the stellar irradiance, and  adopting an albedo of zero and uniform redistribution over the planetary  sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' The values listed for HAT-P-2 and HD 80606 are not from TEP-  Cat, they are the day side temperatures from the time-dependent calcu-  lations of Mayorga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (This Table is published in its entirety in  machine-readable format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' A portion is shown here for guidance regard-  ing its form and content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=')  Planet  Rplanet  Teq  Rstar  Tstar  CoRoT-01  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='551  1915  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='131  5950  CoRoT-02  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='460  1521  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='901  5598  HAT-P-01  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='319  1322  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='174  5975  26  Deming et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Dilution corrections for secondary eclipse depth at both Spitzer wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' Sources: G20 = Garhart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2020), TH =  this paper, SH = Shporer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='  Planet  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='6 µm factor  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='5 µm factor  Source  CoRoT-2b  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0276  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0255  TH  HAT-P-8b  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0109  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0102  TH  HAT-P-16b  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0006  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0006  TH  HAT-P-20b  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0070  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0074  TH  HAT-P-30b  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0121  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0117  G20  HAT-P-33b  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0377  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0332  G20  HAT-P-41b  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0069  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0111  G20  HD 202772b  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='207  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='207  TH  KELT-1b  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0065  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0059  TH  KELT-2b  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='137  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='123  G20  KELT-3b  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0125  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0127  G20  KELT-16b  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0189  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0168  TH  Kepler-5b  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0434  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
+page_content='0384  TH  Kepler-6b  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
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+page_content='3847/1538-3881/aaa458' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf'}
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+Mobility edges and Lyapunov exponents of one dimensional mosaic models with
+nonreciprocal hopping
+Sheng-Lian Jiang,1 Yanxia Liu,2, ∗ and Li-Jun Lang1, †
+1Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials,
+School of Physics and Telecommunication Engineering,
+South China Normal University, Guangzhou 510006, China
+2School of Physics and Astronomy, Yunnan University, Kunming 650091, PR China
+(Dated: January 5, 2023)
+We establish in one dimension a general mapping of mosaic models with nonreciprocal hopping
+to their non-mosaic counterparts of which the critical points of localization are already known.
+The mapping gives birth to the mobility edges of these nonreciprocal mosaic models and even
+the Lyapunov exponents if the non-mosaic counterpart is given. For demonstrations, we take an
+Aubry-Andr´e-like model with complex quasiperiodic potentials and the Ganeshan-Pixley-Das Sarma
+model to show the mapping’s validity, successfully obtaining the mobility edges and the Lyapunov
+exponents of their nonreciprocal mosaic counterparts.
+This general mapping may pave the way
+of studying mobility edges, Lyapunov exponents, and in principle other important quantities of
+nonreciprocal mosaic models.
+I.
+INTRODUCTION
+The effective non-Hermitian Hamiltonians can be used
+to describe open quantum systems [1]. Non-Hermiticity
+can cause intriguing properties, such as exceptional
+points, non-Hermitian skin effects, and the non-Bloch
+band theory [2–14], which have no counterparts in Her-
+mitian systems.
+Meanwhile, non-Hermiticity in disor-
+der or quasiperiodic systems also brings up new phenom-
+ena, for example, the mobility edges can emerge separat-
+ing localized and extended states in the complex plane
+of spectrum in the Hatano-Nelson model, a prototypi-
+cal one-dimensional (1D) non-Hermitian model charac-
+terized by nonreciprocal hopping with random on-site
+disorder [15, 16].
+It is well known that an arbitrarily
+small strength of random disorder leads to the localiza-
+tion of all eigenstates in one- and two-dimensional Her-
+mitian models [17–19], and the effects of disorders in cor-
+responding non-Hermitian models have also been studied
+recently [20, 21]. Besides the non-Hermitian models with
+random disorders, the non-Hermitian quasiperiodic sys-
+tems have also sparked a great deal of interest [22–39],
+especially the non-Hermitian variants [25–27] of the cel-
+ebrated Aubry-Andr´e model [40].
+Recently, the mobility edges and metal-insulator phase
+transition have been studied extensively in quasiperiodic
+systems [41–44], because they can be analytically solved
+by several methods, such as the self-duality symmetry
+[40] and the Avila’s global theory [45]. Another impor-
+tant reason is that quasicrystals can be experimentally
+realized for both Hermitian and non-Hermitian versions,
+which promotes the studies of localization physics and
+mobility edges [46–49]. The mobility edge is one of cen-
+tral concepts in condensed matter physics and can be
+∗ yxliu-china@ynu.edu.cn
+† ljlang@scnu.edu.cn
+obtained in many Hermitian and non-Hermitian disor-
+der or quasiperiodic systems [50–53]. The search for new
+typical models that can be exactly solved with mobility
+edges is still on going case by case.
+In this work, without arduously dealing with models
+case by case, we establish in 1D a general mapping of
+a class of mosaic models with nonreciprocal hopping to
+their non-mosaic counterparts of which the critical points
+of localization are well known. By the mapping, the mo-
+bility edges as well as the Lyapunov exponents can be
+generally obtained in principle for a bunch of nonrecip-
+rocal mosaic models. For demonstrations, we take two
+typical models as examples. One is an Aubry-Andr´e-like
+(AA-like) model with nonreciprocal hopping and com-
+plex quasiperiodic potentials [38]. We can reconstruct the
+mobility edges previously obtained by the Avila’s global
+theory [45], and further obtain the Lyapunov exponents.
+The other one is the mosaic version of Ganeshan-Pixley-
+Das Sarma (GPD) model [54] equipped with nonrecip-
+rocal hopping.
+The analytical mobility edges and the
+Lyapunov exponents can be successfully obtained by the
+general mapping, which can be verified by our numerical
+calculation.
+This general mapping unravels the mechanism of the
+emergence of mobility edges and the localization proper-
+ties characterized by two Lyapunov exponents in mosaic
+models with nonreciprocal hopping, providing a new per-
+spective on this problem. This work is a nonreciprocal
+generalization of the Hermitian mapping in Ref. [55].
+This rest of the paper is organized as follows. We first
+introduce a general 1D mosaic model with nonreciprocal
+hopping in Sec. II, and analytically derive a general map-
+ping between the nonreciprocal mosaic and non-mosaic
+models in Sec. III. Then, we apply this mapping to two
+specific models, the AA-like model and the GPD model,
+in Sec. IV to obtain their exact mobility edges and Lya-
+punov exponents. Finally, a conclusion with some dis-
+cussion is summarized in Sec. V.
+arXiv:2301.01711v1  [cond-mat.dis-nn]  4 Jan 2023
+
+2
+II.
+THE MOSAIC MODEL WITH
+NONRECIPROCAL HOPPING
+We consider a general class of 1D mosaic models, which
+can be described by the Hamiltonian
+H =
+�
+j
+(JL|j⟩⟨j + 1| + JR| j + 1⟩⟨j| + Vj|j⟩⟨j|), (1)
+where JR,L denote the strengths of nonreciprocal hop-
+ping between nearest-neighbor sites with the subscript
+representing the hopping direction, and Vj is the on-site
+potential at site j of the form
+Vj =
+�
+λ∆j, j = κm (m ∈ Z)
+0,
+otherwise
+,
+(2)
+with λ being the strength of mosaic potential, κ ≥ 1
+an integer representing the period of non-mosaic sites,
+and ∆j a model-dependent function that can induce the
+localization.
+Every successive κ sites form a so-called
+quasicell labeled by m, which also denotes the mth non-
+mosaic site that has a nonzero on-site potential.
+For
+convenience, in the following we take N quasicells, i.e.,
+m = 1, · · · , N, and thus the system size L = κN.
+By taking |Ψ⟩ = �
+j ψj|j⟩, the static Schr¨odinger equa-
+tion H|Ψ⟩ = E|Ψ⟩ can be written in terms of the ampli-
+tude ψj as
+�
+Eψκm = JRψκm−1 + JLψκm+1 + λ∆κmψκm
+Eψj = JRψj−1 + JLψj+1,
+(j ̸= mκ)
+. (3)
+III.
+MAPPING BETWEEN NONRECIPROCAL
+MOSAIC AND NON-MOSAIC MODELS
+Since there are (κ − 1) sites that are free of on-site po-
+tentials between two nearest-neighbor non-mosaic sites,
+a reasonable inference is that the localization, if occurs,
+should exist only at the non-mosaic sites. Therefore, one
+may be curious about the effective coupling between the
+non-mosaic sites. To this end, we can resort to the trans-
+fer matrix.
+From the second line of Eq. (3), the system of equa-
+tions for the mosaic sites between two nearest non-mosaic
+sites, say the mth and the (m + 1)th non-mosaic sites,
+can be given explicitly as follows
+�
+�
+�
+Eψκm+1 = JRψκm + JLψκm+2
+· · ·
+Eψκ(m+1)−1 = JRψκ(m+1)−2 + JLψκ(m+1)
+. (4)
+With the aid of the forward transfer matrix F, which
+is defined as
+�
+ψκm+2
+ψκm+1
+�
+=
+�
+E/JL −JR/JL
+1
+0
+� �
+ψκm+1
+ψκm
+�
+≡ F
+�
+ψκm+1
+ψκm
+�
+,
+(5)
+Eq. (4) can be written as
+�
+ψκ(m+1)
+ψκ(m+1)−1
+�
+= F κ−1
+�
+ψκm+1
+ψκm
+�
+,
+(6)
+and thus, the amplitude at mosaic site ψκm+1 can be
+expressed in terms of those of two non-mosaic sites ψκm
+and ψκ(m+1), yielding
+ψκm+1 = Jκ−1
+L
+Aκ
+ψκ(m+1) + JRAκ−1
+Aκ
+ψκm,
+(7)
+with
+Aκ =
+ηκ
++ − ηκ
+−
+√E2 − 4JLJR
+,
+η± = E ± √E2 − 4JLJR
+2
+. (8)
+Likewise, the relation can be also found by considering
+the backward transfer matrices connecting the mth with
+the (m − 1)th non-mosaic sites, and thus the amplitude
+at mosaic site ψκm−1 can be expressed as
+ψκm−1 = Jκ−1
+R
+Aκ
+ψκ(m−1) + JLAκ−1
+Aκ
+ψκm.
+(9)
+The details of derivation for the coefficients can be re-
+ferred to in Appendix .
+Substituting Eqs. (7) and (9) into the first line of Eq.
+(3), one can have a system of equations only involving
+non-mosaic sites, yielding
+Jκ
+Rψκ(m−1) + Jκ
+Lψκ(m+1) + λAκ∆κmψκm = Bκψκm,
+(10)
+with
+Bκ = EAκ − 2JLJRAκ−1,
+(11)
+which shows the effective relation between the nearest-
+neighbor non-mosaic sites. If one regards m as the site
+index with κ being just regarded as a parameter, Eq.
+(10) is nothing but a static Schr¨odinger equation for a
+non-mosaic tight-binding model with Jκ
+L,R being nonre-
+ciprocal hopping, λAκ∆κm the on-site potential, and Bκ
+the effective eigenenergy, which, for brevity, is called the
+scaled model in the following.
+Comparing the scaled model with the non-mosaic
+model described by Eq.
+(3) with κ = 1, we have the
+following mapping:
+JL,R → Jκ
+L,R, λ → λAκfκ, E → Bκ, ψj → ψκm,(12)
+where the introduction of fκ reflects the relation between
+potential functions ∆κm and ∆m in the two models. This
+mapping reveals that if a wave function ψj of energy E
+is localized, say, at site j under the potential function
+∆m of the non-mosaic model, then the localization also
+occurs at site m under the potential function ∆κm of
+the scaled model, described by the wave function ψκm of
+energy Bκ, which also means that the wave function is
+localized at the non-mosaic site κm of the mosaic model
+with finite κ. Thus, given the critical point of localization
+
+3
+determined by a function f(JL, JR, λ, E) = 0 in the non-
+mosaic model, the critical point of the mosaic model can
+be determined by
+f(Jκ
+L, Jκ
+R, λAκfκ, Bκ) = 0,
+(13)
+through the mapping Eq. (12). Generally, the critical
+point of the mosaic model depends on E, i.e., there ex-
+ist mobility edges, even if no mobility edges exist for the
+non-mosaic model, because Aκ is a function of E. In prin-
+ciple, if one knows the critical point of a nonreciprocal
+non-mosaic model, the critical point of the corresponding
+mosaic model can also be obtained by Eq. (13).
+Sometimes, the Lyapunov exponent γ(JL, JR, λ, E) >
+0, characterizing the decaying behavior of a localized
+state, is already established for some non-mosaic models
+with random or quasiperiodic disorders, i.e., the form of
+localized state at site j0 is known as |ψj| ∝ e−γ|j−j0|,
+and then we can also obtain the Lyapunov exponent
+γ(Jκ
+L, Jκ
+R, λAκfκ, Bκ) > 0 of the scaled model (10) with
+respect to the “sites” labeled by m, that is, the state
+|ψκm| ∝ e−γ(m−m0) = e− γ
+κ (κm−κm0) is localized at the
+non-mosaic site κm0, where γ/κ is just the inverse of lo-
+calization length or the Lyapunov exponent [56] of corre-
+sponding mosaic models with respect to non-mosaic sites
+κm. Of course, the critical point obtained from Eq. (13)
+just satisfies the relation γ(Jκ
+L, Jκ
+R, λAκfκ, Bκ) = 0.
+For the case that the nonreciprocal hopping can be
+parameterized as JL = te−g and JR = teg, the scaled
+model Eq. (10) becomes
+teκgψκ(m−1) + te−κgψκ(m+1) + λaκ∆κmψκm = εκψκm,
+(14)
+similar as the reciprocal case in Ref. [55], with only the
+difference in hopping terms caused by the nonreciprocity.
+Here, we replace Aκ and Bκ by
+aκ =
+1
+�
+E2/t2 − 4
+��E/t +
+�
+E2/t2 − 4
+2
+�κ
+−
+�
+E/t −
+�
+E2/t2 − 4
+2
+�κ �
+,
+εκ = Eaκ − 2taκ−1,
+(15)
+respectively, such that εκ has the dimension of energy and
+aκ is dimensionless with t being regarded as the energy
+unit; define a0 = 0 for the consistency of ε1. Specifically,
+a1 = 1,
+ε1 = E,
+a2 = E/t,
+ε2 = E2/t − 2t,
+a3 = E2/t2 − 1, ε3 = E3/t2 − 3E,
+(16)
+which will be used in the following applications. In terms
+of parameters t and g, the mapping Eq. (12) becomes
+t → t, g → κg, λ → λaκfκ, E → εκ, ψj → ψκm.(17)
+Apparently, it can be reduced to the result for reciprocal
+mosaic models (i.e., g = 0) in Ref. [55].
+12
+45
+78
+-50
+-25
+0
+numerical
+analytical
+FIG. 1.
+The fractal dimension Γ with respect to the real
+part of energy Re(E) and the potential strength λ, numer-
+ically calculated for (a) κ = 2 and (b) κ = 3 under PBCs
+with other parameters g = 0.2, φ = 0, L = F12 = 144, and
+β = F11/F12 = 89/144. (c) The existence or not of the imagi-
+nary part of energy with the same parameters as in (a), where
+(non)zeros of Im(E) with accuracy 10−6 are colored in blue
+(yellow). The red solid lines in (a-c) are the mobility edges
+calculated analytically by Eq. (23). (d) Comparison of a lo-
+calized state marked by the red triangle (E = 2.5881, λ = 2.2)
+in (b) with the corresponding analytical form Eq. (22), show-
+ing that the two Lyapunov exponents can well characterize
+the asymmetric localization in the nonreciprocal mosaic AA-
+like model. The dotted line indicates the center site of the
+localization.
+To demonstrate the validity of the general mapping,
+we apply Eq. (17) to two specific nonreciprocal mosaic
+models in the following, where, for convenience, we set
+t = 1 as the energy unit and focus on the cases of g > 0,
+i.e., the hopping is right-biased.
+IV.
+APPLICATIONS
+Aubry-Andr´e-like model
+First, we take the mosaic version of an AA-like model
+with nonreciprocal hopping and complex potentials,
+dubbed nonreciprocal mosaic AA-like model for conve-
+nience in the following, which is described by Hamilto-
+nian (1) with the potential function in Eq. (2) being
+∆j = ∆κm = 2 cos (2πβκm + θ + iφ) ,
+(18)
+where β is an arbitrary irrational number, and (θ, φ) ∈ R
+represent a complex phase shift. The corresponding re-
+ciprocal non-mosaic AA-like model undergoes an Ander-
+son localization at λ = ±e−|φ| [40], which is further gen-
+eralized to λ = ±eg−|φ| [25] for the nonreciprocal non-
+mosaic case; they are all independent of energy E, i.e.,
+
+-6202m161-62020061-62020010614
+there are no mobility edges for non-mosaic AA-like mod-
+els.
+Based on this critical point, we can apply our gen-
+eral mapping (17).
+Since the difference between two
+potential functions ∆κm = 2 cos(2πβκm + θ + iφ) and
+∆m = 2 cos(2πβm+θ+iφ) is only between the irrational
+numbers κβ and β, the fκ in the mapping can be taken
+as 1, because the critical points of Anderson localization
+for quasiperiodic models are independent of the values of
+irrational numbers [57, 58], although the eigenstates and
+eigenvalues under these two potentials are irrelevant in
+details. Thus, by making the replacement of λ → λaκ
+and g → κg in λ = ±eg−|φ|, one can find that the mobil-
+ity edges,
+λκ(E) = ±eκg−|φ|/aκ,
+(19)
+emerge for mosaic AA-like models. This result is identi-
+cal to that obtained in Ref. [38] by using Avila’s global
+theory of quasiperiodic Schr¨odinger operators [45], which
+proves the validity of our general mapping.
+In addition, one can also obtain the Lyapunov expo-
+nents for the nonreciprocal mosaic AA-like models by the
+same mapping. From Ref. [25], we know that two Lya-
+punov exponents γ±g > 0 can be used to characterize an
+asymmetrically localized state of a nonreciprocal model,
+where
+γ = ln |λ| + |φ|
+(20)
+is the Lyapunov exponent of a symmetriccally localized
+state of the corresponding reciprocal model, and the
+smaller one γ − g determines the critical point of Ander-
+son localization for the nonreciprocal model. Therefore,
+with the same replacement, one can get two Lyapunov
+exponents of the scaled model (10) as
+γ(±)
+κ
+(E) = ln |λaκ| + |φ| ± κg,
+(21)
+which depends on the energy E, leading to an asymmet-
+rically localized state,
+|ψκm| ∝
+�
+�
+�
+e− γ(−)
+κ
+κ
+(κm−κm0), m > m0
+e− γ(+)
+κ
+κ
+(κm0−κm), m < m0
+,
+(22)
+with the peak at the non-mosaic site κm0. The right-
+biased asymmetry [i.e., γ(−) < γ(+)] of the localized state
+with respect to the center site κm0 results from the right-
+biased hopping (i.e., g > 0) we set in advance. This can
+be partially checked by setting γ(−)
+κ
+(E) = 0, which just
+gives rise to the critical point in Eq. (19).
+For a further verification, we compare the numerical
+results with the analytical ones for the cases of κ = 2
+and 3 in Fig. 1, where, from Eqs. (19) and (21), the
+critical points and the Lyapunov exponents are specified
+as
+λ2(E) = ±e2g−|φ|
+E
+, γ(−)
+2
+(E)
+2
+= ln |λE| + |φ|
+2
+− g,
+λ3(E) = ±e3g−|φ|
+E2 − 1, γ(−)
+3
+(E)
+3
+= ln |λ(E2 − 1)| + |φ|
+3
+− g.
+(23)
+For the numerical calculation, we take β = Fs−1/Fs as
+a rational approximation of an irrational number β =
+lims→∞ Fs−1/Fs = (
+√
+5 − 1)/2, and the system size L =
+Fs, where Fs (s = 1, 2, · · · ) is the sth Fibonacci number
+defined as Fs = Fs−1 + Fs−2 with the first two numbers
+F1 = F2 = 1. The fractal dimension [19],
+Γ = − ln I/ ln L,
+(24)
+can be used to characterize the localization of a state ψj
+more clearly for a finite system than the commonly used
+inverse participation number (IPR) [19],
+I =
+L
+�
+j=1
+|ψj|4��
+L
+�
+j=1
+|ψj|2�2
+.
+(25)
+To clearly demonstrate the transition of Anderson lo-
+calization, of which the critical points are the same under
+PBCs and OBCs due to the blindness about boundary
+conditions for a bulk-localized state [25], we use PBCs for
+numerical calculations in Fig. 1 to avoid non-Hermitian
+skin effect under OBCs [4] where all bulk states are ag-
+gregated together to one boundary (here the right bound-
+ary for our settings), leading to similar values of fractal
+dimension as bulk-localized states. Meanwhile, accord-
+ing to the bulk-bulk correspondence of a nonreciprocal
+model when φ = 0, i.e., the critical point of Anderson
+localization under OBCs is also a cut between complex
+and real spectra under PBCs [25], as shown in Figs. 1(a)
+and 1(c), one can assume the reality of E and abandon
+the complex-E solutions during the derivation of criti-
+cal points. This can also explain why the mobility edges
+E(λ) are just functions of curves, not surfaces, because
+Im[(E(λ)] = 0. The maximal number 2(κ−1) of mobility
+edges can also be expected, as shown in Figs. 1(a) and
+1(b), from Eq. (19), which can have the highest power
+(κ − 1) of E. The two Lyapunov exponents are demon-
+strated in Fig. 1(d) as two slopes from the peak site to
+the two shoulders. Due to the existence of side peaks on
+two shoulders besides the main peak, the numerical re-
+sult can shift occasionally relative to the analytical result
+for some regions, but the preserved slope still reflects the
+correctness of the Lyapunov exponents.
+Here, we just demonstrate the case of real potential
+function ∆j; of course, the case of complex ∆j in Ref.
+[38] can also be reobtained using this mapping.
+Ganeshan-Pixley-Das Sarma model
+The mobility edges of mosaic AA-like model emerge
+from a constant critical point of the non-mosaic coun-
+terpart. Here, we take as another application the mo-
+saic GPD model [54] with nonreciprocal hopping, whose
+non-mosaic counterpart already has mobility edges. The
+Hamiltonian is described by Eq. (1) with
+∆j =
+2 cos (2πβj)
+1 − b cos (2πβj),
+(26)
+
+5
+8
+24
+40
+-40
+-20
+0
+numerical
+analytical
+FIG. 2. The same meanings and settings as in Fig. 1, except
+for (a,c) κ = 1 and (b,d) κ = 2. The mobility edges [red solid
+lines in (a-c)] are calculated analytically by Eq. (31). The
+localized state marked by the red triangle in (b) is selected as
+E = 3.7328, λ = 2.4. The parameter b = 0.5 is chosen for all
+figures.
+with the same definition of β as the AA-like model and
+the deformation parameter b ∈ (−1, 1).
+For the non-mosaic case with nonreciprocal hopping,
+i.e., κ = 1, the mobility edges [30]
+λ(E) = 1
+2
+�
+− bE ± eg �
+1 +
+√
+1 − b2�
+± e−g �
+1 −
+√
+1 − b2� �
+,
+(27)
+can be obtained by the Avila’s global theory [45] and the
+similarity transformation under OBCs [25]. When g = 0,
+it reduces to the reciprocal case with mobility edges being
+λ(E) = ±1 − bE
+2 ,
+(28)
+which is first obtained by a generalized duality transfor-
+mation [54], and the Lyapunov exponent
+γ(E) = ln
+�����
+|bE + 2λ| +
+�
+(bE + 2λ)2 − 4b2
+2
+�
+1 +
+√
+1 − b2�
+�����
+(29)
+is analytically obtained in Ref. [30, 31] with the aid of the
+global theory; that γ(E) = 0 just gives rise to the critical
+point Eq. (28). As we mentioned before, E should be
+a real number for an Anderson localized state, and thus,
+hereafter the notations (absolute and squared values) in
+Eq. (29) are only defined in the field of real numbers.
+With the aid of the similarity transformation under
+OBCs in Ref. [25], the two Lyapunov exponents, γ(E) ±
+g, for the nonreciprocal non-mosaic case, i.e., κ = 1 and
+g ̸= 0, can be readily obtained, and the mobility edges
+Eq. (27) can also be verified by γ(E) − g = 0.
+Finally for the nonreciprocal mosaic version of GPD
+model, using the mapping in Eq. (17), we get the two
+Lyapunov exponents:
+γ(±)
+κ
+(E)
+κ
+= 1
+κ ln
+�����
+|bεκ + 2λaκ| +
+�
+(bεκ + 2λaκ)2 − 4b2
+2
+�
+1 +
+√
+1 − b2�
+�����
+±g,
+(30)
+where aκ and εκ are defined in Eq. (15), and the mobility
+edges can be determined by γ(−)
+κ
+(E) = 0, or directly by
+the mapping from Eq. (27), yielding
+λκ(E) =
+1
+2aκ
+�
+− bεκ ± eκg �
+1 +
+√
+1 − b2�
+± e−κg �
+1 −
+√
+1 − b2� �
+.
+(31)
+We also take the cases of κ = 1 and 2 for demonstra-
+tion, shown in Fig. 2, where we also use the same setting
+of β and PBCs as the aforementioned AA-like model.
+The mobility edges are also curves separating real spec-
+tra and complex ones; the number can also maximally be
+2(κ − 1) as expected.
+V.
+CONCLUSION AND DISCUSSION
+We establish a general mapping of nonreciprocal mo-
+saic models to their non-mosaic counterparts of which
+the critical points of localization or the Lyapunov expo-
+nents of localized states are known, and can find the an-
+alytical expressions of mobility edges and the Lyapunov
+exponents of nonreciprocal mosaic models. For demon-
+strations, we take an AA-like model [38] and the GPD
+model [54] with nonreciprocal hopping to test the map-
+ping, successfully giving birth to the mobility edges and
+the Lyapunov exponents of these models. This mapping
+is a generalization of the reciprocal version in Ref. [55].
+The mapping Eq.
+(17) can be not only applied to
+quasiperiodic models, but also to randomly disordered
+models and the non-disorder models that have localiza-
+tion transition. However, in one dimension the critical
+points or Lyapunov exponents are less established for
+nonreciprocal models, and thus, we only offer quasiperi-
+odic models as applications in this paper.
+For exam-
+ple, in Ref. [55] the mobility edges of reciprocal mosaic
+Wannier-Stark model can be obtained due to the well-
+known critical point of the reciprocal non-mosaic coun-
+terpart, but the lack of Lyapunov exponent makes us fail
+to obtain the critical point or mobility edges of the non-
+reciprocal mosaic models using the mapping.
+Moreover, without the parameterization, JL = te−g
+and JR = teg, the bulk-bulk principle [25] cannot work
+due to the possible existence of complex spectrum for an
+Anderson localized state even under OBCs. Therefore,
+the energies of critical points could be complex, i.e, the
+mobility edge in general becomes a surface not a curve
+as shown here, which may be more interesting and need
+to further study.
+
+20061-62020lmi6620200
+m-662106
+Appendix: Derivation for the transfer matrix
+For the forward transfer matrix F in Eq. (A.9) of the
+main text, we now calculate F κ−1 as follows.
+First, F can be diagonalized as
+F =
+�
+E/JL −JR/JL
+1
+0
+�
+= UΛU −1,
+(A.1)
+where
+Λ =
+�
+η+/JL
+0
+0
+η−/JL
+�
+(A.2)
+is the diagonal matrix with
+η± = E ± √
+E2 − 4JLJR
+2
+,
+(A.3)
+and
+U =
+�
+η+/JL η−/JL
+1
+1
+�
+(A.4)
+is a unitary matrix with its inverse
+U −1 =
+�
+JL
+−η−
+−JL
+η+
+� ��
+E2 − 4JLJR.
+(A.5)
+Then, one can get
+F κ−1 = UΛκ−1U −1
+=
+�
+Aκ/Jκ−1
+L
+−JRAκ/Jκ−1
+L
+Aκ−1/Jκ−2
+L
+−JRAκ−2/Jκ−2
+L
+�
+,
+(A.6)
+where
+Aκ =
+ηκ
++ − ηκ
+−
+√E2 − 4JLJR
+.
+(A.7)
+Therefore, we have the Eq. (7) in the main text
+ψκm+1 = Jκ−1
+L
+Aκ
+ψκ(m+1) + JRAκ−1
+Aκ
+ψκm.
+(A.8)
+For the backward transfer matrix B, defined as
+�
+ψκm−2
+ψκm−1
+�
+=
+�
+E/JR −JL/JR
+1
+0
+� �
+ψκm−1
+ψκm
+�
+≡ B
+�
+ψκm+1
+ψκm
+�
+,
+(A.9)
+we have
+�
+ψκ(m−1)
+ψκ(m−1)+1
+�
+= Bκ−1
+�
+ψκm−1
+ψκm
+�
+.
+(A.10)
+Likewise, we can get
+Bκ−1 =
+�
+Aκ/Jκ−1
+R
+−JLAκ/Jκ−1
+R
+Aκ−1/Jκ−2
+R
+−JLAκ−2/Jκ−2
+R
+�
+,
+(A.11)
+and thus the Eq. (9) in the main text
+ψκm−1 = Jκ−1
+R
+Aκ
+ψκ(m−1) + JLAκ−1
+Aκ
+ψκm.
+(A.12)
+ACKNOWLEDGMENTS
+This work was supported by the National Natural
+Science Foundation of China (Grant Nos. 11904109,
+12204406), the Guangdong Basic and Applied Basic Re-
+search Foundation (Grant No. 2019A1515111101), the
+Science and Technology Program of Guangzhou (Grant
+No. 2019050001), Guangdong Provincial Key Laboratory
+(Grant No. 2020B1212060066), and NKRDPC (Grant
+No. 2022YFA1405304).
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+
diff --git a/_9AzT4oBgHgl3EQfvv1V/content/tmp_files/load_file.txt b/_9AzT4oBgHgl3EQfvv1V/content/tmp_files/load_file.txt
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@@ -0,0 +1,718 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf,len=717
+page_content='Mobility edges and Lyapunov exponents of one dimensional mosaic models with  nonreciprocal hopping  Sheng-Lian Jiang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='1 Yanxia Liu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' ∗ and Li-Jun Lang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' †  1Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  School of Physics and Telecommunication Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  South China Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Guangzhou 510006,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' China  2School of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Yunnan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Kunming 650091,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' PR China  (Dated: January 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' 2023)  We establish in one dimension a general mapping of mosaic models with nonreciprocal hopping  to their non-mosaic counterparts of which the critical points of localization are already known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  The mapping gives birth to the mobility edges of these nonreciprocal mosaic models and even  the Lyapunov exponents if the non-mosaic counterpart is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' For demonstrations, we take an  Aubry-Andr´e-like model with complex quasiperiodic potentials and the Ganeshan-Pixley-Das Sarma  model to show the mapping’s validity, successfully obtaining the mobility edges and the Lyapunov  exponents of their nonreciprocal mosaic counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  This general mapping may pave the way  of studying mobility edges, Lyapunov exponents, and in principle other important quantities of  nonreciprocal mosaic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  INTRODUCTION  The effective non-Hermitian Hamiltonians can be used  to describe open quantum systems [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Non-Hermiticity  can cause intriguing properties, such as exceptional  points, non-Hermitian skin effects, and the non-Bloch  band theory [2–14], which have no counterparts in Her-  mitian systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  Meanwhile, non-Hermiticity in disor-  der or quasiperiodic systems also brings up new phenom-  ena, for example, the mobility edges can emerge separat-  ing localized and extended states in the complex plane  of spectrum in the Hatano-Nelson model, a prototypi-  cal one-dimensional (1D) non-Hermitian model charac-  terized by nonreciprocal hopping with random on-site  disorder [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  It is well known that an arbitrarily  small strength of random disorder leads to the localiza-  tion of all eigenstates in one- and two-dimensional Her-  mitian models [17–19], and the effects of disorders in cor-  responding non-Hermitian models have also been studied  recently [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Besides the non-Hermitian models with  random disorders, the non-Hermitian quasiperiodic sys-  tems have also sparked a great deal of interest [22–39],  especially the non-Hermitian variants [25–27] of the cel-  ebrated Aubry-Andr´e model [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  Recently, the mobility edges and metal-insulator phase  transition have been studied extensively in quasiperiodic  systems [41–44], because they can be analytically solved  by several methods, such as the self-duality symmetry  [40] and the Avila’s global theory [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Another impor-  tant reason is that quasicrystals can be experimentally  realized for both Hermitian and non-Hermitian versions,  which promotes the studies of localization physics and  mobility edges [46–49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' The mobility edge is one of cen-  tral concepts in condensed matter physics and can be  ∗ yxliu-china@ynu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='cn  † ljlang@scnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='cn  obtained in many Hermitian and non-Hermitian disor-  der or quasiperiodic systems [50–53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' The search for new  typical models that can be exactly solved with mobility  edges is still on going case by case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  In this work, without arduously dealing with models  case by case, we establish in 1D a general mapping of  a class of mosaic models with nonreciprocal hopping to  their non-mosaic counterparts of which the critical points  of localization are well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' By the mapping, the mo-  bility edges as well as the Lyapunov exponents can be  generally obtained in principle for a bunch of nonrecip-  rocal mosaic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' For demonstrations, we take two  typical models as examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' One is an Aubry-Andr´e-like  (AA-like) model with nonreciprocal hopping and com-  plex quasiperiodic potentials [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' We can reconstruct the  mobility edges previously obtained by the Avila’s global  theory [45], and further obtain the Lyapunov exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  The other one is the mosaic version of Ganeshan-Pixley-  Das Sarma (GPD) model [54] equipped with nonrecip-  rocal hopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  The analytical mobility edges and the  Lyapunov exponents can be successfully obtained by the  general mapping, which can be verified by our numerical  calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  This general mapping unravels the mechanism of the  emergence of mobility edges and the localization proper-  ties characterized by two Lyapunov exponents in mosaic  models with nonreciprocal hopping, providing a new per-  spective on this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' This work is a nonreciprocal  generalization of the Hermitian mapping in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  This rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' We first  introduce a general 1D mosaic model with nonreciprocal  hopping in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' II, and analytically derive a general map-  ping between the nonreciprocal mosaic and non-mosaic  models in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Then, we apply this mapping to two  specific models, the AA-like model and the GPD model,  in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' IV to obtain their exact mobility edges and Lya-  punov exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Finally, a conclusion with some dis-  cussion is summarized in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='01711v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='dis-nn]  4 Jan 2023  2  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  THE MOSAIC MODEL WITH  NONRECIPROCAL HOPPING  We consider a general class of 1D mosaic models,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' which  can be described by the Hamiltonian  H =  �  j  (JL|j⟩⟨j + 1| + JR| j + 1⟩⟨j| + Vj|j⟩⟨j|),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (1)  where JR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='L denote the strengths of nonreciprocal hop-  ping between nearest-neighbor sites with the subscript  representing the hopping direction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' and Vj is the on-site  potential at site j of the form  Vj =  �  λ∆j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' j = κm (m ∈ Z)  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  otherwise  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  (2)  with λ being the strength of mosaic potential,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' κ ≥ 1  an integer representing the period of non-mosaic sites,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  and ∆j a model-dependent function that can induce the  localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  Every successive κ sites form a so-called  quasicell labeled by m, which also denotes the mth non-  mosaic site that has a nonzero on-site potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  For  convenience, in the following we take N quasicells, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=',  m = 1, · · · , N, and thus the system size L = κN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  By taking |Ψ⟩ = �  j ψj|j⟩, the static Schr¨odinger equa-  tion H|Ψ⟩ = E|Ψ⟩ can be written in terms of the ampli-  tude ψj as  �  Eψκm = JRψκm−1 + JLψκm+1 + λ∆κmψκm  Eψj = JRψj−1 + JLψj+1,  (j ̸= mκ)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (3)  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  MAPPING BETWEEN NONRECIPROCAL  MOSAIC AND NON-MOSAIC MODELS  Since there are (κ − 1) sites that are free of on-site po-  tentials between two nearest-neighbor non-mosaic sites,  a reasonable inference is that the localization, if occurs,  should exist only at the non-mosaic sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Therefore, one  may be curious about the effective coupling between the  non-mosaic sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' To this end, we can resort to the trans-  fer matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  From the second line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (3), the system of equa-  tions for the mosaic sites between two nearest non-mosaic  sites, say the mth and the (m + 1)th non-mosaic sites,  can be given explicitly as follows  �  �  �  Eψκm+1 = JRψκm + JLψκm+2  · ·  Eψκ(m+1)−1 = JRψκ(m+1)−2 + JLψκ(m+1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (4)  With the aid of the forward transfer matrix F, which  is defined as  �  ψκm+2  ψκm+1  �  =  �  E/JL −JR/JL  1  0  � �  ψκm+1  ψκm  �  ≡ F  �  ψκm+1  ψκm  �  ,  (5)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (4) can be written as  �  ψκ(m+1)  ψκ(m+1)−1  �  = F κ−1  �  ψκm+1  ψκm  �  ,  (6)  and thus, the amplitude at mosaic site ψκm+1 can be  expressed in terms of those of two non-mosaic sites ψκm  and ψκ(m+1), yielding  ψκm+1 = Jκ−1  L  Aκ  ψκ(m+1) + JRAκ−1  Aκ  ψκm,  (7)  with  Aκ =  ηκ  + − ηκ  −  √E2 − 4JLJR  ,  η± = E ± √E2 − 4JLJR  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (8)  Likewise, the relation can be also found by considering  the backward transfer matrices connecting the mth with  the (m − 1)th non-mosaic sites, and thus the amplitude  at mosaic site ψκm−1 can be expressed as  ψκm−1 = Jκ−1  R  Aκ  ψκ(m−1) + JLAκ−1  Aκ  ψκm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  (9)  The details of derivation for the coefficients can be re-  ferred to in Appendix .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  Substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (7) and (9) into the first line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  (3), one can have a system of equations only involving  non-mosaic sites, yielding  Jκ  Rψκ(m−1) + Jκ  Lψκ(m+1) + λAκ∆κmψκm = Bκψκm,  (10)  with  Bκ = EAκ − 2JLJRAκ−1,  (11)  which shows the effective relation between the nearest-  neighbor non-mosaic sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' If one regards m as the site  index with κ being just regarded as a parameter, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  (10) is nothing but a static Schr¨odinger equation for a  non-mosaic tight-binding model with Jκ  L,R being nonre-  ciprocal hopping, λAκ∆κm the on-site potential, and Bκ  the effective eigenenergy, which, for brevity, is called the  scaled model in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  Comparing the scaled model with the non-mosaic  model described by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  (3) with κ = 1, we have the  following mapping:  JL,R → Jκ  L,R, λ → λAκfκ, E → Bκ, ψj → ψκm,(12)  where the introduction of fκ reflects the relation between  potential functions ∆κm and ∆m in the two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' This  mapping reveals that if a wave function ψj of energy E  is localized, say, at site j under the potential function  ∆m of the non-mosaic model, then the localization also  occurs at site m under the potential function ∆κm of  the scaled model, described by the wave function ψκm of  energy Bκ, which also means that the wave function is  localized at the non-mosaic site κm of the mosaic model  with finite κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Thus, given the critical point of localization  3  determined by a function f(JL, JR, λ, E) = 0 in the non-  mosaic model, the critical point of the mosaic model can  be determined by  f(Jκ  L, Jκ  R, λAκfκ, Bκ) = 0,  (13)  through the mapping Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Generally, the critical  point of the mosaic model depends on E, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=', there ex-  ist mobility edges, even if no mobility edges exist for the  non-mosaic model, because Aκ is a function of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' In prin-  ciple, if one knows the critical point of a nonreciprocal  non-mosaic model, the critical point of the corresponding  mosaic model can also be obtained by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  Sometimes, the Lyapunov exponent γ(JL, JR, λ, E) >  0, characterizing the decaying behavior of a localized  state, is already established for some non-mosaic models  with random or quasiperiodic disorders, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=', the form of  localized state at site j0 is known as |ψj| ∝ e−γ|j−j0|,  and then we can also obtain the Lyapunov exponent  γ(Jκ  L, Jκ  R, λAκfκ, Bκ) > 0 of the scaled model (10) with  respect to the “sites” labeled by m, that is, the state  |ψκm| ∝ e−γ(m−m0) = e− γ  κ (κm−κm0) is localized at the  non-mosaic site κm0, where γ/κ is just the inverse of lo-  calization length or the Lyapunov exponent [56] of corre-  sponding mosaic models with respect to non-mosaic sites  κm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Of course, the critical point obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (13)  just satisfies the relation γ(Jκ  L, Jκ  R, λAκfκ, Bκ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  For the case that the nonreciprocal hopping can be  parameterized as JL = te−g and JR = teg, the scaled  model Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (10) becomes  teκgψκ(m−1) + te−κgψκ(m+1) + λaκ∆κmψκm = εκψκm,  (14)  similar as the reciprocal case in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' [55], with only the  difference in hopping terms caused by the nonreciprocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  Here, we replace Aκ and Bκ by  aκ =  1  �  E2/t2 − 4  ��E/t +  �  E2/t2 − 4  2  �κ  −  �  E/t −  �  E2/t2 − 4  2  �κ �  ,  εκ = Eaκ − 2taκ−1,  (15)  respectively, such that εκ has the dimension of energy and  aκ is dimensionless with t being regarded as the energy  unit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' define a0 = 0 for the consistency of ε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Specifically,  a1 = 1,  ε1 = E,  a2 = E/t,  ε2 = E2/t − 2t,  a3 = E2/t2 − 1, ε3 = E3/t2 − 3E,  (16)  which will be used in the following applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' In terms  of parameters t and g, the mapping Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (12) becomes  t → t, g → κg, λ → λaκfκ, E → εκ, ψj → ψκm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (17)  Apparently, it can be reduced to the result for reciprocal  mosaic models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=', g = 0) in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  12  45  78  50  25  0  numerical  analytical  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  The fractal dimension Γ with respect to the real  part of energy Re(E) and the potential strength λ, numer-  ically calculated for (a) κ = 2 and (b) κ = 3 under PBCs  with other parameters g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='2, φ = 0, L = F12 = 144, and  β = F11/F12 = 89/144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (c) The existence or not of the imagi-  nary part of energy with the same parameters as in (a), where  (non)zeros of Im(E) with accuracy 10−6 are colored in blue  (yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' The red solid lines in (a-c) are the mobility edges  calculated analytically by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (d) Comparison of a lo-  calized state marked by the red triangle (E = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='5881, λ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='2)  in (b) with the corresponding analytical form Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (22), show-  ing that the two Lyapunov exponents can well characterize  the asymmetric localization in the nonreciprocal mosaic AA-  like model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' The dotted line indicates the center site of the  localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  To demonstrate the validity of the general mapping,  we apply Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (17) to two specific nonreciprocal mosaic  models in the following, where, for convenience, we set  t = 1 as the energy unit and focus on the cases of g > 0,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=', the hopping is right-biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  APPLICATIONS  Aubry-Andr´e-like model  First, we take the mosaic version of an AA-like model  with nonreciprocal hopping and complex potentials,  dubbed nonreciprocal mosaic AA-like model for conve-  nience in the following, which is described by Hamilto-  nian (1) with the potential function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (2) being  ∆j = ∆κm = 2 cos (2πβκm + θ + iφ) ,  (18)  where β is an arbitrary irrational number, and (θ, φ) ∈ R  represent a complex phase shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' The corresponding re-  ciprocal non-mosaic AA-like model undergoes an Ander-  son localization at λ = ±e−|φ| [40], which is further gen-  eralized to λ = ±eg−|φ| [25] for the nonreciprocal non-  mosaic case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' they are all independent of energy E, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=',  6202m161-62020061-62020010614  there are no mobility edges for non-mosaic AA-like mod-  els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  Based on this critical point, we can apply our gen-  eral mapping (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  Since the difference between two  potential functions ∆κm = 2 cos(2πβκm + θ + iφ) and  ∆m = 2 cos(2πβm+θ+iφ) is only between the irrational  numbers κβ and β, the fκ in the mapping can be taken  as 1, because the critical points of Anderson localization  for quasiperiodic models are independent of the values of  irrational numbers [57, 58], although the eigenstates and  eigenvalues under these two potentials are irrelevant in  details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Thus, by making the replacement of λ → λaκ  and g → κg in λ = ±eg−|φ|, one can find that the mobil-  ity edges,  λκ(E) = ±eκg−|φ|/aκ,  (19)  emerge for mosaic AA-like models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' This result is identi-  cal to that obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' [38] by using Avila’s global  theory of quasiperiodic Schr¨odinger operators [45], which  proves the validity of our general mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  In addition, one can also obtain the Lyapunov expo-  nents for the nonreciprocal mosaic AA-like models by the  same mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' From Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' [25], we know that two Lya-  punov exponents γ±g > 0 can be used to characterize an  asymmetrically localized state of a nonreciprocal model,  where  γ = ln |λ| + |φ|  (20)  is the Lyapunov exponent of a symmetriccally localized  state of the corresponding reciprocal model, and the  smaller one γ − g determines the critical point of Ander-  son localization for the nonreciprocal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Therefore,  with the same replacement, one can get two Lyapunov  exponents of the scaled model (10) as  γ(±)  κ  (E) = ln |λaκ| + |φ| ± κg,  (21)  which depends on the energy E, leading to an asymmet-  rically localized state,  |ψκm| ∝  �  �  �  e− γ(−)  κ  κ  (κm−κm0), m > m0  e− γ(+)  κ  κ  (κm0−κm), m < m0  ,  (22)  with the peak at the non-mosaic site κm0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' The right-  biased asymmetry [i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=', γ(−) < γ(+)] of the localized state  with respect to the center site κm0 results from the right-  biased hopping (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=', g > 0) we set in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' This can  be partially checked by setting γ(−)  κ  (E) = 0, which just  gives rise to the critical point in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  For a further verification, we compare the numerical  results with the analytical ones for the cases of κ = 2  and 3 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' 1, where, from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (19) and (21), the  critical points and the Lyapunov exponents are specified  as  λ2(E) = ±e2g−|φ|  E  , γ(−)  2  (E)  2  = ln |λE| + |φ|  2  − g,  λ3(E) = ±e3g−|φ|  E2 − 1, γ(−)  3  (E)  3  = ln |λ(E2 − 1)| + |φ|  3  − g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  (23)  For the numerical calculation, we take β = Fs−1/Fs as  a rational approximation of an irrational number β =  lims→∞ Fs−1/Fs = (  √  5 − 1)/2, and the system size L =  Fs, where Fs (s = 1, 2, · · · ) is the sth Fibonacci number  defined as Fs = Fs−1 + Fs−2 with the first two numbers  F1 = F2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' The fractal dimension [19],  Γ = − ln I/ ln L,  (24)  can be used to characterize the localization of a state ψj  more clearly for a finite system than the commonly used  inverse participation number (IPR) [19],  I =  L  �  j=1  |ψj|4��  L  �  j=1  |ψj|2�2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  (25)  To clearly demonstrate the transition of Anderson lo-  calization, of which the critical points are the same under  PBCs and OBCs due to the blindness about boundary  conditions for a bulk-localized state [25], we use PBCs for  numerical calculations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' 1 to avoid non-Hermitian  skin effect under OBCs [4] where all bulk states are ag-  gregated together to one boundary (here the right bound-  ary for our settings), leading to similar values of fractal  dimension as bulk-localized states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Meanwhile, accord-  ing to the bulk-bulk correspondence of a nonreciprocal  model when φ = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=', the critical point of Anderson  localization under OBCs is also a cut between complex  and real spectra under PBCs [25], as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' 1(a)  and 1(c), one can assume the reality of E and abandon  the complex-E solutions during the derivation of criti-  cal points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' This can also explain why the mobility edges  E(λ) are just functions of curves, not surfaces, because  Im[(E(λ)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' The maximal number 2(κ−1) of mobility  edges can also be expected, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' 1(a) and  1(b), from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (19), which can have the highest power  (κ − 1) of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' The two Lyapunov exponents are demon-  strated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' 1(d) as two slopes from the peak site to  the two shoulders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Due to the existence of side peaks on  two shoulders besides the main peak, the numerical re-  sult can shift occasionally relative to the analytical result  for some regions, but the preserved slope still reflects the  correctness of the Lyapunov exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  Here, we just demonstrate the case of real potential  function ∆j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' of course, the case of complex ∆j in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  [38] can also be reobtained using this mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  Ganeshan-Pixley-Das Sarma model  The mobility edges of mosaic AA-like model emerge  from a constant critical point of the non-mosaic coun-  terpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Here, we take as another application the mo-  saic GPD model [54] with nonreciprocal hopping, whose  non-mosaic counterpart already has mobility edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' The  Hamiltonian is described by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (1) with  ∆j =  2 cos (2πβj)  1 − b cos (2πβj),  (26)  5  8  24  40  40  20  0  numerical  analytical  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' The same meanings and settings as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' 1, except  for (a,c) κ = 1 and (b,d) κ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' The mobility edges [red solid  lines in (a-c)] are calculated analytically by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' The  localized state marked by the red triangle in (b) is selected as  E = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='7328, λ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' The parameter b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='5 is chosen for all  figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  with the same definition of β as the AA-like model and  the deformation parameter b ∈ (−1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  For the non-mosaic case with nonreciprocal hopping,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=', κ = 1, the mobility edges [30]  λ(E) = 1  2  �  − bE ± eg �  1 +  √  1 − b2�  ± e−g �  1 −  √  1 − b2� �  ,  (27)  can be obtained by the Avila’s global theory [45] and the  similarity transformation under OBCs [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' When g = 0,  it reduces to the reciprocal case with mobility edges being  λ(E) = ±1 − bE  2 ,  (28)  which is first obtained by a generalized duality transfor-  mation [54], and the Lyapunov exponent  γ(E) = ln  �����  |bE + 2λ| +  �  (bE + 2λ)2 − 4b2  2  �  1 +  √  1 − b2�  �����  (29)  is analytically obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' [30, 31] with the aid of the  global theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' that γ(E) = 0 just gives rise to the critical  point Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' As we mentioned before, E should be  a real number for an Anderson localized state, and thus,  hereafter the notations (absolute and squared values) in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (29) are only defined in the field of real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  With the aid of the similarity transformation under  OBCs in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' [25], the two Lyapunov exponents, γ(E) ±  g, for the nonreciprocal non-mosaic case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=', κ = 1 and  g ̸= 0, can be readily obtained, and the mobility edges  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (27) can also be verified by γ(E) − g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  Finally for the nonreciprocal mosaic version of GPD  model, using the mapping in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (17), we get the two  Lyapunov exponents:  γ(±)  κ  (E)  κ  = 1  κ ln  �����  |bεκ + 2λaκ| +  �  (bεκ + 2λaκ)2 − 4b2  2  �  1 +  √  1 − b2�  �����  ±g,  (30)  where aκ and εκ are defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (15), and the mobility  edges can be determined by γ(−)  κ  (E) = 0, or directly by  the mapping from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (27), yielding  λκ(E) =  1  2aκ  �  − bεκ ± eκg �  1 +  √  1 − b2�  ± e−κg �  1 −  √  1 − b2� �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  (31)  We also take the cases of κ = 1 and 2 for demonstra-  tion, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' 2, where we also use the same setting  of β and PBCs as the aforementioned AA-like model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  The mobility edges are also curves separating real spec-  tra and complex ones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' the number can also maximally be  2(κ − 1) as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  CONCLUSION AND DISCUSSION  We establish a general mapping of nonreciprocal mo-  saic models to their non-mosaic counterparts of which  the critical points of localization or the Lyapunov expo-  nents of localized states are known, and can find the an-  alytical expressions of mobility edges and the Lyapunov  exponents of nonreciprocal mosaic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' For demon-  strations, we take an AA-like model [38] and the GPD  model [54] with nonreciprocal hopping to test the map-  ping, successfully giving birth to the mobility edges and  the Lyapunov exponents of these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' This mapping  is a generalization of the reciprocal version in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  The mapping Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  (17) can be not only applied to  quasiperiodic models, but also to randomly disordered  models and the non-disorder models that have localiza-  tion transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' However, in one dimension the critical  points or Lyapunov exponents are less established for  nonreciprocal models, and thus, we only offer quasiperi-  odic models as applications in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  For exam-  ple, in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' [55] the mobility edges of reciprocal mosaic  Wannier-Stark model can be obtained due to the well-  known critical point of the reciprocal non-mosaic coun-  terpart, but the lack of Lyapunov exponent makes us fail  to obtain the critical point or mobility edges of the non-  reciprocal mosaic models using the mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  Moreover, without the parameterization, JL = te−g  and JR = teg, the bulk-bulk principle [25] cannot work  due to the possible existence of complex spectrum for an  Anderson localized state even under OBCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Therefore,  the energies of critical points could be complex, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='e, the  mobility edge in general becomes a surface not a curve  as shown here, which may be more interesting and need  to further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  20061-62020lmi6620200  m-662106  Appendix: Derivation for the transfer matrix  For the forward transfer matrix F in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='9) of the  main text, we now calculate F κ−1 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  First, F can be diagonalized as  F =  �  E/JL −JR/JL  1  0  �  = UΛU −1,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='1)  where  Λ =  �  η+/JL  0  0  η−/JL  �  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='2)  is the diagonal matrix with  η± = E ± √  E2 − 4JLJR  2  ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='3)  and  U =  �  η+/JL η−/JL  1  1  �  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='4)  is a unitary matrix with its inverse  U −1 =  �  JL  −η−  −JL  η+  � ��  E2 − 4JLJR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='5)  Then, one can get  F κ−1 = UΛκ−1U −1  =  �  Aκ/Jκ−1  L  −JRAκ/Jκ−1  L  Aκ−1/Jκ−2  L  −JRAκ−2/Jκ−2  L  �  ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='6)  where  Aκ =  ηκ  + − ηκ  −  √E2 − 4JLJR  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='7)  Therefore, we have the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (7) in the main text  ψκm+1 = Jκ−1  L  Aκ  ψκ(m+1) + JRAκ−1  Aκ  ψκm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='8)  For the backward transfer matrix B, defined as  �  ψκm−2  ψκm−1  �  =  �  E/JR −JL/JR  1  0  � �  ψκm−1  ψκm  �  ≡ B  �  ψκm+1  ψκm  �  ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='9)  we have  �  ψκ(m−1)  ψκ(m−1)+1  �  = Bκ−1  �  ψκm−1  ψκm  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='10)  Likewise, we can get  Bκ−1 =  �  Aκ/Jκ−1  R  −JLAκ/Jκ−1  R  Aκ−1/Jκ−2  R  −JLAκ−2/Jκ−2  R  �  ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='11)  and thus the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' (9) in the main text  ψκm−1 = Jκ−1  R  Aκ  ψκ(m−1) + JLAκ−1  Aκ  ψκm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='12)  ACKNOWLEDGMENTS  This work was supported by the National Natural  Science Foundation of China (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' 11904109,  12204406), the Guangdong Basic and Applied Basic Re-  search Foundation (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' 2019A1515111101), the  Science and Technology Program of Guangzhou (Grant  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' 2019050001), Guangdong Provincial Key Laboratory  (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' 2020B1212060066), and NKRDPC (Grant  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' 2022YFA1405304).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Breuer and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Petruccione, The Theory of Open  Quantum Systems (Oxford University Press, Oxford,  2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  [2] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Gong, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Ashida, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Kawabata, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Takasan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
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+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Budich, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  Bergholtz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
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+page_content=' Wang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
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+page_content=' Thomale, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
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+page_content=' Ueda, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Sato, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
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+page_content=' Deng, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Wang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Zhu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Yi,  and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Xue, Nature Physics 16, 761 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
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+page_content=' Hofmann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Imhof, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Abdelghany,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Kießling, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
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+page_content=' Lee, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
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+page_content=' Thomale, Nature Physics 16, 747  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
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+page_content=' Kawabata, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Shiozaki, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
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+page_content=' Helbig, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
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+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Licciardello, and  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Ramakrishnan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' 42, 673 (1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  [18] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Lee and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
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+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' 57,  287 (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  [19] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Evers and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Mirlin, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' 80, 1355  (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  [20] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
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+page_content=' Tzortzakakis, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
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+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Economou,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' B 101, 014202 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  [21] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Huang and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Shklovskii, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' B 101, 014204  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  [22] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Jazaeri and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Satija, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' E 63, 036222  (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  [23] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Yuce, Physics Letters A 378, 2024 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  [24] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Zeng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Chen, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' L¨u, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' A 95, 062118  (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  [25] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Jiang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Lang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Yang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' B 100, 054301 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  [26] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Longhi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' B 100, 125157 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  [27] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content=' 122, 237601 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  [28] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
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+page_content=' B 101,  020201 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
+page_content='  [29] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
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+page_content=' B  103, 134208 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
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+page_content=' B 103, 174205 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
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+page_content=' 2, 033052 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
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+page_content=' Xia, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
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+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfvv1V/content/2301.01711v1.pdf'}
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+MNRAS 000, 1–17 (20xx)
+Preprint 13 January 2023
+Compiled using MNRAS LATEX style file v3.0
+Rotational modulation in A and F stars: Magnetic stellar spots or
+convective core rotation?
+Andreea I. Henriksen,1★ Victoria Antoci,1† Hideyuki Saio,2 Matteo Cantiello,3 Hans Kjeldsen,4
+Donald W. Kurtz,5,6 Simon J. Murphy,7 Savita Mathur,8,9 Rafael A. García,10 Ângela R. G. Santos 11
+1National Space Institute, Technical University of Denmark, Elektrovej, DK-2800 Kgs. Lyngby, Denmark
+2Astronomical Institute, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan
+3Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA
+4Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, 8000 Aarhus C, Denmark
+5Department of Physics, North-West University, Dr Albert Luthuli Drive, Mahikeng 2735, South Africa
+6Jeremiah Horrocks Institute, University of Central Lancashire, Preston PR1 2HE, UK
+7University of Southern Queensland - Springfield Campus: Springfield, QLD, AU
+8Instituto de Astrofísica de Canarias (IAC), E-38205 La Laguna, Tenerife, Spain
+9Universidad de La Laguna (ULL), Departamento de Astrofísica, E-38206 La Laguna, Tenerife, Spain
+10AIM, CEA, CNRS, Université Paris-Saclay, Université Paris Diderot, Sorbonne Paris Cité, F-91191 Gif-sur-Yvette, France
+11 Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, PT4150-762 Porto, Portugal
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+The Kepler mission revealed a plethora of stellar variability in the light curves of many stars, some associated with magnetic
+activity or stellar oscillations. In this work, we analyse the periodic signal in 162 intermediate-mass stars, interpreted as Rossby
+modes and rotational modulation - the so-called hump & spike feature. We investigate whether the rotational modulation (spike)
+is due to stellar spots caused by magnetic fields or due to Overstable Convective (OsC) modes resonantly exciting g modes,
+with frequencies corresponding to the convective core rotation rate. Assuming that the spikes are created by magnetic spots at
+the stellar surface, we recover the amplitudes of the magnetic fields, which are in good agreement with theoretical predictions.
+Our data show a clear anti-correlation between the spike amplitudes and stellar mass and possibly a correlation with stellar
+age, consistent with the dynamo-generated magnetic fields theory in (sub)-surface convective layers. Investigating the harmonic
+behaviour, we find that for 125 stars neither of the two possible explanations can be excluded. While our results suggest that
+the dynamo-generated magnetic field scenario is more likely to explain the spike feature, we assess further work is needed to
+distinguish between the two scenarios. One method for ruling out one of the two explanations is to directly observe magnetic
+fields in hump & spike stars. Another would be to impose additional constraints through detailed modelling of our stars, regarding
+the rotation requirement in the OsC mode scenario or the presence of a convective-core (stellar age).
+Key words: stars: early-type – stars: rotation – stars: magnetic fields – stars: oscillations
+1 INTRODUCTION
+Around 10 per cent of A-type stars have been discovered to be chem-
+ically and magnetically peculiar. It is widely accepted that the mag-
+netic fields in Ap stars are of fossil origin, i.e. they were formed
+in the pre-main sequence stage of stellar evolution (e.g., Cowling
+1945; Braithwaite & Spruit 2004; Braithwaite & Cantiello 2013).
+A fossil magnetic field means that at the current evolutionary stage,
+the star cannot maintain the magnetic field against spontaneous de-
+cay (Borra et al. 1982). The predicted strengths of fossil-magnetic
+fields, obtained by Braithwaite & Spruit (2004) through numerical
+simulation, were of the order of 10 kG, agreeing with observations.
+Magnetic fields in chemically peculiar stars inhibit the motions of
+ions (which have an electric charge) and force particular elements to
+★ E-mail: andreea@space.dtu.dk
+† E-mail: antoci@space.dtu.dk
+be concentrated in spots, as many observations confirm (e.g., Gray
+& Corbally 2009). As most Ap stars are slow rotators and have
+an overabundance of certain elements (e.g., Kochukhov & Bagnulo
+2006), their spectral lines are very sharp, which allows one to directly
+detect the magnetic fields through magnetic splitting of the lines, or
+by spectro-polarimetric observations measuring the Zeeman effect
+(see, e.g., Donati & Landstreet 2009). However, our understanding
+of how magnetic fields are generated and operate in Ap stars cannot
+be extended to other types of stars that exhibit magnetic activity.
+Magnetic fields have been investigated and measured in other A
+stars, however most of them have weak magnetic field strengths (see
+e.g. Aurière et al. 2007, Lignières et al. 2009, Petit et al. 2011, Blazère
+et al. 2016a,b, Neiner et al. 2017, Seach et al. 2020). Interestingly,
+Aurière et al. (2007) found a lack of stellar magnetic fields with
+strengths between 300 and a few Gauss, which is now commonly
+termed as the ‘magnetic desert’. The strong magnetic fields of Ap
+© 20xx The Authors
+arXiv:2301.04974v1  [astro-ph.SR]  12 Jan 2023
+
+2
+A. I. Henriksen et al.
+stars highlights their distinctive character with respect to non-peculiar
+A stars.
+1.1 Origin of magnetic fields and magnetic activity in
+non-peculiar intermediate-mass stars
+Our initial understanding of stellar magnetic fields is based on studies
+of the Sun. The kinetic energy generated by the motions in the con-
+vective layers, coupled with differential solar rotation, is converted
+by the solar dynamo to magnetic energy. Dynamo generated mag-
+netic fields in the Sun can then be observed at the photosphere (see,
+e.g., the review by Berdyugina 2005 and references therein).
+The magnetic fields at the surface of the Sun can partially in-
+hibit the convective energy transport. Such regions appear as spots,
+visible because they are cooler and therefore darker than the bright
+photosphere (Strassmeier 2009). It is known from both theory and
+helioseismic observations that the convective zone (CZ) of the Sun is
+approximately 29 per cent of its outer radius (e.g. Turck-Chièze et al.
+(1993); Christensen-Dalsgaard (2002)). The solar CZ is efficient and
+deep enough to sustain a dynamo that can generate a magnetic field,
+penetrating the solar photosphere and creating sunspots. Given the
+proximity of the Sun, which allows us to resolve its disk, sunspots
+have been visually observed on its surface for centuries. In order to
+understand stellar and solar dynamo, however, one has to use mag-
+netic activity observations of other stars, which expands the range
+of stellar parameters on which the dynamo theories can be tested
+(Berdyugina 2005; Brandenburg & Subramanian 2005). Magnetic
+activity in late-type stars, which possess deep convective envelopes,
+has been observed abundantly (see Berdyugina 2005, chapter 2).
+In the case of more massive early-type stars, detecting and under-
+standing the origin of their magnetic fields present additional chal-
+lenges. Nevertheless, one can infer information regarding magnetic
+activity by looking for rotational modulation in the photometric light
+curves of stars. The brightness variations can be attributed to tem-
+perature variations caused by stellar spots co-rotating with the stellar
+surface. While the more massive HgMn stars show spots but no sur-
+face magnetic fields have been recorded yet (Kochukhov et al. 2013),
+stellar spots are generally accepted to be a consequence of magnetic
+activity also in intermediate-mass stars (Berdyugina 2005).
+As the stellar structure changes with increasing stellar mass, one
+cannot assume that the mechanism that generates magnetic fields in
+higher mass stars is the same as in the Sun. This also reinforces the
+necessity of studying magnetic fields in more massive stars.
+A debatable aspect regarding the presence of magnetic fields
+in non-chemically peculiar main-sequence higher-mass stars (≥
+1.3 M⊙) comes from the fact that, as opposed to our Sun’s radia-
+tive core and deep convective envelope, the core of more massive
+stars is convective and surrounded by a deep radiative envelope 1. A
+more detailed analysis of the outermost layers of intermediate-mass
+stars reveals, however, that due to high opacity and/or low adiabatic
+gradient (conditions fulfilled in the ionization zones of H, He and/or
+Fe), there are thin convective layers at or below the stellar surface
+(e.g. Cantiello & Braithwaite 2019, hereafter CB19).
+Cantiello & Braithwaite (2011); Cantiello & Braithwaite (2019)
+proposed that despite their small aspect ratio, these convective re-
+gions could sustain a dynamo and generate a magnetic field. Since
+early-type stars tend to be rapidly-rotating, magnetic structures could
+also grow to scales larger than the scale of convective motions.
+1 Depending on the mass, stars with masses lower than 1.3 M⊙, could develop
+a convective core during later stages of the main-sequence lifetime
+The rapid rotation could distinguish stars with this type of dynamo-
+generated magnetic field from those in the slowly rotating Ap stars,
+which, as mentioned above, are believed to have magnetic fields
+generated prior to the main-sequence stage. CB19 also suggest that
+dynamo-generated fields have rapidly evolving features of small-
+scale and that the strength of surface magnetic fields is expected to
+increase as the stars evolve. While it is difficult to predict the ex-
+act size of the magnetic features, it is expected that their sizes are
+correlated to the pressure scale height at the bottom of the shallow
+convective envelope, which is expected to be less than 1 per cent of
+the stellar radius. CB19 predicted surface magnetic fields strengths
+to reach 1kG for the lower mass stars and ≲ 1 G for stars with 𝑇eff
+>10,000 K, noting that these could be just lower limits.
+Both theory and observations suggest that the transition be-
+tween stars with convective surfaces to stars with radiative sur-
+faces is at around 10,000 K (CB19). Cool intermediate-mass stars
+(𝑇eff ≲104 K) likely have a very thin convective surface layer due to
+the ionization of H and He I (see Figure 2 in CB19). The transported
+flux in these convective layers, which separate the radiative envelope
+from the surface, is negligible because they contain very little mass.
+According to CB19, magnetic fields can reach the stellar surface
+through magnetic buoyancy, locally changing the gas pressure and
+lowering the optical depth of the photosphere. Consequently, spots
+will appear brighter than the surrounding photosphere because the
+flux originates from deeper radiative areas in the star, where the
+temperature is higher than at the surface.
+In contrast to the dynamo-generated magnetic fields that CB19
+described, Braithwaite & Cantiello (2013) proposed the failed-fossil
+scenario for the origin of ultra-weak magnetic fields in A stars. The
+failed-fossil magnetic fields are expected to have large-scale, slowly
+evolving features that would decrease as the stars evolve. The pre-
+dictions in Braithwaite & Cantiello (2013) suggest that: (1) the time
+scale of the failed-fossil field’s evolution is around the stellar age,
+(2) the length scale of surface magnetic structures should be at least
+20 per cent of the radius of the star, (3) the higher stellar rotation,
+the stronger the stellar magnetic field. With respect to the large-
+scale and slowly evolving features, the characteristics of failed-fossil
+magnetic fields are opposite to those of dynamo-generated magnetic
+fields predicted by CB19, but for both scenarios the rotation should
+play an important role. One could test both scenarios by performing
+an ensemble analysis on a homogeneous sample of stars that show
+magnetic activity signs.
+For this study, we use the, so-called, hump & spike stars, which
+Saio et al. (2018) suggested could show signs of magnetic activity in
+the form of spots. The common characteristic of these stars is a feature
+in the Fourier spectrum (see Fig. 1 for some examples), interpreted
+to be unresolved Rossby modes (the hump) mechanically excited by
+deviated flows caused by stellar spots (the spike) at intermediate to
+high latitudes (Saio et al. 2018). Rossby modes were first reported in
+intermediate-mass stars by Van Reeth et al. (2016), as they studied
+the core rotation of a sample of 𝛾 Dor stars. They have been reported
+in other works such as: Li et al. 2019, 2020.
+1.2 Alternative explanation for rotational modulation
+Recently, Lee & Saio (2020) and Lee (2021) proposed a different ex-
+planation for rotational modulation of light curves of early-type stars,
+which does not involve the presence of stellar spots generated by mag-
+netic fields. Overstable convective (OsC) modes in rapidly rotating
+early-type stars could resonantly excite low-frequency g modes in the
+envelope. If the stellar core rotates slightly faster than the envelope
+and the amplitudes of the g modes are significant at the photosphere,
+MNRAS 000, 1–17 (20xx)
+
+Rotational modulation in A and F stars
+3
+they would cause brightness variability and therefore be observed as
+rotational modulations (i.e. the spike).
+In addition, Lee (2021) suggested that OsC modes resonating with
+g modes in the envelope can play an essential role in carrying angular
+momentum from the core to the envelope. A small amount of radial
+differential rotation is a critical ingredient in the Lee & Saio (2020)
+and Lee (2021) hypothesis, as OsC modes would not excite g modes in
+uniformly rotating stars (Lee & Saio 2020). Previous studies in which
+core and envelope rotation rates were measured through photometric
+data found that up to about 10 per cent differential rotation between
+convective core and envelope is not rare in 𝛾 Doradus stars (Saio
+et al. 2021).
+In the context of our hump & spike sample of stars, we investi-
+gate the possibility that the spike could in some cases correspond to
+g modes confined near the core and resonantly excited by the con-
+vective core motions. In this case, the light curves of our hump &
+spike stars would be modulated by the convective core rotation.
+In this article, we address both scenarios and discuss the corre-
+sponding implications. More precisely, we investigate whether the
+spike feature could be either: (1) caused by rotational modulation
+due to stellar spots caused by magnetic fields or (2) evidence that
+OsC modes couple with g modes in the envelope, causing the light
+curve to be modulated with the convective core rotation.
+Under the assumption that spots induced by magnetic fields cause
+the spike modulation, the first goal of the present work is to investigate
+whether the stars in our sample could sustain a magnetic field. We
+will test two scenarios: failed-fossil field (Braithwaite & Cantiello
+2013) and dynamo-generated field (CB19).
+The second goal of this study is to test whether the spike is evi-
+dence of convective core rotation, using predictions from Lee (2021).
+If the frequency of the spike is proven to correspond to the frequency
+of a g mode that is resonantly excited by OsC modes, then our re-
+sults would help better understand the convective core rotation in
+intermediate-mass stars as well as the internal mixing. We describe
+observations, data reduction and analysis in Section 2 and discuss
+our results in Section 3.The implications of our results under both
+the magnetic field and the OsC modes hypotheses are presented in
+Section 4. Our conclusions and future work are presented in Sec-
+tion 5.
+2 OBSERVATIONS AND DATA ANALYSIS
+In this work, we use the Kepler long cadence data (Koch et al. 2010),
+which are excellent for our study given the long time coverage, and
+the fact that the signals of interest are found at low frequencies
+(< 20 d−1). The light curves from each quarter were downloaded
+from KASOC (Kepler Asteroseismic Science Operations Center)1.
+For our analysis, we used the PDC (Pre-search Data Conditioning)
+data, as they were free of specific systematic errors, removed by the
+Kepler Science Operations Center Pipeline (Twicken et al. 2010).
+We visually inspected each PDC light curve and compared it to the
+corresponding SAP light curve (Simple Aperture Photometry). We
+concluded that the hump & spike feature remained unaltered after the
+time series were processed by the PDC module of the Kepler data
+analysis pipeline. The data from KASOC is barycentrically corrected,
+and the light curve files have the Barycentric Reduced Julian Date
+(BRJD, BJD-2,400,000.0) as time units (Van Cleve et al. 2016) 2.
+1 kasoc.phys.au.dk
+2 archive.stsci.edu/kepler/manuals/Data_Characteristics_
+Handbook_20110201.pdf
+The flux was converted from units of photo-electrons per second
+(𝑒−s−1) to ppm, after which the quarterly data were stitched into a
+single time series for each star. We computed a Fourier spectrum for
+each stitched time series using the lightkurve package (Lightkurve
+Collaboration et al. 2018).
+2.1 Sample selection
+The target sample initially comprised 94 stars that were visually
+inspected and found to exhibit the hump & spike feature or were
+reported in Balona (2013, 2014, 2017); Balona et al. (2015). The
+temperature range on which we concentrated to find targets was
+6500 < 𝑇eff < 10000 K. An additional set of 115 stars (7500 <
+𝑇eff < 10000 K) from Trust et al. (2020) was included in our study.
+Another four stars were added to our sample from Mathur et al.
+2019 and by visually inspecting the data products available from
+Santos et al. (2021a). A total of 213 stars that exhibited the hump &
+spike feature were analysed. However, in this article, we concentrate
+on 162 stars that did not show signatures of binarity (either from
+photometry or literature). This selection was performed in order to
+study a homogeneous sample of stars. We note that some stars in
+our final sample exhibit a hump after the spike (e.g., KIC 4488313 in
+Fig. 1). This feature is similar to one identified by Saio et al. 2018,
+who suggested that it is probably the result of prograde g modes (see
+figure 8 in Saio et al. 2018). This feature will be addressed in a future
+work that will concentrate more on Rossby modes.
+In Fig. 2, we show an HR diagram illustrating our final sample.
+Stellar evolutionary models for the mass range 1.4 − 3.8M⊙, com-
+puted with Warszaw-New Jersey models (Van Hoolst et al. 1998;
+Pamyatnykh 1999; Pamyatnykh et al. 1998) are shown in the back-
+ground for guidance only, computed with solar metallicity (Asplund
+et al. 2004) and with no rotation. The full sample of stars can be seen
+in Fig. A1 in Appendix A, with the excluded stars marked in orange
+squares and purple triangles while the targets used in the current
+work are marked in green circles.
+The feature of interest is, as mentioned, the hump & spike with
+a few examples shown in Fig. 1, and a more detailed view in the
+left panel of Fig. 3. The following subsection describes how the
+parameters of the spikes were extracted.
+2.2 Spike parameter extraction
+For most stars in our sample, the PDC light curves did not need ad-
+ditional de-trending or corrections. Only ∼10 per cent of PDC light
+curves still contained some non-astrophysical signals, to which some
+minor corrections were applied. A clear example of a light curve that
+needed such corrections can be seen in Fig. 4, where only the jump
+in flux at around BRJD 56 000) was corrected by fitting three low-
+order polynomials to the data. The corrections were done keeping
+in mind that over-fitting might introduce additional and unnecessary
+signals in the data, so the order of the polynomials was kept as low
+as possible. In this example, three second-degree polynomials suf-
+ficed. As seen in the lower panel of Fig. 4, the Fourier spectra of both
+polynomial-corrected data and PDC data are over-plotted, respec-
+tively, the astrophysical signal (hump & spike feature) is unaltered
+after the correction, but the noise at lower frequencies is reduced.
+The applied corrections eliminated only the non-astrophysical sig-
+nals, evidently showing that the peaks at low frequencies are indeed
+associated with the jump in flux that occurs in the time series at
+BRJD ∼56 000 in the upper panel of Fig. 4. We computed also a
+Fourier spectrum without the data that contain the jump in flux (in
+MNRAS 000, 1–17 (20xx)
+
+4
+A. I. Henriksen et al.
+Figure 1. A selection of hump & spike features, illustrating their common
+aspects and amplitude and frequency diversity.
+Figure 2. Stars from our sample placed in an HR diagram. The source for
+luminosity and 𝑇eff values is described in section 2.3. The colour code repre-
+sents the source from which the target originates, as described by the legend.
+More information in section 2.1. Warszaw-New Jersey evolutionary tracks
+(𝑍 = 0.012, Asplund et al. 2004) are displayed in the background for guid-
+ance only.
+the time range 55904 and 56015), coloured in grey in the lower panel
+of Fig. 4. The small differences in the respective Fourier spectra in-
+dicate the non-astrophysical origin of the jump. We conclude that
+the correction applied did not introduce additional non-astrophysical
+signals. Moreover, these minor corrections helped obtain a higher
+signal-to-noise ratio (SNR) for the studied spikes and/or humps.
+For each star in our sample, a simple 1D Gaussian model was fitted
+to the spike (and its harmonics) using astropy (Astropy Collaboration
+et al. 2013, 2018). Spike identification was done semi-automatically.
+We divided the Fourier spectrum into equal sections and identified
+the peaks with scipy (Virtanen et al. 2020). Only peaks with a SNR
+≥ 4 were accepted as significant. After identifying a significant peak,
+we defined a region in the Fourier spectrum (blue shaded region in
+Fig. 5, spike selection of KIC 8385850) and fitted a simple Gaussian.
+Manual spike identification was necessary due to the variety in the
+hump & spike profiles. As seen in Fig. 1, the topology of the hump &
+spike feature changes from star to star. In some cases, the spike was
+Figure 3. Example of a Gaussian fit on a spike feature, KIC 4921184.
+Figure 4. Upper panel: PDC Light curves of KIC 4921184: before the cor-
+rection was applied (purple), after non-astrophysical signal was removed
+(orange). See Section 2.2 for more details. Lower panel: Fourier spectra of
+corrected (orange) and uncorrected (purple) light curves. The grey Fourier
+spectrum was computed with the full data set excluding the region inside the
+two grey vertical lines from upper panel. There are small differences between
+the Fourier spectrum of the corrected data and the one computed without the
+data between BRJD 55904 and 56015 days.
+not well separated from the hump or was a broad feature rather than
+a sharp narrow peak.
+The noise level for the SNR calculations was estimated to be
+the median of the amplitude values between a range selected at
+frequencies higher than ∼15-20 d−1, where no astrophysical signal
+could be identified. The frequency range for SNR calculations was
+selected for each star with the same tool as depicted in Fig. 5. Usually
+the SNR is determined around the extracted peak. However given the
+proximity of the hump (both at lower frequencies and in some case at
+higher frequencies), it was not possible to determine the noise level
+around the spike. We therefore chose to use the noise level at higher
+frequencies which we expect to mirror the white noise in the data set.
+Similar to Trust et al. (2020), we define the value for the spike
+amplitude to be the highest amplitude value in the selected spike
+region. This is a robust way to determine the photometric variability
+in the 4-year Kepler data. The uncertainties in spike amplitudes were
+MNRAS 000, 1–17 (20xx)
+
+10
+KIC2157489
+KIC3240406
+40
+5
+20
+0
+0
+0.65
+0.70
+0.75
+0.80
+0.90
+0.95
+1.00
+1.05
+1.10
+1.15
+40
+KIC3238627
+KIC3459226
+40
+20
+0
+0
+1.45
+1.50
+1.55
+0.85
+0.90
+0.95
+75
+KIC6207232
+KIC10338005
+20
+50
+10
+25
+MMMM
+0
+0
+0.700 0.725
+0.7500.775
+0.800 0.825
+2.00
+2.05
+2.10
+2.15
+15
+20
+KIC4488313
+KIC7842339
+10
+10
+5
+0
+1.10
+1.15
+1.20
+1.10
+1.15
+1.20
+1.25
+1.30
+1.35
+Frequency [d-1]
+Frequency[d-1]3.8 Mo
+3.5 Mo
+102
+3.2 Mo
+2.9 Mo
+2.6 Mo
+2.3 Mo
+2.0 Mo
+101
+1.7 Mo
+Initial sample
+Trust et al. 2020
+Santos et al. 2021
+1.4 Mg
+14000
+12000
+10000
+8000
+6000
+4000
+Teff [K]Gaussian model
+50
+50
+Fourier spectrum
+Fitted Gaussian center
+40
+40
+wddi
+Fourier Gaussian center = 1.3904 d-1
+Spike/Gaussian amplitude = 52.6 ppm
+Gaussian FWHM = 0.0025 d-1
+30
+30
+20
+20
+10
+10
+1.380
+1.4
+1.385
+1.390
+1.395
+1.400
+1.405
+1.410
+Frequency [d-1]
+Frequency [d-1]Excluded data boundaries
+3000
+(for verification only)
+Corrected data
+2000
+Uncorrected data
+[udd]
+1000
+0
+xn
+1000
+-2000
+-3000
+-4000
+55000
+55200
+55400
+55600
+55800
+56000
+56200
+56400
+BRJD[days]Uncorrected data
+70
+Corrected data
+60
+Excluding data from [55,904-56,015] days
+[wdd]
+50
+Amplitude
+40
+30
+20
+10
+0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+Frequency [d-1]Rotational modulation in A and F stars
+5
+Figure 5. Illustration of how a frequency region was selected for the Gaussian
+fitting - KIC 1873552.
+calculated with equation 1, which was adapted from Kjeldsen &
+Bedding 1995.
+𝜎A =
+√𝜋 𝑆𝑁𝑅
+2
+(1)
+where SNR is the noise level in the Fourier spectrum.
+We note that the spike amplitudes could be underestimated. If
+the spike signal is due to stellar spots, the amplitude would depend
+on the angle at which the spot is observed, which in turn depends
+on the stellar inclination and the latitude of the spot. Determining
+the intrinsic value of the spike amplitude requires knowledge of the
+stellar inclination, which is out of the scope of the current work.
+Almost all stars (121) have four years of Kepler data. However a
+few stars have missing quarters, as indicated in Table A1. In the latter
+case, gaps in the data will introduce alias peaks. This will increase
+the uncertainty of 𝑓𝑟𝑜𝑡, as it would not be possible to discern between
+the correct and alias peaks. Furthermore if the length of the data set
+is shorter, the precision of 𝑓𝑟𝑜𝑡 will again decrease. For example,
+KIC 5456027 is the only star that has 12 missing quarters, as seen in
+Table A1 ( 𝑓𝑟𝑜𝑡 = 1.242 d−1, 𝜎 𝑓𝑟𝑜𝑡 = 0.0039 d−1). The uncertainty
+on 𝑓𝑟𝑜𝑡 is on average, an order of magnitude larger than for other stars,
+meaning that the uncertainty in determining the rotation frequency
+increases with the existence of gaps in the data.
+The spike frequency and amplitude and the associated uncertainties
+for each star can be found in Table A1. Figure 6 shows a weak anti-
+correlation between the frequency and amplitude of the spike, sug-
+gesting that rapidly rotating stars tend to have slightly lower spikes.
+The right panel of Fig. 3 shows a fitting example for KIC 4921184.
+Values for the rotation frequency were compared to those in litera-
+ture, as seen in Fig. 7. For details regarding the few differences, see
+Section 3. The upper panel of Fig. 8 depicts a distribution of the
+rotation frequency values, which lie between 0.28 and 2.75 d−1, cor-
+responding to rotation periods between 3.6 and 0.4 d. This highlights
+the short rotation periods of our stars.
+In Fig. A2, panel (d1) depicts the relation between the rotational
+velocity and the spike amplitude for each target in our sample. The
+rotational velocity was calculated as:
+𝑉rot (km s−1) = 2 𝜋 𝑅𝑠 𝐹rot
+86400
+(2)
+where 𝑅𝑠 is the stellar radius in km and 𝐹rot is the rotational fre-
+quency in d−1, corresponding to the frequency of the main spike. For
+93 targets from our sample, the radius values and the associated un-
+certainties were extracted from Gaia DR2 (Gaia Collaboration et al.
+2018). For the remaining 69 stars, the radius values were taken from
+Figure 6. Spike amplitude as a function of spike frequency; if not visible the
+uncertainties in both quantities are smaller than the symbols.
+Figure 7. Comparison between rotation frequency from this work versus
+literature values: Balona (2013, 2014); Balona et al. (2015); Balona (2017);
+Trust et al. (2020); Santos et al. (2021a).
+Berger et al. (2020). The uncertainty in stellar radius has the highest
+contribution to the uncertainty in the rotational velocity. The val-
+ues obtained for the rotational velocities are depicted in a histogram
+in the lower panel of Fig. 8 and are also listed in Table A1 where
+the radius values and the associated uncertainties can also be found.
+Chowdhury et al. (2018) reported rotation periods of 513 stars, out
+of which 394 are within the same 𝑇eff range as our targets. Their
+reported rotational frequencies are similar to our sample, suggesting
+that the hump & spike stars do not rotate differently from other A and
+F stars. Fig. 9 depicts the rotational velocities with respect to their
+location in the HR diagram, confirming that more massive, hotter
+stars rotate faster. We briefly discuss the importance of rapid rotation
+of some of our stars in section 4.3.2, as it is an important ingredient
+in the OsC theory.
+MNRAS 000, 1–17 (20xx)
+
+12.5
+wdd] 
+-10.0
+Amplitude
+7.5
+5.0
+2.5
+0.0
+1.00
+1.05
+1.10
+1.15
+1.20
+1.25
+1.30
+Frequency[d-1]r = -0.2
+102
+101
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Spike frequency [d-1]Balona2013
+Balona2014
+Trust+2020
+Santos+2021
+.
+2.5
+Balona2017
+Balona+2015
+L : 2
+2.0
+1.5
+2:1
+1.0
+5:1
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+Rotation frequency (from this work) [d-1j6
+A. I. Henriksen et al.
+Figure 8. Upper panel: Histogram of the rotation frequencies extracted for
+our sample of stars (spike frequency). Lower panel: Histogram of rotational
+velocities, calculated with equation 2.
+Figure 9. HR diagram with the symbol colour correlated to the rotational
+velocity, calculated using the spike frequency and equation 2. The source for
+luminosity and 𝑇eff values is described in section 2.3. Warszaw-New Jersey
+evolutionary tracks (𝑍 = 0.012, Asplund et al. 2004) are displayed in the
+background for guidance only.
+2.3 Luminosity and 𝑇eff values
+Luminosity values for 156 targets from our sample were extracted
+from Murphy et al. (2019) (supplementary data). For the remaining
+six stars not found in Murphy et al. (2019), we assigned luminosity
+values from Berger et al. (2020) (four stars) and from Gaia DR2 (two
+stars). Last column in Table A1, denotes the source of the stellar
+parameters as explained in the caption.
+For obtaining the luminosity values, Murphy et al. (2019) used
+the Gaia DR2 parallaxes and uncertainties, 𝑔 apparent magnitudes
+(derived from 𝑔KIC and 𝑟KIC magnitudes and calibrated to the SDSS
+scale), extinctions and their uncertainties from the dustmap python
+package (Bayestar 17 reddening map) and bolometric corrections
+with isoclassify python package. For a more in-depth explanation of
+how the values were obtained, see Murphy et al. (2019) and references
+therein.
+The luminosity values for the four stars, extracted from the Berger
+et al. (2020) catalogue, were derived from isochrones and broadband
+photometry, Gaia DR2 parallaxes, and spectroscopic metallicities.
+See Berger et al. (2020) for a more in-depth understanding of how
+these values were obtained. The effective temperatures (𝑇eff) for all
+162 stars were extracted from Gaia DR2.
+Figure 10. HR diagram with the symbol colour indicating the highest har-
+monic detected in the Fourier spectrum. The number in the paranthesis next
+to each item from the legend indicates the number of stars in the sample
+that have the respective highest harmonic. The source for luminosity and 𝑇eff
+values is described in section 2.3. Warszaw-New Jersey evolutionary tracks
+(𝑍 = 0.012, Asplund et al. 2004) are displayed in the background for guidance
+only.
+2.4 Harmonics of the main spike
+The majority of the stars in our sample exhibit one to several har-
+monics of the spike frequency, which can be interpreted as a sign
+of a non-sinusoidal signal. Only 14 stars do not have any detected
+harmonics of the main spike. The colour code in Fig. 10 is dictated by
+the highest harmonic found in the Fourier spectrum. There seems to
+be no apparent correlation between the presence or lack of harmonics
+and the stellar parameters. In the legend of Fig. 10, the number of
+stars that possess specific harmonics can be found. We focus more on
+the significance and importance of harmonics for our stars in Section
+4.3.2.
+2.5 Time-series analysis
+In order to verify the temporal behaviour of the periodic signals, we
+performed a time series analysis on the data, focusing on temporal
+changes of the spike, as shown in Fig. 11, where Fourier spectra of
+subsets of the KIC 4921184 Kepler data are displayed. The average
+duration of each sub-data sets is 800 d and the average overlap 791 d.
+Each Fourier spectrum was computed with the lightkurve package
+(Lightkurve Collaboration et al. 2018). The colour coding in Fig. 11
+represents the time when the photometric data from the data subsets
+were acquired, the yellow Fourier spectrum being computed with the
+most recent Kepler data. The Gaussian centre obtained from fitting
+the spike in the Fourier spectrum of the full data set (see also Fig. 3) is
+depicted in Fig. 11 as a vertical black dotted line at the corresponding
+frequency. Fig. 11 clearly shows the amplitude and structure of the
+spike change in time. The values for computing Fig. 11 (800 d sub-
+data sets, overlapping 791 d) were chosen as a trade-off between
+resolution, SNR and having enough subsets to illustrate the changing
+character of the spike. However, we quantify this temporal change
+using other methods as described in section 2.6.
+The Fourier spectrum computed with the full Kepler data set
+of KIC 4921184 is shown in Fig. 12. The harmonics of the spike
+(f1 = 1.39 d−1) are also visible (2f1, 3f1, 4f1, 5f1 = 2.78, 4.17,
+5.56, 6.95 d−1). The presence of harmonics is indicative of non-
+sinusoidal brightness variations, typical for stellar spots. Further-
+more, the changing shape of the main spike, as seen in Fig. 11, and
+the double-wave light variation in the middle panel of Fig. 13 suggest
+the presence of variable stellar spots (e.g., Chowdhury et al. 2018).
+MNRAS 000, 1–17 (20xx)
+
+300
+3.8Mo
+3.5Mo
+250
+3.2 Mo
+2.9Mo
+2.6Mo
+200
+2.3Md
+150
+2.0M
+101
+1.7 M
+100
+1.4'M0
+50
+14000
+12000
+10000
+8000
+6000
+4000
+Teff [K]3.8 Mo
+3.5 Mo
+102.
+3.2 Mo
+2.9 Mo
+2.6 Mo
+2.3 Mo
+2.0 M
+101
+no harmonics (14)
+6th (7)
+12th (3)
+2nd (59)
+7th (2)
+15th (1)
+7mo
+3rd (28)
+8th (1)
+16th (1)
+4th (24)
+9th (1)
+17th (1)
+5th (18)
+10th (2)
+14000
+12000
+10000
+8000
+6000
+4000
+Teff [K]30
+#
+Count
+20
+10
+0
+0.5
+1.0
+1.5
+2.0
+2.5
+Frequency of the spike [d-1]
+30
+#
+Count
+20
+10
+0
+50
+100
+150
+200
+250
+300
+350
+Rotational velocity [km s-1]Rotational modulation in A and F stars
+7
+Figure 11. KIC 4921184 Fourier spectra computed with subsets of the full
+Kepler data colour coded with respect to time. The yellow Fourier spectrum is
+computed with the most recent Kepler data. Black vertical line at ∼ 1.39d−1
+denotes the frequency of the spike, obtained by fitting a Gaussian to the full
+4-year Kepler data. The amplitude of the spike in the full data set is 52 ± 0.5
+ppm.
+In general, a double-wave variation indicates the presence of spots
+that are in anti-phase, causing a stronger 2nd harmonic (Santos et al.
+2017). However, for KIC 4921184, the amplitude of the 2nd harmonic
+is just as high or slightly lower than that of the main spike in some
+of the Fourier spectra computed with subsets of the full Kepler data,
+suggesting the presence of non-permanent spots.
+2.6 Spike lifetime
+Given the changing nature of the spike (Fig. 11), we searched for
+a way to quantify its evolution in time. It has been suggested that
+the decay time-scales of the autocorrelation functions (ACFs) of
+light curves are related to the lifetimes of stellar spots (Giles et al.
+2017; Trust et al. 2020; Santos et al. 2021b). Given that one of the
+goals of our work was to probe whether the spike is evidence for
+magnetic fields, we decided to compute the ACFs based on which we
+could extract the spike lifetime. We only concentrated on the signal
+induced by the spike and its harmonics, requiring a band-pass filtered
+time series. We calculated the sum of all Fourier components in
+the frequency bands around the spike and the harmonic frequencies,
+𝑛 𝑓rot±Δ 𝑓 , where Δ 𝑓 is the frequency range around 𝑛 times the spike
+frequency. Δ 𝑓 depended on the broadness of the spike/harmonic,
+which varied from case to case; 𝑛 takes values between 1 and the order
+of the highest detected harmonic. For example, for KIC 4921184
+(Fig. 12) 𝑛 = 5. The Fourier components were determined using an
+iterative sine-wave fitting process, which was stopped once all the
+signal within the 𝑛 𝑓rot ± Δ 𝑓 frequency range was recovered.
+Fig. 13 shows an example of the band-pass filtered time series.
+In the upper panel of Fig. 13 the contribution of the spike and its
+harmonics to the brightness variability is illustrated over the com-
+plete Kepler data, while the mid-panel shows a zoom-in on the non-
+sinusoidal signal induced by the spike and its harmonics. The lower
+panel of Fig. 13 shows the difference in Fourier spectra computed
+with the full and band-passed filtered time series. The insert in the
+lower panel depicts a zoom in on the first spike and its hump.
+After computing the ACF of the band-passed filtered time series
+(orange curve in Fig. 14, obtained with numpy package Harris et al.
+2020), the result was fitted with an exponential model (green line),
+described by equation 3, using least-squares minimization (Virtanen
+et al. 2020).
+Figure 12. KIC 4921184 Fourier spectrum with the main spike (f1 = 1.39
+d−1) and its harmonics (2f1, 3f1, 4f1, 5f1 = 2.78, 4.17, 5.56, 6.95 d−1).
+Figure 13. Upper panel: Band-pass filtered time series and the original Kepler
+data for KIC 4921184. Middle panel: Zoom in of Figure a), the shape of the
+non-sinusoidal signal induced by the spike and its harmonics are depicted
+in orange, while the original data are the purple dots. Lower panel: Fourier
+spectrum of both the original data set (purple) and the band passed filtered
+time series (orange), where only the spike and harmonic signal is contained.
+The insert shows a zoom in on the first spike.
+MNRAS 000, 1–17 (20xx)
+
+80
+Gaussian center
+from full data set fit
+Amplitude [ppm]
+Time →
+60
+40
+20
+0
+1.385
+1.390
+1.395
+Frequency[d-1]70
+60
+m
+uddi
+50
+mplitude
+40
+30
+A
+20
+10
+0
+4
+Frequency[d-1]2000
+Band-passed filtered time series
+Non-filtered time series
+1500
+1000
+Flux [ppm]
+500
+0
+-500
+-1000
+-1500
+55000
+55200
+55400
+55600
+55800
+56000
+56200
+56400
+Time [BRJD]600
+Band-passed filtered time series
+Non-filtered time series
+400
+Flux [ppm]
+200
+-200
+-400
+55340
+55342
+55344
+55346
+55348
+55350
+55352
+Time [BRJD]70
+Fourier spectrum - non-filtered data
+Fourier spectrum - band-passed filtered data
+60
+[udd]
+50 
+50
+[wdd]
+40
+Amplitude
+40
+30
+30
+20
+20
+10
+1.25
+1.30
+1.35
+1.40
+1.45
+1.50
+10
+Frequency [d-1]
+0
+0
+2
+4
+6
+8
+10
+Frequency [d-1]8
+A. I. Henriksen et al.
+Figure 14. Autocorrelation function of the band-passed time series
+KIC 4921184. The orange line is the ACF of band-passed filtered time series.
+The green line is a fit obtained with equation 3.
+𝑦(𝑡) =
+�
+𝑒
+−𝑡
+𝜏ACF
+� �
+𝑦0 + 𝐴𝑐𝑜𝑠
+� 2𝜋𝑡
+𝑃𝐴𝐶𝐹
+�
++ 𝐵𝑐𝑜𝑠
+� 4𝜋𝑡
+𝑃𝐴𝐶𝐹
+��
+(3)
+In equation 3, introduced by (Giles et al. 2017) to approximate the
+starspot lifetime of Kepler stars, t represents the correlation time
+lags in days obtained as 𝑡 = 𝑁 Δ𝑇, with Δ𝑇 being the median time
+difference between consecutive observations in the light curve and
+N is the number of observed data points in a light curve. 𝜏ACF is the
+decay time-scale of the magnetic features (stellar spots) and 𝑃ACF is
+the rotation period of the star. 𝐴, 𝐵 and 𝑦0 are not associated with
+physical stellar properties, but constants used in the fitting algorithm
+of above mentioned equation. All parameters in equation 3 were left
+free in the fitting process. Fig. 15 depicts the spike lifetimes distribu-
+tion, obtained using equation 3. The values obtained are similar to
+what Trust et al. (2020) reported for a sample of stars with common
+targets.
+The accuracy of the method was tested on synthetic time series
+with known spike lifetimes. We simulated 400 time series with spike
+lifetimes of 15, 30, 60 and 120 d (100 time series for each value).
+The total length of each time series is 1500 days, with a 30 minutes
+cadence. The synthetic time series did not contain noise. The period
+of the signal was one day for all time series. Amplitude variability
+(excitation and damping) was calculated in a similar way as simulated
+for solar-like p-mode oscillations (see De Ridder et al. 2006). The
+average uncertainty was ∼27 per cent for time series with a spike
+lifetime of 15, 30, 60 d, and ∼42 per cent for 120 d, respectively. In
+terms of accuracy, the method yielded values ∼1 − 2 per cent away
+from the injected ones for 15, 30, and 60 d, and ∼21 per cent for
+120 d, respectively.
+Santos et al. 2021b suggested that the exponential model underes-
+timates decay time-scales in stellar spot simulations. They introduced
+an alternative to the model proposed by Giles et al. 2017. We find,
+however, that the ACFs in this work are better fitted with the Giles
+et al. 2017 model. Nevertheless, in order not to underestimates the
+spike lifetimes, we multiply the values by a factor of two as suggested
+by Santos et al. 2021b. This factor was derived from simulations of
+solar spots, including various number of spots, inclination angles
+and rotation periods. The spike lifetimes were obtained for all targets
+in our sample, and the results are in the HR diagram from Fig. 16,
+where these dictate the colour code. The spike lifetime values are
+also listed in Table A1.
+Figure 15. Spike lifetime distribution (2 𝜏ACF) obtained with equation 3.
+Figure 16. HR diagram with symbols correlated with the spike lifetime as
+calculated in section 2.6. The source for luminosity and𝑇eff values is described
+in section 2.3. Warszaw-New Jersey evolutionary tracks (𝑍 = 0.012, Asplund
+et al. 2004) are displayed in the background for guidance only.
+Figure 17. HR diagram with the colour code illustrating the amplitude of the
+spikes. The source for luminosity and 𝑇eff values is described in section 2.3.
+Warszaw-New Jersey evolutionary tracks (𝑍 = 0.012, Asplund et al. 2004)
+are displayed in the background for guidance only.
+3 RESULTS
+We analysed the time series of 162 hump & spike stars observed
+with the Kepler telescope and extracted the spike frequencies corre-
+sponding to the rotational frequencies, amplitude and corresponding
+uncertainties. We calculated the spike lifetimes, used stellar radius
+to determine the rotational velocity and used luminosity values from
+Murphy et al. (2019), Berger et al. (2020) and Gaia DR2 and 𝑇eff
+values from Gaia DR2 catalogue to place our targets in HR diagrams.
+Fig. 7 shows the difference between the obtained rotational fre-
+quency from this work and values from other sources mentioned in
+the caption and legend; most of our values in the literature agree with
+MNRAS 000, 1–17 (20xx)
+
+1.0
+0.8
+0.6
+0.4
+C
+0.2
+A
+0.0
+-0.2
+Fit: 2 tAcF = 24 +/- 0.1 days
+-0.4
+Observed
+5
+15
+20
+25
+10
+30
+35
+0
+Correlation lags [days]35
+30
+25
+#
+15
+10
+5
+0
+20
+30
+40
+50
+60
+70
+Spike lifetime[days]70
+3.8Mo
+60
+3.5Mo
+3.2 Mo
+50
+2.9Mo
+[days]
+2.6Mo
+40
+2.3M d
+2
+2.0M
+30
+101
+1.7 M
+20
+1.4'M0
+14000
+12000
+10000
+8000
+6000
+4000
+Teff [K]2.2
+3.8Mo
+3.5Mo
+1.8
+3.2 Mo
+amplitude)
+2.9Mo
+1.6
+2.6Mo
+1.4
+2.3M d
+(spike
+1.2
+2.0M.
+101
+0
+1.01
+1.7 M
+log
+0.8
+1.4Mo
+14000
+12000
+10000
+8000
+6000
+4000
+Teff [K]Rotational modulation in A and F stars
+9
+ours. We checked each star that does not have similar rotation fre-
+quency values to the ones previously reported and concluded that the
+values in this work are correctly identified as the rotation frequency.
+The few differences come from misidentifying the main spike with
+one of its harmonics. For example, KIC 3868032 was reported by
+Balona (2013, 2014, 2017) to have a rotation frequency of 0.4 cd−1.
+Trust et al. (2020) reported a frequency of 1.67 cd−1 which is in
+agreement with our value. A more thorough investigation, revealed
+that Kahraman Aliçavuş et al. (2020) reported a v sin i value of 181±8
+km s−1 for this star. A value for the rotational velocity using the 0.4
+cd−1 rotational frequency and a radius value of 2.4+0.07
+−0.06𝑅⊙ (Berger
+et al. 2020) would yield a rotational velocity of ≈ 49 km s−1 . Given
+that the v sin i value is the lower limit for the rotational velocity, this
+value would contradict the result of Kahraman Aliçavuş et al. (2020).
+Assuming the rotation frequency of 1.67 cd−1, the rotational velocity
+would be 206+6
+−5 km s−1, which is consistent with the measured v sin
+i. A star can be represented more than once in Fig. 7, as stars have
+been reported more than once in previous studies.
+We explore different parameters for any significant trends. Fig. A2
+shows the correlations between stellar parameters and various pa-
+rameters extracted in our analysis. In A2.a2, a moderate correlation
+between the rotational velocity and effective temperature is shown.
+This correlation is not surprising as it has been known for decades that
+hotter stars rotate faster than their cooler counterparts (e.g. Tassoul
+1978). A similar correlation can be seen in A2.b2 between luminos-
+ity and rotational velocity, which is interesting. An explanation is
+that the more luminous stars in the sample are the evolved higher-
+mass stars, and the rapid rotation they had while closer to ZAMS
+still dominates over the spin-down as their radii increase with age.
+Yet, we must highlight the bias induced by the target selection when
+discussing these correlations. Targets that originate from Trust et al.
+(2020), are stars initially selected by Balona (2013) to correspond
+roughly to spectral type A9-A0. The temperature range of interest
+was 7500 < 𝑇eff < 10 000 K. This clear 𝑇eff cut off is noticeable also
+in Fig. 2, where stars symbolized by green circles, lie roughly in this
+temperature range (differences may occur due to the fact that Balona
+(2013) used the 𝑇eff from the KIC catalogue, while the values used
+in Fig. 2 were from Gaia DR2). A similar trend, but maybe not as
+significant, can be observed in Fig. A2.c2, where the rotational ve-
+locity is compared to the stellar radius. Considering all of the above
+and that more massive, hotter stars rotate faster (Fig. 9), under the
+assumption that magnetic activity gives rise to the spike frequency,
+these results could also suggest that more massive stars require a
+faster rotation rate to generate observable effects in the form of hump
+& spike.
+In addition to the initial sample of stars with𝑇eff = [6500, 10000] K,
+we also attempted to populate our target list with cooler stars by
+visually inspecting selected stars from Santos et al. (2021a). The
+latter study concentrates on G and late F main-sequence and cool
+subgiant stars (in Fig. 2, stars symbolised with blue circles). Only
+three stars with 𝑇eff = [6310, 7033] K were found, suggesting that the
+hump & spike feature does not occur in cooler stars. The reason may
+be that lower-mass stars rotate on average slower, and/or the stellar
+structure is different, but we stress that a more systematic search
+for hump & spike stars is necessary in order to suggest a possible
+explanation. However, in the OsC modes scenario we do not expect
+cooler hump & spike stars as the convective core does not exist at
+masses lower than roughly 1.3 M⊙. Additionally the thick convective
+envelope of late type stars would not allow g modes to penetrate all
+the way to the surface.
+There seems to be no apparent connection between any parameter
+and the spike lifetime. The average spike lifetime value for the whole
+sample is around 30 d, with ∼95 per cent of the values being less
+than 60 d. The order of magnitude of the spike lifetime values are
+in agreement with values of starspot lifetime reported by Giles et al.
+(2017) regarding F-type stars. Although the F-stars in the Giles et al.
+(2017) sample are not very numerous, they report that the stellar
+spots do not survive very long (unlike in the case M, K and G stars)
+nor reach substantial sizes.
+The missing correlation between the spike lifetimes and stellar
+mass is hard to reconcile in the OsC scenario. As the g modes would
+be modulated by the convective turnover time scales, we expect the
+spike lifetimes to decrease with stellar mass. We address this in more
+detail in section 4.3.
+The weak correlations from Fig. A2.a1, which shows the relation
+between the spike amplitude and 𝑇eff and Fig. A2.b1 that depicts the
+relation between the spike amplitude and luminosity (A2), suggest
+that cooler and less luminous stars have higher spikes. The same
+trend clearly showing that more massive stars have lower spike am-
+plitudes is illustrated in Fig. 17. This correlation is further explored
+and discussed in section 4.2, as it is related to the predictions of
+CB19.
+4 DISCUSSIONS
+4.1 Magnetic field strength
+Cantiello & Braithwaite (2019) argue that the thin envelope convec-
+tion zones of intermediate-mass stars can sustain dynamo-generated
+magnetic fields. If the observed photometric variability is due to
+periodic temperature variations caused by magnetic spots, one can
+estimate the magnetic field strength using the amplitude of the spike
+and some assumptions for the spot filling factor. Following Cantiello
+& Braithwaite (2011), we start from the definition of the radiative
+gradient:
+∇rad =
+� 𝑑𝑙𝑛𝑇
+𝑑𝑙𝑛𝑃
+�
+rad
+≃ 𝑃
+𝑇
+Δ𝑇
+Δ𝑃 ,
+(4)
+which can be re-written as
+Δ𝑇
+𝑇
+= ∇rad
+Δ𝑃
+𝑃 .
+(5)
+We then assume that a magnetic spot is in hydrostatic and thermal
+equilibrium with its surroundings. The latter requires the radius of
+the spot to be significantly smaller than the stellar radius. We also
+assume that the radiative gradient is not affected substantially by the
+presence of the magnetic field. Assuming hydrostatic equilibrium
+implies that a magnetic spot has a smaller gas pressure compared
+to its non-magnetized surroundings, since part of the total pressure
+is provided by the magnetic field. Together with the assumption of
+thermal equilibrium, this implies a lower density in the magnetized
+region. A local depression in the surface density allows photons to
+emerge from deeper regions of the star, so an observer looking down
+into a magnetic spot at the surface of a radiative star is expected to see
+hotter temperatures compare to a non-magnetized region (Cantiello
+& Braithwaite 2011). Using Eq. 5 this temperature contrast can be
+written as
+Δ𝑇
+𝑇
+= ∇rad
+𝑃mag
+𝑃tot
+= ∇rad
+𝛽
+(6)
+where
+𝛽 = 𝑃tot
+𝑃mag
+=
+𝑃tot
+𝐵2/8𝜋 ,
+MNRAS 000, 1–17 (20xx)
+
+10
+A. I. Henriksen et al.
+and 𝑃mag is the component of the pressure due to the presence of the
+magnetic fields, 𝑃tot is the total pressure and 𝐵 the magnetic field
+strength. Assuming that the ratio between the magnetic pressure 𝑃mag
+and the total pressure 𝑃tot (from the gas and radiation) is very small,
+𝛽 ≫ 1, and using 𝐿 = 4𝜋𝑅2𝜎𝑇4
+eff, (where R is the stellar radius),
+we can obtain a relation to estimate the local contrast in luminosity
+between the spot and the rest of the stellar surface (Cantiello &
+Braithwaite 2011):
+Δ𝐿
+𝐿 ≃ 4Δ𝑇
+𝑇
+= 4∇rad
+𝛽
+(7)
+This equation provides the luminosity contrast of a magnetic
+spot compared to the unperturbed stellar luminosity. To compare
+this to observations, one has to take into account the filling factor
+𝑓 = (𝑟/𝑅)2, where 𝑟 is the radius of the spot and R is the stellar
+radius. Note the degeneracy between temperature (or luminosity)
+contrast and filling factor: A large spot with a small luminosity con-
+trast can give a similar brightness variation to a smaller spot with a
+higher luminosity contrast. From a theoretical standpoint, Cantiello
+& Braithwaite 2011 and CB19 suggested that magnetic fields in hot
+stars could produce features on scales that have comparable sizes with
+(or larger than) the pressure scale height of the convective regions
+where they are generated. Lacking firm observational constraints on
+the size of the spots that might be generating the brightness fluctua-
+tions, we will assume the spot size 𝑟 to be comparable to the pressure
+scale height in sub-surface convective layers. As we will discuss be-
+low the spots could actually be larger, so this assumption means our
+estimates for the magnetic fields represent upper limits. Equation 7
+becomes:
+Δ𝐿
+𝐿 = 4∇rad
+𝛽
+� 𝐻𝑝
+𝑅
+�2
+(8)
+where 𝐻𝑝 is the pressure scale height. Using the definition of 𝛽,
+equation 8 can be written as:
+Δ𝐿
+𝐿 = 4∇rad𝐵2
+8𝜋𝑃tot
+� 𝐻𝑝
+𝑅
+�2
+(9)
+The ratio Δ𝐿/𝐿 is then replaced with the observed photometric am-
+plitude (𝐴spike) in the Kepler data. Note that our calculation assumes
+a single spot and we do not correct for the Kepler bandpass. This is
+not necessarily correct, and the data suggest some stars have more
+than one spot, see e.g. the phase plots in Section 4.3.3. The magnetic
+fields amplitude is then recovered using the following relation:
+𝐵2 ≃ 𝐴spike
+2𝜋𝑃tot
+∇rad
+� 𝑅
+𝐻𝑝
+�2
+.
+(10)
+From a grid of ten stellar models in the mass range 1.25 − 4 M⊙,
+obtained using MESA (Paxton et al. 2011, 2013, 2015, 2018, 2019),
+we have extracted through linear interpolation values for the total
+pressure at the surface 𝑃𝑡𝑜𝑡, radiative gradient ∇𝑟𝑎𝑑 and the pressure
+scale height 𝐻𝑝. The MESA models are accessible through github 1.
+The stellar models have solar metallicity (𝑍 = 0.02) and a chemical
+composition mixture taken from Asplund et al. (2005). The mixing
+length parameter was assumed to be 1.6. The 𝑃𝑡𝑜𝑡 values we obtain
+for our sample span from 375 to 17497 Pa, with a median value of
+3092 Pa. The average radiative gradient values for our stars is 0.126,
+1 https://github.com/matteocantiello/hump_spikes_stars
+Figure 18. HR diagram with colour code illustrating the estimated magnetic
+field strengths. The source for luminosity and 𝑇eff values is described in
+section 2.3. Warszaw-New Jersey evolutionary tracks (𝑍 = 0.012, Asplund
+et al. 2004) are displayed in the background for guidance only.
+Figure 19. Estimate values for the magnetic field strength with respect to the
+𝑇eff of the stars.
+spanning from 0.125 to 0.132. An average value for the pressure scale
+height for the stars in our sample was found to be 0.01𝑅⊙, ranging
+from 0.005 to 0.02 𝑅⊙.
+We note that the 𝐻𝑝 values are the lower limits to the sizes of the
+magnetic features. CB19 pointed out that magnetic features expand
+when rising towards the stellar surface depending on the ratio of the
+density in the convective layers and at the photosphere. Moreover,
+intermediate-mass stars tend to be rapidly rotating, which can lead to
+dynamo action on scales larger than the local pressure scale-height
+(Brandenburg & Subramanian 2005; Cantiello & Braithwaite 2011).
+Therefore the recovered values of 𝐵 are upper limits.
+Under all the assumptions made above we computed upper limits
+estimates for the magnetic field strength in our stars sample. The
+results are depicted in Fig. 18. The obtained values are also shown
+with respect to 𝑇eff in Fig. 19 and range from 77 to 2207 G, with an
+average value of 540 G. In the next section we discuss our findings
+in the context of theoretical predictions.
+4.2 Comparison with predictions from literature
+Fig. 17 clearly shows that the spike amplitude is decreasing with
+increasing stellar mass, which in the context of magnetic fields, sug-
+gests stronger magnetic fields for lower-mass stars, which also have
+deeper (sub)surface convective layers.
+Based on the spike amplitude we have estimated the strengths of
+equipartition magnetic fields in the convective envelopes. We note
+again that the intrinsic values of the amplitudes depend on e.g., stellar
+MNRAS 000, 1–17 (20xx)
+
+3.2
+3.8Mo
+3.5Mo
+3.0
+3.2 Mo
+2.8℃
+2.9Mo
+2.6Mo
+2.6
+2.3M d
+2.4
+2.0M
+101
+1.7 Mo
+2.2
+2.0
+1.4M0
+14000
+12000
+10000
+8000
+6000
+4000
+Teff [K]2000
+1500
+B 1000
+500
+0
+10000
+9000
+8000
+7000
+6000
+Teff [K]Rotational modulation in A and F stars
+11
+inclination and spot latitudes, which are unknown. Nevertheless,
+our results are in agreement with the predicted values of average
+equipartition magnetic fields in the sub-surface convective zones (see
+figure 6 from CB19.) By comparing Fig. 18 and figure 8 from CB19,
+a similar general trend can be seen. Both the predicted magnetic field
+strengths in CB19 and the calculated values from this work decrease
+with increasing stellar mass. Furthermore, Fig. 18 shows that more
+evolved stars have stronger calculated magnetic fields, which also
+agrees with the predictions from CB19, as the convective envelope
+gets deeper as the star evolves.
+The strengths of the magnetic fields at the surface that were com-
+puted by CB19 were determined by scaling the equipartition fields
+with the density inside the HeII convective zone, ranging from 2000 G
+to less than 1 G (see figure 9 from CB19). Their findings showed that
+for a star like Vega the expected mean longitudinal field would be
+very small (∼ 10−2 G), which is consistent with the observed value
+of 0.6 ± 0.3 G (Lignières et al. 2009). Therefore the values obtained
+with the density scaling relation CB19 proposed should be taken as
+a lower limit. This also brings forth the fact that the magnetic buoy-
+ancy might not necessarily be the process that brings the magnetic
+field to the surface. Other processes have been proposed such as the
+one proposed by Warnecke & Brandenburg (2010); Warnecke et al.
+(2011), where the magnetic field would reach the surface through
+magnetic tension, in which case the strength at the surface could be
+larger, comparable to the equipartition values.
+The calculated values for the spike lifetimes (Fig. 16) suggest that
+if present, the magnetic features are not very long lived, having life-
+times of tens of days, with only five stars surpassing 60 d. This is in
+agreement with predictions from CB19, regarding the rapidly evolv-
+ing features of the dynamo-generated magnetic fields. In addition,
+the trend that can be observed for the spike amplitude in Fig. 17 is
+not present for the spike lifetime. This suggests that a more complex
+process may determine the spike lifetime.
+To summarize, the spike amplitude trend with mass and 𝑇eff in the
+context of magnetic fields, suggests that the dynamo in the subsurface
+convection hypothesis is favoured over the ‘failed-fossil’ scenario.
+4.3 Convective core rotation
+4.3.1 Convective turnover time
+As described above, Lee & Saio (2020) and Lee (2021) suggest
+that Overstable convective (OsC) modes could, in the presence of
+radial differential rotation, couple with low-frequency g modes in
+the envelope to emerge on the surface. A 20% differential rotation is
+assumed in Lee & Saio (2020), while Lee (2021) assumes the core
+rotation to be 10% higher than the envelope rotation.
+An OsC mode of an azimuthal order 𝑚 would have a frequency 𝑚
+times the core rotation frequency. Here we investigate whether the
+rotational modulation characterised by the spike could be due to OsC
+modes.
+If the spikes corresponded to 𝑚 times the convective core rotation
+rates, it would mean that the spike lifetimes calculated in subsection
+2.6 would be associated with the movement of the convective cells
+inside the core. In other words, we would expect the spike lifetimes to
+be related to the core’s convective turnover time. It is well established
+that the mass of the convective core increases with stellar mass (see,
+e.g., figure 22.7 in Kippenhahn & Weigert 1990). Consequently, the
+spike lifetimes calculated in section 2.6 should correlate with the
+stellar mass. From MESA models, we have extracted a few values for
+the convective turn overtime in the stellar mass regime of interest.
+These values were obtained from models which have solar metallicity
+Table 1. Extracted values of convective turnover time from MESA models
+for a few representative stellar masses.
+ZAMS
+TAMS
+Mass
+𝜏turn
+Mass
+𝜏turn
+[M⊙]
+[days]
+[M⊙]
+[days]
+1.5
+173
+1.5
+44
+2.0
+113
+2.0
+38
+2.4
+83
+2.4
+35
+3.0
+68
+3.0
+32
+3.6
+62
+3.6
+30
+4.0
+59
+4.0
+29
+𝜏turn = convective turnover time
+(𝑍 = 0.02) and a chemical composition mixture taken from Asplund
+et al. (2005). The mixing length parameter was assumed to be 1.6
+(Table 1). As can be seen in Table 1 the convective turnover times
+decrease with stellar mass. This negative correlation is expected to
+be present throughout the entire main-sequence evolutionary stage.
+It is important to point out that our results do not show any corre-
+lation between the spike lifetime and stellar mass, as seen in the HR
+diagram from Fig. 16, suggesting that the broadness of the spike is
+not connected to convective turnover time in the core.
+In the context of OsC modes, a possible explanations for the ob-
+served trend in amplitude seen in Fig. 17, could be the changing
+density of g modes with stellar mass. In other words, the amplitude
+of an OsC mode at a surface should be larger when the resonance
+with a g mode is stronger, which would occur more frequently if the
+density of g modes (in the co-rotating) frame is higher. This would
+take place in less massive stars and could imply that the amplitudes
+of the spikes tend to be lower in more massive stars, as seen in Fig.
+17.
+4.3.2 Harmonic signature and rotation
+As mentioned in section 2.4, only 14 stars do not exhibit detectable
+harmonics of the main spike (see an example in lower right corner of
+Fig. 20). As seen in Fig. 10, the rest show harmonics in their Fourier
+spectra, with KIC 7175896 having the highest amount of harmonics
+(8), with the highest harmonic being at 17 times the spike frequency.
+Lee & Saio 2020 suggested that the expected amplitudes of
+g modes resonantly excited by OsC modes would decrease as the
+azimuthal order (𝑚) increases. The variability induced is expected to
+be sinusoidal and would not cause any additional harmonics in the
+Fourier spectrum. In order to quantify the number of stars that would
+be consistent with this scenario, we subdivided our sample into four
+groups:
+• Group A: the amplitudes of the spike and its harmonics decrease
+as the degree of the harmonic increases (see example in the upper left
+corner of Fig. 20). This group comprises 111 stars and is consistent
+with both scenarios.
+• Group B: a higher order harmonic has a higher amplitude than
+the previous harmonic or the main spike (upper right corner of
+Fig. 20), which can only be explained by non-sinusoidal signals,
+such as spots. KIC 4921184 (Fig. 12) also falls under this category
+comprising 22 stars in total.
+• Group C: the detected harmonic orders are not consecutive, i.e.
+there are missing harmonics, which again is an indication of non-
+sinusoidal signals, that could be induced by stellar spots. This group
+MNRAS 000, 1–17 (20xx)
+
+12
+A. I. Henriksen et al.
+consists of 14 stars. The lower left corner of Fig. 20 shows an example
+where the fourth harmonic is not present while the third and fifth have
+similar amplitudes.
+• Group D: no harmonics detected, only the main spike is signifi-
+cant, not allowing us to draw any conclusion. We find 14 stars falling
+in this category.
+Counting only the harmonic behaviour or the absence of harmonics
+(Group A and D), we find 125 stars for which we cannot strictly
+determine the origin of the spike. In other words, both the OsC
+modes and the stellar spots caused by dynamo-generated magnetic
+fields could explain the spike feature. However, groups B and C
+clearly display a a behaviour that is compatible with stellar spots,
+making it a total of 36 stars.
+We highlight that the OsC modes scenario, as suggested by Lee &
+Saio (2020) and Lee (2021), requires rapid rotation. As seen in Fig.
+8 and 9 more than half of the stars from our sample have a rotational
+velocity higher than 100 km s−1. However, the exact value for the
+required minimum rotation speed depends on model parameters such
+as the degree of radial differential rotation and the super-adiabatic
+temperature gradient in the rotating convective core.
+The evolutionary stage of a star, i.e., whether it is still on the main
+sequence or not, has crucial implications for the OsC scenario. This
+is because our stars beyond the TAMS do not have a convective core.
+Based on their location in the HR diagram (e.g., Fig. 9), we visually
+identified 34 stars that may have left the main-sequence. However,
+we note the following shortcomings to this identification:
+• The visual identification was done without taking into account
+uncertainties in 𝑇eff and luminosity.
+• The evolutionary tracks depicted in our HR diagrams do not
+take into account rotation and core overshooting and only use so-
+lar metallicity. This means that without detailed stellar modelling,
+determining whether a star still has a convective core is imprecise.
+• The effect of gravity darkening increases with rotation . Fig-
+ure 4 from Georgy et al. (2014) and figure 38 from Paxton et al.
+(2019) nicely illustrate how the observed luminosity depends on the
+inclination angle for stars rotating with more than 50-60 per cent of
+their critical velocity. For these stars, the luminosity could be under-
+estimated if observed pole-on or underestimated at the equator. For
+example, the two fast rotators in Fig. 9, which lie around the 3.5 M⊙
+evolutionary track, could still be on the main sequence.
+4.3.3 Phase-folded light curve
+Another piece of evidence that might point towards the spike being
+associated with the stellar surface rotation rather than the convective
+core rotation, is the shape of the phase-folded light curves. Figures
+21 - 26 depict the phase-folded time series of 3 stars from our sample.
+The groups defined in section 4.3.2 to which these stars belong to
+are: KIC 2157489 - B, KIC 4661914 - C, KIC 3440710 - A. All
+the data were folded with the periods derived from the spikes. The
+phase folded light curves were obtained with lightkurve (Lightkurve
+Collaboration et al. 2018) and afterwards binned in phase with the
+values mentioned in the captions. Figures 21, 23 and 25 represent the
+phase-folded light curves of the full Kepler data sets. We note that
+if the median amplitude value is taken in a given bin, the effect of
+amplitude change with time is not visible when displaying the entire
+data set. Figures 22, 24 and 26 contain the phase-folded light curves
+computed with sub-data sets. It is clear that the wiggly shape of the
+light curve changes depending on the time range of the sub-data set.
+Our phase-folded data show a characteristic non-sinusoidal signal
+Figure 20. Harmonics signature of our sample. Group A: the amplitude of
+the spike harmonics decreases with increasing harmonic order. Group B: the
+amplitude of a harmonic is higher than a harmonic of a lower degree or
+the spike. Group C: harmonics are not consecutive, i.e. missing harmonics.
+Group D: no harmonics present. As described in the text in more detail Group
+B and C imply the presence of stellar spots, whereas Group A and D do not
+allow us to distinguish between spots and OsC modes.
+Figure 21. Phase-folded light curve computed with the band-passed filtered
+time series of KIC 4661914 (group C - see section 4.3.2); average bin size
+∼0.003 in phase folded on period ∼1.024 d which corresponds to the fre-
+quency of the main spike ∼0.977 d−1.
+that could be explained by the presence of several evolving stellar
+spots.
+5 CONCLUSIONS AND FUTURE WORK
+In this work, we analysed the Kepler light curves of 213 stars that
+present a similar feature in their Fourier spectra called hump &
+spike. Out of this sample, we have selected 162 stars that did not
+present obvious photometric signs of being in a binary system (e.g.
+no transits or ellipsoidal variation). We aimed to validate the initial
+theoretical interpretation of the origin of the spike (Saio et al. 2018),
+which is assumed to be rotational modulation induced by stellar spots
+co-rotating with the stellar surface. We tested the theory regarding
+the presence, origin and strength of magnetic fields on our sample
+of stars under the assumption that the hump represents unresolved
+Rossby modes and the spike stellar spots. In the present work, we
+have concentrated only on the spike in the hump & spike feature. The
+’hump’ will be the focus of future work currently in preparation.
+In addition, we have also discussed another possible phenomenon
+that could cause rotational modulation in the light curves of our stars,
+as suggested by Lee & Saio (2020) and Lee (2021). In this context,
+the spike would be due to Overstable Convective (OsC) modes that
+MNRAS 000, 1–17 (20xx)
+
+10
+150
+KIC9163520 - Group A
+KIC3240406 - Group B
+8
+6
+100
+4
+ddj
+2
+Amplitude
+0
+0
+0.5
+1.0
+1.5
+2.0
+1
+2
+3
+4
+10
+30
+KIC7959579 - Group C
+KIC8396240 - Group D
+8
+20
+A
+6
+4
+10
+0
+0
+0.5
+1.0
+1.5
+2.0
+2.5
+220
+10
+Flux [ppm]
+0
+-10
+-20
+-30
+-0.4
+-0.2
+0.0
+0.2
+0.4
+PhaseRotational modulation in A and F stars
+13
+Figure 22. Phase-folded light curve computed with subsets of the band-passed
+filtered time series - KIC 4661914 (group C - see section 4.3.2); average bin
+size for each plot ∼0.003 in phase folded on period ∼1.024 d (frequency
+of the main spike ∼0.977d−1). The time range is noted for each plot; the
+epoch time is the same as in Fig. 21 (time of the first Kepler observation for
+KIC 4661914).
+resonantly excite g modes in the co-rotating frame propagating to the
+surface and consequently become observable. The spike frequency
+in this scenario would correspond to the convective core rotation
+frequency.
+We have extracted the spike frequency and amplitude for all stars
+and used stellar parameters such as 𝑇eff, luminosity and the stellar
+radius from various sources (see Table A1 to consult the source
+of the stellar parameters). We determined the rotational velocities
+using stellar radii and spike frequencies. Further, we determined the
+lifetime of the spike using autocorrelation functions and extracted
+the number of spike harmonics present in the Fourier spectra of our
+stars. In favour of the spike being evidence for stellar spots due to
+magnetic fields, we find the following:
+• The spike amplitudes are higher for cooler stars and also slightly
+higher for evolved stars, mirroring predictions regarding the strength
+Figure 23. Phase-folded light curve computed with the full Kepler data -
+KIC 2157489 (group B - see section 4.3.2); average bin size ∼0.003 in phase;
+folded on period ∼1.358 d; which corresponds to the frequency of the main
+spike ∼0.736 d−1.
+Figure 24. Phase-folded light curve computed with subsets of the full time
+series - KIC 2157489 (group B - see section 4.3.2); average bin size ∼0.003 in
+phase; folded on period ∼1.358 d (frequency of the main spike ∼0.736 d−1);
+the time range is noted for each plot; the epoch time is the same as in Fig. 23
+(time of the first Kepler observation for KIC 2157489).
+MNRAS 000, 1–17 (20xx)
+
+20
+[55013-55042] BJD
+10
+o
+-10
+-20
+[55247-55274] BJD
+20
+10
+0
+-10
+[ppm]
+-20
+-30
+[55606-55633] BJD
+20
+10
+-10
+-20
+-30
+20
+[55962-55990] BJD
+10
+0
+-10
+-20
+-30
+0.4
+0.2
+0.0
+0.2
+0.4
+Phase40
+20
+[dd] 
+0
+-20
+-40
+-60
+-0.6
+-0.4
+-0.2
+0.0
+0.2
+0.4
+0.6
+Phase60
+[55048-55096]BJD
+40
+20
+0
+-20
+40
+-60
+[55664-55708]BJD
+40
+20
+[wd
+0
+xnl
+-20
+F
+-40
+-60
+[56329-56373] BJD
+40
+20
+0
+-20
+-40
+-60
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+Phase14
+A. I. Henriksen et al.
+Figure 25. Phase-folded light curve computed with the full Kepler time series
+- KIC 3440710 (group A - see section 4.3.2); average bin size ∼0.003 in phase;
+folded on period ∼0.8 d; which corresponds to the frequency of the main spike
+∼1.25 d−1.
+Figure 26. Phase-folded light curve computed with subsets of the full time
+series - KIC 3440710 (group A - see section 4.3.2); average bin size ∼0.003
+in phase; folded on period ∼0.8 d (frequency of the main spike ∼1.25 d−1);
+the time range is noted for each plot; the epoch time is the same as in Fig. 25
+(time of the first Kepler observation for KIC 3440710).
+of dynamo-generated magnetic fields in (sub) surface convective
+layers.
+• The spike lifetimes are short, of the order of tens of days, sim-
+ilar to what one would expect from magnetic features of dynamo-
+generated magnetic fields.
+• The spike lifetimes do not correlate with stellar mass. This
+would exclude the possibility that the spike frequency would cor-
+respond to the convective core rotation frequency, as it should be
+modulated by the convective turnover time, which increases with
+stellar mass.
+• The harmonic signature of 36 stars suggests a non-sinusoidal
+signal consistent with co-rotating stellar spots.
+• Phase-folded light curves of time series containing the signal
+induced by the spike and its harmonics suggest a non-sinusoidal
+signal specific to changing stellar spots.
+• The estimated magnetic field strengths, however uncertain, are
+of the same order of magnitude as predictions of non-Ap stars.
+• We find 34 stars that could be in a post main-sequence evolu-
+tionary stage, which excludes the OsC modes scenario, as these stars
+do not have a convective core. We note, however, that this identifica-
+tion is visual only. The evolutionary tracks used here do not take into
+account, e.g., core overshooting, rotation and non-solar metallicities.
+In addition, the luminosity values used here are not corrected for the
+gravity-darkening effect, which is considerable in the case of high
+stellar rotation.
+Our results suggest that, despite their very shallow convective en-
+velopes, these stars could have spots likely induced by magnetic
+fields. The characteristics of the features causing the brightness vari-
+ability in the shape of the spike point towards the idea of a dynamo-
+generated magnetic field as suggested by CB19. The short spike
+lifetimes, the estimated magnetic field strengths and presumed mag-
+netic feature sizes favour dynamo-generated fields rather than fossil
+fields. While these stars are unfortunately too faint to qualify for
+spectropolarimetry, recent observations of 𝛽 Cas - a rapidly rotating
+𝛿 Scuti star - indicate that dynamo-generated magnetic fields exist
+in intermediate mass stars (Zwintz et al. 2020). As observations are
+biased towards stars with strong magnetic fields and large magnetic
+features, our work provides a path for future observational efforts
+towards weaker magnetic fields that generate small-scale features.
+Furthermore, ongoing observational efforts from the ground and
+the TESS satellite will allow us to determine whether the spots (if
+present) in the hump & spike stars are bright spots, as suggested
+by CB19. Our work also offers the chance to verify and test the
+predictions and theory of Saio et al. (2018), where it was hypothesised
+that Rossby modes are mechanically excited by deviated flows caused
+by stellar spots. An obvious next step is to find bright hump &
+spike stars observed with TESS, which will allow us to measure the
+magnetic fields directly using ground-based facilities.
+In favour of the spike being evidence of g modes being resonantly
+excited by OsC modes from the convective core we find the following
+arguments:
+• The spike amplitude increasing with mass could be because the
+amplitude of OsC modes is larger when the resonance with g modes
+is stronger. In the case of lower mass stars, the density of g-mode
+frequencies is higher; therefore, it would be more likely that the
+resonance of g modes in the co-rotating frame is stronger.
+• The harmonic signature (amplitude decreases as azimuthal or-
+der decreases) in the case of 111 stars, where the OsC modes scenario
+could be the cause for the rotational modulation. This applies also
+for 14 stars more which do not exhibit detectable harmonics of the
+main spike.
+MNRAS 000, 1–17 (20xx)
+
+30
+C
+20
+10
+[udd]
+0
+-10
+xni
+-20
+30
+5
+-40
+-50
+-0.4
+-0.3
+-0.2
+-0.1
+0.0
+0.1
+0.2
+0.3
+0.4
+Phase100
+[55043-55081]BJD
+75
+50
+25
+0
+25
+50
+75
+100
+[55630-55672]BJD
+60
+40
+[ppm]
+20
+0
+20
+40
+50
+[56215-56258]BJD
+40
+30
+20
+10
+0
+-10
+-20
+-30
+0.4
+0.3
+0.2
+-0.1
+0.0
+0.1
+0.2
+0.3
+0.4
+PhaseRotational modulation in A and F stars
+15
+Based on our analyses and sample of stars, neither scenario can
+be entirely excluded at this stage. It is also possible, although rather
+unlikely, that both stellar spots and OsC modes could generate the
+hump & spike feature. While in this work, we find more arguments
+supporting the idea of stellar spots induced by magnetic fields, as
+summarised above, a full conclusion can be reached only by directly
+measuring magnetic fields in hump & spike stars.
+ACKNOWLEDGEMENTS
+This work was supported by a research grant (00028173) from VIL-
+LUM FONDEN. Funding for the Stellar Astrophysics Centre is pro-
+vided by The Danish National Research Foundation (Grant agree-
+ment no.: DNRF106). This research was supported in part by the
+National Science Foundation under Grant No. NSF PHY-1748958.
+S.M. acknowledges support from the Spanish Ministry of Sci-
+ence and Innovation (MICINN) with the Ramón y Cajal fellowship
+no. RYC-2015-17697 and grant no. PID2019-107187GB-I00, and
+through AEI under the Severo Ochoa Centres of Excellence Pro-
+gramme 2020–2023 (CEX2019-000920-S).
+ARGS acknowledges the supported by FCT through national
+funds and by FEDER through COMPETE2020 by these grants:
+UIDP/04434/2020; PTDC/FIS-AST/30389/2017 & POCI-01-0145-
+FEDER-030389. ARGS is supported by FCT through the work con-
+tract No. 2020.02480.CEECIND/CP1631/CT0001.
+We acknowledge that this research was supported in part by the
+National Science Foundation under Grant No. NSF PHY-1748958.
+R.A.G. acknowledges the support of the PLATO CNES grant.
+DATA AVAILABILITY
+The data underlying this article are available in the article and in its
+online supplementary material.
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+APPENDIX A: EXTRA MATERIAL
+MNRAS 000, 1–17 (20xx)
+
+Rotational modulation in A and F stars
+17
+Figure A1. HR diagram of the hump & spike stars. Green circles depicts the stars studied in the current work. The source for luminosity and 𝑇eff values is
+described in section 2.3. Red pentagons, orange squares and purple triangles represent the stars excluded as they show signs of binarity (see section 2.1). One
+star that was excluded is not shown in the Figure as no value for luminosity was found in literature: KIC 5458880 (𝑇eff Gaia DR2 = 8704 K+782
+−1082), was identified
+as an eclipsing by Kirk et al. 2016 and shows a clear binary signal in Kepler data. All 𝑇eff values are from Gaia DR2. Warszaw-New Jersey evolutionary tracks
+(𝑍 = 0.012, Asplund et al. 2004) are displayed in the background for guidance only.
+Table A1. Extracted spike parameters and stellar parameters. 𝑓rot,, 𝜎 𝑓rot: Spike frequency and associated standard deviation; A, 𝜎A: Spike amplitude and
+its uncertainty; 𝑇eff, 𝑇effp, 𝑇effm: Effective temperature and upper and lower uncertainties - from Gaia DR2; 𝐿, 𝐿p, 𝐿m: Luminosity and upper and lower
+uncertainties; 𝑅, 𝑅p, Rm: Radius value, upper and lower uncertainties; 2𝜏ACF, 𝜎𝜏ACF: Spike lifetime and its uncertainty; No. quarters: number of Kepler
+quarters in which data are available; Source parameter: first letter indicates the source of the luminosity values, second letter indicates the source of the radius
+values, m = Murphy et al. (2019), i = Berger et al. (2020), g = Gaia DR2. The full table is available online; here, the first 10 rows are shown for guidance on
+content and style.
+KIC
+𝑓rot
+𝜎 𝑓rot
+A
+𝜎A
+𝑇eff
+𝑇effp 𝑇effm
+𝐿
+𝐿p
+𝐿m
+𝑅
+𝑅p
+Rm
+2𝜏ACF 𝜎𝜏ACF
+No.
+Sources
+[d−1]
+[d−1]
+[ppm] [ppm]
+[K]
+[K]
+[K]
+[L⊙] [L⊙] [L⊙] [R⊙] [R⊙] [R⊙]
+[d]
+[d]
+quarters
+param.
+1722916 0.553 0.0013
+43.8
+0.65
+7017 260
+98
+5.4
+1.09 1.09
+1.4
+0.04 0.09
+15
+0.1
+18
+m/g
+1873552 1.236 0.0016
+5.1
+0.51
+7826 121
+86
+11.4 1.05 1.06
+1.7
+0.04 0.05
+25
+0.2
+18
+m/g
+2157489 0.737 0.0014
+53.0
+0.65
+7361 140
+254
+9.1
+1.05 1.05
+1.7
+0.13 0.06
+33
+0.2
+18
+m/g
+2158190 1.016 0.0006
+23.7
+0.39
+8048
+42
+219
+35.6 1.05 1.05
+2.8
+0.1
+0.09
+38
+0.1
+18
+m/i
+3002336 1.202 0.0004
+76.3
+0.71
+7115 272
+217
+14.4 1.06 1.07
+2.2
+0.14 0.16
+20
+0.2
+15
+m/g
+3222104
+1.27
+0.0004
+22.8
+0.74
+8821 231
+228
+23.3 1.08 1.08
+2.2
+0.08 0.08
+36
+0.1
+18
+m/i
+3238627 1.537 0.0024
+22.8
+0.96
+7059
+98
+140
+17.4
+1.1
+1.1
+2.5
+0.1
+0.07
+34
+0.2
+17
+m/g
+3240406 1.071 0.0013
+5.7
+0.35
+7865 161
+201
+13.3 1.06 1.06
+1.8
+0.1
+0.07
+30
+0.1
+18
+m/g
+3337124 1.108 0.0004
+22.7
+0.58
+7835
+61
+298
+12.8 1.05 1.05
+1.8
+0.14 0.03
+37
+0.1
+18
+m/g
+3440710
+1.25
+0.0021
+33.3
+1.49
+6531 256
+154
+5.9
+1.06 1.06
+1.6
+0.08 0.12
+28
+0.1
+17
+m/g
+This paper has been typeset from a TEX/LATEX file prepared by the author.
+MNRAS 000, 1–17 (20xx)
+
+3.8 Mo
+3.5 Mo
+102
+3.2 Mo
+2.9 Mo
+2.6 Mo
+2.3 Mo
+Analysed in current work
+2.0 M
+Excluded
+101
+(Luminosity - Murphy et. al 2019)
+Excluded
+1.7 M
+(Luminosity - Gaia DR2)
+Excluded
+(Luminosity - Berger et al. 2021)
+1.4Mc
+14000
+12000
+10000
+8000
+6000
+4000
+Teff [K]18
+A. I. Henriksen et al.
+Figure A2. Correlation between stellar parameters and the parameters extracted from our analysis
+MNRAS 000, 1–17 (20xx)
+
+r = 0.0
+(a2)
+a31
+9000-
+9000-
+9000
+8000-
+8000
+8000
+7000
+7000
+7000
+r = 0.53
+.
+(a1)
++0009
+6000
+6000
+101
+102
+50
+100
+150
+200
+250
+300
+350
+10
+20
+30 
+40
+50
+60
+70
+Spike amplitude [ppm]
+Vrot [kms-1]
+2TAcF [days]
+[L。]
+(b1)
++
+(b2)
+(b3)
+r = -0.06
+r = -0.18
+102,
+102.
+102,
+Luminosity [
+2.0
+101
+101
+101
+r = 0.52
++
+101
+102
+50
+100
+150
+250
+300
+350
+10
+20
+30
+40
+50
+200
+60
+70
+Spike amplitude [ppm]
+Vrot[kms-1]
+2TACF [days]
+(c1)
+(c2)
+5 (c3)
++
+r = 0.43
+r = 0.05
+Radius [Ro]
+r = 0.12
+4
+3
+2
+101
+102
+50
+100
+150
+200
+250
+300
+350
+10
+20 
+30
+40
+50
+60
+70
+Spike amplitude [ppm]
+Spike amplitude [ppm]
+Vrot [kms-1]
+[ppm] 
+2TACF [days]
+r = -0.11
+r = 0.01
+102
+(d2)
+r = 0.01
+102
+d3
+Spike amplitude [
+40 
+D
+101
+101
+(d1)
+2 20
++
++
+50
+100
+150
+200
+250
+300
+350
+50 
+100
+150
+200
+250
+300
+350
+10
+20
+30
+40 
+50
+60 
+70
+Vrot [kms-1]
+Vrot [kms-1]
+2TACF [days]
\ No newline at end of file
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+page_content=' Université Paris Diderot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
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+page_content=' F-91191 Gif-sur-Yvette,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' France  11 Instituto de Astrofísica e Ciências do Espaço,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Universidade do Porto,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' CAUP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Rua das Estrelas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' PT4150-762 Porto,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Portugal  Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' in original form ZZZ  ABSTRACT  The Kepler mission revealed a plethora of stellar variability in the light curves of many stars, some associated with magnetic  activity or stellar oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' In this work, we analyse the periodic signal in 162 intermediate-mass stars, interpreted as Rossby  modes and rotational modulation - the so-called hump & spike feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' We investigate whether the rotational modulation (spike)  is due to stellar spots caused by magnetic fields or due to Overstable Convective (OsC) modes resonantly exciting g modes,  with frequencies corresponding to the convective core rotation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Assuming that the spikes are created by magnetic spots at  the stellar surface, we recover the amplitudes of the magnetic fields, which are in good agreement with theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Our data show a clear anti-correlation between the spike amplitudes and stellar mass and possibly a correlation with stellar  age, consistent with the dynamo-generated magnetic fields theory in (sub)-surface convective layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Investigating the harmonic  behaviour, we find that for 125 stars neither of the two possible explanations can be excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' While our results suggest that  the dynamo-generated magnetic field scenario is more likely to explain the spike feature, we assess further work is needed to  distinguish between the two scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' One method for ruling out one of the two explanations is to directly observe magnetic  fields in hump & spike stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Another would be to impose additional constraints through detailed modelling of our stars, regarding  the rotation requirement in the OsC mode scenario or the presence of a convective-core (stellar age).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Key words: stars: early-type – stars: rotation – stars: magnetic fields – stars: oscillations  1 INTRODUCTION  Around 10 per cent of A-type stars have been discovered to be chem-  ically and magnetically peculiar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' It is widely accepted that the mag-  netic fields in Ap stars are of fossil origin, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' they were formed  in the pre-main sequence stage of stellar evolution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=', Cowling  1945;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Braithwaite & Spruit 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Braithwaite & Cantiello 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  A fossil magnetic field means that at the current evolutionary stage,  the star cannot maintain the magnetic field against spontaneous de-  cay (Borra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The predicted strengths of fossil-magnetic  fields, obtained by Braithwaite & Spruit (2004) through numerical  simulation, were of the order of 10 kG, agreeing with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Magnetic fields in chemically peculiar stars inhibit the motions of  ions (which have an electric charge) and force particular elements to  ★ E-mail: andreea@space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='dtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='dk  † E-mail: antoci@space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='dtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='dk  be concentrated in spots, as many observations confirm (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=', Gray  & Corbally 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' As most Ap stars are slow rotators and have  an overabundance of certain elements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=', Kochukhov & Bagnulo  2006), their spectral lines are very sharp, which allows one to directly  detect the magnetic fields through magnetic splitting of the lines, or  by spectro-polarimetric observations measuring the Zeeman effect  (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=', Donati & Landstreet 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' However, our understanding  of how magnetic fields are generated and operate in Ap stars cannot  be extended to other types of stars that exhibit magnetic activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Magnetic fields have been investigated and measured in other A  stars, however most of them have weak magnetic field strengths (see  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Aurière et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2007, Lignières et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2009, Petit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2011, Blazère  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2016a,b, Neiner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2017, Seach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Interestingly,  Aurière et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' (2007) found a lack of stellar magnetic fields with  strengths between 300 and a few Gauss, which is now commonly  termed as the ‘magnetic desert’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The strong magnetic fields of Ap  © 20xx The Authors  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='04974v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='SR]  12 Jan 2023  2  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Henriksen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  stars highlights their distinctive character with respect to non-peculiar  A stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='1 Origin of magnetic fields and magnetic activity in  non-peculiar intermediate-mass stars  Our initial understanding of stellar magnetic fields is based on studies  of the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The kinetic energy generated by the motions in the con-  vective layers, coupled with differential solar rotation, is converted  by the solar dynamo to magnetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Dynamo generated mag-  netic fields in the Sun can then be observed at the photosphere (see,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=', the review by Berdyugina 2005 and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  The magnetic fields at the surface of the Sun can partially in-  hibit the convective energy transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Such regions appear as spots,  visible because they are cooler and therefore darker than the bright  photosphere (Strassmeier 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' It is known from both theory and  helioseismic observations that the convective zone (CZ) of the Sun is  approximately 29 per cent of its outer radius (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Turck-Chièze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  (1993);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Christensen-Dalsgaard (2002)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The solar CZ is efficient and  deep enough to sustain a dynamo that can generate a magnetic field,  penetrating the solar photosphere and creating sunspots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Given the  proximity of the Sun, which allows us to resolve its disk, sunspots  have been visually observed on its surface for centuries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' In order to  understand stellar and solar dynamo, however, one has to use mag-  netic activity observations of other stars, which expands the range  of stellar parameters on which the dynamo theories can be tested  (Berdyugina 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Brandenburg & Subramanian 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Magnetic  activity in late-type stars, which possess deep convective envelopes,  has been observed abundantly (see Berdyugina 2005, chapter 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  In the case of more massive early-type stars, detecting and under-  standing the origin of their magnetic fields present additional chal-  lenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Nevertheless, one can infer information regarding magnetic  activity by looking for rotational modulation in the photometric light  curves of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The brightness variations can be attributed to tem-  perature variations caused by stellar spots co-rotating with the stellar  surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' While the more massive HgMn stars show spots but no sur-  face magnetic fields have been recorded yet (Kochukhov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2013),  stellar spots are generally accepted to be a consequence of magnetic  activity also in intermediate-mass stars (Berdyugina 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  As the stellar structure changes with increasing stellar mass, one  cannot assume that the mechanism that generates magnetic fields in  higher mass stars is the same as in the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' This also reinforces the  necessity of studying magnetic fields in more massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  A debatable aspect regarding the presence of magnetic fields  in non-chemically peculiar main-sequence higher-mass stars (≥  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3 M⊙) comes from the fact that, as opposed to our Sun’s radia-  tive core and deep convective envelope, the core of more massive  stars is convective and surrounded by a deep radiative envelope 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' A  more detailed analysis of the outermost layers of intermediate-mass  stars reveals, however, that due to high opacity and/or low adiabatic  gradient (conditions fulfilled in the ionization zones of H, He and/or  Fe), there are thin convective layers at or below the stellar surface  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Cantiello & Braithwaite 2019, hereafter CB19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Cantiello & Braithwaite (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Cantiello & Braithwaite (2019)  proposed that despite their small aspect ratio, these convective re-  gions could sustain a dynamo and generate a magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Since  early-type stars tend to be rapidly-rotating, magnetic structures could  also grow to scales larger than the scale of convective motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  1 Depending on the mass, stars with masses lower than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3 M⊙, could develop  a convective core during later stages of the main-sequence lifetime  The rapid rotation could distinguish stars with this type of dynamo-  generated magnetic field from those in the slowly rotating Ap stars,  which, as mentioned above, are believed to have magnetic fields  generated prior to the main-sequence stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' CB19 also suggest that  dynamo-generated fields have rapidly evolving features of small-  scale and that the strength of surface magnetic fields is expected to  increase as the stars evolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' While it is difficult to predict the ex-  act size of the magnetic features, it is expected that their sizes are  correlated to the pressure scale height at the bottom of the shallow  convective envelope, which is expected to be less than 1 per cent of  the stellar radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' CB19 predicted surface magnetic fields strengths  to reach 1kG for the lower mass stars and ≲ 1 G for stars with 𝑇eff  >10,000 K, noting that these could be just lower limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Both theory and observations suggest that the transition be-  tween stars with convective surfaces to stars with radiative sur-  faces is at around 10,000 K (CB19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Cool intermediate-mass stars  (𝑇eff ≲104 K) likely have a very thin convective surface layer due to  the ionization of H and He I (see Figure 2 in CB19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The transported  flux in these convective layers, which separate the radiative envelope  from the surface, is negligible because they contain very little mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  According to CB19, magnetic fields can reach the stellar surface  through magnetic buoyancy, locally changing the gas pressure and  lowering the optical depth of the photosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Consequently, spots  will appear brighter than the surrounding photosphere because the  flux originates from deeper radiative areas in the star, where the  temperature is higher than at the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  In contrast to the dynamo-generated magnetic fields that CB19  described, Braithwaite & Cantiello (2013) proposed the failed-fossil  scenario for the origin of ultra-weak magnetic fields in A stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The  failed-fossil magnetic fields are expected to have large-scale, slowly  evolving features that would decrease as the stars evolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The pre-  dictions in Braithwaite & Cantiello (2013) suggest that: (1) the time  scale of the failed-fossil field’s evolution is around the stellar age,  (2) the length scale of surface magnetic structures should be at least  20 per cent of the radius of the star, (3) the higher stellar rotation,  the stronger the stellar magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' With respect to the large-  scale and slowly evolving features, the characteristics of failed-fossil  magnetic fields are opposite to those of dynamo-generated magnetic  fields predicted by CB19, but for both scenarios the rotation should  play an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' One could test both scenarios by performing  an ensemble analysis on a homogeneous sample of stars that show  magnetic activity signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  For this study, we use the, so-called, hump & spike stars, which  Saio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' (2018) suggested could show signs of magnetic activity in  the form of spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The common characteristic of these stars is a feature  in the Fourier spectrum (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 1 for some examples), interpreted  to be unresolved Rossby modes (the hump) mechanically excited by  deviated flows caused by stellar spots (the spike) at intermediate to  high latitudes (Saio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Rossby modes were first reported in  intermediate-mass stars by Van Reeth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' (2016), as they studied  the core rotation of a sample of 𝛾 Dor stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' They have been reported  in other works such as: Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2019, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2 Alternative explanation for rotational modulation  Recently, Lee & Saio (2020) and Lee (2021) proposed a different ex-  planation for rotational modulation of light curves of early-type stars,  which does not involve the presence of stellar spots generated by mag-  netic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Overstable convective (OsC) modes in rapidly rotating  early-type stars could resonantly excite low-frequency g modes in the  envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' If the stellar core rotates slightly faster than the envelope  and the amplitudes of the g modes are significant at the photosphere,  MNRAS 000, 1–17 (20xx)  Rotational modulation in A and F stars  3  they would cause brightness variability and therefore be observed as  rotational modulations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' the spike).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  In addition, Lee (2021) suggested that OsC modes resonating with  g modes in the envelope can play an essential role in carrying angular  momentum from the core to the envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' A small amount of radial  differential rotation is a critical ingredient in the Lee & Saio (2020)  and Lee (2021) hypothesis, as OsC modes would not excite g modes in  uniformly rotating stars (Lee & Saio 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Previous studies in which  core and envelope rotation rates were measured through photometric  data found that up to about 10 per cent differential rotation between  convective core and envelope is not rare in 𝛾 Doradus stars (Saio  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  In the context of our hump & spike sample of stars, we investi-  gate the possibility that the spike could in some cases correspond to  g modes confined near the core and resonantly excited by the con-  vective core motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' In this case, the light curves of our hump &  spike stars would be modulated by the convective core rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  In this article, we address both scenarios and discuss the corre-  sponding implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' More precisely, we investigate whether the  spike feature could be either: (1) caused by rotational modulation  due to stellar spots caused by magnetic fields or (2) evidence that  OsC modes couple with g modes in the envelope, causing the light  curve to be modulated with the convective core rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Under the assumption that spots induced by magnetic fields cause  the spike modulation, the first goal of the present work is to investigate  whether the stars in our sample could sustain a magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' We  will test two scenarios: failed-fossil field (Braithwaite & Cantiello  2013) and dynamo-generated field (CB19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  The second goal of this study is to test whether the spike is evi-  dence of convective core rotation, using predictions from Lee (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  If the frequency of the spike is proven to correspond to the frequency  of a g mode that is resonantly excited by OsC modes, then our re-  sults would help better understand the convective core rotation in  intermediate-mass stars as well as the internal mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' We describe  observations, data reduction and analysis in Section 2 and discuss  our results in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='The implications of our results under both  the magnetic field and the OsC modes hypotheses are presented in  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Our conclusions and future work are presented in Sec-  tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  2 OBSERVATIONS AND DATA ANALYSIS  In this work, we use the Kepler long cadence data (Koch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2010),  which are excellent for our study given the long time coverage, and  the fact that the signals of interest are found at low frequencies  (< 20 d−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The light curves from each quarter were downloaded  from KASOC (Kepler Asteroseismic Science Operations Center)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  For our analysis, we used the PDC (Pre-search Data Conditioning)  data, as they were free of specific systematic errors, removed by the  Kepler Science Operations Center Pipeline (Twicken et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  We visually inspected each PDC light curve and compared it to the  corresponding SAP light curve (Simple Aperture Photometry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' We  concluded that the hump & spike feature remained unaltered after the  time series were processed by the PDC module of the Kepler data  analysis pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The data from KASOC is barycentrically corrected,  and the light curve files have the Barycentric Reduced Julian Date  (BRJD, BJD-2,400,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0) as time units (Van Cleve et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2016) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  1 kasoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='dk  2 archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='edu/kepler/manuals/Data_Characteristics_  Handbook_20110201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='pdf  The flux was converted from units of photo-electrons per second  (𝑒−s−1) to ppm, after which the quarterly data were stitched into a  single time series for each star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' We computed a Fourier spectrum for  each stitched time series using the lightkurve package (Lightkurve  Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='1 Sample selection  The target sample initially comprised 94 stars that were visually  inspected and found to exhibit the hump & spike feature or were  reported in Balona (2013, 2014, 2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Balona et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The  temperature range on which we concentrated to find targets was  6500 < 𝑇eff < 10000 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' An additional set of 115 stars (7500 <  𝑇eff < 10000 K) from Trust et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' (2020) was included in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Another four stars were added to our sample from Mathur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  2019 and by visually inspecting the data products available from  Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' (2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' A total of 213 stars that exhibited the hump &  spike feature were analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' However, in this article, we concentrate  on 162 stars that did not show signatures of binarity (either from  photometry or literature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' This selection was performed in order to  study a homogeneous sample of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' We note that some stars in  our final sample exhibit a hump after the spike (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=', KIC 4488313 in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' This feature is similar to one identified by Saio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2018,  who suggested that it is probably the result of prograde g modes (see  figure 8 in Saio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' This feature will be addressed in a future  work that will concentrate more on Rossby modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2, we show an HR diagram illustrating our final sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Stellar evolutionary models for the mass range 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='4 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='8M⊙, com-  puted with Warszaw-New Jersey models (Van Hoolst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Pamyatnykh 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Pamyatnykh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 1998) are shown in the back-  ground for guidance only, computed with solar metallicity (Asplund  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2004) and with no rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The full sample of stars can be seen  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' A1 in Appendix A, with the excluded stars marked in orange  squares and purple triangles while the targets used in the current  work are marked in green circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  The feature of interest is, as mentioned, the hump & spike with  a few examples shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 1, and a more detailed view in the  left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The following subsection describes how the  parameters of the spikes were extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2 Spike parameter extraction  For most stars in our sample, the PDC light curves did not need ad-  ditional de-trending or corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Only ∼10 per cent of PDC light  curves still contained some non-astrophysical signals, to which some  minor corrections were applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' A clear example of a light curve that  needed such corrections can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 4, where only the jump  in flux at around BRJD 56 000) was corrected by fitting three low-  order polynomials to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The corrections were done keeping  in mind that over-fitting might introduce additional and unnecessary  signals in the data, so the order of the polynomials was kept as low  as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' In this example, three second-degree polynomials suf-  ficed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' As seen in the lower panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 4, the Fourier spectra of both  polynomial-corrected data and PDC data are over-plotted, respec-  tively, the astrophysical signal (hump & spike feature) is unaltered  after the correction, but the noise at lower frequencies is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  The applied corrections eliminated only the non-astrophysical sig-  nals, evidently showing that the peaks at low frequencies are indeed  associated with the jump in flux that occurs in the time series at  BRJD ∼56 000 in the upper panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' We computed also a  Fourier spectrum without the data that contain the jump in flux (in  MNRAS 000, 1–17 (20xx)  4  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Henriksen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' A selection of hump & spike features, illustrating their common  aspects and amplitude and frequency diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Stars from our sample placed in an HR diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The source for  luminosity and 𝑇eff values is described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The colour code repre-  sents the source from which the target originates, as described by the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  More information in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Warszaw-New Jersey evolutionary tracks  (𝑍 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='012, Asplund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2004) are displayed in the background for guid-  ance only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  the time range 55904 and 56015), coloured in grey in the lower panel  of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The small differences in the respective Fourier spectra in-  dicate the non-astrophysical origin of the jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' We conclude that  the correction applied did not introduce additional non-astrophysical  signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Moreover, these minor corrections helped obtain a higher  signal-to-noise ratio (SNR) for the studied spikes and/or humps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  For each star in our sample, a simple 1D Gaussian model was fitted  to the spike (and its harmonics) using astropy (Astropy Collaboration  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2013, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Spike identification was done semi-automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  We divided the Fourier spectrum into equal sections and identified  the peaks with scipy (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Only peaks with a SNR  ≥ 4 were accepted as significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' After identifying a significant peak,  we defined a region in the Fourier spectrum (blue shaded region in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 5, spike selection of KIC 8385850) and fitted a simple Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Manual spike identification was necessary due to the variety in the  hump & spike profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 1, the topology of the hump &  spike feature changes from star to star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' In some cases, the spike was  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Example of a Gaussian fit on a spike feature, KIC 4921184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Upper panel: PDC Light curves of KIC 4921184: before the cor-  rection was applied (purple), after non-astrophysical signal was removed  (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' See Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Lower panel: Fourier spectra of  corrected (orange) and uncorrected (purple) light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The grey Fourier  spectrum was computed with the full data set excluding the region inside the  two grey vertical lines from upper panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' There are small differences between  the Fourier spectrum of the corrected data and the one computed without the  data between BRJD 55904 and 56015 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  not well separated from the hump or was a broad feature rather than  a sharp narrow peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  The noise level for the SNR calculations was estimated to be  the median of the amplitude values between a range selected at  frequencies higher than ∼15-20 d−1, where no astrophysical signal  could be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The frequency range for SNR calculations was  selected for each star with the same tool as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Usually  the SNR is determined around the extracted peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' However given the  proximity of the hump (both at lower frequencies and in some case at  higher frequencies), it was not possible to determine the noise level  around the spike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' We therefore chose to use the noise level at higher  frequencies which we expect to mirror the white noise in the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Similar to Trust et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' (2020), we define the value for the spike  amplitude to be the highest amplitude value in the selected spike  region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' This is a robust way to determine the photometric variability  in the 4-year Kepler data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The uncertainties in spike amplitudes were  MNRAS 000, 1–17 (20xx)  10  KIC2157489  KIC3240406  40  5  20  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
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+page_content='35  Frequency [d-1]  Frequency[d-1]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='8 Mo  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='5 Mo  102  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2 Mo  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
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+page_content='7 Mo  Initial sample  Trust et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2020  Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2021  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='4 Mg  14000  12000  10000  8000  6000  4000  Teff [K]Gaussian model  50  50  Fourier spectrum  Fitted Gaussian center  40  40  wddi  Fourier Gaussian center = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3904 d-1  Spike/Gaussian amplitude = 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='6 ppm  Gaussian FWHM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0025 d-1  30  30  20  20  10  10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='380  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='385  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='390  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='395  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='400  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='405  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='410  Frequency [d-1]  Frequency [d-1]Excluded data boundaries  3000  (for verification only)  Corrected data  2000  Uncorrected data  [udd]  1000  0  xn  1000  2000  3000  4000  55000  55200  55400  55600  55800  56000  56200  56400  BRJD[days]Uncorrected data  70  Corrected data  60  Excluding data from [55,904-56,015] days  [wdd]  50  Amplitude  40  30  20  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='6  Frequency [d-1]Rotational modulation in A and F stars  5  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Illustration of how a frequency region was selected for the Gaussian  fitting - KIC 1873552.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  calculated with equation 1, which was adapted from Kjeldsen &  Bedding 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  𝜎A =  √𝜋 𝑆𝑁𝑅  2  (1)  where SNR is the noise level in the Fourier spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  We note that the spike amplitudes could be underestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' If  the spike signal is due to stellar spots, the amplitude would depend  on the angle at which the spot is observed, which in turn depends  on the stellar inclination and the latitude of the spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Determining  the intrinsic value of the spike amplitude requires knowledge of the  stellar inclination, which is out of the scope of the current work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Almost all stars (121) have four years of Kepler data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' However a  few stars have missing quarters, as indicated in Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' In the latter  case, gaps in the data will introduce alias peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' This will increase  the uncertainty of 𝑓𝑟𝑜𝑡, as it would not be possible to discern between  the correct and alias peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Furthermore if the length of the data set  is shorter, the precision of 𝑓𝑟𝑜𝑡 will again decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' For example,  KIC 5456027 is the only star that has 12 missing quarters, as seen in  Table A1 ( 𝑓𝑟𝑜𝑡 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='242 d−1, 𝜎 𝑓𝑟𝑜𝑡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0039 d−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The uncertainty  on 𝑓𝑟𝑜𝑡 is on average, an order of magnitude larger than for other stars,  meaning that the uncertainty in determining the rotation frequency  increases with the existence of gaps in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  The spike frequency and amplitude and the associated uncertainties  for each star can be found in Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Figure 6 shows a weak anti-  correlation between the frequency and amplitude of the spike, sug-  gesting that rapidly rotating stars tend to have slightly lower spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  The right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 3 shows a fitting example for KIC 4921184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Values for the rotation frequency were compared to those in litera-  ture, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' For details regarding the few differences, see  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The upper panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 8 depicts a distribution of the  rotation frequency values, which lie between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='28 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='75 d−1, cor-  responding to rotation periods between 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='6 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='4 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' This highlights  the short rotation periods of our stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' A2, panel (d1) depicts the relation between the rotational  velocity and the spike amplitude for each target in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The  rotational velocity was calculated as:  𝑉rot (km s−1) = 2 𝜋 𝑅𝑠 𝐹rot  86400  (2)  where 𝑅𝑠 is the stellar radius in km and 𝐹rot is the rotational fre-  quency in d−1, corresponding to the frequency of the main spike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' For  93 targets from our sample, the radius values and the associated un-  certainties were extracted from Gaia DR2 (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' For the remaining 69 stars, the radius values were taken from  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Spike amplitude as a function of spike frequency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' if not visible the  uncertainties in both quantities are smaller than the symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Comparison between rotation frequency from this work versus  literature values: Balona (2013, 2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Balona et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Balona (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Trust et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' (2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The uncertainty in stellar radius has the highest  contribution to the uncertainty in the rotational velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The val-  ues obtained for the rotational velocities are depicted in a histogram  in the lower panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 8 and are also listed in Table A1 where  the radius values and the associated uncertainties can also be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Chowdhury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' (2018) reported rotation periods of 513 stars, out  of which 394 are within the same 𝑇eff range as our targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Their  reported rotational frequencies are similar to our sample, suggesting  that the hump & spike stars do not rotate differently from other A and  F stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 9 depicts the rotational velocities with respect to their  location in the HR diagram, confirming that more massive, hotter  stars rotate faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' We briefly discuss the importance of rapid rotation  of some of our stars in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2, as it is an important ingredient  in the OsC theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  MNRAS 000, 1–17 (20xx)  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='5  wdd]  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  Amplitude  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='30  Frequency[d-1]r = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2  102  101  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  Spike frequency [d-1]Balona2013  Balona2014  Trust+2020  Santos+2021  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='5  Balona2017  Balona+2015  L : 2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='5  2:1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  5:1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='5  Rotation frequency (from this work) [d-1j6  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Henriksen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Upper panel: Histogram of the rotation frequencies extracted for  our sample of stars (spike frequency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Lower panel: Histogram of rotational  velocities, calculated with equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' HR diagram with the symbol colour correlated to the rotational  velocity, calculated using the spike frequency and equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The source for  luminosity and 𝑇eff values is described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Warszaw-New Jersey  evolutionary tracks (𝑍 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='012, Asplund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2004) are displayed in the  background for guidance only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3 Luminosity and 𝑇eff values  Luminosity values for 156 targets from our sample were extracted  from Murphy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' (2019) (supplementary data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' For the remaining  six stars not found in Murphy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' (2019), we assigned luminosity  values from Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' (2020) (four stars) and from Gaia DR2 (two  stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Last column in Table A1, denotes the source of the stellar  parameters as explained in the caption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  For obtaining the luminosity values, Murphy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' (2019) used  the Gaia DR2 parallaxes and uncertainties, 𝑔 apparent magnitudes  (derived from 𝑔KIC and 𝑟KIC magnitudes and calibrated to the SDSS  scale), extinctions and their uncertainties from the dustmap python  package (Bayestar 17 reddening map) and bolometric corrections  with isoclassify python package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' For a more in-depth explanation of  how the values were obtained, see Murphy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' (2019) and references  therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  The luminosity values for the four stars, extracted from the Berger  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' (2020) catalogue, were derived from isochrones and broadband  photometry, Gaia DR2 parallaxes, and spectroscopic metallicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  See Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' (2020) for a more in-depth understanding of how  these values were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The effective temperatures (𝑇eff) for all  162 stars were extracted from Gaia DR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' HR diagram with the symbol colour indicating the highest har-  monic detected in the Fourier spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The number in the paranthesis next  to each item from the legend indicates the number of stars in the sample  that have the respective highest harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The source for luminosity and 𝑇eff  values is described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Warszaw-New Jersey evolutionary tracks  (𝑍 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='012, Asplund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2004) are displayed in the background for guidance  only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='4 Harmonics of the main spike  The majority of the stars in our sample exhibit one to several har-  monics of the spike frequency, which can be interpreted as a sign  of a non-sinusoidal signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Only 14 stars do not have any detected  harmonics of the main spike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The colour code in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 10 is dictated by  the highest harmonic found in the Fourier spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' There seems to  be no apparent correlation between the presence or lack of harmonics  and the stellar parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' In the legend of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 10, the number of  stars that possess specific harmonics can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' We focus more on  the significance and importance of harmonics for our stars in Section  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='5 Time-series analysis  In order to verify the temporal behaviour of the periodic signals, we  performed a time series analysis on the data, focusing on temporal  changes of the spike, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 11, where Fourier spectra of  subsets of the KIC 4921184 Kepler data are displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The average  duration of each sub-data sets is 800 d and the average overlap 791 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Each Fourier spectrum was computed with the lightkurve package  (Lightkurve Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The colour coding in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 11  represents the time when the photometric data from the data subsets  were acquired, the yellow Fourier spectrum being computed with the  most recent Kepler data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The Gaussian centre obtained from fitting  the spike in the Fourier spectrum of the full data set (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 3) is  depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 11 as a vertical black dotted line at the corresponding  frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 11 clearly shows the amplitude and structure of the  spike change in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The values for computing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 11 (800 d sub-  data sets, overlapping 791 d) were chosen as a trade-off between  resolution, SNR and having enough subsets to illustrate the changing  character of the spike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' However, we quantify this temporal change  using other methods as described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  The Fourier spectrum computed with the full Kepler data set  of KIC 4921184 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The harmonics of the spike  (f1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='39 d−1) are also visible (2f1, 3f1, 4f1, 5f1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='78, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='17,  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='56, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='95 d−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The presence of harmonics is indicative of non-  sinusoidal brightness variations, typical for stellar spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Further-  more, the changing shape of the main spike, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 11, and  the double-wave light variation in the middle panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 13 suggest  the presence of variable stellar spots (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=', Chowdhury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  MNRAS 000, 1–17 (20xx)  300  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='8Mo  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='5Mo  250  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2 Mo  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='9Mo  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='6Mo  200  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3Md  150  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0M  101  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='7 M  100  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content="4'M0  50  14000  12000  10000  8000  6000  4000  Teff [K]3." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='8 Mo  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='5 Mo  102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2 Mo  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='9 Mo  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='6 Mo  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3 Mo  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0 M  101  no harmonics (14)  6th (7)  12th (3)  2nd (59)  7th (2)  15th (1)  7mo  3rd (28)  8th (1)  16th (1)  4th (24)  9th (1)  17th (1)  5th (18)  10th (2)  14000  12000  10000  8000  6000  4000  Teff [K]30  #  Count  20  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='5  Frequency of the spike [d-1]  30  #  Count  20  10  0  50  100  150  200  250  300  350  Rotational velocity [km s-1]Rotational modulation in A and F stars  7  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' KIC 4921184 Fourier spectra computed with subsets of the full  Kepler data colour coded with respect to time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The yellow Fourier spectrum is  computed with the most recent Kepler data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Black vertical line at ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='39d−1  denotes the frequency of the spike, obtained by fitting a Gaussian to the full  4-year Kepler data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The amplitude of the spike in the full data set is 52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='5  ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  In general, a double-wave variation indicates the presence of spots  that are in anti-phase, causing a stronger 2nd harmonic (Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' However, for KIC 4921184, the amplitude of the 2nd harmonic  is just as high or slightly lower than that of the main spike in some  of the Fourier spectra computed with subsets of the full Kepler data,  suggesting the presence of non-permanent spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='6 Spike lifetime  Given the changing nature of the spike (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 11), we searched for  a way to quantify its evolution in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' It has been suggested that  the decay time-scales of the autocorrelation functions (ACFs) of  light curves are related to the lifetimes of stellar spots (Giles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Trust et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Given that one of the  goals of our work was to probe whether the spike is evidence for  magnetic fields, we decided to compute the ACFs based on which we  could extract the spike lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' We only concentrated on the signal  induced by the spike and its harmonics, requiring a band-pass filtered  time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' We calculated the sum of all Fourier components in  the frequency bands around the spike and the harmonic frequencies,  𝑛 𝑓rot±Δ 𝑓 , where Δ 𝑓 is the frequency range around 𝑛 times the spike  frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Δ 𝑓 depended on the broadness of the spike/harmonic,  which varied from case to case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 𝑛 takes values between 1 and the order  of the highest detected harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' For example, for KIC 4921184  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 12) 𝑛 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The Fourier components were determined using an  iterative sine-wave fitting process, which was stopped once all the  signal within the 𝑛 𝑓rot ± Δ 𝑓 frequency range was recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 13 shows an example of the band-pass filtered time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  In the upper panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 13 the contribution of the spike and its  harmonics to the brightness variability is illustrated over the com-  plete Kepler data, while the mid-panel shows a zoom-in on the non-  sinusoidal signal induced by the spike and its harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The lower  panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 13 shows the difference in Fourier spectra computed  with the full and band-passed filtered time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The insert in the  lower panel depicts a zoom in on the first spike and its hump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  After computing the ACF of the band-passed filtered time series  (orange curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 14, obtained with numpy package Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  2020), the result was fitted with an exponential model (green line),  described by equation 3, using least-squares minimization (Virtanen  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' KIC 4921184 Fourier spectrum with the main spike (f1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='39  d−1) and its harmonics (2f1, 3f1, 4f1, 5f1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='78, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='17, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='56, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='95 d−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Upper panel: Band-pass filtered time series and the original Kepler  data for KIC 4921184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Middle panel: Zoom in of Figure a), the shape of the  non-sinusoidal signal induced by the spike and its harmonics are depicted  in orange, while the original data are the purple dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Lower panel: Fourier  spectrum of both the original data set (purple) and the band passed filtered  time series (orange), where only the spike and harmonic signal is contained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  The insert shows a zoom in on the first spike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  MNRAS 000, 1–17 (20xx)  80  Gaussian center  from full data set fit  Amplitude [ppm]  Time →  60  40  20  0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
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+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='Amplitude  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='30  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='35  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='40  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='45  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='50  10  Frequency [d-1]  0  0  2  4  6  8  10  Frequency [d-1]8  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Henriksen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Autocorrelation function of the band-passed time series  KIC 4921184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The orange line is the ACF of band-passed filtered time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  The green line is a fit obtained with equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  𝑦(𝑡) =  �  𝑒  −𝑡  𝜏ACF  � �  𝑦0 + 𝐴𝑐𝑜𝑠  � 2𝜋𝑡  𝑃𝐴𝐶𝐹  �  + 𝐵𝑐𝑜𝑠  � 4𝜋𝑡  𝑃𝐴𝐶𝐹  ��  (3)  In equation 3, introduced by (Giles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2017) to approximate the  starspot lifetime of Kepler stars, t represents the correlation time  lags in days obtained as 𝑡 = 𝑁 Δ𝑇, with Δ𝑇 being the median time  difference between consecutive observations in the light curve and  N is the number of observed data points in a light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 𝜏ACF is the  decay time-scale of the magnetic features (stellar spots) and 𝑃ACF is  the rotation period of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 𝐴, 𝐵 and 𝑦0 are not associated with  physical stellar properties, but constants used in the fitting algorithm  of above mentioned equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' All parameters in equation 3 were left  free in the fitting process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 15 depicts the spike lifetimes distribu-  tion, obtained using equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The values obtained are similar to  what Trust et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' (2020) reported for a sample of stars with common  targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  The accuracy of the method was tested on synthetic time series  with known spike lifetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' We simulated 400 time series with spike  lifetimes of 15, 30, 60 and 120 d (100 time series for each value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  The total length of each time series is 1500 days, with a 30 minutes  cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The synthetic time series did not contain noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The period  of the signal was one day for all time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Amplitude variability  (excitation and damping) was calculated in a similar way as simulated  for solar-like p-mode oscillations (see De Ridder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The  average uncertainty was ∼27 per cent for time series with a spike  lifetime of 15, 30, 60 d, and ∼42 per cent for 120 d, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' In  terms of accuracy, the method yielded values ∼1 − 2 per cent away  from the injected ones for 15, 30, and 60 d, and ∼21 per cent for  120 d, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2021b suggested that the exponential model underes-  timates decay time-scales in stellar spot simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' They introduced  an alternative to the model proposed by Giles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' We find,  however, that the ACFs in this work are better fitted with the Giles  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2017 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Nevertheless, in order not to underestimates the  spike lifetimes, we multiply the values by a factor of two as suggested  by Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2021b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' This factor was derived from simulations of  solar spots, including various number of spots, inclination angles  and rotation periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The spike lifetimes were obtained for all targets  in our sample, and the results are in the HR diagram from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 16,  where these dictate the colour code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The spike lifetime values are  also listed in Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Spike lifetime distribution (2 𝜏ACF) obtained with equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' HR diagram with symbols correlated with the spike lifetime as  calculated in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The source for luminosity and𝑇eff values is described  in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Warszaw-New Jersey evolutionary tracks (𝑍 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='012, Asplund  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2004) are displayed in the background for guidance only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' HR diagram with the colour code illustrating the amplitude of the  spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The source for luminosity and 𝑇eff values is described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Warszaw-New Jersey evolutionary tracks (𝑍 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='012, Asplund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2004)  are displayed in the background for guidance only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  3 RESULTS  We analysed the time series of 162 hump & spike stars observed  with the Kepler telescope and extracted the spike frequencies corre-  sponding to the rotational frequencies, amplitude and corresponding  uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' We calculated the spike lifetimes, used stellar radius  to determine the rotational velocity and used luminosity values from  Murphy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' (2019), Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' (2020) and Gaia DR2 and 𝑇eff  values from Gaia DR2 catalogue to place our targets in HR diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 7 shows the difference between the obtained rotational fre-  quency from this work and values from other sources mentioned in  the caption and legend;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' most of our values in the literature agree with  MNRAS 000, 1–17 (20xx)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='4  C  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2  A  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2  Fit: 2 tAcF = 24 +/- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='1 days  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='4  Observed  5  15  20  25  10  30  35  0  Correlation lags [days]35  30  25  #  15  10  5  0  20  30  40  50  60  70  Spike lifetime[days]70  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='8Mo  60  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='5Mo  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2 Mo  50  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='9Mo  [days]  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='6Mo  40  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3M d  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0M  30  101  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='7 M  20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content="4'M0  14000  12000  10000  8000  6000  4000  Teff [K]2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='8Mo  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='5Mo  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='8  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2 Mo  amplitude)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='9Mo  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='6Mo  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3M d  (spike  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  101  0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='01  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='7 M  log  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='4Mo  14000  12000  10000  8000  6000  4000  Teff [K]Rotational modulation in A and F stars  9  ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' We checked each star that does not have similar rotation fre-  quency values to the ones previously reported and concluded that the  values in this work are correctly identified as the rotation frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  The few differences come from misidentifying the main spike with  one of its harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' For example, KIC 3868032 was reported by  Balona (2013, 2014, 2017) to have a rotation frequency of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='4 cd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Trust et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' (2020) reported a frequency of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='67 cd−1 which is in  agreement with our value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' A more thorough investigation, revealed  that Kahraman Aliçavuş et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' (2020) reported a v sin i value of 181±8  km s−1 for this star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' A value for the rotational velocity using the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='4  cd−1 rotational frequency and a radius value of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='4+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='07  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='06𝑅⊙ (Berger  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2020) would yield a rotational velocity of ≈ 49 km s−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Given  that the v sin i value is the lower limit for the rotational velocity, this  value would contradict the result of Kahraman Aliçavuş et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Assuming the rotation frequency of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='67 cd−1, the rotational velocity  would be 206+6  −5 km s−1, which is consistent with the measured v sin  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' A star can be represented more than once in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 7, as stars have  been reported more than once in previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  We explore different parameters for any significant trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' A2  shows the correlations between stellar parameters and various pa-  rameters extracted in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' In A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='a2, a moderate correlation  between the rotational velocity and effective temperature is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  This correlation is not surprising as it has been known for decades that  hotter stars rotate faster than their cooler counterparts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Tassoul  1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' A similar correlation can be seen in A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='b2 between luminos-  ity and rotational velocity, which is interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' An explanation is  that the more luminous stars in the sample are the evolved higher-  mass stars, and the rapid rotation they had while closer to ZAMS  still dominates over the spin-down as their radii increase with age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Yet, we must highlight the bias induced by the target selection when  discussing these correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Targets that originate from Trust et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  (2020), are stars initially selected by Balona (2013) to correspond  roughly to spectral type A9-A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The temperature range of interest  was 7500 < 𝑇eff < 10 000 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' This clear 𝑇eff cut off is noticeable also  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2, where stars symbolized by green circles, lie roughly in this  temperature range (differences may occur due to the fact that Balona  (2013) used the 𝑇eff from the KIC catalogue, while the values used  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2 were from Gaia DR2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' A similar trend, but maybe not as  significant, can be observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='c2, where the rotational ve-  locity is compared to the stellar radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Considering all of the above  and that more massive, hotter stars rotate faster (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 9), under the  assumption that magnetic activity gives rise to the spike frequency,  these results could also suggest that more massive stars require a  faster rotation rate to generate observable effects in the form of hump  & spike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  In addition to the initial sample of stars with𝑇eff = [6500, 10000] K,  we also attempted to populate our target list with cooler stars by  visually inspecting selected stars from Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' (2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The  latter study concentrates on G and late F main-sequence and cool  subgiant stars (in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2, stars symbolised with blue circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Only  three stars with 𝑇eff = [6310, 7033] K were found, suggesting that the  hump & spike feature does not occur in cooler stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The reason may  be that lower-mass stars rotate on average slower, and/or the stellar  structure is different, but we stress that a more systematic search  for hump & spike stars is necessary in order to suggest a possible  explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' However, in the OsC modes scenario we do not expect  cooler hump & spike stars as the convective core does not exist at  masses lower than roughly 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Additionally the thick convective  envelope of late type stars would not allow g modes to penetrate all  the way to the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  There seems to be no apparent connection between any parameter  and the spike lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The average spike lifetime value for the whole  sample is around 30 d, with ∼95 per cent of the values being less  than 60 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The order of magnitude of the spike lifetime values are  in agreement with values of starspot lifetime reported by Giles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  (2017) regarding F-type stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Although the F-stars in the Giles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  (2017) sample are not very numerous, they report that the stellar  spots do not survive very long (unlike in the case M, K and G stars)  nor reach substantial sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  The missing correlation between the spike lifetimes and stellar  mass is hard to reconcile in the OsC scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' As the g modes would  be modulated by the convective turnover time scales, we expect the  spike lifetimes to decrease with stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' We address this in more  detail in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  The weak correlations from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='a1, which shows the relation  between the spike amplitude and 𝑇eff and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='b1 that depicts the  relation between the spike amplitude and luminosity (A2), suggest  that cooler and less luminous stars have higher spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The same  trend clearly showing that more massive stars have lower spike am-  plitudes is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' This correlation is further explored  and discussed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2, as it is related to the predictions of  CB19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  4 DISCUSSIONS  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='1 Magnetic field strength  Cantiello & Braithwaite (2019) argue that the thin envelope convec-  tion zones of intermediate-mass stars can sustain dynamo-generated  magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' If the observed photometric variability is due to  periodic temperature variations caused by magnetic spots, one can  estimate the magnetic field strength using the amplitude of the spike  and some assumptions for the spot filling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Following Cantiello  & Braithwaite (2011), we start from the definition of the radiative  gradient:  ∇rad =  � 𝑑𝑙𝑛𝑇  𝑑𝑙𝑛𝑃  �  rad  ≃ 𝑃  𝑇  Δ𝑇  Δ𝑃 ,  (4)  which can be re-written as  Δ𝑇  𝑇  = ∇rad  Δ𝑃  𝑃 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  (5)  We then assume that a magnetic spot is in hydrostatic and thermal  equilibrium with its surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The latter requires the radius of  the spot to be significantly smaller than the stellar radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' We also  assume that the radiative gradient is not affected substantially by the  presence of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Assuming hydrostatic equilibrium  implies that a magnetic spot has a smaller gas pressure compared  to its non-magnetized surroundings, since part of the total pressure  is provided by the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Together with the assumption of  thermal equilibrium, this implies a lower density in the magnetized  region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' A local depression in the surface density allows photons to  emerge from deeper regions of the star, so an observer looking down  into a magnetic spot at the surface of a radiative star is expected to see  hotter temperatures compare to a non-magnetized region (Cantiello  & Braithwaite 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 5 this temperature contrast can be  written as  Δ𝑇  𝑇  = ∇rad  𝑃mag  𝑃tot  = ∇rad  𝛽  (6)  where  𝛽 = 𝑃tot  𝑃mag  =  𝑃tot  𝐵2/8𝜋 ,  MNRAS 000, 1–17 (20xx)  10  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Henriksen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  and 𝑃mag is the component of the pressure due to the presence of the  magnetic fields, 𝑃tot is the total pressure and 𝐵 the magnetic field  strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Assuming that the ratio between the magnetic pressure 𝑃mag  and the total pressure 𝑃tot (from the gas and radiation) is very small,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  𝛽 ≫ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' and using 𝐿 = 4𝜋𝑅2𝜎𝑇4  eff,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' (where R is the stellar radius),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  we can obtain a relation to estimate the local contrast in luminosity  between the spot and the rest of the stellar surface (Cantiello &  Braithwaite 2011):  Δ𝐿  𝐿 ≃ 4Δ𝑇  𝑇  = 4∇rad  𝛽  (7)  This equation provides the luminosity contrast of a magnetic  spot compared to the unperturbed stellar luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' To compare  this to observations, one has to take into account the filling factor  𝑓 = (𝑟/𝑅)2, where 𝑟 is the radius of the spot and R is the stellar  radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Note the degeneracy between temperature (or luminosity)  contrast and filling factor: A large spot with a small luminosity con-  trast can give a similar brightness variation to a smaller spot with a  higher luminosity contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' From a theoretical standpoint, Cantiello  & Braithwaite 2011 and CB19 suggested that magnetic fields in hot  stars could produce features on scales that have comparable sizes with  (or larger than) the pressure scale height of the convective regions  where they are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Lacking firm observational constraints on  the size of the spots that might be generating the brightness fluctua-  tions, we will assume the spot size 𝑟 to be comparable to the pressure  scale height in sub-surface convective layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' As we will discuss be-  low the spots could actually be larger, so this assumption means our  estimates for the magnetic fields represent upper limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Equation 7  becomes:  Δ𝐿  𝐿 = 4∇rad  𝛽  � 𝐻𝑝  𝑅  �2  (8)  where 𝐻𝑝 is the pressure scale height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Using the definition of 𝛽,  equation 8 can be written as:  Δ𝐿  𝐿 = 4∇rad𝐵2  8𝜋𝑃tot  � 𝐻𝑝  𝑅  �2  (9)  The ratio Δ𝐿/𝐿 is then replaced with the observed photometric am-  plitude (𝐴spike) in the Kepler data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Note that our calculation assumes  a single spot and we do not correct for the Kepler bandpass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' This is  not necessarily correct, and the data suggest some stars have more  than one spot, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' the phase plots in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The magnetic  fields amplitude is then recovered using the following relation:  𝐵2 ≃ 𝐴spike  2𝜋𝑃tot  ∇rad  � 𝑅  𝐻𝑝  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  (10)  From a grid of ten stellar models in the mass range 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='25 − 4 M⊙,  obtained using MESA (Paxton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2011, 2013, 2015, 2018, 2019),  we have extracted through linear interpolation values for the total  pressure at the surface 𝑃𝑡𝑜𝑡, radiative gradient ∇𝑟𝑎𝑑 and the pressure  scale height 𝐻𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The MESA models are accessible through github 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  The stellar models have solar metallicity (𝑍 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='02) and a chemical  composition mixture taken from Asplund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The mixing  length parameter was assumed to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The 𝑃𝑡𝑜𝑡 values we obtain  for our sample span from 375 to 17497 Pa, with a median value of  3092 Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The average radiative gradient values for our stars is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='126,  1 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='com/matteocantiello/hump_spikes_stars  Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' HR diagram with colour code illustrating the estimated magnetic  field strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The source for luminosity and 𝑇eff values is described in  section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Warszaw-New Jersey evolutionary tracks (𝑍 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='012, Asplund  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2004) are displayed in the background for guidance only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Estimate values for the magnetic field strength with respect to the  𝑇eff of the stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  spanning from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='125 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' An average value for the pressure scale  height for the stars in our sample was found to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='01𝑅⊙, ranging  from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='005 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='02 𝑅⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  We note that the 𝐻𝑝 values are the lower limits to the sizes of the  magnetic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' CB19 pointed out that magnetic features expand  when rising towards the stellar surface depending on the ratio of the  density in the convective layers and at the photosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Moreover,  intermediate-mass stars tend to be rapidly rotating, which can lead to  dynamo action on scales larger than the local pressure scale-height  (Brandenburg & Subramanian 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Cantiello & Braithwaite 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Therefore the recovered values of 𝐵 are upper limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Under all the assumptions made above we computed upper limits  estimates for the magnetic field strength in our stars sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The  results are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The obtained values are also shown  with respect to 𝑇eff in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 19 and range from 77 to 2207 G, with an  average value of 540 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' In the next section we discuss our findings  in the context of theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2 Comparison with predictions from literature  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 17 clearly shows that the spike amplitude is decreasing with  increasing stellar mass, which in the context of magnetic fields, sug-  gests stronger magnetic fields for lower-mass stars, which also have  deeper (sub)surface convective layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Based on the spike amplitude we have estimated the strengths of  equipartition magnetic fields in the convective envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' We note  again that the intrinsic values of the amplitudes depend on e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=', stellar  MNRAS 000, 1–17 (20xx)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='8Mo  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='5Mo  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2 Mo  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='8℃  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='9Mo  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='6Mo  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3M d  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0M  101  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='7 Mo  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='4M0  14000  12000  10000  8000  6000  4000  Teff [K]2000  1500  B 1000  500  0  10000  9000  8000  7000  6000  Teff [K]Rotational modulation in A and F stars  11  inclination and spot latitudes, which are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Nevertheless,  our results are in agreement with the predicted values of average  equipartition magnetic fields in the sub-surface convective zones (see  figure 6 from CB19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=') By comparing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 18 and figure 8 from CB19,  a similar general trend can be seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Both the predicted magnetic field  strengths in CB19 and the calculated values from this work decrease  with increasing stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Furthermore, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 18 shows that more  evolved stars have stronger calculated magnetic fields, which also  agrees with the predictions from CB19, as the convective envelope  gets deeper as the star evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  The strengths of the magnetic fields at the surface that were com-  puted by CB19 were determined by scaling the equipartition fields  with the density inside the HeII convective zone, ranging from 2000 G  to less than 1 G (see figure 9 from CB19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Their findings showed that  for a star like Vega the expected mean longitudinal field would be  very small (∼ 10−2 G), which is consistent with the observed value  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3 G (Lignières et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Therefore the values obtained  with the density scaling relation CB19 proposed should be taken as  a lower limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' This also brings forth the fact that the magnetic buoy-  ancy might not necessarily be the process that brings the magnetic  field to the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Other processes have been proposed such as the  one proposed by Warnecke & Brandenburg (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Warnecke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  (2011), where the magnetic field would reach the surface through  magnetic tension, in which case the strength at the surface could be  larger, comparable to the equipartition values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  The calculated values for the spike lifetimes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 16) suggest that  if present, the magnetic features are not very long lived, having life-  times of tens of days, with only five stars surpassing 60 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' This is in  agreement with predictions from CB19, regarding the rapidly evolv-  ing features of the dynamo-generated magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' In addition,  the trend that can be observed for the spike amplitude in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 17 is  not present for the spike lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' This suggests that a more complex  process may determine the spike lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  To summarize, the spike amplitude trend with mass and 𝑇eff in the  context of magnetic fields, suggests that the dynamo in the subsurface  convection hypothesis is favoured over the ‘failed-fossil’ scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3 Convective core rotation  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='1 Convective turnover time  As described above, Lee & Saio (2020) and Lee (2021) suggest  that Overstable convective (OsC) modes could, in the presence of  radial differential rotation, couple with low-frequency g modes in  the envelope to emerge on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' A 20% differential rotation is  assumed in Lee & Saio (2020), while Lee (2021) assumes the core  rotation to be 10% higher than the envelope rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  An OsC mode of an azimuthal order 𝑚 would have a frequency 𝑚  times the core rotation frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Here we investigate whether the  rotational modulation characterised by the spike could be due to OsC  modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  If the spikes corresponded to 𝑚 times the convective core rotation  rates, it would mean that the spike lifetimes calculated in subsection  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='6 would be associated with the movement of the convective cells  inside the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' In other words, we would expect the spike lifetimes to  be related to the core’s convective turnover time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' It is well established  that the mass of the convective core increases with stellar mass (see,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=', figure 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='7 in Kippenhahn & Weigert 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Consequently, the  spike lifetimes calculated in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='6 should correlate with the  stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' From MESA models, we have extracted a few values for  the convective turn overtime in the stellar mass regime of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  These values were obtained from models which have solar metallicity  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Extracted values of convective turnover time from MESA models  for a few representative stellar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  ZAMS  TAMS  Mass  𝜏turn  Mass  𝜏turn  [M⊙]  [days]  [M⊙]  [days]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='5  173  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='5  44  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  113  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  38  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='4  83  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='4  35  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  68  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  32  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='6  62  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='6  30  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  59  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  29  𝜏turn = convective turnover time  (𝑍 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='02) and a chemical composition mixture taken from Asplund  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The mixing length parameter was assumed to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='6  (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' As can be seen in Table 1 the convective turnover times  decrease with stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' This negative correlation is expected to  be present throughout the entire main-sequence evolutionary stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  It is important to point out that our results do not show any corre-  lation between the spike lifetime and stellar mass, as seen in the HR  diagram from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 16, suggesting that the broadness of the spike is  not connected to convective turnover time in the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  In the context of OsC modes, a possible explanations for the ob-  served trend in amplitude seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 17, could be the changing  density of g modes with stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' In other words, the amplitude  of an OsC mode at a surface should be larger when the resonance  with a g mode is stronger, which would occur more frequently if the  density of g modes (in the co-rotating) frame is higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' This would  take place in less massive stars and could imply that the amplitudes  of the spikes tend to be lower in more massive stars, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2 Harmonic signature and rotation  As mentioned in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='4, only 14 stars do not exhibit detectable  harmonics of the main spike (see an example in lower right corner of  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 10, the rest show harmonics in their Fourier  spectra, with KIC 7175896 having the highest amount of harmonics  (8), with the highest harmonic being at 17 times the spike frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Lee & Saio 2020 suggested that the expected amplitudes of  g modes resonantly excited by OsC modes would decrease as the  azimuthal order (𝑚) increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The variability induced is expected to  be sinusoidal and would not cause any additional harmonics in the  Fourier spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' In order to quantify the number of stars that would  be consistent with this scenario, we subdivided our sample into four  groups:  Group A: the amplitudes of the spike and its harmonics decrease  as the degree of the harmonic increases (see example in the upper left  corner of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' This group comprises 111 stars and is consistent  with both scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Group B: a higher order harmonic has a higher amplitude than  the previous harmonic or the main spike (upper right corner of  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 20), which can only be explained by non-sinusoidal signals,  such as spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' KIC 4921184 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 12) also falls under this category  comprising 22 stars in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Group C: the detected harmonic orders are not consecutive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  there are missing harmonics, which again is an indication of non-  sinusoidal signals, that could be induced by stellar spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' This group  MNRAS 000, 1–17 (20xx)  12  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Henriksen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  consists of 14 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The lower left corner of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 20 shows an example  where the fourth harmonic is not present while the third and fifth have  similar amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Group D: no harmonics detected, only the main spike is signifi-  cant, not allowing us to draw any conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' We find 14 stars falling  in this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Counting only the harmonic behaviour or the absence of harmonics  (Group A and D), we find 125 stars for which we cannot strictly  determine the origin of the spike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' In other words, both the OsC  modes and the stellar spots caused by dynamo-generated magnetic  fields could explain the spike feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' However, groups B and C  clearly display a a behaviour that is compatible with stellar spots,  making it a total of 36 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  We highlight that the OsC modes scenario, as suggested by Lee &  Saio (2020) and Lee (2021), requires rapid rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  8 and 9 more than half of the stars from our sample have a rotational  velocity higher than 100 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' However, the exact value for the  required minimum rotation speed depends on model parameters such  as the degree of radial differential rotation and the super-adiabatic  temperature gradient in the rotating convective core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  The evolutionary stage of a star, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=', whether it is still on the main  sequence or not, has crucial implications for the OsC scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' This  is because our stars beyond the TAMS do not have a convective core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Based on their location in the HR diagram (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 9), we visually  identified 34 stars that may have left the main-sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' However,  we note the following shortcomings to this identification:  The visual identification was done without taking into account  uncertainties in 𝑇eff and luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  The evolutionary tracks depicted in our HR diagrams do not  take into account rotation and core overshooting and only use so-  lar metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' This means that without detailed stellar modelling,  determining whether a star still has a convective core is imprecise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  The effect of gravity darkening increases with rotation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Fig-  ure 4 from Georgy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' (2014) and figure 38 from Paxton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  (2019) nicely illustrate how the observed luminosity depends on the  inclination angle for stars rotating with more than 50-60 per cent of  their critical velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' For these stars, the luminosity could be under-  estimated if observed pole-on or underestimated at the equator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' For  example, the two fast rotators in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 9, which lie around the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='5 M⊙  evolutionary track, could still be on the main sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3 Phase-folded light curve  Another piece of evidence that might point towards the spike being  associated with the stellar surface rotation rather than the convective  core rotation, is the shape of the phase-folded light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Figures  21 - 26 depict the phase-folded time series of 3 stars from our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  The groups defined in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2 to which these stars belong to  are: KIC 2157489 - B, KIC 4661914 - C, KIC 3440710 - A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' All  the data were folded with the periods derived from the spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The  phase folded light curves were obtained with lightkurve (Lightkurve  Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2018) and afterwards binned in phase with the  values mentioned in the captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Figures 21, 23 and 25 represent the  phase-folded light curves of the full Kepler data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' We note that  if the median amplitude value is taken in a given bin, the effect of  amplitude change with time is not visible when displaying the entire  data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Figures 22, 24 and 26 contain the phase-folded light curves  computed with sub-data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' It is clear that the wiggly shape of the  light curve changes depending on the time range of the sub-data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Our phase-folded data show a characteristic non-sinusoidal signal  Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Harmonics signature of our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Group A: the amplitude of  the spike harmonics decreases with increasing harmonic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Group B: the  amplitude of a harmonic is higher than a harmonic of a lower degree or  the spike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Group C: harmonics are not consecutive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' missing harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Group D: no harmonics present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' As described in the text in more detail Group  B and C imply the presence of stellar spots, whereas Group A and D do not  allow us to distinguish between spots and OsC modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Phase-folded light curve computed with the band-passed filtered  time series of KIC 4661914 (group C - see section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' average bin size  ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='003 in phase folded on period ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='024 d which corresponds to the fre-  quency of the main spike ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='977 d−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  that could be explained by the presence of several evolving stellar  spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  5 CONCLUSIONS AND FUTURE WORK  In this work, we analysed the Kepler light curves of 213 stars that  present a similar feature in their Fourier spectra called hump &  spike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Out of this sample, we have selected 162 stars that did not  present obvious photometric signs of being in a binary system (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  no transits or ellipsoidal variation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' We aimed to validate the initial  theoretical interpretation of the origin of the spike (Saio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2018),  which is assumed to be rotational modulation induced by stellar spots  co-rotating with the stellar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' We tested the theory regarding  the presence, origin and strength of magnetic fields on our sample  of stars under the assumption that the hump represents unresolved  Rossby modes and the spike stellar spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' In the present work, we  have concentrated only on the spike in the hump & spike feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The  ’hump’ will be the focus of future work currently in preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  In addition, we have also discussed another possible phenomenon  that could cause rotational modulation in the light curves of our stars,  as suggested by Lee & Saio (2020) and Lee (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' In this context,  the spike would be due to Overstable Convective (OsC) modes that  MNRAS 000, 1–17 (20xx)  10  150  KIC9163520 - Group A  KIC3240406 - Group B  8  6  100  4  ddj  2  Amplitude  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  1  2  3  4  10  30  KIC7959579 - Group C  KIC8396240 - Group D  8  20  A  6  4  10  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='5  220  10  Flux [ppm]  0  10  20  30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='4  PhaseRotational modulation in A and F stars  13  Figure 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Phase-folded light curve computed with subsets of the band-passed  filtered time series - KIC 4661914 (group C - see section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' average bin  size for each plot ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='003 in phase folded on period ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='024 d (frequency  of the main spike ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='977d−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The time range is noted for each plot;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' the  epoch time is the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 21 (time of the first Kepler observation for  KIC 4661914).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  resonantly excite g modes in the co-rotating frame propagating to the  surface and consequently become observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The spike frequency  in this scenario would correspond to the convective core rotation  frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  We have extracted the spike frequency and amplitude for all stars  and used stellar parameters such as 𝑇eff, luminosity and the stellar  radius from various sources (see Table A1 to consult the source  of the stellar parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' We determined the rotational velocities  using stellar radii and spike frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Further, we determined the  lifetime of the spike using autocorrelation functions and extracted  the number of spike harmonics present in the Fourier spectra of our  stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' In favour of the spike being evidence for stellar spots due to  magnetic fields, we find the following:  The spike amplitudes are higher for cooler stars and also slightly  higher for evolved stars, mirroring predictions regarding the strength  Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Phase-folded light curve computed with the full Kepler data -  KIC 2157489 (group B - see section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' average bin size ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='003 in phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  folded on period ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='358 d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' which corresponds to the frequency of the main  spike ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='736 d−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Figure 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Phase-folded light curve computed with subsets of the full time  series - KIC 2157489 (group B - see section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' average bin size ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='003 in  phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' folded on period ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='358 d (frequency of the main spike ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='736 d−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  the time range is noted for each plot;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' the epoch time is the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 23  (time of the first Kepler observation for KIC 2157489).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  MNRAS 000, 1–17 (20xx)  20  [55013-55042] BJD  10  o  10  20  [55247-55274] BJD  20  10  0  10  [ppm]  20  30  [55606-55633] BJD  20  10  10  20  30  20  [55962-55990] BJD  10  0  10  20  30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='4  Phase40  20  [dd]  0  20  40  60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='6  Phase60  [55048-55096]BJD  40  20  0  20  40  60  [55664-55708]BJD  40  20  [wd  0  xnl  20  F  40  60  [56329-56373] BJD  40  20  0  20  40  60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='6  Phase14  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Henriksen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Figure 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Phase-folded light curve computed with the full Kepler time series  KIC 3440710 (group A - see section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' average bin size ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='003 in phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  folded on period ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='8 d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' which corresponds to the frequency of the main spike  ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='25 d−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Figure 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Phase-folded light curve computed with subsets of the full time  series - KIC 3440710 (group A - see section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' average bin size ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='003  in phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' folded on period ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='8 d (frequency of the main spike ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='25 d−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  the time range is noted for each plot;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' the epoch time is the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 25  (time of the first Kepler observation for KIC 3440710).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  of dynamo-generated magnetic fields in (sub) surface convective  layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  The spike lifetimes are short, of the order of tens of days, sim-  ilar to what one would expect from magnetic features of dynamo-  generated magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  The spike lifetimes do not correlate with stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' This  would exclude the possibility that the spike frequency would cor-  respond to the convective core rotation frequency, as it should be  modulated by the convective turnover time, which increases with  stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  The harmonic signature of 36 stars suggests a non-sinusoidal  signal consistent with co-rotating stellar spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Phase-folded light curves of time series containing the signal  induced by the spike and its harmonics suggest a non-sinusoidal  signal specific to changing stellar spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  The estimated magnetic field strengths, however uncertain, are  of the same order of magnitude as predictions of non-Ap stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  We find 34 stars that could be in a post main-sequence evolu-  tionary stage, which excludes the OsC modes scenario, as these stars  do not have a convective core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' We note, however, that this identifica-  tion is visual only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The evolutionary tracks used here do not take into  account, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=', core overshooting, rotation and non-solar metallicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  In addition, the luminosity values used here are not corrected for the  gravity-darkening effect, which is considerable in the case of high  stellar rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Our results suggest that, despite their very shallow convective en-  velopes, these stars could have spots likely induced by magnetic  fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The characteristics of the features causing the brightness vari-  ability in the shape of the spike point towards the idea of a dynamo-  generated magnetic field as suggested by CB19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The short spike  lifetimes, the estimated magnetic field strengths and presumed mag-  netic feature sizes favour dynamo-generated fields rather than fossil  fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' While these stars are unfortunately too faint to qualify for  spectropolarimetry, recent observations of 𝛽 Cas - a rapidly rotating  𝛿 Scuti star - indicate that dynamo-generated magnetic fields exist  in intermediate mass stars (Zwintz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' As observations are  biased towards stars with strong magnetic fields and large magnetic  features, our work provides a path for future observational efforts  towards weaker magnetic fields that generate small-scale features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Furthermore, ongoing observational efforts from the ground and  the TESS satellite will allow us to determine whether the spots (if  present) in the hump & spike stars are bright spots, as suggested  by CB19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Our work also offers the chance to verify and test the  predictions and theory of Saio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' (2018), where it was hypothesised  that Rossby modes are mechanically excited by deviated flows caused  by stellar spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' An obvious next step is to find bright hump &  spike stars observed with TESS, which will allow us to measure the  magnetic fields directly using ground-based facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  In favour of the spike being evidence of g modes being resonantly  excited by OsC modes from the convective core we find the following  arguments:  The spike amplitude increasing with mass could be because the  amplitude of OsC modes is larger when the resonance with g modes  is stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' In the case of lower mass stars, the density of g-mode  frequencies is higher;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' therefore, it would be more likely that the  resonance of g modes in the co-rotating frame is stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  The harmonic signature (amplitude decreases as azimuthal or-  der decreases) in the case of 111 stars, where the OsC modes scenario  could be the cause for the rotational modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' This applies also  for 14 stars more which do not exhibit detectable harmonics of the  main spike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  MNRAS 000, 1–17 (20xx)  30  C  20  10  [udd]  0  10  xni  20  30  5  40  50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='4  Phase100  [55043-55081]BJD  75  50  25  0  25  50  75  100  [55630-55672]BJD  60  40  [ppm]  20  0  20  40  50  [56215-56258]BJD  40  30  20  10  0  10  20  30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='4  PhaseRotational modulation in A and F stars  15  Based on our analyses and sample of stars, neither scenario can  be entirely excluded at this stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' It is also possible, although rather  unlikely, that both stellar spots and OsC modes could generate the  hump & spike feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' While in this work, we find more arguments  supporting the idea of stellar spots induced by magnetic fields, as  summarised above, a full conclusion can be reached only by directly  measuring magnetic fields in hump & spike stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  This work was supported by a research grant (00028173) from VIL-  LUM FONDEN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Funding for the Stellar Astrophysics Centre is pro-  vided by The Danish National Research Foundation (Grant agree-  ment no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' : DNRF106).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' This research was supported in part by the  National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' NSF PHY-1748958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' acknowledges support from the Spanish Ministry of Sci-  ence and Innovation (MICINN) with the Ramón y Cajal fellowship  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' RYC-2015-17697 and grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' PID2019-107187GB-I00, and  through AEI under the Severo Ochoa Centres of Excellence Pro-  gramme 2020–2023 (CEX2019-000920-S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  ARGS acknowledges the supported by FCT through national  funds and by FEDER through COMPETE2020 by these grants:  UIDP/04434/2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' PTDC/FIS-AST/30389/2017 & POCI-01-0145-  FEDER-030389.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' ARGS is supported by FCT through the work con-  tract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='02480.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='CEECIND/CP1631/CT0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  We acknowledge that this research was supported in part by the  National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' NSF PHY-1748958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' acknowledges the support of the PLATO CNES grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  DATA AVAILABILITY  The data underlying this article are available in the article and in its  online supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  REFERENCES  Asplund M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
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+page_content=' HR diagram of the hump & spike stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Green circles depicts the stars studied in the current work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' The source for luminosity and 𝑇eff values is  described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Red pentagons, orange squares and purple triangles represent the stars excluded as they show signs of binarity (see section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' One  star that was excluded is not shown in the Figure as no value for luminosity was found in literature: KIC 5458880 (𝑇eff Gaia DR2 = 8704 K+782  −1082), was identified  as an eclipsing by Kirk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2016 and shows a clear binary signal in Kepler data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' All 𝑇eff values are from Gaia DR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Warszaw-New Jersey evolutionary tracks  (𝑍 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='012, Asplund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2004) are displayed in the background for guidance only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Extracted spike parameters and stellar parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 𝑓rot,, 𝜎 𝑓rot: Spike frequency and associated standard deviation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' A, 𝜎A: Spike amplitude and  its uncertainty;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 𝑇eff, 𝑇effp, 𝑇effm: Effective temperature and upper and lower uncertainties - from Gaia DR2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 𝐿, 𝐿p, 𝐿m: Luminosity and upper and lower  uncertainties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 𝑅, 𝑅p, Rm: Radius value, upper and lower uncertainties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2𝜏ACF, 𝜎𝜏ACF: Spike lifetime and its uncertainty;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' quarters: number of Kepler  quarters in which data are available;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Source parameter: first letter indicates the source of the luminosity values, second letter indicates the source of the radius  values, m = Murphy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' (2019), i = Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
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+page_content='8 Mo  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='5 Mo  102  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='2 Mo  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
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+page_content='6 Mo  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='3 Mo  Analysed in current work  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0 M  Excluded  101  (Luminosity - Murphy et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' al 2019)  Excluded  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='7 M  (Luminosity - Gaia DR2)  Excluded  (Luminosity - Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' 2021)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='4Mc  14000  12000  10000  8000  6000  4000  Teff [K]18  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Henriksen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  Figure A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=' Correlation between stellar parameters and the parameters extracted from our analysis  MNRAS 000, 1–17 (20xx)  r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  (a2)  a31  9000-  9000-  9000  8000-  8000  8000  7000  7000  7000  r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='53  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  (a1)  +0009  6000  6000  101  102  50  100  150  200  250  300  350  10  20  30  40  50  60  70  Spike amplitude [ppm]  Vrot [kms-1]  2TAcF [days]  [L。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content=']  (b1)  +  (b2)  (b3)  r = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='06  r = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='18  102,  102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='  102,  Luminosity [  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='0  101  101  101  r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='52  +  101  102  50  100  150  250  300  350  10  20  30  40  50  200  60  70  Spike amplitude [ppm]  Vrot[kms-1]  2TACF [days]  (c1)  (c2)  5 (c3)  +  r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='43  r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='05  Radius [Ro]  r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='12  4  3  2  101  102  50  100  150  200  250  300  350  10  20  30  40  50  60  70  Spike amplitude [ppm]  Spike amplitude [ppm]  Vrot [kms-1]  [ppm]  2TACF [days]  r = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='11  r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='01  102  (d2)  r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
+page_content='01  102  d3  Spike amplitude [  40  D  101  101  (d1)  2 20  +  +  50  100  150  200  250  300  350  50  100  150  200  250  300  350  10  20  30  40  50  60  70  Vrot [kms-1]  Vrot [kms-1]  2TACF [days]' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfPgyg/content/2301.04974v1.pdf'}
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+arXiv:2301.03870v1  [math.PR]  10 Jan 2023
+Discrete mixture representations of spherical distributions
+Ludwig Baringhaus∗ and Rudolf Gr¨ubel
+Institute of Actuarial and Financial Mathematics
+Leibniz Universit¨at Hannover
+Postfach 60 09, D-30060 Hannover, Germany
+lbaring@stochastik.uni-hannover.de
+rgrubel@stochastik.uni-hannover.de
+Abstract
+We obtain discrete mixture representations for parametric families of probability
+distributions on Euclidean spheres, such as the von Mises–Fisher, the Watson and the
+angular Gaussian families. In addition to several special results we present a general
+approach to isotropic distribution families that is based on density expansions in terms
+of special surface harmonics. We discuss the connections to stochastic processes on
+spheres, in particular random walks, discrete mixture representations derived from
+spherical diffusions, and the use of Markov representations for the mixing base to
+obtain representations for families of spherical distributions.
+Keywords: Mixture distribution, isotropy, surface harmonics, self-mixing stable dis-
+tribution families, almost sure representations, skew product decomposition
+MSC2020-Mathematics Subject Classification: Primary 62H11; secondary 60E05
+1
+Introduction
+A discrete mixture representation for a parametric family {Pθ : θ ∈ Θ} of probability
+measures in terms of another family {Qn : n ∈ N0} of probability measures, the mixing
+base, all defined on the same measurable space, is of the form
+Pθ =
+∞
+�
+n=0
+wθ(n) Qn
+for all θ ∈ Θ.
+(1)
+Here, for each θ ∈ Θ, the mixing coefficients (wθ(n))n∈N0 are the individual probabilities
+of a distribution Wθ, the mixing distribution, on (the set of subsets of) N0. A classical
+case is the representation of non-central chisquared distributions with k degrees of freedom,
+Pθ = χ2
+k(θ2) with non-centrality parameter θ2 > 0, as Poisson mixtures of central chisquared
+distributions, where Qn = χ2
+2n+1 := χ2
+2n+1(0) and where Wθ is the Poisson distribution with
+mean λ = θ2/2; see e.g. the books of Liese and Miescke (2008) and M¨orters and Peres (2010)
+where the representation appears in statistics in connection with the power of statistical tests
+and in probability theory in connection with the local times of Markov processes respectively.
+∗Corresponding author
+1
+
+Such mixture representations can be related to two-stage experiments: In order to obtain a
+value x with distribution Pθ we first choose n according to Wθ and then choose x according
+to Qn. This leads to an immediate application of discrete mixture representations in the
+context of simulation methodology.
+In the present paper we continue our previous investigations (see Baringhaus and Gr¨ubel
+(2021a,b)), and now specifically consider distributions on the Euclidean sphere Sd := {x ∈
+Rd+1 : ∥x∥ = 1} of (d + 1)-dimensional real vectors of unit length. This case seems to us to
+deserve some interest, in particular if specific properties of spheres are taken into account:
+The group O(d+1) of orthogonal transformations of the ambient space Rd+1 acts transitively
+on Sd, and there is a ‘polar decomposition’ (or ‘tangent-normal decomposition’, see Section
+4.3) that relates Sd to [−1, 1] × Sd−1 if d > 1.
+We generally assume that the distribution parameters in the above general setup are of
+the form θ = (η, ρ), where η ∈ Sd may be seen as a location parameter; instead of Pθ we also
+write Pη,ρ. We obtain mixture representations that split the dependence on the two parts
+of the parameter in the sense that
+Pη,ρ =
+∞
+�
+n=0
+wρ(n) Qn,η
+for all θ = (η, ρ) ∈ Θ.
+(2)
+In particular, the mixing distributions depend on ρ only. For fixed ρ on the left, or fixed
+n ∈ N0 on the right hand side of (2), the families {Pη,ρ : η ∈ Sd} respectively {Qn,η : η ∈ Sd}
+are parametrized by the sphere and are defined on its Borel subsets B(Sd). We assume
+that these families interact with the group action mentioned above in the sense that they
+are isotropic; see (8) below. In particular, their elements are then rotationally symmetric
+about the axis specified by η. As a simple application of the representation (2) we mention
+that with the finite sums Rη,ρ(A) := �n
+k=0 wθ(n) Qk(A) we have monotonically increasing
+approximations of the probabilities Pη,ρ(A), A ∈ B(Sd), with uniform error bounds in the
+sense that
+0 ≤ Pη,ρ(A) − Rη,ρ(A) ≤
+∞
+�
+k=n+1
+wρ(n)
+for all η ∈ Sd, A ∈ B(Sd).
+The literature contains several other applications; see for example the relation to nonpara-
+metric Bayesian inference in Baringhaus and Gr¨ubel (2021b, Section 5.1).
+In Section 2 we collect some basic notation and obtain mixture representations for the
+von Mises–Fisher family and two spherical Cauchy families in Theorem 2, the Watson
+family in Theorem 3, and an angular Gaussian family in Theorem 5. The mixing bases
+are chosen specifically for the respective family, with a view towards reflecting its prop-
+erties. A different base will generally lead to a different representation, as demonstrated
+by Baringhaus and Gr¨ubel (2021a) in the context of non-central chisquared distributions.
+In Section 3 we present a general approach that uses expansions of densities in terms of
+special surface harmonics. The resulting mixing base has a structural property that we call
+self-mixing stability. This property makes it comparably easy to relate different expansions
+to each other. We obtain representations with this mixing base for the wrapped Cauchy
+and the wrapped normal families in Theorem 8, and for the von Mises–Fisher families in
+Theorem 9. These results only hold under conditions on the parameter ρ that ensure that
+the respective distribution is not too far away from the uniform distribution on the sphere;
+in Example 11 we work out a possibility for extending this range.
+For fixed ρ or n we may regard Pη,ρ and Qn,η as probability kernels via (η, A) �→ Pη,ρ(A),
+(η, A) �→ Qn,η(A). This provides a general connection with Markov processes. We briefly
+2
+
+return to the classical mixture representation of non-central one-dimensional chisquared
+distributions, which may be written as
+(X + θ)2 =D X2 + 2
+N(θ)
+�
+j=1
+Ej.
+(3)
+Here the random variables N(θ), X, E1, E2, . . . are independent, X has the standard normal
+distribution, E1, E2, . . . are exponentially distributed with mean 1, and N(θ) has the Poisson
+distribution with parameter θ2/2. The path-wise point of view displays the distributions
+χ2
+1(θ), θ ≥ 0, as the distributions of randomly stopped partial sums of independent random
+variables, and (3) may be used to read off stochastic monotonicity and infinite divisibility
+of non-central chisquared distributions. Note that the representation only covers the one-
+dimensional marginal distributions of the process ((X + θ)2)θ≥0, as the left hand side of (3)
+is obviously not pathwise monotone in the ‘time parameter’ θ. Quite generally, (1) can be
+related to randomly stopped stochastic processes: If X = (Xn)n∈N0 is such that Xn has
+distribution Qn for all n ∈ N0 then Xτ has distribution Pθ if τ is independent of X and has
+distribution Wθ.
+In Section 4 we discuss several connections between families of spherical distributions
+and stochastic processes on spheres. We consider random walks on spheres in Section 4.1,
+distribution families that arise in connection with diffusion processes in Section 4.2, and the
+use of Markov representations of the mixing base in connection with almost sure representa-
+tions for distribution families in Section 4.3. The ultraspherical mixing base from Section 3
+will be useful at various stages.
+For a single transition kernel we obtain a family (Xη
+n)n∈N0 of Markov chains indexed
+by their initial state η, meaning that Xη
+0 = η with probability 1. Isotropy of the kernel
+then extends to isotropy of the corresponding distributions on the path space.
+For the
+elements of the mixing base in Section 3 the marginal distributions of these chains have
+a particular simple description.
+Further, isotropy relates a family {Pη : η ∈ Sd} to a
+single distribution on [−1, 1] via the latitude projection x → ηtx, with η as ‘north pole’.
+For the chain Xη = (Xη
+n)n∈N0 we obtain an associated latitude process Y = (Yn)n∈N0 via
+Yn := ηtXn for all n ∈ N0. For isotropic kernels this is again a Markov chain, now on
+[−1, 1] and with start at 1. Finally, for the von Mises–Fisher distributions we show that a
+homogeneous Markov process on the sphere with these as marginal distributions does not
+exist, see Theorem 16, and we obtain a result similar to (3), see Example 18.
+Proofs are collected in Section 5.
+Mixing of distributions is a standard topic in probability theory and statistics, see
+e.g. Lindsay (1995). Spherical data and families of spherical distributions have similarly
+been investigated for a long time and by many researchers; standard references are the clas-
+sic monograph of Watson (1983) and, more recently, the book of Mardia and Jupp (2000).
+For a review of distributions on spheres we refer to Pewsey and Garc´ıa–Portugu´es (2021),
+see also Watson (1982). Of particular interest for the topics treated here is the very recent
+paper of Mijatovi´c et al. (2020) where a discrete mixture representation for the marginal
+distributions of spherical Brownian motion is developed. More specific references will be
+given at the appropriate places below.
+2
+Generalities and some special results
+We need some basic notions and definitions. We write X ∼ µ if X is a random variable on
+some background probability space (Ω, F, P) with distribution µ. Formally, let (Ω, A) and
+3
+
+(Ω′, A′) be measurable spaces and suppose that T : Ω → Ω′ is (A, A′)-measurable. Then the
+push-forward P T of a probability measure P on (Ω, A) under T is the probability measure
+on (Ω′, A′) given by P T (A) = P(T −1(A)), A ∈ A′, and X ∼ µ is the same as PX = µ. For
+many of the measurable spaces considered below there is a canonical uniform distribution,
+often defined by invariance under a group operation. To avoid tiresome repetitions we agree
+that densities refer to the respective uniform distribution if not specified otherwise.
+We fix a dimension d ≥ 1, but instead of d we often use
+λ = λ(d) := (d − 1)/2,
+(4)
+as this is common in connection with families of special functions. In particular, whenever
+d and λ appear together, they are related by (4). The group O(d + 1) of orthogonal (d +
+1) × (d + 1)-matrices U acts on Sd via x �→ Ux, and the uniform distribution unif(Sd) on
+the sphere is the unique probability measure on the Borel subsets of Sd that is invariant
+under all such transformations. For a fixed η ∈ Sd the push-forward νd of unif(Sd) under
+the mapping x �→ ηtx has density hd with respect to the uniform distribution unif(−1, 1) on
+the interval [−1, 1], where
+hd(y) :=
+Γ(λ + 1)
+Γ(1/2)Γ(λ + 1/2) (1 − y2)λ−1/2,
+−1 < y < 1.
+(5)
+Note that this does not depend on η ∈ Sd. Further, if X ∼ unif(Sd) and Y = ηtX ∼ νd,
+then the conditional distribution of X given Y = y is the uniform distribution on
+Cd(η, y) := {x ∈ Sd : ηtx = y},
+(6)
+with unif(Cd(η, y)) the unique probability measure on this set that is invariant under the
+subgroup {U ∈ O(d + 1) : Uη = η} of O(d + 1). This may be seen in the context of the
+polar decomposition mentioned in the introduction.
+Conversely, given a probability measure ν on [−1, 1] and a parameter η ∈ Sd, we can con-
+struct a distribution µ = µη on Sd via the kernel (y, A) �→ unif(Cd(η, y))(A). In particular,
+for bounded and measurable functions φ : Sd → R,
+�
+φ(x) µ(dx) =
+�
+[−1,1]
+�
+Cd(η,y)
+φ(x) unif(Cd(η, y))(dx) ν(dy).
+For η ∈ Sd and a measurable function g : [−1, 1] → R the function fη : Sd → R given by
+fη(x) = g(ηtx),
+x ∈ Sd,
+(7)
+is unif(Sd)-integrable if and only if g is νd-integrable, and then
+�
+fη(x) unif(Sd)(dx) =
+�
+g(t) νd(dt).
+Thus fη is the density of a probability measure on Sd if and only if g is the νd-density of
+a probability measure on [−1, 1]. In particular, a probability density g on [−1, 1] generates
+a family {Qη : η ∈ Sd} of spherical distributions via (7), and such families are isotropic in
+the sense that
+QU
+η = QUη
+for all U ∈ O(d + 1), η ∈ Sd.
+(8)
+In particular, each Qη is invariant under all rotations with axis η. As the function η �→
+�
+A g(ηtx) unif(Sd)(dx) is B(Sd)-measurable for all A ∈ B(Sd), Q· : Sd × B(Sd) → R defined
+4
+
+by Q·(η, A) = Qη(A), (η, A) ∈ Sd×B(Sd), is a Markov kernel from (Sd, B(Sd)) to (Sd, B(Sd)).
+Further, if Q is a Markov kernel from (Sd, B(Sd)) to (Sd, B(Sd)) and U ∈ O(d + 1), then the
+kernel QU : Sd × B(Sd) → R defined by QU(η, A) = Q(η, U tA), (η, A) ∈ Sd × B(Sd), with
+U tA := {U tx : x ∈ A}, is the push-forward of Q under U. The kernel Q is isotropic if
+QU(η, ·) = Q(Uη, ·)
+for all η ∈ Sd and all U ∈ O(d + 1).
+(9)
+Some classical special functions will be needed below. Let
+(α)n := Γ(α + n)
+Γ(α)
+,
+α > 0, n ∈ N0,
+be the ascending factorials. The modified Bessel functions Iα of the first kind are given by
+Iα(x) =
+∞
+�
+n=0
+1
+n! Γ(n + α + 1)
+�x
+2
+�2n+α
+,
+x ≥ 0,
+(10)
+with real nonnegative parameter α, the confluent hypergeometric functions are
+1F1(α; β; x) =
+∞
+�
+n=0
+(α)n
+(β)n n! xn,
+x ∈ R,
+with real positive parameters α and β, and the hypergeometric functions are
+2F1(α, β; γ; x) =
+∞
+�
+n=0
+(α)n(β)n
+(γ)n n! xn,
+−1 < x < 1,
+with real positive parameters α, β, and γ.
+Example 1. (a) Let p > −1/2 and let ν be the distribution on [−1, 1] with unif(−1, 1)-density
+y �→
+Γ(p+λ+1)
+Γ(p+1/2)Γ(λ+1/2)|y|2p(1 − y2)λ−1/2. Then ν has νd-density y �→ Γ(1/2)Γ(p+λ+1)
+Γ(p+1/2)Γ(λ+1) |y|2p, and
+we obtain the spherical power distribution SPd(η, p) with density
+f SP
+d (x|η, p) = Γ(1/2)Γ(p + λ + 1)
+Γ(p + 1/2)Γ(λ + 1) |ηtx|2p,
+x ∈ Sd.
+We mainly use this with p = n ∈ N0, and then have
+f SP
+d (x|η, n) = (λ + 1)n
+(1/2)n
+(ηtx)2n,
+x ∈ Sd.
+(b) Starting with ν = Beta[−1,1](p+λ−1/2, q+λ−1/2), p, q > 1/2−λ, the beta distributions
+on [−1, 1] with densities y �→ c(p + λ − 1/2, q + λ − 1/2)(1 − y)p+λ−3/2(1 + y)q+λ−3/2, where
+c(p + λ − 1/2, q + λ − 1/2) = Γ(p + q + 2λ − 1)/
+�
+2p+q+2(λ−1)Γ(p + λ − 1/2)Γ(q + λ − 1/2)
+�
+, we
+obtain the spherical beta distributions SBetad(η, p, q), with densities
+f SBeta
+d
+(x|η, p, q) = cd(p, q) (1 − ηtx)p−1(1 + ηtx)q−1,
+x ∈ Sd,
+where the norming constants are given by
+cd(p, q) = 2−(p+q+2(λ−1))
+Γ(1/2)Γ(λ + 1/2)Γ(p + q + 2λ − 1)
+Γ(λ + 1)Γ(p + λ − 1/2)Γ(q + λ − 1/2).
+(11)
+5
+
+(c) The von Mises–Fisher distributions, which we denote by MFd(η, ρ), ρ > 0, arise if
+we start with νd-density proportional to y �→ exp(ρy), −1 ≤ y ≤ 1. The continuous density
+of the associated spherical distribution is
+f MF
+d
+(x|η, ρ) = cd(ρ) exp(ρ ηtx),
+x ∈ Sd,
+where the norming constants are given by
+cd(ρ) =
+ρλ
+2λΓ(λ + 1)Iλ(ρ) .
+Further, MFd(η, 0) = unif(Sd).
+(d) The Watson distributions Watd(η, ρ), ρ ∈ R, arise if we begin with νd-density pro-
+portional to y �→ exp(ρy2), −1 ≤ y ≤ 1. The continuous density of the associated spherical
+distribution is
+f Wat
+d
+(x|η, ρ) = cd(ρ) exp
+�
+ρ (ηtx)2�
+,
+x ∈ Sd,
+with norming constants cd(ρ) =
+�
+1F1(1/2; λ + 1; ρ)
+�−1. Clearly, Watd(η, 0) = unif(Sd).
+(e) The angular Gaussian distributions are the distributions of X = Z/∥Z∥, where Z
+has the (d + 1)-variate normal distribution Nd+1(a, Σ) with mean vector a ∈ Rd+1 \ {0} and
+symmetric positive definite covariance matrix Σ; see, e.g. Watson (1983, p. 108). Here, we
+exclusively deal with the case where Σ is the identity matrix Id+1, as the radial parts then
+lead to isotropic families. The distributions arising in this special case seem to have first
+been studied in detail by Saw (1978). Putting η = a/∥a∥ and ρ = ( 1
+2∥a∥2)
+1/2 we denote
+by AGd(η, ρ) the distribution of X and speak of the angular Gaussian distribution with
+parameters η and ρ. Its density is represented by the infinite series
+f AG
+d
+(x|η, ρ) = e−ρ2
+∞
+�
+k=0
+(2ρ ηtx)k Γ((d + 1 + k)/2)
+k!Γ((d + 1)/2) ,
+x ∈ Sd, ρ > 0.
+We refer to Saw (1978), where it is also pointed out that, with a random variable S ∼ χ2
+d+1,
+the density can be written as
+f AG
+d
+(x|η, ρ) = E
+�
+e−ρ2 exp(2
+1/2ρ S
+1/2 ηtx)
+�
+.
+(12)
+(f) We consider two types of spherical Cauchy distributions: The spherical Cauchy dis-
+tributions of type I have the unif(Sd)-densities
+f CI
+d (x|η, ρ) =
+�
+1 − ρ2
+1 − 2ρ ηt x + ρ2
+�d
+,
+x ∈ Sd,
+and the spherical Cauchy distributions of type II have the unif(Sd)-densities
+f CII
+d
+(x|η, ρ) =
+1 − ρ2
+(1 − 2ρ ηt x + ρ2)(d+1)/2 ,
+x ∈ Sd,
+both with parameters η ∈ Sd and ρ ∈ (0, 1).
+We denote by CId(η, ρ) the distribution
+with unif(Sd)-density f CI
+d ( · |η, ρ), and by CIId(η, ρ) the distribution with unif(Sd)-density
+f CII
+d
+( · |η, ρ). Clearly, CId(η, 0) = CIId(η, 0) = unif(Sd).
+6
+
+The distributions in Example 1 (a) - (e) and their push-forwards under x �→ ηtx, re-
+spectively, are all classical; for basic as well as specific properties and interesting historical
+comments we refer to Watson (1982, 1983) and Mardia and Jupp (2000). It is well known,
+for example, that the von Mises–Fisher family in part (c) arises from the multivariate nor-
+mal distributions in part (e) by conditioning on ∥Z∥. A well known relation with Brownian
+motion on Sd is addressed in Section 4.2 below. In the special case d = 1 = (d+1)/2 the dis-
+tributions WC1(η, ρ) := CI1(η, ρ) = CII1(η, ρ) in Example 1 (f) are known as the wrapped
+Cauchy or circular Cauchy distributions; see, e.g. Mardia and Jupp (2000) and Section 3
+below. Hence, with the two types of spherical Cauchy distributions given above we have
+two different extensions of this distribution family to higher dimensions. For distinction, we
+added the name supplement ‘of type I’ and ‘of type II’, respectively. The spherical Cauchy
+distributions of type I were introduced and studied by Kato and McCullagh (2020). Gen-
+eralizing results obtained by McCullagh (1996) for d = 1, the authors especially deal with
+the behavior of the spherical Cauchy distributions of type I under M¨obius transformations.
+The densities f CII
+d
+( · |η, ρ) were considered by McCullagh (1989), though the author does
+not speak of spherical Cauchy distributions but, with the push-forward of CII(η, ρ) under
+x �→ ηtx, of a noncentral version of the univariate symmetric beta distribution.
+We recall from the introductory remarks that for a discrete mixture representation we
+need a mixing base (Qn,η)n∈N0, η ∈ Sd, where each Qn,η is a probability measure on the
+sphere, and mixing distributions on N0 that depend on ρ only; see (2). Of special interest
+in the latter context are the the negative binomial distributions NB(r, p) with parameters
+r > 0, p ∈ (0, 1), and probability mass function
+nb(n|r, p) = (r)n
+n! (1 − p)npr,
+n ∈ N0,
+the confluent hypergeometric series distributions CHS(α, β, τ) on N0 with parameters α, β, τ >
+0 and probability mass functions
+chs(n|α, β, τ) =
+�
+1F1(α; β; τ)
+�−1 (α)n
+(β)n
+τ n
+n! ,
+n ∈ N0,
+and the hypergeometric series distributions HS(α, β, γ, τ) on N0 with parameters α, β, γ, τ >
+0 and probability mass functions
+hs(n|α, β, γ, τ) =
+�
+2F1(α, β; γ; τ)
+�−1 (α)n(β)n
+(γ)n
+τ n
+n! ,
+n ∈ N0.
+For τ = 0 we take CHS(α, β, τ) and HS(α, β, τ) to be the one-point mass at 0. The dis-
+tribution CHS(α, β, τ) arises as the stationary distribution of a birth-death process with
+birth rates (α + i)τ and death rates i(β + i − 1), i ∈ N0; see Hall (1956). Note that these
+three distribution families are subclasses of the family of generalized hypergeometric distri-
+butions considered recently by Themangani et al (2020). We also require that each family
+{Qn,η : η ∈ Sd} is isotropic.
+We can now state our first results. Let d ∈ N be fixed and let λ be as in (4).
+Theorem 2. (a) The family {MFd(η, ρ) : η ∈ Sd, ρ ≥ 0} has a unique discrete mixture
+representation with mixing base SBetad(η, 1, n + 1), n ∈ N0. This representation is given by
+MFd(η, ρ) =
+∞
+�
+n=0
+chs(n|λ + 1/2, 2λ + 1, 2ρ) SBetad(η, 1, n + 1).
+(13)
+7
+
+(b) The family {CId(η, ρ) : η ∈ Sd, ρ ∈ (0, 1)} has a unique discrete mixture representation
+with mixing base SBetad(η, 1, n + 1), n ∈ N0. This representation is given by
+CId(η, ρ) =
+∞
+�
+n=0
+nb
+�
+n|λ + 1/2, 4ρ/(1 + ρ)2�
+SBetad(η, 1, n + 1).
+(14)
+(c) The family {CIId(η, ρ) : η ∈ Sd, ρ ∈ (0, 1)} has a unique discrete mixture representation
+with mixing base SBetad(η, 1, n + 1), n ∈ N0. This representation is given by
+CIId(η, ρ) =
+∞
+�
+n=0
+hs
+�
+n|λ + 1/2, λ + 1, 2λ + 1, 4ρ/(1 + ρ)2�
+SBetad(η, 1, n + 1).
+(15)
+In our next result the mixing base depends on the value of ρ.
+Theorem 3. (a) The family {Watd(η, ρ) : η ∈ Sd, ρ ≥ 0} has a unique discrete mixture
+representation with mixing base SPd(η, n), n ∈ N0. This representation is given by
+Watd(η, ρ) =
+∞
+�
+n=0
+chs(n|1/2, λ + 1, ρ) SPd(η, n).
+(16)
+(b) The family {Wat(η, ρ) : η ∈ Sd, ρ ≤ 0} has a unique discrete mixture representation with
+mixing base SBetad(η, n + 1, n + 1), n ∈ N0. This representation is given by
+Watd(η, ρ) =
+∞
+�
+n=0
+chs(n|λ + 1/2, λ + 1, −ρ) SBetad(η, n + 1, n + 1).
+(17)
+In order to obtain a similar representation for the family of {AGd(η, ρ) : η ∈ Sd, ρ > 0}
+of angular Gaussian distributions we make use of the integral representation
+Dν(z) = e−z2/4
+Γ(−ν)
+� ∞
+0
+t−ν−1e−zt−t2/2 dt,
+z ∈ R,
+(18)
+of the parabolic cylinder functions Dν with real index ν < 0; see Magnus et al. (1966, p. 328).
+Lemma 4. Let δ > 1/2 and τ > 0. Then dpc( · |δ, τ) with
+dpc(k|δ, τ) = 1
+k!(2
+1/2τ)k2k+δ−1 (k + 2δ − 1)Γ(k + δ − 1/2)
+Γ(1/2)
+e−τ 2/2D−(k+2δ)(2
+1/2τ),
+(19)
+k ∈ N0, is a probability mass function.
+We write DPC(δ, τ) for the associated discrete parabolic cylinder distribution with pa-
+rameters δ > 1/2 and τ > 0. For the special values δ = λ + 1 = (d + 1)/2 with d ∈ N the
+statement of the lemma also follows from (20) below as the values on the right hand side
+of (19) are all nonnegative.
+Theorem 5. The family {AGd(η, ρ) : η ∈ Sd, ρ > 0} has a unique discrete mixture repre-
+sentation with mixing base SBetad(η, 1, n + 1), n ∈ N0. This representation is given by
+AGd(η, ρ) =
+∞
+�
+n=0
+dpc(n|λ + 1, ρ) SBetad(η, 1, n + 1).
+(20)
+8
+
+Remark 6. (a) Regarding probability measures as real functions on a set of events, we may
+define the series in (13) - (17) and (20) as referring to pointwise convergence of functions.
+In fact, as the distributions involved all have smooth densities and compact domain, con-
+vergence even holds with respect to uniform convergence in spaces of continuous functions.
+(b) The representations are minimal in the sense that the respective mixing base cannot
+be reduced. This follows from the uniqueness and the fact that the mixing probabilities are
+strictly positive.
+(c) In connection with the base in Theorem 2 all mixtures have densities that are increas-
+ing in ηtx, and in Theorem 3 and Theorem 5 all mixtures are invariant under the reflection
+x �→ −x.
+(d) Interestingly, the von Mises–Fisher family, the spherical Cauchy families and the
+angular Gaussian family have the same mixing base of spherical beta distributions. So, these
+families are obtained by picking at random (the index n of) the element SBetad(η, 1, n + 1)
+according to the respective mixing distributions. Another family with this mixing base is
+the family of spherical normal distributions; see Section 4.2.
+(e) For a discussion of other similarities as well as differences between the von Mises–
+Fisher family and the spherical Cauchy family of type I we refer to Kato and McCullagh
+(2020). The von Mises–Fisher family and the spherical Cauchy family of type II both have
+representations in terms of multivariate Brownian motion. To be specific, let X = (Xt)t≥0
+be a standard Brownian motion in Rd+1, let Y = (Yt)t≥0 with Yt = ρηt + Xt for t ≥ 0 be
+the drifted standard Brownian motion with constant drift vector ρη, where ρ ≥ 0, η ∈ Sd,
+and let Z = (Zt)t≥0 with Zt = ρη + Xt for t ≥ 0, where 0 ≤ ρ < 1, η ∈ Sd, be the Brownian
+motion starting at ρη. With TY := inf{t ≥ 0 : ∥Yt∥ ≥ 1} as the first time that Y exits the
+Euclidean unit ball Bd = {x ∈ Rd+1 : ∥x∥ < 1} it then holds that YTY ∼ MFd(η, ρ); see
+Gatto (2013) for a more recent proof and historical remarks on this result. Further, with
+TZ := inf{t ≥ 0 : ∥Zt∥ ≥ 1} the first time that Z exits Bd it holds that ZTZ ∼ CIId(η, ρ);
+see Chung (1982, p. 170) and McCullagh (1989).
+3
+Ultraspherical mixing bases
+Our aim in this section is a mixing base that is applicable for general distribution families
+where, as before, we consider distributions Pθ on (Sd, B(Sd)), with θ = (η, ρ) ∈ Sd × I and
+I ⊂ R+ an interval, that have densities fθ of the form
+fθ(x) = gρ(ηtx),
+x ∈ Sd.
+(21)
+We will occasionally omit d or λ from the notation. Recall that λ = (d − 1)/2 and that νd
+is the push-forward of unif(Sd) under the mapping x �→ ηtx.
+We assume that the functions gρ in (21) are elements of
+Hλ := L2�
+[−1, 1], B([−1, 1]), νd
+�
+,
+and on Hλ we use the inner product
+⟨f, g⟩λ =
+�
+f(t)g(t) νd(dt)
+and the norm ∥f∥λ = ⟨f, f⟩1/2
+λ . Then (Hλ, ⟨·, ·⟩λ) is a Hilbert space. We deal with a special
+complete sequence of orthogonal polynomials in this space. For d = 1 and λ = 0 this is
+the sequence of Chebyshev polynomials Tn of the first kind of degree n ∈ N0, for d > 1
+9
+
+and λ > 0 we use the sequence of Gegenbauer or ultraspherical polynomials Cλ
+n of degree
+n ∈ N0; see Erd´elyi et al. (1953, Chs.
+X, XI). These functions play an important role
+in directional statistics, especially nonparametric directional statistics; see e.g. the papers
+of Bingham (1972), Gin´e (1975), Prentice (1978), Baringhaus (1991), Jupp (2008), and
+Garc´ıa–Portugu´es et al. (2021). The functions are standardized such that
+Cλ
+n(1) = Γ(2λ + n)
+Γ(2λ)n!
+= (2λ)n
+n!
+for all n ∈ N0
+if λ > 0; further, Tn(1) = 1 for all n ∈ N0. In particular, Cλ
+0 ≡ 1 ≡ T0. Of course, for n > 0
+none of these functions is a probability density with respect to νd. However, it is known
+that the Chebyshev polynomials and, for λ > 0, the Gegenbauer polynomials attain their
+absolute maximum on [−1, 1] at t = 1; see Erd´elyi et al. (1953, p. 206, formula (7)) and
+Abramowitz and Stegun (1964, p. 786). Hence the standardization
+Dλ
+n(t) :=
+�
+Tn(t)
+if λ = 0,
+Cλ
+n(1)−1 Cλ
+n(t)
+if λ > 0,
+provides a sequence (Dλ
+n)n∈N0 of orthogonal polynomials that are bounded in absolute value
+on [−1, +1] by their value 1 in t = 1. As (Dλ
+n)n∈N0 is complete in Hλ, we have the series
+expansion converging in Hλ
+gρ =
+∞
+�
+n=0
+⟨gρ, Dλ
+n⟩λ ∥Dn∥−2
+λ Dλ
+n = 1 +
+∞
+�
+n=1
+βn(ρ) Dλ
+n,
+(22)
+with βn(ρ) := ⟨gρ, Dλ
+n⟩λ ∥Dn∥−2
+λ
+for n ∈ N. We assume throughout this section that gρ is
+such that
+β(ρ) :=
+∞
+�
+n=1
+|βn(ρ)| < ∞.
+(23)
+For fixed n ∈ N0 and η ∈ Sd the function Hλ
+n,η : Sd → R defined by
+Hλ
+n,η(x) = Dλ
+n(ηtx), x ∈ Sd,
+is the unique surface harmonic of degree n that depends only on ηtx and that satisfies
+Hλ
+n,η(η) = 1; see Erd´elyi et al. (1953, p. 238). Obviously, Hλ
+n,η is invariant under the sub-
+group {U ∈ O(d + 1) : Uη = η} of O(d + 1), i.e. for all elements U of the subgroup it holds
+that Hλ
+n,η(Ux) = Hλ
+n,η(x) for all x ∈ Sd.
+Let
+γ0
+0 := 1, γ0
+n := 1/2
+for all n ∈ N,
+γλ
+n :=
+1
+(1 + n/λ) Cλn(1)
+for all λ > 0, n ∈ N0.
+(24)
+We will repeatedly make use of the basic formulas
+�
+Dλ
+m(ηtx)Dλ
+n(ξtx) unif(Sd)(dx) = 0,
+η, ξ ∈ Sd, m, n ∈ N0, m ̸= n,
+(25)
+and
+�
+Dλ
+n(ηtx)Dλ
+n(ξtx) unif(Sd)(dx) = γλ
+nDλ
+n(ηtξ),
+η, ξ ∈ Sd, n ∈ N0;
+(26)
+10
+
+see e.g. Erd´elyi et al. (1953, p. 245) and Saw (1984, formula (1.14)).
+The mixing bases considered in what follows are of a very simple structure: The densities
+of the base distributions are built with only two special surface harmonics. To be precise,
+for n ∈ N and real numbers −1 ≤ α ≤ +1 the functions
+x �→ 1 + αDλ
+n(ηtx), x ∈ Sd,
+(27)
+are unif(Sd)-densities of probability distributions ∆λ
+n,η,α on Sd. These distributions can be
+regarded as a multivariate generalization of the cardioid distributions introduced by Jeffreys
+(1948, p. 302); see also Mardia and Jupp (2000, Section 3.5.5). Let ∆λ
+0,η,α := unif(Sd). Here
+we mainly deal with the special distributions ∆λ
+n,η := ∆λ
+n,η,1, but see also Remark 10 (a)
+and Proposition 12 below. So, for n ∈ N the unif(Sd)-density of ∆λ
+n,η is simply the sum of
+the two special surface harmonics Hλ
+0,η ≡ 1 and Hλ
+n,η.
+With the mixing base ∆λ
+n,η, n ∈ N0, given for each η ∈ Sd, we obtain discrete mixture
+representations for all spherical distributions with densities of the form (21) that are not
+too far away from unif(Sd). This may be seen as an instance of the perturbation approach
+discussed in the survey paper of Pewsey and Garc´ıa–Portugu´es (2021) and, indeed, the value
+of β(ρ) may be interpreted as a distance between νd and the measure with νd-density gρ.
+By an ultraspherical mixing base we mean a family {∆λ
+n,η : n ∈ N0}.
+The following general formula is an immediate consequence of the above definitions and
+the expansion in (22).
+Proposition 7. Suppose that βn(ρ) ≥ 0 for all n ∈ N and that β(ρ) ≤ 1. Let η ∈ Sd. Then
+Pη,ρ has the discrete mixture expansion
+Pη,ρ =
+∞
+�
+n=0
+wρ(n) ∆λ
+n,η,
+(28)
+with wρ(0) = 1 − β(ρ) and wρ(n) = βn(ρ) for all n ∈ N.
+Applying this construction to several specific families we have to take care of the crucial
+condition β(ρ) ≤ 1, equivalently wρ(0) ≥ 0. In each case, we obtain the mixing distribution
+and a range of ρ-values for the validity of the representation. Any distribution on N0 may
+be written as a mixture of unit mass at 0 and a distribution on N and it turns out that
+the latter are occasionally from a standard family.
+For a distribution on N0 with mass
+function w on N0 such that w(0) < 1 we call the distribution on N with mass function
+n �→ w(n)/(1 − w(0)), n ∈ N, its zero-truncated counterpart. Some of the results stated in
+what follows turn out to be simple consequences of Proposition 7.
+In the first theorem we consider two families of wrapped distributions, hence d = 1 and
+λ = 0. There are different notational conventions in the literature; here, we regard the
+wrapped distribution associated with a given distribution µ on (the Borel subsets of) the
+real line as the push-forward µT of µ under the mapping T : R → S1, x �→ (cos(x), sin(x))t.
+This is often applied to location-scale families. Alternatively, the interval [−π, π) is used
+instead of S1 as the base set for the wrapped distribution. This means that with X ∼ µ one
+deals with the [−π, π)-valued random variable X0 as the variable X reduced modulo 2π. If
+X has the density f with respect to the Lebesgue measure, then X0 has the unif ([−π, +π))-
+density 2π �+∞
+n=−∞ f(s+2πn), s ∈ [−π, π). If the characteristic function ϕ of X is absolutely
+integrable, then f is continuous and the Poisson summation formula applies, i.e.
+2π
++∞
+�
+n=−∞
+f(s + 2πn) =
++∞
+�
+n=−∞
+ϕ(n) e−ins,
+s ∈ [−π, π);
+(29)
+11
+
+see Feller (1971, p. 632). For example, the wrapped normal distribution WN1(η, ρ) arises from
+the normal distribution N(α, σ2) with mean α and variance σ2, where η = (cos(α), sin(α))t
+and ρ = σ2. Note that some authors use ρ = 2σ2, see, e.g. Hartmann and Watson (1974).
+With X ∼ N
+�
+α, σ2�
+we deduce from (29) that X0 has the density
+1 + 2
+∞
+�
+n=1
+e−n2ρ/2 cos n(s − α),
+s ∈ [−π, +π),
+which means that T ◦ X ∼ WN1(η, ρ) has the unif(S1)-density
+f WN
+1
+(x|η, ρ) = 1 + 2
+∞
+�
+n=1
+e−n2ρ/2 Tn(ηtx),
+x ∈ S1.
+(30)
+It is worthwhile to note that as ρ → 0 the distribution WN1(η, ρ) converges weakly to
+the one-point mass distribution at η. This is in contrast to other distributions considered
+here.
+For example, MF1(η, ρ), Wat1(η, ρ), AG1(η, ρ) all converge weakly to the uniform
+distribution unif(S1) as ρ → 0.
+Also, as ρ → ∞, the distribution WN1(η, ρ) converges
+weakly to unif(S1).
+For the wrapped Cauchy distribution WC1(η, ρ) we follow the definition given by Pewsey and Garc´ıa–Portugu´es
+(2021): If X has a standard Cauchy distribution with density x �→ 1/(π(1 + x2)), x ∈ R,
+then we apply the wrapping procedure to Y := σX + α, where σ > 0, α ∈ R, and take η as
+in the wrapped normal case. For the scaling we use the parametrization ρ := e−σ ∈ (0, 1)
+and augment this with the limiting uniform distribution at ρ = 0. Using (29) again we
+obtain that the distribution WC1(η, ρ) has the density
+f WC
+1
+(x|η, ρ) = 1 + 2
+∞
+�
+n=1
+Tn(ηtx)ρn =
+1 − ρ2
+1 − 2ρ ηtx + ρ2 ,
+x ∈ S1;
+(31)
+see also Pewsey and Garc´ıa–Portugu´es (2021).
+We write geoN0(n|p) = p(1 − p)n, n ∈ N0, for the probability mass function of the
+geometric distribution on N0 with parameter p ∈ (0, 1), and geoN for the mass function of
+its zero-truncated counterpart. Recall that the function β for a given distribution family is
+defined in (23).
+Theorem 8. (a) For the wrapped Cauchy distributions we have β(ρ) = 2ρ/(1 − ρ) and, for
+0 ≤ ρ ≤ 1/3,
+WC1(η, ρ) = (1 − β(ρ)) unif(S1) + β(ρ)
+∞
+�
+n=1
+geoN(n|1 − ρ) ∆0
+n,η.
+(b) For the wrapped normal distributions we have β(ρ) = 2 �∞
+n=1 e−n2ρ/2. Let ρ0 ≈ 1.570818
+be the unique solution of the equation β(ρ) = 1. Then, with
+br(n|ρ) := 2β(ρ)−1e−n2ρ/2
+for all n ∈ N,
+and ρ ≥ ρ0, it holds that
+WN1(η, ρ) = (1 − β(ρ)) unif(S1) + β(ρ)
+∞
+�
+n=1
+br(n|ρ) ∆0
+n,η.
+12
+
+We deal with the von Mises–Fisher families next. For these, we need variants of the
+Skellam distribution with parameter ρ. This distribution arises as the distribution of N1−N2
+where N1, N2 are independent random variables that both have the Poisson distribution with
+parameter ρ/2; see Irwin (1937), and see Skellam (1946) where the more general case with
+possibly different means for N1 and N2 is considered. In the one-dimensional case we need
+the positive Skellam distribution, which is the conditional distribution of |N1 − N2| given
+that N1 ̸= N2. The associated probability mass function is given by
+psk(n|ρ) =
+2e−ρIn(ρ)
+1 − e−ρI0(ρ),
+n ∈ N.
+(32)
+For d > 1 we use the generalized positive Skellam distribution with parameters κ > 0 and
+τ > 0, with mass function
+gpsk(n|κ, τ) :=
+�
+1 + n
+κ
+� (2κ)n
+n!
+2κΓ(κ + 1)τ−κe−τIκ(τ)
+1 − 2κΓ(κ + 1)τ −κe−τIκ(τ)
+Iκ+n(τ)
+Iκ(τ) ,
+n ∈ N.
+It will turn out as part of the proof of the next result that this is indeed a probability
+mass function. We have limκ→0 1
+κ
+(2κ)n
+n!
+=
+2
+n, which gives limκ→0 gpsk(n|κ, τ) = psk(n|τ)
+for all n ∈ N. Hence the positive Skellam distribution appears as the limiting case of the
+generalized positive Skellam distribution as κ → 0.
+Theorem 9. We consider the von Mises–Fisher families {MFd(η, ρ) : ρ ≥ 0}, η ∈ Sd.
+(a) If d = 1 then β(ρ) = eρ/I0(ρ) − 1, the equation β(ρ) = 1 has a unique finite positive
+solution ρ0 ≈ 0.876842, and for 0 ≤ ρ ≤ ρ0 it holds that
+MF1(η, ρ) = (1 − β(ρ)) unif(S1) + β(ρ)
+∞
+�
+n=1
+psk(n|ρ) ∆0
+n,η.
+(b) If d > 1 then
+β(ρ) = βλ(ρ) :=
+ρλeρ
+2λΓ(λ + 1)Iλ(ρ) − 1,
+the equation βλ(ρ) = 1 has a unique finite positive solution ρ0(λ), and for 0 ≤ ρ ≤ ρ0(λ) it
+holds that
+MFd(η, ρ) = (1 − βλ(ρ)) unif(Sd) + βλ(ρ)
+∞
+�
+n=1
+gpsk(n|λ, ρ)∆λ
+n,η.
+Remark 10. (a) The condition that βn(ρ) ≥ 0 for all n ∈ N in Proposition 7 is satisfied
+in all of the above families, but it can easily be removed by an appropriate extension of the
+mixing base. For this, let ∆λ,−
+n,η be the distribution with density x �→ 1 − Dλ
+n(ηtx), x ∈ Sd.
+Then the representation (28) continues to hold if we take Qn,η = ∆λ,−
+n,η and wρ(n) = −βn(ρ)
+whenever βn(ρ) < 0.
+(b) The condition β(ρ) ≤ 1 in Proposition 7 holds if Pη,ρ is sufficiently close to the uniform
+distribution on the sphere. If instead of Pη,ρ we consider a mixture of this distribution with
+the uniform, with enough weight on the latter, then the result is close enough to the uniform,
+and we again obtain a mixture representation. In the von Mises–Fisher case with d = 1, for
+example, we get for ρ > ρ0 and with α(ρ) := (1 − 2e−ρI0(ρ))/(1 − e−ρI0(ρ)),
+α(ρ) unif(S1) + (1 − α(ρ)) MF1(η, ρ) =
+∞
+�
+n=1
+psk(n|ρ) ∆0
+n,η.
+13
+
+(c) Is there a countable mixing base that represents all isotropic spherical distributions
+with densities of the form x �→ g(ηtx) with η ∈ Sd and g ∈ Hλ? This may be rephrased
+in terms of the set of extremal points of a convex set in an infinite dimensional space. We
+refer to Baringhaus and Gr¨ubel (2021b) for such geometric aspects in general, and for the
+construction of tree-based mixing bases that would lead to a positive answer for the set of
+all g ∈ Hλ that are Riemann integrable.
+The passage from an L2-expansion (22) of gρ to the mixture representation (28) heavily
+relies on the nonnegativity of the functions 1 + Dλ
+n (respectively 1 − Dλ
+n in part (a) above).
+More generally, we may consider a mixing base (Qn,η)n∈N where the density of Qn,η is a
+polynomial of degree n in ηtx. This leads to the consideration of general linear combina-
+tions of ultraspherical polynomials; indeed, finding conditions for such polynomials to be
+nonnegative (on a given interval) is an ongoing research topic, see Askey (1975).
+We confine ourselves to an example with λ = 0 and the Chebyshev polynomials. A
+change of mixing base will obviously lead to a change in the sequence of mixing coefficients.
+It turns out that this may lead to a representation of wider applicability.
+Example 11. We define functions gρ : [−1, 1] → R+, 0 ≤ ρ < 1, by
+gρ(t) := (1 − ρt + φρ(t))1/2
+21/2φρ(t)
+,
+−1 ≤ t ≤ 1,
+(33)
+where φρ(t) := (1 − 2ρt + ρ2)1/2. These can be written as
+gρ(t) =
+∞
+�
+n=0
+(1/2)n
+n!
+Tn(t) ρn,
+(34)
+see Magnus et al. (1966, p. 259). In particular,
+� 1
+−1 gρ(t) dt = 1, so that we may define a
+family of distributions Pη,ρ on S1 via their densities x �→ gρ(ηtx), x ∈ S1.
+We first derive an expansion in terms of the distributions ∆0
+nη. With t = 1 we get
+β(ρ) =
+∞
+�
+n=1
+(1/2)n
+n!
+ρn = (1 − ρ)−1/2 − 1,
+and it follows that β(ρ) ≤ 1 if and only if ρ ≤ 1 − 2−1/2 =: ρ0. We now introduce the zero-
+truncated negative binomial distribution with parameters r > 0, p ∈ (0, 1), and probability
+mass function
+znb(n|r, p) = (r)n
+n! (1 − p)n
+pr
+1 − pr ,
+n ∈ N.
+Then (34) leads to the discrete mixture representations
+Pη,ρ =
+�
+1 − β(ρ)
+�
+unif(S1) + β(ρ)
+∞
+�
+n=1
+znb(n|1/2, 1 − ρ) ∆0
+n,η,
+(35)
+for all η ∈ S1, 0 < ρ ≤ ρ0.
+On the other hand, Turan (1953) proved that, for all n ∈ N0,
+n
+�
+k=0
+(1/2)k
+k!
+cos(kϑ) > 0
+for 0 < ϑ < π.
+14
+
+In view of Tk(cos ϑ) = cos(kϑ) this can be used to obtain an alternative representation.
+For this, let Σn,η be the distribution on S1 with density x �→ �n
+k=0 αkTk(ηtx), where
+αk := (1/2)k/k! and n ∈ N0. Then (34), together with a summation by parts, leads to
+Pη,ρ =
+∞
+�
+n=0
+geoN0(n|1 − ρ) Σn,η = (1 − ρ) unif(S1) + ρ
+∞
+�
+n=1
+geoN(n|1 − ρ) Σn,η,
+(36)
+where geoN0 and geoN are as in Theorem 8. Both (35) and (36) hold for all η ∈ S1, but
+note that the range of permissible ρ-values has increased from 0 ≤ ρ ≤ 1 − 2−1/2 to the full
+interval 0 ≤ ρ < 1.
+As pointed out earlier mixing families constructed with surface harmonics can be used
+with all distributions of the form (21) as long as these are sufficiently close to the uniform.
+The following result gives a property which we interpret as self-mixing stability of the dis-
+tribution families Dλ
+0 := {unif(Sd)}, Dλ
+n := {∆λ
+n,η,α : η ∈ Sd, −1 ≤ α ≤ +1}, n ∈ N,
+and Dλ := �
+n∈N0 Dλ
+n: Mixing two elements of the same family results in a distribution
+that belongs to this family as well. Additionally, mixing any two elements moves the mix-
+ing distribution closer to the uniform distribution. Generally, the mixing operation relates
+distributions with different location parameter η ∈ Sd to each other.
+Proposition 12. Let γλ
+n be the constants defined in (24).
+(a) For all n ∈ N0 and η ∈ Sd, −1 ≤ α, β ≤ +1,
+�
+∆λ
+n,ζ,α(A) ∆λ
+n,η,β(dζ) = ∆λ
+n,η,γλ
+nαβ(A)
+= (1 − γλ
+n) unif(Sd)(A) + γλ
+n ∆λ
+n,η,αβ(A)
+for all A ∈ B(Sd).
+(b) For all n, m ∈ N0 with n ̸= m, and all η ∈ Sd, −1 ≤ α, β ≤ +1,
+�
+∆λ
+m,ζ,α(A) ∆λ
+n,η,β(dζ) = unif(Sd)(A)
+for all A ∈ B(Sd).
+We recall that a probability kernel from a measurable space (E, E) to another measurable
+space (F, F) is a function Q : E × F → R that is E-measurable in its first and a probability
+measure on (F, F) in its second argument. Given a probability measure P on (E, E) and a
+kernel Q from (E, E) to (F, F) we define a probability measure P ◦ Q on (F, F) by
+P ◦ Q(A) =
+�
+Q(x, A) P(dx)
+for all A ∈ F.
+(37)
+For a family {Qx : x ∈ E} of probability measures Qx on (F, F) with the property that the
+map x �→ Qx(A) is E-measurable for all A ∈ F, we may regard Q· : E × F → R defined by
+Q·(x, A) = Qx(A), (x, A) ∈ E × F, as a Markov kernel from (E, E) to (F, F). Then (37)
+reads
+P ◦ Q·(A) =
+�
+Qx(A) P(dx)
+for all A ∈ F,
+which may be interpreted as a mixing operation.
+For use in the next section we note
+that for a family {Pη : η ∈ Sd} of probability measures Pη on (Sd, B(Sd)) and a kernel Q
+from (Sd, B(Sd)) to (Sd, B(Sd)) that are both isotropic in the sense defined by (8) and (9)
+respectively, the mixing results in an isotropic family again,
+�
+Pη ◦ Q
+�U = PUη ◦ Q
+for all η ∈ Sd and all U ∈ O(d + 1).
+(38)
+15
+
+Further, for α = β = 1 the statements in Proposition 12 can be simply rephrased as
+∆λ
+n,η ◦ ∆λ
+n,· = (1 − γλ
+n) unif(Sd) + γλ
+n ∆λ
+n,η,
+∆λ
+n,η ◦ ∆λ
+m,· = unif(Sd)
+if n ̸= m.
+Using the self-mixing stability and the bilinearity of the operation defined in (37) we ob-
+tain a discrete mixture representation for the composition of two families that both have a
+representation in terms of the ultraspherical mixing base.
+Proposition 13. Suppose that Pη and P ′
+η, η ∈ Sd, are distribution families on Sd with
+discrete mixture representations Pη = �∞
+n=0 w(n)∆λ
+n,η and P ′
+η = �∞
+n=0 w′(n)∆λ
+n,η respec-
+tively. Then
+Pη ◦ P ′
+· =
+∞
+�
+n=0
+˜w(n) ∆λ
+n,η,
+where
+˜w(0) := 1 − �∞
+n=1 ˜w(n) and
+˜w(n) := γλ
+n w(n)w′(n) for all n ∈ N.
+This can be used to obtain the composition of wrapped Cauchy distributions. Indeed,
+taken together, Theorem 8 (a) and Proposition 13 lead to
+WC1(η, ρ) ◦ WC1(·, ρ′) = WC1(η, ρρ′)
+for all ρ, ρ′ ≤ 1/3.
+(39)
+It is worthwhile to point out that the restriction ρ, ρ′ ≤ 1/3 in (39) can be omitted. In fact,
+for all 0 ≤ ρ, ρ′ < 1, using (24), (25), (26), and (31), the density of WC1(η, ρ) ◦ WC1(·, ρ′)
+is easily calculated to be
+�
+f WC
+1
+(x|ξ, ρ′) f WC
+1
+(ξ|η, ρ) unif(S1)(dξ) = f WC
+1
+(x|ξ, ρρ′),
+x ∈ S1.
+In the next section this will be put into a wider context.
+4
+Discrete mixture representations and Markov pro-
+cesses
+Let {Pη,ρ : η ∈ Sd, ρ ∈ I}, I ⊂ R+ an interval, be a family of distributions of the type
+considered in the previous sections. Below we briefly discuss three different connections to
+Markov processes. First, for ρ ∈ I fixed, the corresponding subfamily may be regarded
+as a probability kernel and thus induces a Markov chain on spheres. Second, an isotropic
+diffusion process on Sd leads to a family of the above type via its one-dimensional marginal
+distributions, where η and ρ take over the role of starting point and (transformed) time
+parameter respectively. Third, starting with a discrete mixture representation we may find
+a discrete time Markov chain with marginal distributions equal to elements of the mixing
+base, and thus obtain an almost sure representation of the family as the distributions of the
+chain at random times.
+4.1
+Random walks on spheres
+Let {Qη : η ∈ Sd} be a family of probability distributions that leads to a Markov kernel as
+described at the end of Section 3. Such kernels arise as transition probabilities of Markov
+processes. We may, for example, fix an η ∈ Sd and define a Markov chain (Xn)n∈N0 with
+state space (Sd, B(Sd)) by the requirements that X0 ≡ η and that the distribution of Xn+1
+16
+
+conditionally on Xn = ξ is given by Qξ. For isotropic kernels each transition can be divided
+into two steps that make use of the representation of Sd by [−1, 1]×Sd−1 that also appeared
+in connection with (5) and (6). In the geometrically most familiar case we consider the
+current position as the ‘north pole’, then first choose a latitude and thereafter, indepen-
+dently, a longitude uniformly at random. The result is regarded as the new north pole. A
+generalization of this setup has been considered by Bingham (1972), see also the references
+given there.
+The case d = 1 is somewhat special as the wrapping procedure is a group homomorphism
+from the additive group of real numbers into the multiplicative group S1, regarded as a subset
+of C and endowed with complex multiplication. Wrapping a random walk or a L´evy process
+thus leads to processes with values in S1 that have stationary and independent increments,
+where the latter are now to be understood as ratios rather than differences. In fact, the
+location-scale family of Cauchy distributions arises as the one-dimensional marginals of a
+specific L´evy process, which gives (39) after an appropriate rescaling of the variance param-
+eter. A similar approach, now using Brownian motion on the real line, gives a corresponding
+statement for the wrapped normal distributions.
+We collect some observations in the following result. Recall that the distribution L(X)
+of a Markov chain X = (Xn)n∈N0 with state space (Sd, B(Sd)) is a probability measure on
+the path space (SN0
+d , B
+�
+SN0
+d
+�
+) of the chain, where the σ-field B
+�
+SN0
+d
+�
+is generated by the
+projections πk : SN0
+d
+→ Sd, (xn)n∈N0 �→ xk, k ∈ N0. Any measurable mapping T : Sd → Sd
+may be lifted to a mapping from and to paths by componentwise application.
+Proposition 14. Suppose that Q is an isotropic kernel on Sd and that Xη = (Xη
+n)n∈N0 is
+a Markov chain with start at η ∈ Sd and transition kernel Q.
+(a) The family {L(Xη) : η ∈ Sd} of probability measures on the path space is isotropic in
+the sense that L(Xη)U = L(XUη) for all η ∈ Sd, U ∈ O(d + 1).
+(b) The latitude process Y = (Yn)n∈N0, with Yn := ηtXη
+n for all n ∈ N0, is a Markov chain
+with state space [−1, 1] and start at 1.
+(c) Suppose that Q(η, · ) = �∞
+k=0 w(k)∆λ
+k,η for all η ∈ Sd. Then the representation
+L(Xη
+n) =
+∞
+�
+k=0
+wn(k)∆λ
+k,η
+for all n ∈ N, η ∈ Sd,
+(40)
+holds with w1(k) := w(k) for all k ∈ N0 and, for n > 1,
+wn(k) := (γλ
+k )n−1w(k)n, k ∈ N,
+wn(0) := 1 −
+∞
+�
+k=1
+wn(k).
+The fact that the geodesic distances from the starting point are again a Markov chain
+is an instance of lumpability, see Rogers and Pitman (1981) for a general discussion. That
+the dependence on η is lost in the lumping transition is part of the assertion of part (b).
+Further, it follows from the cosine theorem for spherical triangles that the transition kernel
+QY of Y may be written as
+QY (y, · ) = L
+�
+yZ + U(1 − y2)1/2(1 − Z2)1/2�
+(41)
+with Z, U independent, L(Z) = L(ηtXη
+1 ) and L(U) = νd−1; see also Bingham (1972).
+Part (c) shows that the mixing base introduced in Section 3 is useful in the Markov chain
+context, and (40) may be seen as a discrete mixture representation of the family {Pη,n : η ∈
+17
+
+Sd, n ∈ N}, with Pη,n = L(Xη
+n). It implies that the marginal distributions of the associated
+latitude process are given by
+L(Yn) =
+∞
+�
+k=0
+wn(k) µλ
+k
+for all n ∈ N,
+where µλ
+k is the push-forward of ∆λ
+k,η under x �→ ηtx.
+4.2
+Diffusion processes
+Let X = (Xt)t≥0 be a homogeneous continuous time Markov process on the sphere with
+start at η, i.e. P(X0 = η) = 1, and transition densities pt(x, y), t > 0, x, y ∈ Sd, that are
+isotropic in the sense that pt(Ux, Uy) = pt(x, y) for all t > 0, x, y ∈ Sd and U ∈ O(d + 1).
+Then the marginal distributions L(Xt), t ≥ 0, of the process may have a discrete mixture
+representation of the type considered above, with ρ related to time t.
+We sketch the basic argument, see also Karlin and McGregor (1960), and then apply this
+in the context of the spherical Brownian motion (Bt)t≥0. For a general discussion of the
+latter we refer to Ito and McKean (1997, Section 7.15) and Hsu (2002, Example 3.3.2). We
+note in passing that the marginal distributions of X characterize the full distribution L(X)
+of the process, in view of the Chapman–Kolmogorov equations and the invariance of the
+transition mechanism under orthogonal transformations (clearly, O(d + 1) acts transitively
+on Sd).
+Suppose that X has transition densities pt(x, y) and that its infinitesimal generator A
+has a discrete spectrum. As transitions are isotropic it is enough to consider one specific
+starting value x = η. For the Kolmogorov forward equations
+� ∂
+∂tp.(η, y)
+�
+(t) =
+�
+Apt(η, ·)
+�
+(y)
+we may try to find a family of basic solutions φ by a separation ansatz φ(t, y) = f(t)g(y).
+This leads to
+f ′(t)
+f(t) = (Ag)(y)
+g(y)
+.
+As the left and right hand side respectively depend on t and y only, we may hope that
+pt(η, y) =
+∞
+�
+n=0
+e−ωnt φn,η(y),
+where ωn, n ∈ N0, are the eigenvalues of the operator A, with eigenfunctions φn,η.
+Recall that λ = (d − 1)/2.
+Theorem 15. Let (Bt)t≥0 be the spherical Brownian motion on Sd, d > 1, with start at
+η ∈ Sd. Let
+βλ(t) :=
+∞
+�
+n=1
+�
+1 + n
+λ
+�(2λ)n
+n!
+e−n(n+2λ)t/2
+and let
+brλ
+t (n) : = βλ(t)−1�
+1 + n
+λ
+�(2λ)n
+n!
+e−n(n+2λ)t/2,
+n ∈ N.
+Further, let tλ
+0 be the unique solution of the equation βλ(t) = 1. Then, for t ≥ tλ
+0,
+L(Bt) = (1 − βλ(t)) unif(Sd) + βλ(t)
+∞
+�
+n=1
+brλ
+t (n) ∆λ
+n,η.
+18
+
+In view of the wrapping representation mentioned above the corresponding result for
+d = 1 is contained in part (b) of Theorem 8.
+Let B = (Bt)t≥0 be as in the theorem and let Y = (Yt)t≥0 with Yt = ηtBt, t ≥ 0, be
+the associated latitude process. Then a polar decomposition shows that, for t > 0 fixed, Bt
+can be synthesized (in distribution) from Yt and an independent random variable Z that is
+uniformly distributed on Sd−1. On the level of processes the conditional distribution of B
+given Y is a result known as the skew product decomposition of spherical Brownian motion;
+see Ito and McKean (1997, p. 270) and Mijatovi´c et al. (2020).
+A representation with a different mixing base, closer in spirit to the representations in
+Section 2, has very recently been obtained by Mijatovi´c et al. (2020). The result is based
+on the authors’ observation that for a spherical Brownian motion (Bλ
+t )t≥0 with start at
+η ∈ Sd, d > 1, the rescaled latitude process (Yt)t≥0, Yt := (1 − ηtBλ
+t )/2 for all t ≥ 0, is
+a neutral Wright-Fisher diffusion with both mutation parameters equal to λ. For these,
+a discrete mixture representation had earlier been given by Jenkins and Spano (2017), see
+also Griffiths and Spano (2010). Taken together, this leads to
+L(Bλ
+t ) =
+∞
+�
+n=0
+wλ
+t (n) SBetad(η, 1, n + 1)
+for all t > 0,
+(42)
+with
+wλ
+t (n) =
+∞
+�
+k=n
+(−1)k−n (d + 2k − 1)(d + n)k−1
+n!(k − n)!
+e−k(k+d−1)t/2
+for all n ∈ N0, t > 0,
+where the term (d)−1 appearing with n = k = 0 is defined to be 1/(d−1); see Jenkins and Spano
+(2017, formula (5)). Interestingly, the mixture coefficients turn out to be the individual
+probabilities associated with the marginal distributions of a particular pure death process
+(Zt)t>0, i.e. wλ
+t (n) = P(Zt = n). In contrast to our representation in Theorem 15 via sur-
+face harmonics no further restrictions on the time parameter are needed. Moreover, there
+is also a fascinating probabilistic interpretation, relating neutral Wright-Fisher diffusions to
+Kingman’s coalescent via moment duality; see Mijatovi´c et al. (2020) for the details.
+Mardia and Jupp (2000) call the L(Bλ
+t ) Brownian motion distributions on Sd; Kent
+(1977) regards L(ηtBλ
+t ) as a spherical normal distribution. We adopt the notation of the
+latter. Remembering that Bλ starts in η ∈ Sd, we denote by SNd(η, ρ) = L
+�
+Bλ
+1/ρ
+�
+the
+spherical normal distribution with parameters η ∈ Sd and ρ > 0. Then, interestingly, by
+(42) we have a discrete mixture representation for the family { SNd(η, ρ) : η ∈ Sd, ρ > 0}
+with the same mixing base of spherical beta distributions SBetad(η, 1, n + 1), n ∈ N0,
+as obtained in Section 2 for the von Mises–Fisher, the spherical Cauchy, and the angular
+Gaussian families.
+As explained at the beginning of this subsection, isotropic diffusion processes on the
+sphere may lead to a discrete mixture representation for the family of their marginal dis-
+tributions. Conversely, for a given family of the type considered in the previous sections,
+one might ask for a representation of its elements as the marginals of some diffusion pro-
+cess with values in Sd. The following result answers this question for the von Mises–Fisher
+distributions.
+Theorem 16. There is no homogeneous Markov process X = (Xt)t≥0 on Sd with the
+property that, for all η ∈ Sd,
+L(Xt|X0 = η) ∈
+�
+MFd(η, ρ) : ρ > 0
+�
+for all t > 0.
+(43)
+19
+
+Alternatively one might start with a diffusion on the ambient space Rd+1 and then use
+the transition x �→ x/∥x∥ from Rd+1 to Sd. For example, if B = (Bt)t≥0 is a Brownian
+motion on Rd+1 with start at η ∈ Sd, then X = (Xt)t≥0, with Xt := ∥Bt∥−1Bt for all t ≥ 0,
+represents the family {AGd(η, ρ) : ρ > 0} in the sense that
+L(Xt) = AGd
+�
+η, ρ(t)
+�
+for all t > 0.
+Here the bijection ρ : (0, ∞) → (0, ∞) is given by ρ(t) := (2t)−1/2, t > 0.
+Remark 17. We relate Theorem 16 to the infinite divisibility statement for the von Mises–
+Fisher distributions obtained by Kent (1977). Interestingly, in both cases the proofs are
+based on the same series expansions (57), (59) of the densities f MF
+d
+( · |η, ρ) in terms of
+ultraspherical polynomials. An important ingredient of Kent’s approach is the associative
+convolution algebra (F, ◦d) on the space F of probability measures on [−1, 1] introduced
+by Bingham (1972), where the convolution F1 ◦d F2 of F1, F2 ∈ F is defined to be the
+distribution of S1S2 + Λ(1 − S1)
+1/2(1 − S2)
+1/2, see also (41). Here S1, S2, Λ are independent,
+Si ∼ Fi for i = 1, 2, and Λ ∼ νd−1.
+For d = 0 we take ν0 to be the discrete uniform
+distribution on S0 := {−1, 1}. With this definition a probability measure F ∈ F is said to
+be ◦p-infinitely divisible if for each m ∈ N there exists an Fm ∈ F such that F is equal
+to the m-fold convolution F ◦dm
+m
+of Fm. As F ∈ F is uniquely determined by its Fourier
+transform ϕF : N0 → R defined by
+ϕF (m) =
+�
+Dλ
+m(t) F(dt)
+for m ∈ N0,
+and ϕF1◦dF2 = ϕF1ϕF2 for all F1, F2 ∈ F, it holds that F is ◦d-infinitely divisible if and
+only if for each m ∈ N there is a Fourier transform ϕFm : N0 → R of an Fm ∈ F such that
+ϕF = ϕm
+Fm for all m ∈ N. The distribution MF∗
+d(ρ) of ηtX, with X ∼ MFd(η, ρ), has the
+νd-density
+ρd
+2dΓ(λ+1)Iλ(ρ) exp(ρt), t ∈ [−1, 1]. Kent (1977) showed that MF∗
+d(ρ) is ◦d-infinitely
+divisible. In fact, Kent even gives an interesting representation of the distributions Fm such
+that MF∗
+d(ρ) = F ◦dm
+m
+based on the spherical Brownian motion (Bt)t≥0 on Sd with start at
+η ∈ Sd: He showed that for each m ∈ N there exists an absolutely continuous distribution
+Gm on the positive half-line such that ηtBTm ∼ Fm, where Tm ∼ Gm is independent of
+(Bt)t≥0. Consequently,
+MF∗
+d(ρ) = L
+�
+ηtBTm
+�◦dm
+for all m ∈ N.
+(44)
+In order to lift this from the unit interval to the sphere let η ∈ Sd be fixed and let Pη be the
+family of distributions P on Sd that are axially symmetric with respect to η, i.e. P U = P for
+each U ∈ O(d + 1) with Uη = η, and P U the push-forward of P under U. In order to carry
+over the convolution operation ◦d to Pη, a spherical addition ⊕ on Sd is defined. Assign
+to each x ∈ Sd an element Ux ∈ O(d + 1) in such a way that Uxη = x for all x ∈ Sd and
+that x → Ux is a measurable injection (a measurable embedding) from Sd into O(d + 1).
+Then, for x, y ∈ Sd, let x ⊕ y := Uxy. For P1, P2 ∈ Pη, with independent Sd-valued random
+vectors Xi ∼ Pi, i = 1, 2, the ⋆d-convolution P1 ⋆d P2 of P1 and P2 is defined to be the
+distribution of X1 ⊕ X2. Kent showed that P1 ⋆d P2 ∈ Pη, and with Fi as the distribution
+of ηtXi, i = 1, 2, the cosine theorem for spherical triangles leads to ηt(X1 ⊕ X2) ∼ F1 ◦d F2,
+i.e.
+L
+�
+ηt(X1 ⊕ X2)
+�
+= L(ηtX1) ◦d L(ηtX2).
+From (44) it now follows that MFd(η, ρ) = L
+�
+BTm
+�⋆dm for all m ∈ N.
+20
+
+4.3
+Almost sure representations
+The basic relation (2) connects {Pθ : θ ∈ Θ} to the distributions Wρ on N0 and Qn,η on Sd.
+Isotropy means that we may consider the η-part of the parameter as fixed. In this situation,
+if (Nρ)ρ≥0 and (Xn)n∈N0 are independent stochastic processes such that
+L(Nρ) = Wρ for all ρ ≥ 0,
+L(Xn) = Qn,η for all n ∈ N0,
+then
+Pη,ρ = L(XNρ)
+for all ρ ≥ 0.
+(45)
+Equation (45) may be regarded as an almost sure representation of the distributional equa-
+tion (2). Classical examples of such almost sure representations are the Skorohod coupling
+in connection with distributional convergence, see e.g. Kallenberg (1997, Theorem 3.30),
+and the representation of a sequence of uniformly distributed permutations on the sets
+{0, 1 . . ., n}, n ∈ N, by the Chinese restaurant process, see e.g. Pitman (2006, Section 3.1).
+Of course, such representations are most useful if the successive variables are close to each
+other (and not simply chosen to be independent). Our aim here are representations of the
+type (45) for the distribution families considered above. For this, we first formalize the polar
+decomposition.
+Let η ∈ Sd, we assume that d > 1. Recall from (6) that Cd(η, y) = {z ∈ Sd : ηtz = y}
+and let
+nη : Sd \ {−η, η} → Cd(η, 0),
+x �→
+x − (ηtx)η
+∥x − (ηtx)η∥,
+be the normalized projection onto the orthogonal complement of the linear subspace of Rd+1
+spanned by η. This can be extended to the whole of the sphere by choosing some arbitrary of
+ξ ∈ Sd as the value of nη(±η). Then an inverse of the polar decomposition x �→ (ηtx, nη(x))
+is given by
+Ψη : [−1, 1] × Cd(η, 0) → Sd,
+(y, z) �→ yη + (1 − y2)1/2z,
+in the sense that Ψη(ηtx, nη(x)) = x for all x ∈ Sd, and a random vector X with values in
+Sd may be written as
+X = Ψη
+�
+ηtX, nη(X)
+�
+.
+(46)
+Let Qη be a distribution on (Sd, B(Sd)) that has a density f with respect to unif(Sd) which
+can be written as fη(x) = g(ηtx), x ∈ Sd, see (7). If X ∼ Qη then ηtX and nη(X) are
+independent, ηtX has the distribution νd;g with νd-density g, and nη(X) ∼ unif (Cd(η, 0));
+see Watson (1983, p. 92) and Mardia and Jupp (2000, p. 169). So, conversely, if we have
+independent random variables Y ∼ νd;g and Z ∼ unif (Cd(η, 0)), then
+X =D Ψη(Y, Z) = Y η + (1 − Y 2)1/2Z.
+(47)
+Mardia and Jupp (2000, p. 161,p. 169) call (46) and the distributional version (47) the
+tangent-normal decomposition.
+In the past this decomposition has been successfully ap-
+plied by many authors treating different problems in directional statistics, see e.g.
+Saw
+(1978, 1983, 1984), Garc´ıa–Portugu´es et al. (2020), and Ulrich (1984). In practice, a ran-
+dom variable X with distribution Qη is simply obtained as follows. Suppose first that η is
+equal to e1 = (1, 0, . . . , 0)t, the first unit vector in the canonical basis of Rd+1. With e1 as
+‘north pole’ the polar representation takes on a particularly simple form: From y ∈ [−1, 1]
+and z = (z1, . . . , zd)t ∈ Sd−1 we get x ∈ Sd by x = Φ(y, z) with
+Φ(y, z) :=
+�
+y , (1 − y2)1/2 z1, . . . , (1 − y2)1/2 zd
+�t .
+(48)
+21
+
+Then, starting with a random variable Y ∼ νd;g and another random variable Z ∼ unif(Sd−1)
+independent of Y , we obtain an Sd-valued random variable X with distribution Qe1 via
+X := Φ(Y, Z). For a general η ∈ Sd we use that Qη is the push-forward QU
+e1 of Qe1 under
+the mapping x �→ Ux, where U ∈ O(d + 1) is such that η = Ue1, i.e. U has η as its first
+column. Then, defining Φη(y, z) = UΦ(y, z) it follows that X := Φη(Y, Z) ∼ Qη; see also
+Saw (1978) and Ulrich (1984) for this construction.
+Starting with a random variable Y that is almost surely equal to 1, it follows that the
+random variable X = Φη(Y, Z) is almost surely equal to η. This means that from a Markov
+chain (Yn)n∈N0 with state space [−1, 1] starting in 1, where for n ∈ N the distribution of Yn
+is the push-forward of Qn,η under x �→ ηtx, and a single random variable Z ∼ unif(Sd−1),
+we obtain a Markov chain (Xn)n∈N0 with the desired one-dimensional marginal distributions
+by putting Xn := Φη(Yn, Z) for all n ∈ N0. Clearly, (Yn)n∈N0 is then the latitude process
+associated with (Xn)n∈N0.
+This reduces the first step in an almost sure construction (45) to finding a Markov chain
+on [−1, 1] with prescribed marginals. For the second step we require a suitable integer-valued
+process N = (Nρ)ρ≥0 with marginal distributions Wρ. One general possibility is the quantile
+transformation, which can also be used to construct a Skorohod coupling for real random
+variables: With U ∼ unif(0, 1) we obtain a random variable X with distribution function F
+via X := F −1(U), where F −1(u) := inf{t ∈ R; F(t) ≥ u} for 0 < u < 1. If F = Fρ is the
+distribution function associated with Wρ the paths of the process N = (Nρ)ρ≥0 constructed
+in this way depend on the relations between the distribution functions for different ρ’s. In
+particular, if the distributions Wρ are stochastically monotone, meaning that
+1 − Fρ(x) ≤ 1 − Fρ′(x)
+for all x ∈ R
+(49)
+whenever ρ ≤ ρ′, then the paths of N are increasing. It is well known that this stochastic
+monotonicity applies to arbitrary distributions Wρ, Wρ′ with monotone likelihood ratio. To
+be precise, defining a likelihood ratio of Wρ′ with respect to Wρ to be a B(R)-measurable
+function Lρ,ρ′ : R → [0, ∞] such that Wρ(Lρ,ρ′ < ∞) = 1 and
+Wρ′(A) =
+�
+A
+Lρ,ρ′(x) Wρ(dx) + Wρ′({Lρ,ρ′ = ∞} ∩ A)
+for all A ∈ B(R),
+the distributions Wρ, Wρ′ with ρ < ρ′ have monotone likelihood ratio if there exists an
+increasing function hρ,ρ′ : R → [0, ∞] such that
+Lρ,ρ′ = hρ,ρ′ (Wρ + Wρ′)-almost everywhere.
+Note that if fρ, fρ′ are densities of Wρ, Wρ′ with respect to some σ-finite measure, then
+L∗
+ρ,ρ′ = f ′
+ρ
+fρ
+1(fρ > 0) + ∞ 1(fρ = 0, fρ′ > 0)
+is a special version of the likelihood ratio of Wρ′ with respect to Wρ; here 1(·) denotes the
+indicator function. In fact, (49) holds if Wρ, Wρ′ with ρ < ρ′ have monotone likelihood ratio;
+see, e.g. Witting (1985, Satz 2.28).
+Again, we consider the special case of the von Mises–Fisher distributions in some detail.
+Example 18. Let η ∈ Sd. In order to translate Theorem 2, see also Example 1 (c), into
+an almost sure representation for the family {MFd(η, ρ) : ρ ≥ 0} we need random vari-
+ables Yn with Yn ∼ Beta[−1,1](d/2, d/2 + n) for all n ∈ N0.
+To this end let V , Wi,
+i ∈ N0, be independent random variables with V
+∼ Γ(d/2, 1), W0 ∼ Γ(d/2, 1), and
+22
+
+Wi ∼ Exp(1) for all i ∈ N.
+Then, using the well known connection between beta and
+gamma distributions, see e.g. Johnson and Kotz (1970), we obtain independent random
+variables B0 ∼ Beta(d/2, d/2), Bn ∼ Beta(d + n − 1, 1), n ∈ N, via
+B0 :=
+V
+V + W0
+,
+Bn := V + W0 + . . . + Wn−1
+V + W0 + · · · + Wn
+for all n ∈ N,
+and products
+˜Yn :=
+n
+�
+i=0
+Bi =
+V
+V + W0 + . . . + Wn
+∼ Beta(d/2, d/2 + n)
+for all n ∈ N0.
+The transformation Yn := 1 − 2 ˜Yn, n ∈ N0, now gives the desired sequence Y = (Yn)n∈N0.
+Moreover, ˜Yn+1 = ˜YnBn+1 implies that
+Yn+1 = 1 − (1 − Yn) Bn+1.
+(50)
+As Bn+1 is independent of Yn this shows that Y is a Markov chain.
+Suppose now that (Nρ)ρ≥0 is a stochastic process with Nρ ∼ CHS(d/2, d, 2ρ) for all
+ρ ≥ 0. Let Z ∼ unif(Sd−1) be independent of the variables V and Wi, i ∈ N. Then (Xρ)ρ≥0
+with
+Xρ := Φη
+�
+YNρ, Z
+�
+for all ρ ≥ 0,
+(51)
+and Φη as in the remarks preceding the example, has the desired property that Xρ ∼
+MFd(η, ρ) for all ρ ≥ 0.
+For the construction of the counting process we use the quantile transformation. The
+likelihood ratios turn out to be
+chs(n|d/2, d, 2ρ′)
+chs(n|d/2, d, 2ρ) =
+1F1(d/2; d; ρ)
+1F1(d/2; d; ρ′)
+(2ρ′)n
+(2ρ)n
+which, as a function of n, is increasing whenever ρ < ρ′. As explained above, this shows
+that the paths of (Nρ)ρ≥0 are increasing. From (50) it follows that Yn ≥ Yn−1 for all n ∈ N.
+Taken together we see that we have found an almost sure representation (51) with a process
+(YNρ)ρ≥0 that has increasing paths.
+Some comments are in order. Obviously, almost sure representations are generally not
+unique. In the first step of Example 18, we could use a sequence (Yn)n∈N0 of independent
+random variables with Yn ∼ Beta[−1,1](d/2, d/2+n) for all n ∈ N0, or we could use the quan-
+tile transformation to obtain suitable variables Yn as functions of one single U ∼ unif(0, 1)
+(in fact, the corresponding likelihood ratios would be increasing).
+Similar to (3) in the
+classical case, the representation (51) strikes a structural middle ground in this spectrum
+from no dependence at all to total dependence between the variables of interest. Also, the
+Markov chain featuring in the denominator of
+YNρ = 1 −
+2V
+V + �Nρ
+n=0 Wn
+has some resemblance to the sum appearing in (3).
+In Baringhaus and Gr¨ubel (2021a,
+Remark 4 (a)) we found a discrete mixture representation for the non-central family of
+hyperbolic secant distributions that may similarly written as a function of a Markov chain
+of this type.
+23
+
+Returning to the general situation, we may regard the right hand side of (3) or (45) as a
+representation of a continuous time stochastic process Z = (Zt)t≥0 by independent processes
+X = (Xn)n∈N0 and N = (Nt)t≥0 via Zt = XNt for all t ≥ 0. As already mentioned in the
+introduction, equality of the marginal distributions is considerably weaker than equality of
+the distributions of the processes. For example, the representation in Example 18 leads to
+a process that moves by jumps from η on a fixed great circle through η towards the equator
+in a piecewise constant manner. Loosely speaking, a discrete mixture representation on
+the process level is only possible for processes of the pure jump type.
+However, if the
+time parameter of the base process is X continuous too then we obtain a connection to a
+famous group of results, known as skew product decompositions. For example, with (Xt)t≥0
+a Brownian motion on Rd+1 starting at a ̸= 0 and Rt := ∥Xt∥, t ≥ 0, we have R−1
+t Xt = BNt
+for all t ≥ 0, where (Bt)t≥0 is a Brownian motion on Sd starting at η := ∥a∥−1a, Nt is
+given implicitly by
+� Nt
+0
+R−2
+s
+ds = t, and B and N are independent. In particular, this leads
+to a representation of the distribution of Xt/∥Xt∥, t ≥ 0, which is a family of spherical
+distributions, as a continuous mixture. In contrast to discrete mixture representations these
+seem to be less suitable for simulation.
+5
+Proofs
+5.1
+Proof of Theorem 2
+(a) We first simplify the norming constant in (11) for the d-dimensional spherical beta
+distribution with parameters p = 1 and q = n + 1, n ∈ N0. Using the duplication formula
+Γ(1/2)Γ(2z) = 22z−1Γ(z)Γ(z + 1/2),
+z > 0,
+(52)
+for the gamma function we obtain
+cd(1, n + 1) = 2−(n+2λ)
+Γ(1/2)Γ(n + 2λ + 1)
+Γ(λ + 1)Γ(n + λ + 1/2)
+= 2−(n+2λ−1) (2λ + 1)n
+(λ + 1/2)n
+Γ(1/2)Γ(2λ)
+Γ(λ)Γ(λ + 1/2)
+(53)
+= 2−n (2λ + 1)n
+(λ + 1/2)n
+,
+and it follows that the density of SBetad(η, 1, n + 1) can be written as
+f SBeta
+d
+(x|η, 1, n + 1) = 2−n (2λ + 1)n
+(λ + 1/2)n
+�
+1 + ηtx
+�n ,
+x ∈ Sd.
+(54)
+We now write
+exp
+�
+ρ ηtx
+�
+= exp(−ρ) exp
+�
+ρ (1 + ηtx)
+�
+and use the expansion
+exp
+�
+ρ (1 + ηtx)
+�
+=
+∞
+�
+n=0
+ρn
+n!
+�
+1 + ηtx
+�n
+=
+∞
+�
+n=0
+(2ρ)n
+n!
+(λ + 1/2)n
+(2λ + 1)n
+f SBeta
+d
+(x|η, 1, n + 1)(x),
+x ∈ Sd,
+24
+
+together with the identity
+e−ρ
+1F1(λ + 1/2; 2λ + 1; 2ρ) = 2λΓ(λ + 1)Iλ(ρ)/ρλ
+to obtain
+f MF
+d
+(x|η, ρ) =
+∞
+�
+n=0
+chs(n|λ + 1/2, 2λ + 1, 2ρ) f SBeta
+d
+(x|η, 1, n + 1),
+x ∈ Sd.
+In order to prove uniqueness suppose that
+∞
+�
+n=0
+vn SBetad(η, 1, n + 1) =
+∞
+�
+n=0
+wn SBetad(η, 1, n + 1)
+for two sequences v = (vn)n∈N0, w = (wn)n∈N0 of non-negative real numbers with sum 1.
+Passing to the respective push-forwards under x �→ ηtx this leads to the equality of the
+(continuous) densities,
+∞
+�
+n=0
+vn
+n + 1
+2n+1 (1 + y)n =
+∞
+�
+n=0
+wn
+n + 1
+2n+1 (1 + y)n,
+−1 < y < 1,
+hence the sequences v and w are equal to each other.
+(b) From d = 2λ + 1, 1 −
+4ρ
+(1+ρ)2 = (1−ρ)2
+(1+ρ)2 , and (53) it follows that
+f CII
+d
+(x|η, ρ) =
+� 1 − ρ2
+(1 + ρ)2
+�2λ+1 �
+1 −
+2ρ
+(1 + ρ)2
+�
+1 + ηtx
+��−(2λ+1)
+=
+�1 − ρ
+1 + ρ
+�2λ+1
+∞
+�
+n=0
+(2λ + 1)n
+n!
+�
+2ρ
+(1 + ρ)2
+�n �
+1 + ηtx
+�n
+=
+�(1 − ρ)2
+(1 + ρ)2
+�λ+1/2
+∞
+�
+n=0
+(2λ + 1)n
+n!
+�
+4ρ
+(1 + ρ)2
+�n (λ + 1/2)n
+(2λ + 1)n
+f SBeta
+d
+(x|η, 1, n + 1)
+=
+∞
+�
+n=0
+nb
+�
+n|λ + 1/2, (1 − ρ)2/(1 + ρ)2�
+f SBeta
+d
+(x|η, 1, n + 1),
+x ∈ Sd.
+The proof of the uniqueness is similar to that of part (a).
+(c) Noticing (d + 1)/2 = λ + 1, and
+2F1
+�
+λ + 1/2, λ + 1; 2λ + 1; 4ρ/(1 + ρ)2�
+= (1 + ρ)2λ+1
+1 − ρ
+,
+see Magnus et al. (1966, p. 39), we get arguing as in part (b) that
+f CII
+d
+(x|η, ρ) =
+1 − ρ2
+(1 + ρ)2(λ+1)
+�
+1 −
+2ρ
+(1 + ρ)2
+�
+1 + ηtx
+��−(λ+1)
+=
+1 − ρ
+(1 + ρ)2λ+1
+∞
+�
+n=0
+(λ + 1)n
+n!
+�
+4ρ
+(1 + ρ)2
+�n (λ + 1/2)n
+(2λ + 1)n
+f SBeta
+d
+(x|η, 1, n + 1)
+=
+∞
+�
+n=0
+hs
+�
+n|λ + 1/2, λ + 1, 2λ + 1, 4ρ/(1 + ρ)2�
+f SBeta
+d
+(x|η, 1, n + 1),
+x ∈ Sd.
+25
+
+5.2
+Proof of Theorem 3
+(a) Straightforward manipulations give
+f Wat
+d
+(x|η, ρ) =
+1
+1F1(1/2; λ + 1; ρ)
+∞
+�
+n=0
+ρn(ηtx)2n
+n!
+=
+∞
+�
+n=0
+chs(n|1/2, λ + 1, ρ) f SP
+d (x|η, n).
+(b) As at the beginning of the proof of Theorem 2, with p = q = n + 1, n ∈ N0, the
+general expression (11) for the norming constants can be simplified to
+cd(n + 1, n + 1) =
+(λ + 1)n
+(λ + 1/2)n
+.
+Hence the density for the associated spherical beta distribution may be written as
+f SBeta
+d
+(x|η, n + 1, n + 1) =
+(λ + 1)n
+(λ + 1/2)n
+�
+1 − (ηtx)2�n,
+x ∈ Sd.
+(55)
+For ρ < 0 and η ∈ Sd we have
+eρ (ηtx)2 = eρe−ρ(1−(ηtx)2) = eρ
+∞
+�
+n=0
+(−ρ)n
+n!
+(λ + 1/2)n
+(λ + 1)n
+f SBeta
+d
+(x|η, n + 1, n + 1)
+for all x ∈ Sd. Because of
+e−ρ1F1(1/2; λ + 1; ρ) = 1F1(λ + 1/2; λ + 1; −ρ)
+(see, e.g. Magnus et al. (1966, p. 267)) it follows that
+f Wat
+d
+(x|η, ρ) =
+∞
+�
+n=0
+chs(n|λ + 1/2, λ + 1, −ρ) f SBeta
+d
+(x|η, n + 1, n + 1)
+for all η ∈ Sd, ρ < 0.
+In both cases, it is easy to adapt the uniqueness argument from the von Mises–Fisher
+context to the Watson situation.
+5.3
+Proof of Lemma 4
+We write Γ(δ, α) for the gamma distribution with shape parameter δ > 0, scale parameter
+α > 0 and Lebesgue density t �→ Γ(δ)−1αδtδ−1 exp(−αt), t > 0. Let V be a positive random
+variable with V ∼ Γ(δ, 1/2). Then, for k ∈ N0,
+E
+�
+V k/2 exp
+�
+−2
+1/2 τ V
+1/2��
+= Γ(k + 2δ)
+2δ−1Γ(δ) eτ 2/2 D−(k+2δ)(2
+1/2τ).
+Writing
+(δ − 1/2)k
+(2δ − 1)k
+= Γ(δ − 1/2 + k)
+Γ(δ − 1/2)
+Γ(2δ − 1)
+Γ(2δ − 1 + k)
+26
+
+and using the duplication formula (52) we obtain
+dpc(k|δ, τ) = 1
+k!(2
+1/2τ)k2k (δ − 1/2)k
+(2δ − 1)k
+e−τ 2E
+�
+V k/2 exp
+�
+−2
+1/2 τ V
+1/2��
+.
+(56)
+Thus,
+∞
+�
+k=0
+dpc(k|δ, τ) = e−τ 2
+∞
+�
+k=0
+1
+k!(2
+1/2τ)k2k (δ − 1/2)k
+(2δ − 1)k
+E
+�
+V k/2 exp
+�
+−2
+1/2 τV 1/2��
+= e−τ 2E
+�
+exp
+�
+−2
+1/2 τV 1/2�
+1F1(δ − 1/2; 2δ − 1; 2
+3/2τV 1/2)
+�
+.
+Using
+e−z 1F1(δ − 1/2; 2δ − 1; 2z) = Iδ−1(z)Γ(δ)(z/2)−(δ−1),
+for z ∈ R, see e.g. Erd´elyi et al. (1953, p. 265, formula (10)), and
+Iδ−1(z)Γ(δ)(z/2)−(δ−1) =
+∞
+�
+k=0
+(z/2)2k
+k!Γ(k + δ)
+we obtain
+∞
+�
+k=0
+dpc(k|δ, τ) = Γ(δ)e−τ 2
+∞
+�
+k=0
+(τ 2/2)k
+k!Γ(k + δ)E(V k),
+which in view of E(V k) = Γ(k+δ)
+Γ(δ) 2k gives �∞
+k=0 dpc(k|τ, δ) = 1.
+5.4
+Proof of Theorem 5
+Using (12) with S ∼ χ2
+d+1 and writing
+exp
+�
+2
+1/2ρS
+1/2 ηtx
+�
+= exp
+�
+2
+1/2ρS
+1/2 (1 + ηtx)
+�
+exp
+�
+−2
+1/2ρS
+1/2�
+we see that
+f AG
+d
+(x|η, ρ) =
+∞
+�
+n=0
+(1 + ηtx)n
+�
+2
+1/2ρ
+�n
+n!
+e−ρ2E
+�
+Sn/2 exp(−2
+1/2ρS
+1/2)
+�
+=
+∞
+�
+n=0
+f SBeta
+d
+(x|ρ, 1, n + 1)
+�
+2
+1/2ρ
+�n
+n!
+2n (λ + 1/2)n
+(2λ + 1)n
+e−ρ2E
+�
+Sn/2 exp(−2
+1/2ρS
+1/2)
+�
+.
+As χ2
+d+1 = Γ(λ + 1, 1/2), the asserted discrete mixture representation now follows with (56).
+The uniqueness of the representation is obtained as in the proofs of Theorems 2 and 3.
+5.5
+Proof of Theorem 8
+(a) We use Proposition 7. We have β(ρ) = �∞
+n=1 2ρn =
+2ρ
+1−ρ ≤ 1 if and only if ρ ≤ 1/3. By
+(31), the density f WC
+1
+(·|η, ρ) of the wrapped Cauchy distribution WC1(η, ρ) can be written
+as
+f WC
+1
+(x|η, ρ) = 1 − β(ρ) + 2
+∞
+�
+n=1
+�
+1 + Tn(ηtx)
+�
+ρn,
+x ∈ S1.
+27
+
+The representation now follows easily.
+(b) Using (30) we can write the density f WN
+1
+(·|η, ρ) of the wrapped normal distribution
+WN1(η, ρ) as
+f WN
+1
+(x|η, ρ) = 1 − β(ρ) + 2
+∞
+�
+n=1
+e−n2ρ/2 �
+1 + Tn(ηtx)
+�
+,
+x ∈ S1.
+The function β is easily seen to be continuous and strictly decreasing, with unique solution
+ρ0 ≈ 1.570818 of the equation β(ρ) = 1. From Proposition 7 we thus obtain a discrete
+mixture expansion for the family {WN1(η, ρ) : η ∈ S1, ρ ≥ ρ0} with mixing base elements
+∆0
+n,η and weights wρ(n) = 2e−n2ρ/2, n ∈ N.
+5.6
+Proof of Theorem 9
+(a) The density f MF
+1
+(·|η, ρ) of the von Mises–Fisher distribution MF1(η, ρ) can be written
+as
+f MF
+1
+(x|η, ρ) = exp(ρηtx)
+I0(ρ)
+= 1 + 2
+∞
+�
+n=1
+In(ρ)
+I0(ρ) Tn(ηtx),
+x ∈ S1,
+(57)
+see e.g. Abramowitz and Stegun (1964, formula 9.6.34). In fact, the representation (57)
+corresponds to the Fourier series expansion of the density of the unique random angle in
+[−π, π) associated with X ∼ MF1(η, ρ). To be precise, let for simplicity η = (1, 0)t and let
+Θ ∈ [−π, π) be the unique random angle such that X = (cos Θ, sin Θ)t. Then the density
+hρ(θ) = exp(ρ cos θ)
+2πI0(ρ) , θ ∈ [−π, π), of Θ with respect to the Lebesgue measure on [−π, π) has
+the absolutely convergent Fourier series expansion
+hρ(θ) = 1 + 2
+∞
+�
+n=1
+In(ρ)
+I0(ρ) cos(nθ),
+θ ∈ [−π, π);
+see, Kent (1977, formula (1.1a)). In particular, β(ρ) = 2 �∞
+n=1
+In(ρ)
+I0(ρ) = eρ/I0(ρ)−1. We have
+limρ→0 β(ρ) = 0, limρ→∞ β(ρ) = ∞, see Abramowitz and Stegun (1964, formula 9.7.1), and
+d
+dρβ(ρ) = I0(ρ)−2eρ (I0(ρ) − I1(ρ)) > 0
+for all ρ > 0,
+see Soni (1965), hence the function β is strictly increasing. Taken together this implies that
+the equation β(ρ) = 1 has a unique finite positive root ρ = ρ0 ≈ 0.876842. Proposition 7
+now leads to the discrete mixture representation
+MF1(η, ρ) = (2 − eρ/I0(ρ)) unif(S1) + (eρ/I0(ρ) − 1)
+∞
+�
+n=1
+2e−ρIn(ρ)
+1 − e−ρI0(ρ)∆0
+n,η
+= (1 − β(ρ)) unif(S1) + β(ρ)
+∞
+�
+n=1
+psk(n|ρ) ∆0
+n,η .
+(b) For κ > 0, τ > 0, and −1 ≤ t ≤ 1, we have
+eτt = 2κΓ(κ)τ −κ
+∞
+�
+n=0
+(κ + n)Iκ+n(τ)Cκ
+n(t),
+(58)
+28
+
+see Magnus et al. (1966, p. 227) or Kent (1977, Section 7). Hence the density f MF
+d
+(·|η, ρ) of
+MFd(η, ρ) can be written as
+f MF
+d
+(x|η, ρ) = 1 +
+∞
+�
+n=1
+�
+1 + n
+λ
+�(2λ)n
+n!
+Iλ+n(ρ)
+Iλ(ρ) Dλ
+n(ηtx),
+x ∈ Sd.
+(59)
+With t = 1 in (58) we get, after some algebra,
+1 =
+∞
+�
+n=1
+�
+1 + n
+κ
+�(2κ)n
+n!
+2κΓ(κ + 1)τ−κe−τIκ(τ)
+1 − 2κΓ(κ + 1)τ−κe−τIκ(τ)
+Iκ+n(τ)
+Iκ(τ) .
+This implies that gpsk(·|κ, τ) is a probability mass function. Further,
+βλ(ρ) =
+ρλeρ
+2λΓ(λ + 1)Iλ(ρ) − 1 =
+∞
+�
+n=1
+�
+1 + n
+λ
+�Iλ+n(ρ)
+Iλ(ρ) Cλ
+n(1).
+Arguing as in part (a) we find that the equation βλ(ρ) = 1 has a unique finite positive root
+ρ = ρ0(λ). For the family {MFd(η, ρ) : η ∈ Sd, 0 < ρ ≤ ρ0(λ)} this finally gives the discrete
+mixture representation
+MFd(η, ρ) = (1 − βλ(ρ)) unif(Sd) + βλ(ρ)
+∞
+�
+n=1
+gpsk(n|λ, ρ)∆λ
+n,η.
+5.7
+Proof of Proposition 12
+The statements are a consequence of the basic formulas (25) and (26).
+5.8
+Proof of Proposition 13
+All terms involved are positive, so there are no convergence issues, and the representation
+follows from the bilinearity of the mixture operation.
+5.9
+Proof of Proposition 14
+(a) The distribution of X is determined by the distributions of the vectors (X0, . . . , Xn),
+n ∈ N0. We may thus use induction, and (38) provides the necessary argument for the
+induction step.
+(b) Given a north pole η we obtain a partitioning of Sd into the sets Cη(y) = {x ∈
+Sd :
+ηtx = y} with the same latitude y ∈ [−1, 1]. Isotropy implies that the transition
+mechanism interacts with the function that maps the points of Sd to their latitude in the
+manner required by Dynkin’s criterion, see Dynkin (1965, Theorem 10.13). As the action
+of O(d + 1) on Sd is doubly transitive, for unit vectors η1, η2 ∈ Sd with the property that
+ηtη1 = ηtη2 there exists some U ∈ O(d + 1) such that Uη = η and Uη1 = η2. The isotropy
+of Q then implies that for all A ∈ B([−1, 1])
+Q(η2, {x ∈ Sd : ηtx ∈ A}) = Q(Uη1, {x ∈ Sd : ηtx ∈ A})
+= Q(η1, {U tx ∈ Sd : ηtx ∈ A})
+= Q(η1, {y ∈ Sd : ηtUy ∈ A})
+= Q(η1, {y ∈ Sd : ηty ∈ A}).
+(c) This follows easily on using induction and Proposition 13.
+29
+
+5.10
+Proof of Theorem 15
+The transition probabilities for the spherical Brownian motion in dimension d ≥ 2 are given
+in Hartmann and Watson (1974),
+pt(x, y) = 1 +
+∞
+�
+n=1
+�
+1 + n
+λ
+�(2λ)n
+n!
+e−n(n+2λ)t/2 Dλ
+n(xty),
+x, y ∈ Sd, t > 0,
+see also Karlin and McGregor (1960). Let Pt(·, x) be the distribution on Sd with density
+y → pt(x, y), let
+βλ(t) :=
+∞
+�
+n=1
+�
+1 + n
+λ
+�(2λ)n
+n!
+e−n(n+2λ)t/2
+for t > 0,
+and let tλ > 0 be the unique positive real number such that βλ(tλ) = 1. Then, with brd(·|t)
+as in the theorem, we get the discrete mixture representations
+Pt(·, η) = (1 − βλ(t)) unif(Sd) + βλ(t)
+∞
+�
+n=1
+brd(n|t)∆λ
+n,η.
+5.11
+Proof of Theorem 16
+Suppose that, on the contrary, (Xt)t≥0 is a homogeneous Markov process with the property
+that there exists a function κ : (0, ∞) → (0, ∞) such that for all ξ ∈ Sd it holds that if
+P(X0 = ξ) = 1 then
+L(Xt) = MFd (ξ, κ(t))
+for all t > 0.
+Because of the Markov property we would then have positive parameters ρ, ρ′, ρ′′ such that
+MFd(η, ρ) ◦ MFd(·, ρ′) = MFd(η, ρ′′).
+(60)
+To see that this cannot be true, we note that the density of MFd(η, ρ) ◦ MFd(·, ρ′) is given
+by
+h(x) :=
+�
+f MF
+d
+(x|ξ, ρ′) f MF
+d
+(ξ|η, ρ) unif(Sd)(dξ),
+x ∈ Rd.
+We now refer to Kent (1977, Section 7) and the proof of Theorem 9, where it is shown that
+the density f MF
+d
+(·|η, ρ) can be written as
+f MF
+d
+(x|η, ρ) =
+∞
+�
+n=0
+1
+γλn
+Iλ+n(ρ)
+Iλ(ρ)
+Dλ
+n(ηtx)
+for all x ∈ Sd.
+Expressing f MF
+d
+(·|ξ, ρ′) correspondingly, and using (25) and (26), we obtain
+h(x) =
+∞
+�
+n=0
+1
+γλn
+Iλ+n(ρ)
+Iλ(ρ)
+Iλ+n(ρ′)
+Iλ(ρ′)
+Dλ
+n(ηtx)
+for all x ∈ Sd.
+Hence by (60) we would have
+Iλ+n(ρ)
+Iλ(ρ)
+Iλ+n(ρ′)
+Iλ(ρ′)
+= Iλ+n(ρ′′)
+Iλ(ρ′′)
+for all n ∈ N0.
+(61)
+30
+
+From (10) it is easily seen that for each τ > 0 the asymptotic relation
+Iλ+n(τ) ∼ 2−(λ+n)Γ(λ + n + 1)−1τ λ+n
+as n → ∞
+holds. Thus,
+log Iλ+n(τ) = −(λ + n) log 2 − log Γ(λ + n + 1) + (λ + n) log τ + o(1)
+as n → ∞.
+By Stirling’s formula,
+log Γ(λ + n + 1) = (λ + n + 1/2) log n − n + 1
+2 log (2π) + o(1)
+as n → ∞.
+From this we deduce that
+lim
+n→∞
+log Iλ+n(τ)
+Iλ(τ)
+n log n
+= −1,
+which is in contradiction to (61).
+Acknowledgment
+The authors would like to thank the reviewers for helpful suggestions.
+Statements and Declarations
+The authors declare that they have no conflict of interest.
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diff --git a/bNE2T4oBgHgl3EQfaAeA/content/tmp_files/load_file.txt b/bNE2T4oBgHgl3EQfaAeA/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,860 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf,len=859
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='03870v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='PR]  10 Jan 2023  Discrete mixture representations of spherical distributions  Ludwig Baringhaus∗ and Rudolf Gr¨ubel  Institute of Actuarial and Financial Mathematics  Leibniz Universit¨at Hannover  Postfach 60 09, D-30060 Hannover, Germany  lbaring@stochastik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='uni-hannover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='de  rgrubel@stochastik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='uni-hannover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='de  Abstract  We obtain discrete mixture representations for parametric families of probability  distributions on Euclidean spheres, such as the von Mises–Fisher, the Watson and the  angular Gaussian families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' In addition to several special results we present a general  approach to isotropic distribution families that is based on density expansions in terms  of special surface harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' We discuss the connections to stochastic processes on  spheres, in particular random walks, discrete mixture representations derived from  spherical diffusions, and the use of Markov representations for the mixing base to  obtain representations for families of spherical distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Keywords: Mixture distribution, isotropy, surface harmonics, self-mixing stable dis-  tribution families, almost sure representations, skew product decomposition  MSC2020-Mathematics Subject Classification: Primary 62H11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' secondary 60E05  1  Introduction  A discrete mixture representation for a parametric family {Pθ : θ ∈ Θ} of probability  measures in terms of another family {Qn : n ∈ N0} of probability measures, the mixing  base, all defined on the same measurable space, is of the form  Pθ =  ∞  �  n=0  wθ(n) Qn  for all θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (1)  Here, for each θ ∈ Θ, the mixing coefficients (wθ(n))n∈N0 are the individual probabilities  of a distribution Wθ, the mixing distribution, on (the set of subsets of) N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' A classical  case is the representation of non-central chisquared distributions with k degrees of freedom,  Pθ = χ2  k(θ2) with non-centrality parameter θ2 > 0, as Poisson mixtures of central chisquared  distributions, where Qn = χ2  2n+1 := χ2  2n+1(0) and where Wθ is the Poisson distribution with  mean λ = θ2/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' the books of Liese and Miescke (2008) and M¨orters and Peres (2010)  where the representation appears in statistics in connection with the power of statistical tests  and in probability theory in connection with the local times of Markov processes respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  ∗Corresponding author  1  Such mixture representations can be related to two-stage experiments: In order to obtain a  value x with distribution Pθ we first choose n according to Wθ and then choose x according  to Qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' This leads to an immediate application of discrete mixture representations in the  context of simulation methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  In the present paper we continue our previous investigations (see Baringhaus and Gr¨ubel  (2021a,b)), and now specifically consider distributions on the Euclidean sphere Sd := {x ∈  Rd+1 : ∥x∥ = 1} of (d + 1)-dimensional real vectors of unit length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' This case seems to us to  deserve some interest, in particular if specific properties of spheres are taken into account:  The group O(d+1) of orthogonal transformations of the ambient space Rd+1 acts transitively  on Sd, and there is a ‘polar decomposition’ (or ‘tangent-normal decomposition’, see Section  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='3) that relates Sd to [−1, 1] × Sd−1 if d > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  We generally assume that the distribution parameters in the above general setup are of  the form θ = (η, ρ), where η ∈ Sd may be seen as a location parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' instead of Pθ we also  write Pη,ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' We obtain mixture representations that split the dependence on the two parts  of the parameter in the sense that  Pη,ρ =  ∞  �  n=0  wρ(n) Qn,η  for all θ = (η, ρ) ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (2)  In particular, the mixing distributions depend on ρ only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' For fixed ρ on the left, or fixed  n ∈ N0 on the right hand side of (2), the families {Pη,ρ : η ∈ Sd} respectively {Qn,η : η ∈ Sd}  are parametrized by the sphere and are defined on its Borel subsets B(Sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' We assume  that these families interact with the group action mentioned above in the sense that they  are isotropic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' see (8) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' In particular, their elements are then rotationally symmetric  about the axis specified by η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' As a simple application of the representation (2) we mention  that with the finite sums Rη,ρ(A) := �n  k=0 wθ(n) Qk(A) we have monotonically increasing  approximations of the probabilities Pη,ρ(A), A ∈ B(Sd), with uniform error bounds in the  sense that  0 ≤ Pη,ρ(A) − Rη,ρ(A) ≤  ∞  �  k=n+1  wρ(n)  for all η ∈ Sd, A ∈ B(Sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  The literature contains several other applications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' see for example the relation to nonpara-  metric Bayesian inference in Baringhaus and Gr¨ubel (2021b, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  In Section 2 we collect some basic notation and obtain mixture representations for the  von Mises–Fisher family and two spherical Cauchy families in Theorem 2, the Watson  family in Theorem 3, and an angular Gaussian family in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' The mixing bases  are chosen specifically for the respective family, with a view towards reflecting its prop-  erties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' A different base will generally lead to a different representation, as demonstrated  by Baringhaus and Gr¨ubel (2021a) in the context of non-central chisquared distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  In Section 3 we present a general approach that uses expansions of densities in terms of  special surface harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' The resulting mixing base has a structural property that we call  self-mixing stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' This property makes it comparably easy to relate different expansions  to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' We obtain representations with this mixing base for the wrapped Cauchy  and the wrapped normal families in Theorem 8, and for the von Mises–Fisher families in  Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' These results only hold under conditions on the parameter ρ that ensure that  the respective distribution is not too far away from the uniform distribution on the sphere;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  in Example 11 we work out a possibility for extending this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  For fixed ρ or n we may regard Pη,ρ and Qn,η as probability kernels via (η, A) �→ Pη,ρ(A),  (η, A) �→ Qn,η(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' This provides a general connection with Markov processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' We briefly  2  return to the classical mixture representation of non-central one-dimensional chisquared  distributions, which may be written as  (X + θ)2 =D X2 + 2  N(θ)  �  j=1  Ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (3)  Here the random variables N(θ), X, E1, E2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' are independent, X has the standard normal  distribution, E1, E2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' are exponentially distributed with mean 1, and N(θ) has the Poisson  distribution with parameter θ2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' The path-wise point of view displays the distributions  χ2  1(θ), θ ≥ 0, as the distributions of randomly stopped partial sums of independent random  variables, and (3) may be used to read off stochastic monotonicity and infinite divisibility  of non-central chisquared distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Note that the representation only covers the one-  dimensional marginal distributions of the process ((X + θ)2)θ≥0, as the left hand side of (3)  is obviously not pathwise monotone in the ‘time parameter’ θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Quite generally, (1) can be  related to randomly stopped stochastic processes: If X = (Xn)n∈N0 is such that Xn has  distribution Qn for all n ∈ N0 then Xτ has distribution Pθ if τ is independent of X and has  distribution Wθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  In Section 4 we discuss several connections between families of spherical distributions  and stochastic processes on spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' We consider random walks on spheres in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='1,  distribution families that arise in connection with diffusion processes in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='2, and the  use of Markov representations of the mixing base in connection with almost sure representa-  tions for distribution families in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' The ultraspherical mixing base from Section 3  will be useful at various stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  For a single transition kernel we obtain a family (Xη  n)n∈N0 of Markov chains indexed  by their initial state η, meaning that Xη  0 = η with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Isotropy of the kernel  then extends to isotropy of the corresponding distributions on the path space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  For the  elements of the mixing base in Section 3 the marginal distributions of these chains have  a particular simple description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Further, isotropy relates a family {Pη : η ∈ Sd} to a  single distribution on [−1, 1] via the latitude projection x → ηtx, with η as ‘north pole’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  For the chain Xη = (Xη  n)n∈N0 we obtain an associated latitude process Y = (Yn)n∈N0 via  Yn := ηtXn for all n ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' For isotropic kernels this is again a Markov chain, now on  [−1, 1] and with start at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Finally, for the von Mises–Fisher distributions we show that a  homogeneous Markov process on the sphere with these as marginal distributions does not  exist, see Theorem 16, and we obtain a result similar to (3), see Example 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Proofs are collected in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Mixing of distributions is a standard topic in probability theory and statistics, see  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Lindsay (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Spherical data and families of spherical distributions have similarly  been investigated for a long time and by many researchers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' standard references are the clas-  sic monograph of Watson (1983) and, more recently, the book of Mardia and Jupp (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  For a review of distributions on spheres we refer to Pewsey and Garc´ıa–Portugu´es (2021),  see also Watson (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Of particular interest for the topics treated here is the very recent  paper of Mijatovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' (2020) where a discrete mixture representation for the marginal  distributions of spherical Brownian motion is developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' More specific references will be  given at the appropriate places below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  2  Generalities and some special results  We need some basic notions and definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' We write X ∼ µ if X is a random variable on  some background probability space (Ω, F, P) with distribution µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Formally, let (Ω, A) and  3  (Ω′, A′) be measurable spaces and suppose that T : Ω → Ω′ is (A, A′)-measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Then the  push-forward P T of a probability measure P on (Ω, A) under T is the probability measure  on (Ω′, A′) given by P T (A) = P(T −1(A)), A ∈ A′, and X ∼ µ is the same as PX = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' For  many of the measurable spaces considered below there is a canonical uniform distribution,  often defined by invariance under a group operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' To avoid tiresome repetitions we agree  that densities refer to the respective uniform distribution if not specified otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  We fix a dimension d ≥ 1, but instead of d we often use  λ = λ(d) := (d − 1)/2,  (4)  as this is common in connection with families of special functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' In particular, whenever  d and λ appear together, they are related by (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' The group O(d + 1) of orthogonal (d +  1) × (d + 1)-matrices U acts on Sd via x �→ Ux, and the uniform distribution unif(Sd) on  the sphere is the unique probability measure on the Borel subsets of Sd that is invariant  under all such transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' For a fixed η ∈ Sd the push-forward νd of unif(Sd) under  the mapping x �→ ηtx has density hd with respect to the uniform distribution unif(−1, 1) on  the interval [−1, 1], where  hd(y) :=  Γ(λ + 1)  Γ(1/2)Γ(λ + 1/2) (1 − y2)λ−1/2,  −1 < y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (5)  Note that this does not depend on η ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Further, if X ∼ unif(Sd) and Y = ηtX ∼ νd,  then the conditional distribution of X given Y = y is the uniform distribution on  Cd(η, y) := {x ∈ Sd : ηtx = y},  (6)  with unif(Cd(η, y)) the unique probability measure on this set that is invariant under the  subgroup {U ∈ O(d + 1) : Uη = η} of O(d + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' This may be seen in the context of the  polar decomposition mentioned in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Conversely, given a probability measure ν on [−1, 1] and a parameter η ∈ Sd, we can con-  struct a distribution µ = µη on Sd via the kernel (y, A) �→ unif(Cd(η, y))(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' In particular,  for bounded and measurable functions φ : Sd → R,  �  φ(x) µ(dx) =  �  [−1,1]  �  Cd(η,y)  φ(x) unif(Cd(η, y))(dx) ν(dy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  For η ∈ Sd and a measurable function g : [−1, 1] → R the function fη : Sd → R given by  fη(x) = g(ηtx),  x ∈ Sd,  (7)  is unif(Sd)-integrable if and only if g is νd-integrable, and then  �  fη(x) unif(Sd)(dx) =  �  g(t) νd(dt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Thus fη is the density of a probability measure on Sd if and only if g is the νd-density of  a probability measure on [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' In particular, a probability density g on [−1, 1] generates  a family {Qη : η ∈ Sd} of spherical distributions via (7), and such families are isotropic in  the sense that  QU  η = QUη  for all U ∈ O(d + 1), η ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (8)  In particular, each Qη is invariant under all rotations with axis η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' As the function η �→  �  A g(ηtx) unif(Sd)(dx) is B(Sd)-measurable for all A ∈ B(Sd), Q· : Sd × B(Sd) → R defined  4  by Q·(η, A) = Qη(A), (η, A) ∈ Sd×B(Sd), is a Markov kernel from (Sd, B(Sd)) to (Sd, B(Sd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Further, if Q is a Markov kernel from (Sd, B(Sd)) to (Sd, B(Sd)) and U ∈ O(d + 1), then the  kernel QU : Sd × B(Sd) → R defined by QU(η, A) = Q(η, U tA), (η, A) ∈ Sd × B(Sd), with  U tA := {U tx : x ∈ A}, is the push-forward of Q under U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' The kernel Q is isotropic if  QU(η, ·) = Q(Uη, ·)  for all η ∈ Sd and all U ∈ O(d + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (9)  Some classical special functions will be needed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Let  (α)n := Γ(α + n)  Γ(α)  ,  α > 0, n ∈ N0,  be the ascending factorials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' The modified Bessel functions Iα of the first kind are given by  Iα(x) =  ∞  �  n=0  1  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Γ(n + α + 1)  �x  2  �2n+α  ,  x ≥ 0,  (10)  with real nonnegative parameter α, the confluent hypergeometric functions are  1F1(α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' x) =  ∞  �  n=0  (α)n  (β)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' xn,  x ∈ R,  with real positive parameters α and β, and the hypergeometric functions are  2F1(α, β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' x) =  ∞  �  n=0  (α)n(β)n  (γ)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' xn,  −1 < x < 1,  with real positive parameters α, β, and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' (a) Let p > −1/2 and let ν be the distribution on [−1, 1] with unif(−1, 1)-density  y �→  Γ(p+λ+1)  Γ(p+1/2)Γ(λ+1/2)|y|2p(1 − y2)λ−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Then ν has νd-density y �→ Γ(1/2)Γ(p+λ+1)  Γ(p+1/2)Γ(λ+1) |y|2p, and  we obtain the spherical power distribution SPd(η, p) with density  f SP  d (x|η, p) = Γ(1/2)Γ(p + λ + 1)  Γ(p + 1/2)Γ(λ + 1) |ηtx|2p,  x ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  We mainly use this with p = n ∈ N0, and then have  f SP  d (x|η, n) = (λ + 1)n  (1/2)n  (ηtx)2n,  x ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (b) Starting with ν = Beta[−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='1](p+λ−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' q+λ−1/2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' q > 1/2−λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' the beta distributions  on [−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' 1] with densities y �→ c(p + λ − 1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' q + λ − 1/2)(1 − y)p+λ−3/2(1 + y)q+λ−3/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' where  c(p + λ − 1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' q + λ − 1/2) = Γ(p + q + 2λ − 1)/  �  2p+q+2(λ−1)Γ(p + λ − 1/2)Γ(q + λ − 1/2)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' we  obtain the spherical beta distributions SBetad(η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' q),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' with densities  f SBeta  d  (x|η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' q) = cd(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' q) (1 − ηtx)p−1(1 + ηtx)q−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  x ∈ Sd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  where the norming constants are given by  cd(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' q) = 2−(p+q+2(λ−1))  Γ(1/2)Γ(λ + 1/2)Γ(p + q + 2λ − 1)  Γ(λ + 1)Γ(p + λ − 1/2)Γ(q + λ − 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (11)  5  (c) The von Mises–Fisher distributions, which we denote by MFd(η, ρ), ρ > 0, arise if  we start with νd-density proportional to y �→ exp(ρy), −1 ≤ y ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' The continuous density  of the associated spherical distribution is  f MF  d  (x|η, ρ) = cd(ρ) exp(ρ ηtx),  x ∈ Sd,  where the norming constants are given by  cd(ρ) =  ρλ  2λΓ(λ + 1)Iλ(ρ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Further, MFd(η, 0) = unif(Sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (d) The Watson distributions Watd(η, ρ), ρ ∈ R, arise if we begin with νd-density pro-  portional to y �→ exp(ρy2), −1 ≤ y ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' The continuous density of the associated spherical  distribution is  f Wat  d  (x|η, ρ) = cd(ρ) exp  �  ρ (ηtx)2�  ,  x ∈ Sd,  with norming constants cd(ρ) =  �  1F1(1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' λ + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' ρ)  �−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Clearly, Watd(η, 0) = unif(Sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (e) The angular Gaussian distributions are the distributions of X = Z/∥Z∥, where Z  has the (d + 1)-variate normal distribution Nd+1(a, Σ) with mean vector a ∈ Rd+1 \\ {0} and  symmetric positive definite covariance matrix Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Watson (1983, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' 108).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Here, we  exclusively deal with the case where Σ is the identity matrix Id+1, as the radial parts then  lead to isotropic families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' The distributions arising in this special case seem to have first  been studied in detail by Saw (1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Putting η = a/∥a∥ and ρ = ( 1  2∥a∥2)  1/2 we denote  by AGd(η, ρ) the distribution of X and speak of the angular Gaussian distribution with  parameters η and ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Its density is represented by the infinite series  f AG  d  (x|η, ρ) = e−ρ2  ∞  �  k=0  (2ρ ηtx)k Γ((d + 1 + k)/2)  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='Γ((d + 1)/2) ,  x ∈ Sd, ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  We refer to Saw (1978), where it is also pointed out that, with a random variable S ∼ χ2  d+1,  the density can be written as  f AG  d  (x|η, ρ) = E  �  e−ρ2 exp(2  1/2ρ S  1/2 ηtx)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (12)  (f) We consider two types of spherical Cauchy distributions: The spherical Cauchy dis-  tributions of type I have the unif(Sd)-densities  f CI  d (x|η, ρ) =  �  1 − ρ2  1 − 2ρ ηt x + ρ2  �d  ,  x ∈ Sd,  and the spherical Cauchy distributions of type II have the unif(Sd)-densities  f CII  d  (x|η, ρ) =  1 − ρ2  (1 − 2ρ ηt x + ρ2)(d+1)/2 ,  x ∈ Sd,  both with parameters η ∈ Sd and ρ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  We denote by CId(η, ρ) the distribution  with unif(Sd)-density f CI  d ( · |η, ρ), and by CIId(η, ρ) the distribution with unif(Sd)-density  f CII  d  ( · |η, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Clearly, CId(η, 0) = CIId(η, 0) = unif(Sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  6  The distributions in Example 1 (a) - (e) and their push-forwards under x �→ ηtx, re-  spectively, are all classical;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' for basic as well as specific properties and interesting historical  comments we refer to Watson (1982, 1983) and Mardia and Jupp (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' It is well known,  for example, that the von Mises–Fisher family in part (c) arises from the multivariate nor-  mal distributions in part (e) by conditioning on ∥Z∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' A well known relation with Brownian  motion on Sd is addressed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' In the special case d = 1 = (d+1)/2 the dis-  tributions WC1(η, ρ) := CI1(η, ρ) = CII1(η, ρ) in Example 1 (f) are known as the wrapped  Cauchy or circular Cauchy distributions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Mardia and Jupp (2000) and Section 3  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Hence, with the two types of spherical Cauchy distributions given above we have  two different extensions of this distribution family to higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' For distinction, we  added the name supplement ‘of type I’ and ‘of type II’, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' The spherical Cauchy  distributions of type I were introduced and studied by Kato and McCullagh (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Gen-  eralizing results obtained by McCullagh (1996) for d = 1, the authors especially deal with  the behavior of the spherical Cauchy distributions of type I under M¨obius transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  The densities f CII  d  ( · |η, ρ) were considered by McCullagh (1989), though the author does  not speak of spherical Cauchy distributions but, with the push-forward of CII(η, ρ) under  x �→ ηtx, of a noncentral version of the univariate symmetric beta distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  We recall from the introductory remarks that for a discrete mixture representation we  need a mixing base (Qn,η)n∈N0, η ∈ Sd, where each Qn,η is a probability measure on the  sphere, and mixing distributions on N0 that depend on ρ only;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' see (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Of special interest  in the latter context are the the negative binomial distributions NB(r, p) with parameters  r > 0, p ∈ (0, 1), and probability mass function  nb(n|r, p) = (r)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' (1 − p)npr,  n ∈ N0,  the confluent hypergeometric series distributions CHS(α, β, τ) on N0 with parameters α, β, τ >  0 and probability mass functions  chs(n|α, β, τ) =  �  1F1(α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' τ)  �−1 (α)n  (β)n  τ n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' ,  n ∈ N0,  and the hypergeometric series distributions HS(α, β, γ, τ) on N0 with parameters α, β, γ, τ >  0 and probability mass functions  hs(n|α, β, γ, τ) =  �  2F1(α, β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' τ)  �−1 (α)n(β)n  (γ)n  τ n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' ,  n ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  For τ = 0 we take CHS(α, β, τ) and HS(α, β, τ) to be the one-point mass at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' The dis-  tribution CHS(α, β, τ) arises as the stationary distribution of a birth-death process with  birth rates (α + i)τ and death rates i(β + i − 1), i ∈ N0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' see Hall (1956).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Note that these  three distribution families are subclasses of the family of generalized hypergeometric distri-  butions considered recently by Themangani et al (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' We also require that each family  {Qn,η : η ∈ Sd} is isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  We can now state our first results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Let d ∈ N be fixed and let λ be as in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' (a) The family {MFd(η, ρ) : η ∈ Sd, ρ ≥ 0} has a unique discrete mixture  representation with mixing base SBetad(η, 1, n + 1), n ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' This representation is given by  MFd(η, ρ) =  ∞  �  n=0  chs(n|λ + 1/2, 2λ + 1, 2ρ) SBetad(η, 1, n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (13)  7  (b) The family {CId(η, ρ) : η ∈ Sd, ρ ∈ (0, 1)} has a unique discrete mixture representation  with mixing base SBetad(η, 1, n + 1), n ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' This representation is given by  CId(η, ρ) =  ∞  �  n=0  nb  �  n|λ + 1/2, 4ρ/(1 + ρ)2�  SBetad(η, 1, n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (14)  (c) The family {CIId(η, ρ) : η ∈ Sd, ρ ∈ (0, 1)} has a unique discrete mixture representation  with mixing base SBetad(η, 1, n + 1), n ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' This representation is given by  CIId(η, ρ) =  ∞  �  n=0  hs  �  n|λ + 1/2, λ + 1, 2λ + 1, 4ρ/(1 + ρ)2�  SBetad(η, 1, n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (15)  In our next result the mixing base depends on the value of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' (a) The family {Watd(η, ρ) : η ∈ Sd, ρ ≥ 0} has a unique discrete mixture  representation with mixing base SPd(η, n), n ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' This representation is given by  Watd(η, ρ) =  ∞  �  n=0  chs(n|1/2, λ + 1, ρ) SPd(η, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (16)  (b) The family {Wat(η, ρ) : η ∈ Sd, ρ ≤ 0} has a unique discrete mixture representation with  mixing base SBetad(η, n + 1, n + 1), n ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' This representation is given by  Watd(η, ρ) =  ∞  �  n=0  chs(n|λ + 1/2, λ + 1, −ρ) SBetad(η, n + 1, n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (17)  In order to obtain a similar representation for the family of {AGd(η, ρ) : η ∈ Sd, ρ > 0}  of angular Gaussian distributions we make use of the integral representation  Dν(z) = e−z2/4  Γ(−ν)  � ∞  0  t−ν−1e−zt−t2/2 dt,  z ∈ R,  (18)  of the parabolic cylinder functions Dν with real index ν < 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' see Magnus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' (1966, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' 328).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Let δ > 1/2 and τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Then dpc( · |δ, τ) with  dpc(k|δ, τ) = 1  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' (2  1/2τ)k2k+δ−1 (k + 2δ − 1)Γ(k + δ − 1/2)  Γ(1/2)  e−τ 2/2D−(k+2δ)(2  1/2τ),  (19)  k ∈ N0, is a probability mass function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  We write DPC(δ, τ) for the associated discrete parabolic cylinder distribution with pa-  rameters δ > 1/2 and τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' For the special values δ = λ + 1 = (d + 1)/2 with d ∈ N the  statement of the lemma also follows from (20) below as the values on the right hand side  of (19) are all nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' The family {AGd(η, ρ) : η ∈ Sd, ρ > 0} has a unique discrete mixture repre-  sentation with mixing base SBetad(η, 1, n + 1), n ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' This representation is given by  AGd(η, ρ) =  ∞  �  n=0  dpc(n|λ + 1, ρ) SBetad(η, 1, n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (20)  8  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' (a) Regarding probability measures as real functions on a set of events, we may  define the series in (13) - (17) and (20) as referring to pointwise convergence of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  In fact, as the distributions involved all have smooth densities and compact domain, con-  vergence even holds with respect to uniform convergence in spaces of continuous functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (b) The representations are minimal in the sense that the respective mixing base cannot  be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' This follows from the uniqueness and the fact that the mixing probabilities are  strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (c) In connection with the base in Theorem 2 all mixtures have densities that are increas-  ing in ηtx, and in Theorem 3 and Theorem 5 all mixtures are invariant under the reflection  x �→ −x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (d) Interestingly, the von Mises–Fisher family, the spherical Cauchy families and the  angular Gaussian family have the same mixing base of spherical beta distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' So, these  families are obtained by picking at random (the index n of) the element SBetad(η, 1, n + 1)  according to the respective mixing distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Another family with this mixing base is  the family of spherical normal distributions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (e) For a discussion of other similarities as well as differences between the von Mises–  Fisher family and the spherical Cauchy family of type I we refer to Kato and McCullagh  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' The von Mises–Fisher family and the spherical Cauchy family of type II both have  representations in terms of multivariate Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' To be specific, let X = (Xt)t≥0  be a standard Brownian motion in Rd+1, let Y = (Yt)t≥0 with Yt = ρηt + Xt for t ≥ 0 be  the drifted standard Brownian motion with constant drift vector ρη, where ρ ≥ 0, η ∈ Sd,  and let Z = (Zt)t≥0 with Zt = ρη + Xt for t ≥ 0, where 0 ≤ ρ < 1, η ∈ Sd, be the Brownian  motion starting at ρη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' With TY := inf{t ≥ 0 : ∥Yt∥ ≥ 1} as the first time that Y exits the  Euclidean unit ball Bd = {x ∈ Rd+1 : ∥x∥ < 1} it then holds that YTY ∼ MFd(η, ρ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' see  Gatto (2013) for a more recent proof and historical remarks on this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Further, with  TZ := inf{t ≥ 0 : ∥Zt∥ ≥ 1} the first time that Z exits Bd it holds that ZTZ ∼ CIId(η, ρ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  see Chung (1982, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' 170) and McCullagh (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  3  Ultraspherical mixing bases  Our aim in this section is a mixing base that is applicable for general distribution families  where, as before, we consider distributions Pθ on (Sd, B(Sd)), with θ = (η, ρ) ∈ Sd × I and  I ⊂ R+ an interval, that have densities fθ of the form  fθ(x) = gρ(ηtx),  x ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (21)  We will occasionally omit d or λ from the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Recall that λ = (d − 1)/2 and that νd  is the push-forward of unif(Sd) under the mapping x �→ ηtx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  We assume that the functions gρ in (21) are elements of  Hλ := L2�  [−1, 1], B([−1, 1]), νd  �  ,  and on Hλ we use the inner product  ⟨f, g⟩λ =  �  f(t)g(t) νd(dt)  and the norm ∥f∥λ = ⟨f, f⟩1/2  λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Then (Hλ, ⟨·, ·⟩λ) is a Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' We deal with a special  complete sequence of orthogonal polynomials in this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' For d = 1 and λ = 0 this is  the sequence of Chebyshev polynomials Tn of the first kind of degree n ∈ N0, for d > 1  9  and λ > 0 we use the sequence of Gegenbauer or ultraspherical polynomials Cλ  n of degree  n ∈ N0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' see Erd´elyi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' (1953, Chs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  X, XI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' These functions play an important role  in directional statistics, especially nonparametric directional statistics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' the papers  of Bingham (1972), Gin´e (1975), Prentice (1978), Baringhaus (1991), Jupp (2008), and  Garc´ıa–Portugu´es et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' The functions are standardized such that  Cλ  n(1) = Γ(2λ + n)  Γ(2λ)n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  = (2λ)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  for all n ∈ N0  if λ > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' further, Tn(1) = 1 for all n ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' In particular, Cλ  0 ≡ 1 ≡ T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Of course, for n > 0  none of these functions is a probability density with respect to νd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' However, it is known  that the Chebyshev polynomials and, for λ > 0, the Gegenbauer polynomials attain their  absolute maximum on [−1, 1] at t = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' see Erd´elyi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' (1953, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' 206, formula (7)) and  Abramowitz and Stegun (1964, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' 786).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Hence the standardization  Dλ  n(t) :=  �  Tn(t)  if λ = 0,  Cλ  n(1)−1 Cλ  n(t)  if λ > 0,  provides a sequence (Dλ  n)n∈N0 of orthogonal polynomials that are bounded in absolute value  on [−1, +1] by their value 1 in t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' As (Dλ  n)n∈N0 is complete in Hλ, we have the series  expansion converging in Hλ  gρ =  ∞  �  n=0  ⟨gρ, Dλ  n⟩λ ∥Dn∥−2  λ Dλ  n = 1 +  ∞  �  n=1  βn(ρ) Dλ  n,  (22)  with βn(ρ) := ⟨gρ, Dλ  n⟩λ ∥Dn∥−2  λ  for n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' We assume throughout this section that gρ is  such that  β(ρ) :=  ∞  �  n=1  |βn(ρ)| < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (23)  For fixed n ∈ N0 and η ∈ Sd the function Hλ  n,η : Sd → R defined by  Hλ  n,η(x) = Dλ  n(ηtx), x ∈ Sd,  is the unique surface harmonic of degree n that depends only on ηtx and that satisfies  Hλ  n,η(η) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' see Erd´elyi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' (1953, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' 238).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Obviously, Hλ  n,η is invariant under the sub-  group {U ∈ O(d + 1) : Uη = η} of O(d + 1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' for all elements U of the subgroup it holds  that Hλ  n,η(Ux) = Hλ  n,η(x) for all x ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Let  γ0  0 := 1, γ0  n := 1/2  for all n ∈ N,  γλ  n :=  1  (1 + n/λ) Cλn(1)  for all λ > 0, n ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (24)  We will repeatedly make use of the basic formulas  �  Dλ  m(ηtx)Dλ  n(ξtx) unif(Sd)(dx) = 0,  η, ξ ∈ Sd, m, n ∈ N0, m ̸= n,  (25)  and  �  Dλ  n(ηtx)Dλ  n(ξtx) unif(Sd)(dx) = γλ  nDλ  n(ηtξ),  η, ξ ∈ Sd, n ∈ N0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (26)  10  see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Erd´elyi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' (1953, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' 245) and Saw (1984, formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='14)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  The mixing bases considered in what follows are of a very simple structure: The densities  of the base distributions are built with only two special surface harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' To be precise,  for n ∈ N and real numbers −1 ≤ α ≤ +1 the functions  x �→ 1 + αDλ  n(ηtx), x ∈ Sd,  (27)  are unif(Sd)-densities of probability distributions ∆λ  n,η,α on Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' These distributions can be  regarded as a multivariate generalization of the cardioid distributions introduced by Jeffreys  (1948, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' 302);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' see also Mardia and Jupp (2000, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Let ∆λ  0,η,α := unif(Sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Here  we mainly deal with the special distributions ∆λ  n,η := ∆λ  n,η,1, but see also Remark 10 (a)  and Proposition 12 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' So, for n ∈ N the unif(Sd)-density of ∆λ  n,η is simply the sum of  the two special surface harmonics Hλ  0,η ≡ 1 and Hλ  n,η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  With the mixing base ∆λ  n,η, n ∈ N0, given for each η ∈ Sd, we obtain discrete mixture  representations for all spherical distributions with densities of the form (21) that are not  too far away from unif(Sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' This may be seen as an instance of the perturbation approach  discussed in the survey paper of Pewsey and Garc´ıa–Portugu´es (2021) and, indeed, the value  of β(ρ) may be interpreted as a distance between νd and the measure with νd-density gρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  By an ultraspherical mixing base we mean a family {∆λ  n,η : n ∈ N0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  The following general formula is an immediate consequence of the above definitions and  the expansion in (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Suppose that βn(ρ) ≥ 0 for all n ∈ N and that β(ρ) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Let η ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Then  Pη,ρ has the discrete mixture expansion  Pη,ρ =  ∞  �  n=0  wρ(n) ∆λ  n,η,  (28)  with wρ(0) = 1 − β(ρ) and wρ(n) = βn(ρ) for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Applying this construction to several specific families we have to take care of the crucial  condition β(ρ) ≤ 1, equivalently wρ(0) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' In each case, we obtain the mixing distribution  and a range of ρ-values for the validity of the representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Any distribution on N0 may  be written as a mixture of unit mass at 0 and a distribution on N and it turns out that  the latter are occasionally from a standard family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  For a distribution on N0 with mass  function w on N0 such that w(0) < 1 we call the distribution on N with mass function  n �→ w(n)/(1 − w(0)), n ∈ N, its zero-truncated counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Some of the results stated in  what follows turn out to be simple consequences of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  In the first theorem we consider two families of wrapped distributions, hence d = 1 and  λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' There are different notational conventions in the literature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' here, we regard the  wrapped distribution associated with a given distribution µ on (the Borel subsets of) the  real line as the push-forward µT of µ under the mapping T : R → S1, x �→ (cos(x), sin(x))t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  This is often applied to location-scale families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Alternatively, the interval [−π, π) is used  instead of S1 as the base set for the wrapped distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' This means that with X ∼ µ one  deals with the [−π, π)-valued random variable X0 as the variable X reduced modulo 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' If  X has the density f with respect to the Lebesgue measure, then X0 has the unif ([−π, +π))-  density 2π �+∞  n=−∞ f(s+2πn), s ∈ [−π, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' If the characteristic function ϕ of X is absolutely  integrable, then f is continuous and the Poisson summation formula applies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  2π  +∞  �  n=−∞  f(s + 2πn) =  +∞  �  n=−∞  ϕ(n) e−ins,  s ∈ [−π, π);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (29)  11  see Feller (1971, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' 632).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' For example, the wrapped normal distribution WN1(η, ρ) arises from  the normal distribution N(α, σ2) with mean α and variance σ2, where η = (cos(α), sin(α))t  and ρ = σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Note that some authors use ρ = 2σ2, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Hartmann and Watson (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  With X ∼ N  �  α, σ2�  we deduce from (29) that X0 has the density  1 + 2  ∞  �  n=1  e−n2ρ/2 cos n(s − α),  s ∈ [−π, +π),  which means that T ◦ X ∼ WN1(η, ρ) has the unif(S1)-density  f WN  1  (x|η, ρ) = 1 + 2  ∞  �  n=1  e−n2ρ/2 Tn(ηtx),  x ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (30)  It is worthwhile to note that as ρ → 0 the distribution WN1(η, ρ) converges weakly to  the one-point mass distribution at η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' This is in contrast to other distributions considered  here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  For example, MF1(η, ρ), Wat1(η, ρ), AG1(η, ρ) all converge weakly to the uniform  distribution unif(S1) as ρ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Also, as ρ → ∞, the distribution WN1(η, ρ) converges  weakly to unif(S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  For the wrapped Cauchy distribution WC1(η, ρ) we follow the definition given by Pewsey and Garc´ıa–Portugu´es  (2021): If X has a standard Cauchy distribution with density x �→ 1/(π(1 + x2)), x ∈ R,  then we apply the wrapping procedure to Y := σX + α, where σ > 0, α ∈ R, and take η as  in the wrapped normal case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' For the scaling we use the parametrization ρ := e−σ ∈ (0, 1)  and augment this with the limiting uniform distribution at ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Using (29) again we  obtain that the distribution WC1(η, ρ) has the density  f WC  1  (x|η, ρ) = 1 + 2  ∞  �  n=1  Tn(ηtx)ρn =  1 − ρ2  1 − 2ρ ηtx + ρ2 ,  x ∈ S1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (31)  see also Pewsey and Garc´ıa–Portugu´es (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  We write geoN0(n|p) = p(1 − p)n, n ∈ N0, for the probability mass function of the  geometric distribution on N0 with parameter p ∈ (0, 1), and geoN for the mass function of  its zero-truncated counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Recall that the function β for a given distribution family is  defined in (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' (a) For the wrapped Cauchy distributions we have β(ρ) = 2ρ/(1 − ρ) and, for  0 ≤ ρ ≤ 1/3,  WC1(η, ρ) = (1 − β(ρ)) unif(S1) + β(ρ)  ∞  �  n=1  geoN(n|1 − ρ) ∆0  n,η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (b) For the wrapped normal distributions we have β(ρ) = 2 �∞  n=1 e−n2ρ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Let ρ0 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='570818  be the unique solution of the equation β(ρ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Then, with  br(n|ρ) := 2β(ρ)−1e−n2ρ/2  for all n ∈ N,  and ρ ≥ ρ0, it holds that  WN1(η, ρ) = (1 − β(ρ)) unif(S1) + β(ρ)  ∞  �  n=1  br(n|ρ) ∆0  n,η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  12  We deal with the von Mises–Fisher families next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' For these, we need variants of the  Skellam distribution with parameter ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' This distribution arises as the distribution of N1−N2  where N1, N2 are independent random variables that both have the Poisson distribution with  parameter ρ/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' see Irwin (1937), and see Skellam (1946) where the more general case with  possibly different means for N1 and N2 is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' In the one-dimensional case we need  the positive Skellam distribution, which is the conditional distribution of |N1 − N2| given  that N1 ̸= N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' The associated probability mass function is given by  psk(n|ρ) =  2e−ρIn(ρ)  1 − e−ρI0(ρ),  n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (32)  For d > 1 we use the generalized positive Skellam distribution with parameters κ > 0 and  τ > 0, with mass function  gpsk(n|κ, τ) :=  �  1 + n  κ  � (2κ)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  2κΓ(κ + 1)τ−κe−τIκ(τ)  1 − 2κΓ(κ + 1)τ −κe−τIκ(τ)  Iκ+n(τ)  Iκ(τ) ,  n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  It will turn out as part of the proof of the next result that this is indeed a probability  mass function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' We have limκ→0 1  κ  (2κ)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  =  2  n, which gives limκ→0 gpsk(n|κ, τ) = psk(n|τ)  for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Hence the positive Skellam distribution appears as the limiting case of the  generalized positive Skellam distribution as κ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' We consider the von Mises–Fisher families {MFd(η, ρ) : ρ ≥ 0}, η ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (a) If d = 1 then β(ρ) = eρ/I0(ρ) − 1, the equation β(ρ) = 1 has a unique finite positive  solution ρ0 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='876842, and for 0 ≤ ρ ≤ ρ0 it holds that  MF1(η, ρ) = (1 − β(ρ)) unif(S1) + β(ρ)  ∞  �  n=1  psk(n|ρ) ∆0  n,η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (b) If d > 1 then  β(ρ) = βλ(ρ) :=  ρλeρ  2λΓ(λ + 1)Iλ(ρ) − 1,  the equation βλ(ρ) = 1 has a unique finite positive solution ρ0(λ), and for 0 ≤ ρ ≤ ρ0(λ) it  holds that  MFd(η, ρ) = (1 − βλ(ρ)) unif(Sd) + βλ(ρ)  ∞  �  n=1  gpsk(n|λ, ρ)∆λ  n,η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' (a) The condition that βn(ρ) ≥ 0 for all n ∈ N in Proposition 7 is satisfied  in all of the above families, but it can easily be removed by an appropriate extension of the  mixing base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' For this, let ∆λ,−  n,η be the distribution with density x �→ 1 − Dλ  n(ηtx), x ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Then the representation (28) continues to hold if we take Qn,η = ∆λ,−  n,η and wρ(n) = −βn(ρ)  whenever βn(ρ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (b) The condition β(ρ) ≤ 1 in Proposition 7 holds if Pη,ρ is sufficiently close to the uniform  distribution on the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' If instead of Pη,ρ we consider a mixture of this distribution with  the uniform, with enough weight on the latter, then the result is close enough to the uniform,  and we again obtain a mixture representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' In the von Mises–Fisher case with d = 1, for  example, we get for ρ > ρ0 and with α(ρ) := (1 − 2e−ρI0(ρ))/(1 − e−ρI0(ρ)),  α(ρ) unif(S1) + (1 − α(ρ)) MF1(η, ρ) =  ∞  �  n=1  psk(n|ρ) ∆0  n,η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  13  (c) Is there a countable mixing base that represents all isotropic spherical distributions  with densities of the form x �→ g(ηtx) with η ∈ Sd and g ∈ Hλ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' This may be rephrased  in terms of the set of extremal points of a convex set in an infinite dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' We  refer to Baringhaus and Gr¨ubel (2021b) for such geometric aspects in general, and for the  construction of tree-based mixing bases that would lead to a positive answer for the set of  all g ∈ Hλ that are Riemann integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  The passage from an L2-expansion (22) of gρ to the mixture representation (28) heavily  relies on the nonnegativity of the functions 1 + Dλ  n (respectively 1 − Dλ  n in part (a) above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  More generally, we may consider a mixing base (Qn,η)n∈N where the density of Qn,η is a  polynomial of degree n in ηtx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' This leads to the consideration of general linear combina-  tions of ultraspherical polynomials;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' indeed, finding conditions for such polynomials to be  nonnegative (on a given interval) is an ongoing research topic, see Askey (1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  We confine ourselves to an example with λ = 0 and the Chebyshev polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' A  change of mixing base will obviously lead to a change in the sequence of mixing coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  It turns out that this may lead to a representation of wider applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Example 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' We define functions gρ : [−1, 1] → R+, 0 ≤ ρ < 1, by  gρ(t) := (1 − ρt + φρ(t))1/2  21/2φρ(t)  ,  −1 ≤ t ≤ 1,  (33)  where φρ(t) := (1 − 2ρt + ρ2)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' These can be written as  gρ(t) =  ∞  �  n=0  (1/2)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Tn(t) ρn,  (34)  see Magnus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' (1966, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' 259).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' In particular,  � 1  −1 gρ(t) dt = 1, so that we may define a  family of distributions Pη,ρ on S1 via their densities x �→ gρ(ηtx), x ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  We first derive an expansion in terms of the distributions ∆0  nη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' With t = 1 we get  β(ρ) =  ∞  �  n=1  (1/2)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  ρn = (1 − ρ)−1/2 − 1,  and it follows that β(ρ) ≤ 1 if and only if ρ ≤ 1 − 2−1/2 =: ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' We now introduce the zero-  truncated negative binomial distribution with parameters r > 0, p ∈ (0, 1), and probability  mass function  znb(n|r, p) = (r)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' (1 − p)n  pr  1 − pr ,  n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Then (34) leads to the discrete mixture representations  Pη,ρ =  �  1 − β(ρ)  �  unif(S1) + β(ρ)  ∞  �  n=1  znb(n|1/2, 1 − ρ) ∆0  n,η,  (35)  for all η ∈ S1, 0 < ρ ≤ ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  On the other hand, Turan (1953) proved that, for all n ∈ N0,  n  �  k=0  (1/2)k  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  cos(kϑ) > 0  for 0 < ϑ < π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  14  In view of Tk(cos ϑ) = cos(kϑ) this can be used to obtain an alternative representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  For this, let Σn,η be the distribution on S1 with density x �→ �n  k=0 αkTk(ηtx), where  αk := (1/2)k/k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' and n ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Then (34), together with a summation by parts, leads to  Pη,ρ =  ∞  �  n=0  geoN0(n|1 − ρ) Σn,η = (1 − ρ) unif(S1) + ρ  ∞  �  n=1  geoN(n|1 − ρ) Σn,η,  (36)  where geoN0 and geoN are as in Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Both (35) and (36) hold for all η ∈ S1, but  note that the range of permissible ρ-values has increased from 0 ≤ ρ ≤ 1 − 2−1/2 to the full  interval 0 ≤ ρ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  As pointed out earlier mixing families constructed with surface harmonics can be used  with all distributions of the form (21) as long as these are sufficiently close to the uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  The following result gives a property which we interpret as self-mixing stability of the dis-  tribution families Dλ  0 := {unif(Sd)}, Dλ  n := {∆λ  n,η,α : η ∈ Sd, −1 ≤ α ≤ +1}, n ∈ N,  and Dλ := �  n∈N0 Dλ  n: Mixing two elements of the same family results in a distribution  that belongs to this family as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Additionally, mixing any two elements moves the mix-  ing distribution closer to the uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Generally, the mixing operation relates  distributions with different location parameter η ∈ Sd to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Let γλ  n be the constants defined in (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (a) For all n ∈ N0 and η ∈ Sd, −1 ≤ α, β ≤ +1,  �  ∆λ  n,ζ,α(A) ∆λ  n,η,β(dζ) = ∆λ  n,η,γλ  nαβ(A)  = (1 − γλ  n) unif(Sd)(A) + γλ  n ∆λ  n,η,αβ(A)  for all A ∈ B(Sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (b) For all n, m ∈ N0 with n ̸= m, and all η ∈ Sd, −1 ≤ α, β ≤ +1,  �  ∆λ  m,ζ,α(A) ∆λ  n,η,β(dζ) = unif(Sd)(A)  for all A ∈ B(Sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  We recall that a probability kernel from a measurable space (E, E) to another measurable  space (F, F) is a function Q : E × F → R that is E-measurable in its first and a probability  measure on (F, F) in its second argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Given a probability measure P on (E, E) and a  kernel Q from (E, E) to (F, F) we define a probability measure P ◦ Q on (F, F) by  P ◦ Q(A) =  �  Q(x, A) P(dx)  for all A ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (37)  For a family {Qx : x ∈ E} of probability measures Qx on (F, F) with the property that the  map x �→ Qx(A) is E-measurable for all A ∈ F, we may regard Q· : E × F → R defined by  Q·(x, A) = Qx(A), (x, A) ∈ E × F, as a Markov kernel from (E, E) to (F, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Then (37)  reads  P ◦ Q·(A) =  �  Qx(A) P(dx)  for all A ∈ F,  which may be interpreted as a mixing operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  For use in the next section we note  that for a family {Pη : η ∈ Sd} of probability measures Pη on (Sd, B(Sd)) and a kernel Q  from (Sd, B(Sd)) to (Sd, B(Sd)) that are both isotropic in the sense defined by (8) and (9)  respectively, the mixing results in an isotropic family again,  �  Pη ◦ Q  �U = PUη ◦ Q  for all η ∈ Sd and all U ∈ O(d + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (38)  15  Further, for α = β = 1 the statements in Proposition 12 can be simply rephrased as  ∆λ  n,η ◦ ∆λ  n,· = (1 − γλ  n) unif(Sd) + γλ  n ∆λ  n,η,  ∆λ  n,η ◦ ∆λ  m,· = unif(Sd)  if n ̸= m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Using the self-mixing stability and the bilinearity of the operation defined in (37) we ob-  tain a discrete mixture representation for the composition of two families that both have a  representation in terms of the ultraspherical mixing base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Suppose that Pη and P ′  η, η ∈ Sd, are distribution families on Sd with  discrete mixture representations Pη = �∞  n=0 w(n)∆λ  n,η and P ′  η = �∞  n=0 w′(n)∆λ  n,η respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Then  Pη ◦ P ′  =  ∞  �  n=0  ˜w(n) ∆λ  n,η,  where  ˜w(0) := 1 − �∞  n=1 ˜w(n) and  ˜w(n) := γλ  n w(n)w′(n) for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  This can be used to obtain the composition of wrapped Cauchy distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Indeed,  taken together, Theorem 8 (a) and Proposition 13 lead to  WC1(η, ρ) ◦ WC1(·, ρ′) = WC1(η, ρρ′)  for all ρ, ρ′ ≤ 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (39)  It is worthwhile to point out that the restriction ρ, ρ′ ≤ 1/3 in (39) can be omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' In fact,  for all 0 ≤ ρ, ρ′ < 1, using (24), (25), (26), and (31), the density of WC1(η, ρ) ◦ WC1(·, ρ′)  is easily calculated to be  �  f WC  1  (x|ξ, ρ′) f WC  1  (ξ|η, ρ) unif(S1)(dξ) = f WC  1  (x|ξ, ρρ′),  x ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  In the next section this will be put into a wider context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  4  Discrete mixture representations and Markov pro-  cesses  Let {Pη,ρ : η ∈ Sd, ρ ∈ I}, I ⊂ R+ an interval, be a family of distributions of the type  considered in the previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Below we briefly discuss three different connections to  Markov processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' First, for ρ ∈ I fixed, the corresponding subfamily may be regarded  as a probability kernel and thus induces a Markov chain on spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Second, an isotropic  diffusion process on Sd leads to a family of the above type via its one-dimensional marginal  distributions, where η and ρ take over the role of starting point and (transformed) time  parameter respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Third, starting with a discrete mixture representation we may find  a discrete time Markov chain with marginal distributions equal to elements of the mixing  base, and thus obtain an almost sure representation of the family as the distributions of the  chain at random times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='1  Random walks on spheres  Let {Qη : η ∈ Sd} be a family of probability distributions that leads to a Markov kernel as  described at the end of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Such kernels arise as transition probabilities of Markov  processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' We may, for example, fix an η ∈ Sd and define a Markov chain (Xn)n∈N0 with  state space (Sd, B(Sd)) by the requirements that X0 ≡ η and that the distribution of Xn+1  16  conditionally on Xn = ξ is given by Qξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' For isotropic kernels each transition can be divided  into two steps that make use of the representation of Sd by [−1, 1]×Sd−1 that also appeared  in connection with (5) and (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' In the geometrically most familiar case we consider the  current position as the ‘north pole’, then first choose a latitude and thereafter, indepen-  dently, a longitude uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' The result is regarded as the new north pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' A  generalization of this setup has been considered by Bingham (1972), see also the references  given there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  The case d = 1 is somewhat special as the wrapping procedure is a group homomorphism  from the additive group of real numbers into the multiplicative group S1, regarded as a subset  of C and endowed with complex multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Wrapping a random walk or a L´evy process  thus leads to processes with values in S1 that have stationary and independent increments,  where the latter are now to be understood as ratios rather than differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' In fact, the  location-scale family of Cauchy distributions arises as the one-dimensional marginals of a  specific L´evy process, which gives (39) after an appropriate rescaling of the variance param-  eter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' A similar approach, now using Brownian motion on the real line, gives a corresponding  statement for the wrapped normal distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  We collect some observations in the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Recall that the distribution L(X)  of a Markov chain X = (Xn)n∈N0 with state space (Sd, B(Sd)) is a probability measure on  the path space (SN0  d , B  �  SN0  d  �  ) of the chain, where the σ-field B  �  SN0  d  �  is generated by the  projections πk : SN0  d  → Sd, (xn)n∈N0 �→ xk, k ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Any measurable mapping T : Sd → Sd  may be lifted to a mapping from and to paths by componentwise application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Proposition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Suppose that Q is an isotropic kernel on Sd and that Xη = (Xη  n)n∈N0 is  a Markov chain with start at η ∈ Sd and transition kernel Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (a) The family {L(Xη) : η ∈ Sd} of probability measures on the path space is isotropic in  the sense that L(Xη)U = L(XUη) for all η ∈ Sd, U ∈ O(d + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (b) The latitude process Y = (Yn)n∈N0, with Yn := ηtXη  n for all n ∈ N0, is a Markov chain  with state space [−1, 1] and start at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (c) Suppose that Q(η, · ) = �∞  k=0 w(k)∆λ  k,η for all η ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Then the representation  L(Xη  n) =  ∞  �  k=0  wn(k)∆λ  k,η  for all n ∈ N, η ∈ Sd,  (40)  holds with w1(k) := w(k) for all k ∈ N0 and, for n > 1,  wn(k) := (γλ  k )n−1w(k)n, k ∈ N,  wn(0) := 1 −  ∞  �  k=1  wn(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  The fact that the geodesic distances from the starting point are again a Markov chain  is an instance of lumpability, see Rogers and Pitman (1981) for a general discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' That  the dependence on η is lost in the lumping transition is part of the assertion of part (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Further, it follows from the cosine theorem for spherical triangles that the transition kernel  QY of Y may be written as  QY (y, · ) = L  �  yZ + U(1 − y2)1/2(1 − Z2)1/2�  (41)  with Z, U independent, L(Z) = L(ηtXη  1 ) and L(U) = νd−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' see also Bingham (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Part (c) shows that the mixing base introduced in Section 3 is useful in the Markov chain  context, and (40) may be seen as a discrete mixture representation of the family {Pη,n : η ∈  17  Sd, n ∈ N}, with Pη,n = L(Xη  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' It implies that the marginal distributions of the associated  latitude process are given by  L(Yn) =  ∞  �  k=0  wn(k) µλ  k  for all n ∈ N,  where µλ  k is the push-forward of ∆λ  k,η under x �→ ηtx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='2  Diffusion processes  Let X = (Xt)t≥0 be a homogeneous continuous time Markov process on the sphere with  start at η, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' P(X0 = η) = 1, and transition densities pt(x, y), t > 0, x, y ∈ Sd, that are  isotropic in the sense that pt(Ux, Uy) = pt(x, y) for all t > 0, x, y ∈ Sd and U ∈ O(d + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Then the marginal distributions L(Xt), t ≥ 0, of the process may have a discrete mixture  representation of the type considered above, with ρ related to time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  We sketch the basic argument, see also Karlin and McGregor (1960), and then apply this  in the context of the spherical Brownian motion (Bt)t≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' For a general discussion of the  latter we refer to Ito and McKean (1997, Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='15) and Hsu (2002, Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' We  note in passing that the marginal distributions of X characterize the full distribution L(X)  of the process, in view of the Chapman–Kolmogorov equations and the invariance of the  transition mechanism under orthogonal transformations (clearly, O(d + 1) acts transitively  on Sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Suppose that X has transition densities pt(x, y) and that its infinitesimal generator A  has a discrete spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' As transitions are isotropic it is enough to consider one specific  starting value x = η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' For the Kolmogorov forward equations  � ∂  ∂tp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' (η, y)  �  (t) =  �  Apt(η, ·)  �  (y)  we may try to find a family of basic solutions φ by a separation ansatz φ(t, y) = f(t)g(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  This leads to  f ′(t)  f(t) = (Ag)(y)  g(y)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  As the left and right hand side respectively depend on t and y only, we may hope that  pt(η, y) =  ∞  �  n=0  e−ωnt φn,η(y),  where ωn, n ∈ N0, are the eigenvalues of the operator A, with eigenfunctions φn,η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Recall that λ = (d − 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Theorem 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Let (Bt)t≥0 be the spherical Brownian motion on Sd, d > 1, with start at  η ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Let  βλ(t) :=  ∞  �  n=1  �  1 + n  λ  �(2λ)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  e−n(n+2λ)t/2  and let  brλ  t (n) : = βλ(t)−1�  1 + n  λ  �(2λ)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  e−n(n+2λ)t/2,  n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Further, let tλ  0 be the unique solution of the equation βλ(t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Then, for t ≥ tλ  0,  L(Bt) = (1 − βλ(t)) unif(Sd) + βλ(t)  ∞  �  n=1  brλ  t (n) ∆λ  n,η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  18  In view of the wrapping representation mentioned above the corresponding result for  d = 1 is contained in part (b) of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Let B = (Bt)t≥0 be as in the theorem and let Y = (Yt)t≥0 with Yt = ηtBt, t ≥ 0, be  the associated latitude process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Then a polar decomposition shows that, for t > 0 fixed, Bt  can be synthesized (in distribution) from Yt and an independent random variable Z that is  uniformly distributed on Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' On the level of processes the conditional distribution of B  given Y is a result known as the skew product decomposition of spherical Brownian motion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  see Ito and McKean (1997, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' 270) and Mijatovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  A representation with a different mixing base, closer in spirit to the representations in  Section 2, has very recently been obtained by Mijatovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' The result is based  on the authors’ observation that for a spherical Brownian motion (Bλ  t )t≥0 with start at  η ∈ Sd, d > 1, the rescaled latitude process (Yt)t≥0, Yt := (1 − ηtBλ  t )/2 for all t ≥ 0, is  a neutral Wright-Fisher diffusion with both mutation parameters equal to λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' For these,  a discrete mixture representation had earlier been given by Jenkins and Spano (2017), see  also Griffiths and Spano (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Taken together, this leads to  L(Bλ  t ) =  ∞  �  n=0  wλ  t (n) SBetad(η, 1, n + 1)  for all t > 0,  (42)  with  wλ  t (n) =  ∞  �  k=n  (−1)k−n (d + 2k − 1)(d + n)k−1  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' (k − n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  e−k(k+d−1)t/2  for all n ∈ N0, t > 0,  where the term (d)−1 appearing with n = k = 0 is defined to be 1/(d−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' see Jenkins and Spano  (2017, formula (5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Interestingly, the mixture coefficients turn out to be the individual  probabilities associated with the marginal distributions of a particular pure death process  (Zt)t>0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' wλ  t (n) = P(Zt = n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' In contrast to our representation in Theorem 15 via sur-  face harmonics no further restrictions on the time parameter are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Moreover, there  is also a fascinating probabilistic interpretation, relating neutral Wright-Fisher diffusions to  Kingman’s coalescent via moment duality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' see Mijatovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' (2020) for the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Mardia and Jupp (2000) call the L(Bλ  t ) Brownian motion distributions on Sd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Kent  (1977) regards L(ηtBλ  t ) as a spherical normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' We adopt the notation of the  latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Remembering that Bλ starts in η ∈ Sd, we denote by SNd(η, ρ) = L  �  Bλ  1/ρ  �  the  spherical normal distribution with parameters η ∈ Sd and ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Then, interestingly, by  (42) we have a discrete mixture representation for the family { SNd(η, ρ) : η ∈ Sd, ρ > 0}  with the same mixing base of spherical beta distributions SBetad(η, 1, n + 1), n ∈ N0,  as obtained in Section 2 for the von Mises–Fisher, the spherical Cauchy, and the angular  Gaussian families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  As explained at the beginning of this subsection, isotropic diffusion processes on the  sphere may lead to a discrete mixture representation for the family of their marginal dis-  tributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Conversely, for a given family of the type considered in the previous sections,  one might ask for a representation of its elements as the marginals of some diffusion pro-  cess with values in Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' The following result answers this question for the von Mises–Fisher  distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Theorem 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' There is no homogeneous Markov process X = (Xt)t≥0 on Sd with the  property that, for all η ∈ Sd,  L(Xt|X0 = η) ∈  �  MFd(η, ρ) : ρ > 0  �  for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (43)  19  Alternatively one might start with a diffusion on the ambient space Rd+1 and then use  the transition x �→ x/∥x∥ from Rd+1 to Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' For example, if B = (Bt)t≥0 is a Brownian  motion on Rd+1 with start at η ∈ Sd, then X = (Xt)t≥0, with Xt := ∥Bt∥−1Bt for all t ≥ 0,  represents the family {AGd(η, ρ) : ρ > 0} in the sense that  L(Xt) = AGd  �  η, ρ(t)  �  for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Here the bijection ρ : (0, ∞) → (0, ∞) is given by ρ(t) := (2t)−1/2, t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Remark 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' We relate Theorem 16 to the infinite divisibility statement for the von Mises–  Fisher distributions obtained by Kent (1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Interestingly, in both cases the proofs are  based on the same series expansions (57), (59) of the densities f MF  d  ( · |η, ρ) in terms of  ultraspherical polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' An important ingredient of Kent’s approach is the associative  convolution algebra (F, ◦d) on the space F of probability measures on [−1, 1] introduced  by Bingham (1972), where the convolution F1 ◦d F2 of F1, F2 ∈ F is defined to be the  distribution of S1S2 + Λ(1 − S1)  1/2(1 − S2)  1/2, see also (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Here S1, S2, Λ are independent,  Si ∼ Fi for i = 1, 2, and Λ ∼ νd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  For d = 0 we take ν0 to be the discrete uniform  distribution on S0 := {−1, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' With this definition a probability measure F ∈ F is said to  be ◦p-infinitely divisible if for each m ∈ N there exists an Fm ∈ F such that F is equal  to the m-fold convolution F ◦dm  m  of Fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' As F ∈ F is uniquely determined by its Fourier  transform ϕF : N0 → R defined by  ϕF (m) =  �  Dλ  m(t) F(dt)  for m ∈ N0,  and ϕF1◦dF2 = ϕF1ϕF2 for all F1, F2 ∈ F, it holds that F is ◦d-infinitely divisible if and  only if for each m ∈ N there is a Fourier transform ϕFm : N0 → R of an Fm ∈ F such that  ϕF = ϕm  Fm for all m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' The distribution MF∗  d(ρ) of ηtX, with X ∼ MFd(η, ρ), has the  νd-density  ρd  2dΓ(λ+1)Iλ(ρ) exp(ρt), t ∈ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Kent (1977) showed that MF∗  d(ρ) is ◦d-infinitely  divisible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' In fact, Kent even gives an interesting representation of the distributions Fm such  that MF∗  d(ρ) = F ◦dm  m  based on the spherical Brownian motion (Bt)t≥0 on Sd with start at  η ∈ Sd: He showed that for each m ∈ N there exists an absolutely continuous distribution  Gm on the positive half-line such that ηtBTm ∼ Fm, where Tm ∼ Gm is independent of  (Bt)t≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Consequently,  MF∗  d(ρ) = L  �  ηtBTm  �◦dm  for all m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (44)  In order to lift this from the unit interval to the sphere let η ∈ Sd be fixed and let Pη be the  family of distributions P on Sd that are axially symmetric with respect to η, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' P U = P for  each U ∈ O(d + 1) with Uη = η, and P U the push-forward of P under U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' In order to carry  over the convolution operation ◦d to Pη, a spherical addition ⊕ on Sd is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Assign  to each x ∈ Sd an element Ux ∈ O(d + 1) in such a way that Uxη = x for all x ∈ Sd and  that x → Ux is a measurable injection (a measurable embedding) from Sd into O(d + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Then, for x, y ∈ Sd, let x ⊕ y := Uxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' For P1, P2 ∈ Pη, with independent Sd-valued random  vectors Xi ∼ Pi, i = 1, 2, the ⋆d-convolution P1 ⋆d P2 of P1 and P2 is defined to be the  distribution of X1 ⊕ X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Kent showed that P1 ⋆d P2 ∈ Pη, and with Fi as the distribution  of ηtXi, i = 1, 2, the cosine theorem for spherical triangles leads to ηt(X1 ⊕ X2) ∼ F1 ◦d F2,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  L  �  ηt(X1 ⊕ X2)  �  = L(ηtX1) ◦d L(ηtX2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  From (44) it now follows that MFd(η, ρ) = L  �  BTm  �⋆dm for all m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  20  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='3  Almost sure representations  The basic relation (2) connects {Pθ : θ ∈ Θ} to the distributions Wρ on N0 and Qn,η on Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Isotropy means that we may consider the η-part of the parameter as fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' In this situation,  if (Nρ)ρ≥0 and (Xn)n∈N0 are independent stochastic processes such that  L(Nρ) = Wρ for all ρ ≥ 0,  L(Xn) = Qn,η for all n ∈ N0,  then  Pη,ρ = L(XNρ)  for all ρ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (45)  Equation (45) may be regarded as an almost sure representation of the distributional equa-  tion (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Classical examples of such almost sure representations are the Skorohod coupling  in connection with distributional convergence, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Kallenberg (1997, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='30),  and the representation of a sequence of uniformly distributed permutations on the sets  {0, 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=', n}, n ∈ N, by the Chinese restaurant process, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Pitman (2006, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Of course, such representations are most useful if the successive variables are close to each  other (and not simply chosen to be independent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Our aim here are representations of the  type (45) for the distribution families considered above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' For this, we first formalize the polar  decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Let η ∈ Sd, we assume that d > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Recall from (6) that Cd(η, y) = {z ∈ Sd : ηtz = y}  and let  nη : Sd \\ {−η, η} → Cd(η, 0),  x �→  x − (ηtx)η  ∥x − (ηtx)η∥,  be the normalized projection onto the orthogonal complement of the linear subspace of Rd+1  spanned by η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' This can be extended to the whole of the sphere by choosing some arbitrary of  ξ ∈ Sd as the value of nη(±η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Then an inverse of the polar decomposition x �→ (ηtx, nη(x))  is given by  Ψη : [−1, 1] × Cd(η, 0) → Sd,  (y, z) �→ yη + (1 − y2)1/2z,  in the sense that Ψη(ηtx, nη(x)) = x for all x ∈ Sd, and a random vector X with values in  Sd may be written as  X = Ψη  �  ηtX, nη(X)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (46)  Let Qη be a distribution on (Sd, B(Sd)) that has a density f with respect to unif(Sd) which  can be written as fη(x) = g(ηtx), x ∈ Sd, see (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' If X ∼ Qη then ηtX and nη(X) are  independent, ηtX has the distribution νd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='g with νd-density g, and nη(X) ∼ unif (Cd(η, 0));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  see Watson (1983, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' 92) and Mardia and Jupp (2000, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' 169).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' So, conversely, if we have  independent random variables Y ∼ νd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='g and Z ∼ unif (Cd(η, 0)), then  X =D Ψη(Y, Z) = Y η + (1 − Y 2)1/2Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (47)  Mardia and Jupp (2000, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' 161,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' 169) call (46) and the distributional version (47) the  tangent-normal decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  In the past this decomposition has been successfully ap-  plied by many authors treating different problems in directional statistics, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Saw  (1978, 1983, 1984), Garc´ıa–Portugu´es et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' (2020), and Ulrich (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' In practice, a ran-  dom variable X with distribution Qη is simply obtained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Suppose first that η is  equal to e1 = (1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' , 0)t, the first unit vector in the canonical basis of Rd+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' With e1 as  ‘north pole’ the polar representation takes on a particularly simple form: From y ∈ [−1, 1]  and z = (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' , zd)t ∈ Sd−1 we get x ∈ Sd by x = Φ(y, z) with  Φ(y, z) :=  �  y , (1 − y2)1/2 z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' , (1 − y2)1/2 zd  �t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (48)  21  Then, starting with a random variable Y ∼ νd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='g and another random variable Z ∼ unif(Sd−1)  independent of Y , we obtain an Sd-valued random variable X with distribution Qe1 via  X := Φ(Y, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' For a general η ∈ Sd we use that Qη is the push-forward QU  e1 of Qe1 under  the mapping x �→ Ux, where U ∈ O(d + 1) is such that η = Ue1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' U has η as its first  column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Then, defining Φη(y, z) = UΦ(y, z) it follows that X := Φη(Y, Z) ∼ Qη;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' see also  Saw (1978) and Ulrich (1984) for this construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Starting with a random variable Y that is almost surely equal to 1, it follows that the  random variable X = Φη(Y, Z) is almost surely equal to η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' This means that from a Markov  chain (Yn)n∈N0 with state space [−1, 1] starting in 1, where for n ∈ N the distribution of Yn  is the push-forward of Qn,η under x �→ ηtx, and a single random variable Z ∼ unif(Sd−1),  we obtain a Markov chain (Xn)n∈N0 with the desired one-dimensional marginal distributions  by putting Xn := Φη(Yn, Z) for all n ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Clearly, (Yn)n∈N0 is then the latitude process  associated with (Xn)n∈N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  This reduces the first step in an almost sure construction (45) to finding a Markov chain  on [−1, 1] with prescribed marginals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' For the second step we require a suitable integer-valued  process N = (Nρ)ρ≥0 with marginal distributions Wρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' One general possibility is the quantile  transformation, which can also be used to construct a Skorohod coupling for real random  variables: With U ∼ unif(0, 1) we obtain a random variable X with distribution function F  via X := F −1(U), where F −1(u) := inf{t ∈ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' F(t) ≥ u} for 0 < u < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' If F = Fρ is the  distribution function associated with Wρ the paths of the process N = (Nρ)ρ≥0 constructed  in this way depend on the relations between the distribution functions for different ρ’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' In  particular, if the distributions Wρ are stochastically monotone, meaning that  1 − Fρ(x) ≤ 1 − Fρ′(x)  for all x ∈ R  (49)  whenever ρ ≤ ρ′, then the paths of N are increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' It is well known that this stochastic  monotonicity applies to arbitrary distributions Wρ, Wρ′ with monotone likelihood ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' To  be precise, defining a likelihood ratio of Wρ′ with respect to Wρ to be a B(R)-measurable  function Lρ,ρ′ : R → [0, ∞] such that Wρ(Lρ,ρ′ < ∞) = 1 and  Wρ′(A) =  �  A  Lρ,ρ′(x) Wρ(dx) + Wρ′({Lρ,ρ′ = ∞} ∩ A)  for all A ∈ B(R),  the distributions Wρ, Wρ′ with ρ < ρ′ have monotone likelihood ratio if there exists an  increasing function hρ,ρ′ : R → [0, ∞] such that  Lρ,ρ′ = hρ,ρ′ (Wρ + Wρ′)-almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Note that if fρ, fρ′ are densities of Wρ, Wρ′ with respect to some σ-finite measure, then  L∗  ρ,ρ′ = f ′  ρ  fρ  1(fρ > 0) + ∞ 1(fρ = 0, fρ′ > 0)  is a special version of the likelihood ratio of Wρ′ with respect to Wρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' here 1(·) denotes the  indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' In fact, (49) holds if Wρ, Wρ′ with ρ < ρ′ have monotone likelihood ratio;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Witting (1985, Satz 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Again, we consider the special case of the von Mises–Fisher distributions in some detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Example 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Let η ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' In order to translate Theorem 2, see also Example 1 (c), into  an almost sure representation for the family {MFd(η, ρ) : ρ ≥ 0} we need random vari-  ables Yn with Yn ∼ Beta[−1,1](d/2, d/2 + n) for all n ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  To this end let V , Wi,  i ∈ N0, be independent random variables with V  ∼ Γ(d/2, 1), W0 ∼ Γ(d/2, 1), and  22  Wi ∼ Exp(1) for all i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Then, using the well known connection between beta and  gamma distributions, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Johnson and Kotz (1970), we obtain independent random  variables B0 ∼ Beta(d/2, d/2), Bn ∼ Beta(d + n − 1, 1), n ∈ N, via  B0 :=  V  V + W0  ,  Bn := V + W0 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' + Wn−1  V + W0 + · · · + Wn  for all n ∈ N,  and products  ˜Yn :=  n  �  i=0  Bi =  V  V + W0 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' + Wn  ∼ Beta(d/2, d/2 + n)  for all n ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  The transformation Yn := 1 − 2 ˜Yn, n ∈ N0, now gives the desired sequence Y = (Yn)n∈N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Moreover, ˜Yn+1 = ˜YnBn+1 implies that  Yn+1 = 1 − (1 − Yn) Bn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (50)  As Bn+1 is independent of Yn this shows that Y is a Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Suppose now that (Nρ)ρ≥0 is a stochastic process with Nρ ∼ CHS(d/2, d, 2ρ) for all  ρ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Let Z ∼ unif(Sd−1) be independent of the variables V and Wi, i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Then (Xρ)ρ≥0  with  Xρ := Φη  �  YNρ, Z  �  for all ρ ≥ 0,  (51)  and Φη as in the remarks preceding the example, has the desired property that Xρ ∼  MFd(η, ρ) for all ρ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  For the construction of the counting process we use the quantile transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' The  likelihood ratios turn out to be  chs(n|d/2, d, 2ρ′)  chs(n|d/2, d, 2ρ) =  1F1(d/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' ρ)  1F1(d/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' ρ′)  (2ρ′)n  (2ρ)n  which, as a function of n, is increasing whenever ρ < ρ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' As explained above, this shows  that the paths of (Nρ)ρ≥0 are increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' From (50) it follows that Yn ≥ Yn−1 for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Taken together we see that we have found an almost sure representation (51) with a process  (YNρ)ρ≥0 that has increasing paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Some comments are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Obviously, almost sure representations are generally not  unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' In the first step of Example 18, we could use a sequence (Yn)n∈N0 of independent  random variables with Yn ∼ Beta[−1,1](d/2, d/2+n) for all n ∈ N0, or we could use the quan-  tile transformation to obtain suitable variables Yn as functions of one single U ∼ unif(0, 1)  (in fact, the corresponding likelihood ratios would be increasing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Similar to (3) in the  classical case, the representation (51) strikes a structural middle ground in this spectrum  from no dependence at all to total dependence between the variables of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Also, the  Markov chain featuring in the denominator of  YNρ = 1 −  2V  V + �Nρ  n=0 Wn  has some resemblance to the sum appearing in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  In Baringhaus and Gr¨ubel (2021a,  Remark 4 (a)) we found a discrete mixture representation for the non-central family of  hyperbolic secant distributions that may similarly written as a function of a Markov chain  of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  23  Returning to the general situation, we may regard the right hand side of (3) or (45) as a  representation of a continuous time stochastic process Z = (Zt)t≥0 by independent processes  X = (Xn)n∈N0 and N = (Nt)t≥0 via Zt = XNt for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' As already mentioned in the  introduction, equality of the marginal distributions is considerably weaker than equality of  the distributions of the processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' For example, the representation in Example 18 leads to  a process that moves by jumps from η on a fixed great circle through η towards the equator  in a piecewise constant manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Loosely speaking, a discrete mixture representation on  the process level is only possible for processes of the pure jump type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  However, if the  time parameter of the base process is X continuous too then we obtain a connection to a  famous group of results, known as skew product decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' For example, with (Xt)t≥0  a Brownian motion on Rd+1 starting at a ̸= 0 and Rt := ∥Xt∥, t ≥ 0, we have R−1  t Xt = BNt  for all t ≥ 0, where (Bt)t≥0 is a Brownian motion on Sd starting at η := ∥a∥−1a, Nt is  given implicitly by  � Nt  0  R−2  s  ds = t, and B and N are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' In particular, this leads  to a representation of the distribution of Xt/∥Xt∥, t ≥ 0, which is a family of spherical  distributions, as a continuous mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' In contrast to discrete mixture representations these  seem to be less suitable for simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  5  Proofs  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='1  Proof of Theorem 2  (a) We first simplify the norming constant in (11) for the d-dimensional spherical beta  distribution with parameters p = 1 and q = n + 1, n ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Using the duplication formula  Γ(1/2)Γ(2z) = 22z−1Γ(z)Γ(z + 1/2),  z > 0,  (52)  for the gamma function we obtain  cd(1, n + 1) = 2−(n+2λ)  Γ(1/2)Γ(n + 2λ + 1)  Γ(λ + 1)Γ(n + λ + 1/2)  = 2−(n+2λ−1) (2λ + 1)n  (λ + 1/2)n  Γ(1/2)Γ(2λ)  Γ(λ)Γ(λ + 1/2)  (53)  = 2−n (2λ + 1)n  (λ + 1/2)n  ,  and it follows that the density of SBetad(η, 1, n + 1) can be written as  f SBeta  d  (x|η, 1, n + 1) = 2−n (2λ + 1)n  (λ + 1/2)n  �  1 + ηtx  �n ,  x ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (54)  We now write  exp  �  ρ ηtx  �  = exp(−ρ) exp  �  ρ (1 + ηtx)  �  and use the expansion  exp  �  ρ (1 + ηtx)  �  =  ∞  �  n=0  ρn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  �  1 + ηtx  �n  =  ∞  �  n=0  (2ρ)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (λ + 1/2)n  (2λ + 1)n  f SBeta  d  (x|η, 1, n + 1)(x),  x ∈ Sd,  24  together with the identity  e−ρ  1F1(λ + 1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' 2λ + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' 2ρ) = 2λΓ(λ + 1)Iλ(ρ)/ρλ  to obtain  f MF  d  (x|η, ρ) =  ∞  �  n=0  chs(n|λ + 1/2, 2λ + 1, 2ρ) f SBeta  d  (x|η, 1, n + 1),  x ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  In order to prove uniqueness suppose that  ∞  �  n=0  vn SBetad(η, 1, n + 1) =  ∞  �  n=0  wn SBetad(η, 1, n + 1)  for two sequences v = (vn)n∈N0, w = (wn)n∈N0 of non-negative real numbers with sum 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Passing to the respective push-forwards under x �→ ηtx this leads to the equality of the  (continuous) densities,  ∞  �  n=0  vn  n + 1  2n+1 (1 + y)n =  ∞  �  n=0  wn  n + 1  2n+1 (1 + y)n,  −1 < y < 1,  hence the sequences v and w are equal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (b) From d = 2λ + 1, 1 −  4ρ  (1+ρ)2 = (1−ρ)2  (1+ρ)2 , and (53) it follows that  f CII  d  (x|η, ρ) =  � 1 − ρ2  (1 + ρ)2  �2λ+1 �  1 −  2ρ  (1 + ρ)2  �  1 + ηtx  ��−(2λ+1)  =  �1 − ρ  1 + ρ  �2λ+1  ∞  �  n=0  (2λ + 1)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  �  2ρ  (1 + ρ)2  �n �  1 + ηtx  �n  =  �(1 − ρ)2  (1 + ρ)2  �λ+1/2  ∞  �  n=0  (2λ + 1)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  �  4ρ  (1 + ρ)2  �n (λ + 1/2)n  (2λ + 1)n  f SBeta  d  (x|η, 1, n + 1)  =  ∞  �  n=0  nb  �  n|λ + 1/2, (1 − ρ)2/(1 + ρ)2�  f SBeta  d  (x|η, 1, n + 1),  x ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  The proof of the uniqueness is similar to that of part (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (c) Noticing (d + 1)/2 = λ + 1, and  2F1  �  λ + 1/2, λ + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' 2λ + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' 4ρ/(1 + ρ)2�  = (1 + ρ)2λ+1  1 − ρ  ,  see Magnus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' (1966, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' 39), we get arguing as in part (b) that  f CII  d  (x|η, ρ) =  1 − ρ2  (1 + ρ)2(λ+1)  �  1 −  2ρ  (1 + ρ)2  �  1 + ηtx  ��−(λ+1)  =  1 − ρ  (1 + ρ)2λ+1  ∞  �  n=0  (λ + 1)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  �  4ρ  (1 + ρ)2  �n (λ + 1/2)n  (2λ + 1)n  f SBeta  d  (x|η, 1, n + 1)  =  ∞  �  n=0  hs  �  n|λ + 1/2, λ + 1, 2λ + 1, 4ρ/(1 + ρ)2�  f SBeta  d  (x|η, 1, n + 1),  x ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  25  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='2  Proof of Theorem 3  (a) Straightforward manipulations give  f Wat  d  (x|η, ρ) =  1  1F1(1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' λ + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' ρ)  ∞  �  n=0  ρn(ηtx)2n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  =  ∞  �  n=0  chs(n|1/2, λ + 1, ρ) f SP  d (x|η, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (b) As at the beginning of the proof of Theorem 2, with p = q = n + 1, n ∈ N0, the  general expression (11) for the norming constants can be simplified to  cd(n + 1, n + 1) =  (λ + 1)n  (λ + 1/2)n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Hence the density for the associated spherical beta distribution may be written as  f SBeta  d  (x|η, n + 1, n + 1) =  (λ + 1)n  (λ + 1/2)n  �  1 − (ηtx)2�n,  x ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (55)  For ρ < 0 and η ∈ Sd we have  eρ (ηtx)2 = eρe−ρ(1−(ηtx)2) = eρ  ∞  �  n=0  (−ρ)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (λ + 1/2)n  (λ + 1)n  f SBeta  d  (x|η, n + 1, n + 1)  for all x ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Because of  e−ρ1F1(1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' λ + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' ρ) = 1F1(λ + 1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' λ + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' −ρ)  (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Magnus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' (1966, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' 267)) it follows that  f Wat  d  (x|η, ρ) =  ∞  �  n=0  chs(n|λ + 1/2, λ + 1, −ρ) f SBeta  d  (x|η, n + 1, n + 1)  for all η ∈ Sd, ρ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  In both cases, it is easy to adapt the uniqueness argument from the von Mises–Fisher  context to the Watson situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='3  Proof of Lemma 4  We write Γ(δ, α) for the gamma distribution with shape parameter δ > 0, scale parameter  α > 0 and Lebesgue density t �→ Γ(δ)−1αδtδ−1 exp(−αt), t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Let V be a positive random  variable with V ∼ Γ(δ, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Then, for k ∈ N0,  E  �  V k/2 exp  �  −2  1/2 τ V  1/2��  = Γ(k + 2δ)  2δ−1Γ(δ) eτ 2/2 D−(k+2δ)(2  1/2τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Writing  (δ − 1/2)k  (2δ − 1)k  = Γ(δ − 1/2 + k)  Γ(δ − 1/2)  Γ(2δ − 1)  Γ(2δ − 1 + k)  26  and using the duplication formula (52) we obtain  dpc(k|δ, τ) = 1  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' (2  1/2τ)k2k (δ − 1/2)k  (2δ − 1)k  e−τ 2E  �  V k/2 exp  �  −2  1/2 τ V  1/2��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (56)  Thus,  ∞  �  k=0  dpc(k|δ, τ) = e−τ 2  ∞  �  k=0  1  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' (2  1/2τ)k2k (δ − 1/2)k  (2δ − 1)k  E  �  V k/2 exp  �  −2  1/2 τV 1/2��  = e−τ 2E  �  exp  �  −2  1/2 τV 1/2�  1F1(δ − 1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' 2δ − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' 2  3/2τV 1/2)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Using  e−z 1F1(δ − 1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' 2δ − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' 2z) = Iδ−1(z)Γ(δ)(z/2)−(δ−1),  for z ∈ R, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Erd´elyi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' (1953, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' 265, formula (10)), and  Iδ−1(z)Γ(δ)(z/2)−(δ−1) =  ∞  �  k=0  (z/2)2k  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='Γ(k + δ)  we obtain  ∞  �  k=0  dpc(k|δ, τ) = Γ(δ)e−τ 2  ∞  �  k=0  (τ 2/2)k  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='Γ(k + δ)E(V k),  which in view of E(V k) = Γ(k+δ)  Γ(δ) 2k gives �∞  k=0 dpc(k|τ, δ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='4  Proof of Theorem 5  Using (12) with S ∼ χ2  d+1 and writing  exp  �  2  1/2ρS  1/2 ηtx  �  = exp  �  2  1/2ρS  1/2 (1 + ηtx)  �  exp  �  −2  1/2ρS  1/2�  we see that  f AG  d  (x|η, ρ) =  ∞  �  n=0  (1 + ηtx)n  �  2  1/2ρ  �n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  e−ρ2E  �  Sn/2 exp(−2  1/2ρS  1/2)  �  =  ∞  �  n=0  f SBeta  d  (x|ρ, 1, n + 1)  �  2  1/2ρ  �n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  2n (λ + 1/2)n  (2λ + 1)n  e−ρ2E  �  Sn/2 exp(−2  1/2ρS  1/2)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  As χ2  d+1 = Γ(λ + 1, 1/2), the asserted discrete mixture representation now follows with (56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  The uniqueness of the representation is obtained as in the proofs of Theorems 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='5  Proof of Theorem 8  (a) We use Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' We have β(ρ) = �∞  n=1 2ρn =  2ρ  1−ρ ≤ 1 if and only if ρ ≤ 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' By  (31), the density f WC  1  (·|η, ρ) of the wrapped Cauchy distribution WC1(η, ρ) can be written  as  f WC  1  (x|η, ρ) = 1 − β(ρ) + 2  ∞  �  n=1  �  1 + Tn(ηtx)  �  ρn,  x ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  27  The representation now follows easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (b) Using (30) we can write the density f WN  1  (·|η, ρ) of the wrapped normal distribution  WN1(η, ρ) as  f WN  1  (x|η, ρ) = 1 − β(ρ) + 2  ∞  �  n=1  e−n2ρ/2 �  1 + Tn(ηtx)  �  ,  x ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  The function β is easily seen to be continuous and strictly decreasing, with unique solution  ρ0 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='570818 of the equation β(ρ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' From Proposition 7 we thus obtain a discrete  mixture expansion for the family {WN1(η, ρ) : η ∈ S1, ρ ≥ ρ0} with mixing base elements  ∆0  n,η and weights wρ(n) = 2e−n2ρ/2, n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='6  Proof of Theorem 9  (a) The density f MF  1  (·|η, ρ) of the von Mises–Fisher distribution MF1(η, ρ) can be written  as  f MF  1  (x|η, ρ) = exp(ρηtx)  I0(ρ)  = 1 + 2  ∞  �  n=1  In(ρ)  I0(ρ) Tn(ηtx),  x ∈ S1,  (57)  see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Abramowitz and Stegun (1964, formula 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' In fact, the representation (57)  corresponds to the Fourier series expansion of the density of the unique random angle in  [−π, π) associated with X ∼ MF1(η, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' To be precise, let for simplicity η = (1, 0)t and let  Θ ∈ [−π, π) be the unique random angle such that X = (cos Θ, sin Θ)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Then the density  hρ(θ) = exp(ρ cos θ)  2πI0(ρ) , θ ∈ [−π, π), of Θ with respect to the Lebesgue measure on [−π, π) has  the absolutely convergent Fourier series expansion  hρ(θ) = 1 + 2  ∞  �  n=1  In(ρ)  I0(ρ) cos(nθ),  θ ∈ [−π, π);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  see, Kent (1977, formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='1a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' In particular, β(ρ) = 2 �∞  n=1  In(ρ)  I0(ρ) = eρ/I0(ρ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' We have  limρ→0 β(ρ) = 0, limρ→∞ β(ρ) = ∞, see Abramowitz and Stegun (1964, formula 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='1), and  d  dρβ(ρ) = I0(ρ)−2eρ (I0(ρ) − I1(ρ)) > 0  for all ρ > 0,  see Soni (1965), hence the function β is strictly increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Taken together this implies that  the equation β(ρ) = 1 has a unique finite positive root ρ = ρ0 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='876842.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Proposition 7  now leads to the discrete mixture representation  MF1(η, ρ) = (2 − eρ/I0(ρ)) unif(S1) + (eρ/I0(ρ) − 1)  ∞  �  n=1  2e−ρIn(ρ)  1 − e−ρI0(ρ)∆0  n,η  = (1 − β(ρ)) unif(S1) + β(ρ)  ∞  �  n=1  psk(n|ρ) ∆0  n,η .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (b) For κ > 0, τ > 0, and −1 ≤ t ≤ 1, we have  eτt = 2κΓ(κ)τ −κ  ∞  �  n=0  (κ + n)Iκ+n(τ)Cκ  n(t),  (58)  28  see Magnus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' (1966, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' 227) or Kent (1977, Section 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Hence the density f MF  d  (·|η, ρ) of  MFd(η, ρ) can be written as  f MF  d  (x|η, ρ) = 1 +  ∞  �  n=1  �  1 + n  λ  �(2λ)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Iλ+n(ρ)  Iλ(ρ) Dλ  n(ηtx),  x ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (59)  With t = 1 in (58) we get, after some algebra,  1 =  ∞  �  n=1  �  1 + n  κ  �(2κ)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  2κΓ(κ + 1)τ−κe−τIκ(τ)  1 − 2κΓ(κ + 1)τ−κe−τIκ(τ)  Iκ+n(τ)  Iκ(τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  This implies that gpsk(·|κ, τ) is a probability mass function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Further,  βλ(ρ) =  ρλeρ  2λΓ(λ + 1)Iλ(ρ) − 1 =  ∞  �  n=1  �  1 + n  λ  �Iλ+n(ρ)  Iλ(ρ) Cλ  n(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Arguing as in part (a) we find that the equation βλ(ρ) = 1 has a unique finite positive root  ρ = ρ0(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' For the family {MFd(η, ρ) : η ∈ Sd, 0 < ρ ≤ ρ0(λ)} this finally gives the discrete  mixture representation  MFd(η, ρ) = (1 − βλ(ρ)) unif(Sd) + βλ(ρ)  ∞  �  n=1  gpsk(n|λ, ρ)∆λ  n,η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='7  Proof of Proposition 12  The statements are a consequence of the basic formulas (25) and (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='8  Proof of Proposition 13  All terms involved are positive, so there are no convergence issues, and the representation  follows from the bilinearity of the mixture operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='9  Proof of Proposition 14  (a) The distribution of X is determined by the distributions of the vectors (X0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' , Xn),  n ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' We may thus use induction, and (38) provides the necessary argument for the  induction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (b) Given a north pole η we obtain a partitioning of Sd into the sets Cη(y) = {x ∈  Sd :  ηtx = y} with the same latitude y ∈ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Isotropy implies that the transition  mechanism interacts with the function that maps the points of Sd to their latitude in the  manner required by Dynkin’s criterion, see Dynkin (1965, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' As the action  of O(d + 1) on Sd is doubly transitive, for unit vectors η1, η2 ∈ Sd with the property that  ηtη1 = ηtη2 there exists some U ∈ O(d + 1) such that Uη = η and Uη1 = η2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' The isotropy  of Q then implies that for all A ∈ B([−1, 1])  Q(η2, {x ∈ Sd : ηtx ∈ A}) = Q(Uη1, {x ∈ Sd : ηtx ∈ A})  = Q(η1, {U tx ∈ Sd : ηtx ∈ A})  = Q(η1, {y ∈ Sd : ηtUy ∈ A})  = Q(η1, {y ∈ Sd : ηty ∈ A}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (c) This follows easily on using induction and Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  29  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='10  Proof of Theorem 15  The transition probabilities for the spherical Brownian motion in dimension d ≥ 2 are given  in Hartmann and Watson (1974),  pt(x, y) = 1 +  ∞  �  n=1  �  1 + n  λ  �(2λ)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  e−n(n+2λ)t/2 Dλ  n(xty),  x, y ∈ Sd, t > 0,  see also Karlin and McGregor (1960).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Let Pt(·, x) be the distribution on Sd with density  y → pt(x, y), let  βλ(t) :=  ∞  �  n=1  �  1 + n  λ  �(2λ)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  e−n(n+2λ)t/2  for t > 0,  and let tλ > 0 be the unique positive real number such that βλ(tλ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Then, with brd(·|t)  as in the theorem, we get the discrete mixture representations  Pt(·, η) = (1 − βλ(t)) unif(Sd) + βλ(t)  ∞  �  n=1  brd(n|t)∆λ  n,η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='11  Proof of Theorem 16  Suppose that, on the contrary, (Xt)t≥0 is a homogeneous Markov process with the property  that there exists a function κ : (0, ∞) → (0, ∞) such that for all ξ ∈ Sd it holds that if  P(X0 = ξ) = 1 then  L(Xt) = MFd (ξ, κ(t))  for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Because of the Markov property we would then have positive parameters ρ, ρ′, ρ′′ such that  MFd(η, ρ) ◦ MFd(·, ρ′) = MFd(η, ρ′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (60)  To see that this cannot be true, we note that the density of MFd(η, ρ) ◦ MFd(·, ρ′) is given  by  h(x) :=  �  f MF  d  (x|ξ, ρ′) f MF  d  (ξ|η, ρ) unif(Sd)(dξ),  x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  We now refer to Kent (1977, Section 7) and the proof of Theorem 9, where it is shown that  the density f MF  d  (·|η, ρ) can be written as  f MF  d  (x|η, ρ) =  ∞  �  n=0  1  γλn  Iλ+n(ρ)  Iλ(ρ)  Dλ  n(ηtx)  for all x ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Expressing f MF  d  (·|ξ, ρ′) correspondingly, and using (25) and (26), we obtain  h(x) =  ∞  �  n=0  1  γλn  Iλ+n(ρ)  Iλ(ρ)  Iλ+n(ρ′)  Iλ(ρ′)  Dλ  n(ηtx)  for all x ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Hence by (60) we would have  Iλ+n(ρ)  Iλ(ρ)  Iλ+n(ρ′)  Iλ(ρ′)  = Iλ+n(ρ′′)  Iλ(ρ′′)  for all n ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  (61)  30  From (10) it is easily seen that for each τ > 0 the asymptotic relation  Iλ+n(τ) ∼ 2−(λ+n)Γ(λ + n + 1)−1τ λ+n  as n → ∞  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content=' Thus,  log Iλ+n(τ) = −(λ + n) log 2 − log Γ(λ + n + 1) + (λ + n) log τ + o(1)  as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  By Stirling’s formula,  log Γ(λ + n + 1) = (λ + n + 1/2) log n − n + 1  2 log (2π) + o(1)  as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  From this we deduce that  lim  n→∞  log Iλ+n(τ)  Iλ(τ)  n log n  = −1,  which is in contradiction to (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Acknowledgment  The authors would like to thank the reviewers for helpful suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
+page_content='  Statements and Declarations  The authors declare that they have no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
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+page_content=' J Math and Phys 44: 406–407  33  Themangani  R,  Porwal  S,  Magesh  N  (2020)  Generalized  hypergeometric  dis-  tribution  and  its  applications  on  univalent  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfaAeA/content/2301.03870v1.pdf'}
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+Max-Min Diversification with Fairness Constraints:
+Exact and Approximation Algorithms
+Yanhao Wang∗
+Michael Mathioudakis†
+Jia Li‡
+Francesco Fabbri§
+Abstract
+Diversity maximization aims to select a diverse and representative subset of items from a large dataset. It is
+a fundamental optimization task that finds applications in data summarization, feature selection, web search,
+recommender systems, and elsewhere. However, in a setting where data items are associated with different
+groups according to sensitive attributes like sex or race, it is possible that algorithmic solutions for this task,
+if left unchecked, will under- or over-represent some of the groups. Therefore, we are motivated to address
+the problem of max-min diversification with fairness constraints, aiming to select k items to maximize the
+minimum distance between any pair of selected items while ensuring that the number of items selected from
+each group falls within predefined lower and upper bounds. In this work, we propose an exact algorithm based
+on integer linear programming that is suitable for small datasets as well as a
+1−ε
+5 -approximation algorithm
+for any ε ∈ (0, 1) that scales to large datasets. Extensive experiments on real-world datasets demonstrate the
+superior performance of our proposed algorithms over existing ones.
+Keywords: max-min diversification, algorithmic fairness
+1
+Introduction
+In recent years, algorithms have been increasingly used for data-driven automated decision-making in many
+domains of everyday life. This has raised concerns about the possibility that algorithms may produce unfair and
+discriminatory decisions for specific population groups, particularly in sensitive socio-computational domains such
+as voting, hiring, banking, education, and criminal justice [12, 25]. To alleviate such concerns, there has been
+a lot of research devoted to incorporating fairness into the algorithms for automated decision tasks, including
+classification [14], clustering [10], ranking [24,32], matching [28], and data summarization [8,20].
+This paper considers the diversity maximization problem and addresses its fairness-aware variant.
+The
+problem consists in selecting a diverse subset of items from a given dataset and is encountered in data
+summarization [8, 23], web search [2], recommendation [21], feature selection [31], and elsewhere [34]. Existing
+literature on the problem of diversity maximization primarily focuses on two objectives, namely max-min
+diversification (MMD), which aims to maximize the minimum distance between any pair of selected items, and
+max-sum diversification (MSD), which seeks to maximize the sum of pairwise distances between selected items.
+As shown in Figure 1, MMD tends to cover the data range uniformly, while MSD tends to pick “outliers” and may
+include highly similar items in the solution. Since the notion of diversity captured by MMD better represents the
+property that data summarization, feature selection, and many other tasks target with their solutions, we will
+only consider MMD in this paper. To be precise, given a set V of n items in a metric space and a positive integer
+k ≤ n, MMD asks for a size-k subset S of V to maximize the minimum pairwise distance within S.
+In particular, we study the fair max-min diversification (FMMD) problem, a variant of MMD that aims not
+only to maximize the diversity measure defined above but also to guarantee the satisfaction of group fairness
+constraints as described below. Let all the items in V be divided into C disjoint groups V1, . . . , VC by a sensitive
+attribute such as sex or race. To ensure a fair representation, the number of items selected from each group Vc,
+where c ∈ [1, . . . , C], is limited to be between lower and upper bounds specified as input. This definition of group
+fairness constraints captures and generalizes several existing notions of fairness for groups, including proportional
+representation [7, 17], equal representation [19, 20], and statistical parity [14, 33], and has been widely used in
+optimization problems such as top-k ranking [9], submodular maximization [17], and multiwinner voting [7].
+∗East China Normal University. yhwang@dase.ecnu.edu.cn
+†University of Helsinki. michael.mathioudakis@helsinki.fi
+‡East China Normal University. jiali@stu.ecnu.edu.cn
+§Spotify. francescof@spotify.com
+arXiv:2301.02053v1  [cs.DS]  5 Jan 2023
+
+(a) MMD
+(b) MSD
+Figure 1: An illustration of max-min diversification (MMD) vs. max-sum diversification (MSD). The items in the
+solutions are marked in blue color.
+1.1
+Related Work Erkut [15] proved that the MMD problem is NP-hard in metric spaces. Ravi et al. [26]
+proposed a
+1
+2-approximation greedy algorithm [16] for MMD and proved that no polynomial algorithm could
+achieve a better approximation factor unless P=NP. Recently, many different algorithms have been proposed for
+MMD in various settings. Indyk et al. [18] proposed a 1
+3-approximation distributed algorithm for MMD based
+on the notion of coresets. Drosou and Pitoura [13] designed a b−1
+2b2 -approximation cover tree-based algorithm for
+MMD on dynamic data, where b is the base of the cover tree. Ceccarello et al. [6] proposed ( 1
+2 − ε)-approximate
+MapReduce and streaming algorithms for MMD in metric spaces of bounded doubling dimension. Borassi et
+al. [5] proposed a sliding-window algorithm for MMD. Nevertheless, none of the above algorithms are applicable
+to FMMD because they cannot guarantee the fulfillment of fairness constraints.
+Moumoulidou et al. [23] first proposed approximation algorithms for the fair variant of MMD. Addanki et
+al. [1] improved the approximation ratios of the algorithms in [23]. Wang et al. [30] proposed two streaming
+algorithms for the fair variant of MMD. However, these algorithms work for exact-size group fairness constraints,
+a special case of our bounded-size group fairness constraints. Moreover, as shown empirically, these algorithms
+provide lower-quality solutions than ours.
+Besides MMD, many other optimization problems have similar group fairness-aware variants – e.g.,
+determinantal point processes [8], k-centers [11, 19, 20], top-k ranking [9], submodular maximization [17, 29],
+and multiwinner voting [7]. However, since their objectives differ from MMD, the algorithms proposed for their
+fair variants are not directly applicable to FMMD.
+1.2
+Our Results The main results of this paper are two novel algorithms for the fair max-min diversification
+(FMMD) problem, which selects a size-k subset S from a dataset V that maximizes the diversity value while
+satisfying group-fairness constraints.
+We first propose FMMD-E, an exact algorithm that is suitable for solving FMMD on small datasets, despite
+the NP-hardness of the problem.
+This algorithm exploits the connection between the MMD and maximum
+independent set (MIS) problems. It formulates FMMD as the problem of finding an independent set of vertices
+with group fairness constraints on an undirected graph. Then, the optimal solution of FMMD can be obtained
+in O(nk log n) time by solving the reduced problem via integer-linear programming (ILP).
+Since FMMD-E cannot scale to large datasets, we propose FMMD-S, a more scalable approximation algorithm
+for FMMD. Specifically, for any ε ∈ (0, 1), FMMD-S provides
+1−ε
+5 -approximate solutions for FMMD in
+O
+�
+Ckn + Ck log 1
+ε
+�
+time.
+Under the assumptions that k = o(log n) and C = O(1), the time complexity of
+FMMD-S is reduced to O
+�
+n(k + log 1
+ε)
+�
+. The basic idea of FMMD-S is, in the first step, to limit the computation
+to a considerably smaller subset of the original dataset (i.e., coreset) by running the greedy algorithm [16, 26]
+and, in the second step, to use an ILP-based method similar to FMMD-E to obtain an approximate solution to
+FMMD from the subset.
+Finally, we compare the performance of our algorithms with the state-of-the-art algorithms in [1, 23, 30]
+for the FMMD problem on real-world datasets. The results show that i) FMMD-E provides exact solutions in
+reasonable time on small datasets (e.g., n = 1, 000); ii) FMMD-S returns solutions of higher quality than existing
+approximation algorithms in comparable time while scaling to large datasets with millions of items.
+
+2
+Preliminaries
+In this section, we first formally define the FMMD problem, a fairness-aware variant of max-min diversification
+(MMD), and then provide its hardness result.
+Max-Min Diversification (MMD). Let V be a set of n items and d : V × V → R≥0 be a distance
+metric that captures the dissimilarities between items in V . We remind that, by definition, d(·, ·) satisfies the
+following properties for any u, v, w ∈ V : i) d(u, v) = 0 ⇔ u = v (identity of indiscernibles); ii) d(u, v) = d(v, u)
+(symmetry); iii) d(u, v) + d(v, w) ≥ d(u, w) (triangle inequality).
+For MMD, the diversity value div(S) of a
+subset S ⊆ V is defined as the minimum among all pairwise distances between distinct items in S – i.e.,
+div(S) = minu,v∈S : u̸=v d(u, v).
+Given a set V , a distance function d(·, ·), and a positive integer k ≤ n, the
+MMD problem asks for a size-k subset S of V such that div(S) is maximized.
+Fair Max-Min Diversification (FMMD). Let the set V be divided into C disjoint groups V1, . . . , VC by
+a sensitive categorical attribute, such as sex or race. For FMMD, the fairness-aware variant of MMD, the group
+fairness constraints restrict the selection of items from each group Vc for c ∈ [C] so that the number of items
+selected from Vc lies within a range of values from lc to hc (both inclusive). Meanwhile, it also requires that the
+total number of selected items is k. Formally, the collection F of all feasible solutions for FMMD is
+F = {S ⊆ V : |S| = k ∧ lc ≤ |S ∩ Vc| ≤ hc, ∀c ∈ [C]}
+and, to discard from consideration trivially empty sets F, we will further assume that lc ≤ hc ≤ |Vc| and
+�C
+c=1 lc ≤ k ≤ �C
+c=1 hc.
+The FMMD problem asks for a subset S of V so that S satisfies the group
+fairness constraints (i.e., S ∈ F) and div(S) is maximized, or formally, S∗ = arg maxS∈F div(S), where S∗
+and OPT = div(S∗) denote the optimal solution of FMMD and its diversity value, respectively.
+Hardness of FMMD. By using a reduction from the Clique problem, MMD is proven to be NP-hard
+for general metric spaces and cannot be approximated within any factor greater than 1
+2 unless P=NP [15, 26].
+Nevertheless, a greedy algorithm provides the best possible 1
+2-approximate solution in O(kn) time [16]. Although
+the greedy algorithm does not work for FMMD directly, as it may provide solutions that do not fall within F (i.e.,
+are not “fair”), it will be used as a subroutine for our FMMD-S algorithm in Section 3.2 for data reduction. Since
+MMD is a special case of FMMD when C = 1 and l1 ≤ k ≤ h1, the hardness result for MMD can be generalized
+to FMMD as follows:
+Theorem 2.1. FMMD is NP-hard and cannot be approximated by a factor of 1
+2 + ε for any parameter ε > 0
+unless P=NP.
+3
+Algorithms
+In this section, we describe our proposed algorithms for FMMD. First, we propose FMMD-E, an exact algorithm
+that runs in O(nk log n) time (Section 3.1). Second, we propose FMMD-S, a 1−ε
+5 -approximation algorithm that
+runs in O
+�
+Ckn + Ck log 1
+ε
+�
+time for any error parameter ε ∈ (0, 1) (Section 3.2).
+3.1
+An Exact ILP-Based Algorithm To build an exact algorithm for FMMD, we use ideas similar to [3] for
+the reduction from MMD to maximum independent set (MIS). Given the set of feasible solutions F and a positive
+real number δ, the decision version of FMMD asks whether there is a set S ⊆ V such that S ∈ F and div(S) ≥ δ.
+Given an instance of the FMMD decision problem, we build an undirected graph G = (V, E) as follows: the set
+of vertices in G is identical to V and there is an edge between two vertices u, v ∈ V if and only if d(u, v) < δ. We
+remind that a vertex set S is called an independent set iff no two vertices in S are adjacent. Moreover, we define
+the Fair Independent Set (FIS) problem that determines whether there exists an independent vertex set S ∈ F
+on the graph G. Based on the above definitions, the lemma below asserts the equivalence between FMMD and
+FIS.
+Lemma 3.1. FMMD is equivalent to FIS.
+Proof. In the one direction, assume that the answer to FMMD is ‘yes’ – i.e., there is a subset S ∈ F of V with
+div(S) ≥ δ. Then, we have d(u, v) ≥ δ for any u, v ∈ S. Thus, by construction, there is no edge (u, v) ∈ E, and S
+is an independent vertex set of G. Therefore, the answer to FIS is ‘yes’ as well. In the opposite direction, assume
+that the answer to FIS is ‘yes’ – i.e., S ∈ F is an independent set. By definition, there is no edge between any of
+
+𝛿
+(a) FMMD
+(b) FIS
+Figure 2: Example for the reduction from Fair Max-Min Diversification (FMMD) to Fair Independent Set (FIS)
+on a dataset with n = 10 points and C = 2 groups in blue and red. An FMMD instance with k = 5, lc = 2
+and hc = 3 for c = 1, 2 is reduced to an FIS instance where a fair independent set of vertices is represented by
+triangles.
+its vertices in G, which by construction means that d(u, v) ≥ δ for any u, v ∈ S and, therefore, we have div(S) ≥ δ
+for the given S ∈ F. Therefore, the answer to FMMD is also ‘yes’. We thus prove that the answer to FMMD is
+‘yes’ if and only if the answer to FIS is ‘yes’, which concludes the proof.
+Additionally, we have two observations for FMMD, which are easy to verify from its definition.
+Fact 1. (Monotonicity) If there exists a set S ∈ F with div(S) ≥ δ, then such a set will exist for any δ′ ≤ δ;
+If there does not exist any set S ∈ F with div(S) ≥ δ, then such a set will not exist for any δ′ ≥ δ.
+Fact 2. (Discontinuity) The optimal diversity value OPT for FMMD is always equal to the distance d(u, v)
+between some pair of items u, v ∈ V .
+From all the above results, the following theorem asserts that FMMD is reducible to FIS.
+Theorem 3.1. An exact solution of FMMD is obtained by solving O(log n) FIS instances.
+Proof. Let us consider the following algorithm. First, compute and sort the distances between all pairs of items
+in V . Then, use a binary search on the sorted array of pairwise distances to find the largest d∗ such that the
+answer to its corresponding FIS instance is ‘yes’. The binary search finds d∗ in O(log n) steps, as the number
+of pairwise distances is O(n2). And it holds that div(S∗) ≥ d∗ from Lemma 3.1. Observation 1 guarantees that
+there does not exist any S ∈ F with div(S) > d∗ due to the maximality of d∗. Observation 2 ensures that d∗ is
+exactly equal to OPT. Thus, the above procedure identifies the exact solution to FMMD.
+The ILP Formulation of FMMD. In light of Theorem 3.1, what remains to obtain an exact algorithm
+for FMMD is to design an exact algorithm for FIS. We note that FIS without fairness constraints is equivalent to
+the maximum independent set (MIS) problem. We thus adapt the edge-based integer-linear programming (ILP)
+formulation of MIS by adding fairness constraints to define an FIS instance, as shown in Eq. 3.1–3.5.
+max
+z =
+n
+�
+i=1
+xi
+(3.1)
+s.t.
+xi + xj ≤ 1, ∀(vi, vj) ∈ E
+(3.2)
+n
+�
+i=1
+xi ≤ k
+(3.3)
+lc ≤
+�
+vi∈Vc
+xi ≤ hc, ∀c ∈ [C]
+(3.4)
+xi ∈ {0, 1}, ∀i ∈ [n]
+(3.5)
+
+Algorithm 1 FMMD-E
+Input: Dataset V = �C
+c=1 Vc with n = |V |; lower and upper bounds lc, hc ∈ Z+ for c ∈ [C]; size constraint k ∈ Z+.
+Output: A set S∗ ⊆ V such that S∗ ∈ F.
+1: Compute the distances of all pairs of items in V and sort them ascendingly as D[1, . . . , n(n−1)
+2
+]
+2: Let L ← 1, H ← n(n−1)
+2
+, cur ← L+H
+2
+, S∗ ← ∅
+3: while H > L do
+4:
+Build an undirected graph G = (V, E) where E = {(u, v) ∈ V × V | d(u, v) < D[cur]}
+5:
+Compute the solution x of the ILP in Eq. 3.1–3.5
+6:
+Find a set S = {vi ∈ V : xi = 1, i ∈ [n]} based on x
+7:
+if |S| = k then
+8:
+If div(S) > div(S∗) or S∗ = ∅, then S∗ ← S
+9:
+Let L ← cur + 1 and cur ← L+H
+2
+10:
+else
+11:
+Let H ← cur − 1 and cur ← L+H
+2
+12: return S∗
+where xi is a binary variable to indicate whether vi ∈ V is included in the solution or not, the objective function
+in Eq. 3.1 and the first constraint in Eq. 3.2 are the same as the edge-based ILP formulation of MIS, the second
+constraint in Eq. 3.3 limits the solution size to at most k, and the third constraint in Eq. 3.4 is on the upper and
+lower bounds of the number of items chosen from each group Vc. By solving the ILP in Eq. 3.1–3.5 optimally, we
+will either find a fair independent set S = {vi ∈ V : xi = 1, i ∈ [n]} of G if z = k or confirm that there does not
+exist such a set if z < k.
+Algorithm Description and Complexity. By combining the constructive proof of Theorem 3.1 and the
+ILP formulation of FMMD, we obtain FMMD-E, an exact algorithm for FMMD, as presented in Algorithm 1.
+First, it computes the distances of all n(n−1)
+2
+pairs of distinct items in V in O(n2) steps and sorts them in ascending
+order in an array D[1, . . . , n(n−1)
+2
+] in O(n2 log n) steps. Then, a binary search is performed on D to find OPT in
+O(log n) steps. For each guess D[cur] of OPT, it builds an undirected graph G in O(n2) steps and finds a set S
+by solving the ILP in Eq. 3.1–3.5 in
+�n
+k
+�
+= O(nk) steps. If |S| = k, then S ∈ F and div(S) ≥ D[cur]. In this
+case, the search space is narrowed to the upper half to check whether there is a better solution. Otherwise, or if
+|S| < k, then OPT < D[cur] and the search space is narrowed to the lower half. Finally, when the binary search
+is terminated, the algorithm has found the exact solution S∗ to FMMD. The time complexity of FMMD-E is
+O(nk log n). Moreover, since |D| = |E| = O(n2), its space complexity is O(n2).
+3.2
+A More Scalable Approximation Algorithm The main drawback of FMMD-E is that it cannot handle
+large datasets due to exponential complexity. Standard optimization libraries, such as CPLEX1 and Gurobi2,
+can only solve ILPs with up to several thousand variables optimally in a reasonable time. A natural approach to
+addressing this challenge is to identify a “coreset”, i.e., a small subset of the original dataset on which the exact
+algorithm is run to look for approximate solutions. Formally, a subset V ′ ⊆ V is called an α-coreset (0 ≤ α ≤ 1)
+of V for FMMD if OPT[V ′] ≥ α · OPT, where OPT[V ′] is the optimal diversity value for FMMD on V ′.
+It now remains to answer i) how such a coreset is built and ii) what approximation factor is obtained. For
+i), we are inspired by the notion of composable coresets [18,31] for MMD in streaming and distributed settings.
+The basic idea is first to partition the dataset and run the greedy algorithm of [16] on each partition to obtain a
+partial solution and then compute a final solution from the union of partial solutions. In the context of FMMD,
+the dataset is naturally divided into C groups. Thus, we first find a solution from each group, then consider the
+union of all group-specific solutions as our coreset, and finally use FMMD-E to obtain a solution from the coreset,
+which is feasible since the coreset size is small. We refer to the resulting algorithm as FMMD-S. For ii), we prove
+that the obtained solution offers an approximation factor of 1−ε
+5
+for any ε ∈ (0, 1).
+Algorithm Description. FMMD-S is described in Algorithm 2. Initially, it invokes the greedy algorithm on
+V without fairness constraints to compute an initial solution U (Lines 1-3). Note that the greedy algorithm is 1
+2-
+approximate for MMD, and any feasible solution of FMMD must also be feasible for MMD. Therefore, the optimal
+1www.ibm.com/products/ilog-cplex-optimization-studio
+2www.gurobi.com/products/gurobi-optimizer/
+
+Algorithm 2 FMMD-S
+Input: Dataset V = �C
+c=1 Vc with n = |V |; lower and upper bounds lc, hc ∈ Z+ for c ∈ [C]; size constraint k ∈ Z+; error
+parameter ε ∈ (0, 1).
+Output: A set S ⊆ V such that S ∈ F.
+1: Pick an arbitrary item u1 from V and set U = {u1}
+2: for i ← 2, . . . , k do
+3:
+ui ← arg maxv∈V minu∈U d(u, v) and U ← U ∪ {ui}
+4: Set Uc ← Vc ∩ U, d′ ← 2 · div(U)
+5: repeat
+6:
+for c ← 1, . . . , C do
+7:
+while |Uc| < k and there exists some item v ∈ Vc such that div(Uc ∪ {v}) ≥ d′ do
+8:
+u′
+c ← arg maxv∈Vc minu∈Uc d(u, v)
+9:
+Uc ← Uc ∪ {u′
+c}
+10:
+Build an undirected graph G = (V ′, E), where V ′ = �
+c Uc and E = {(u, v) ∈ V ′ × V ′ | d(u, v) < d′
+2 }
+11:
+Compute the solution x of the ILP in Eq. 3.1–3.5
+12:
+Find a set S = {vi ∈ V : xi = 1, i ∈ [n]} based on x
+13:
+if |S| < k then
+14:
+S ← ∅ and d′ ← (1 − ε) · d′
+15: until S ̸= ∅
+16: return S
+diversity OPT of FMMD is bounded by 2·div(U). Subsequently, the algorithm divides U by group into U1, . . . , UC
+and guesses OPT as its upper bound d′ = 2 · div(U) (Line 4). For each c ∈ [C], it runs the greedy algorithm to add
+new items from Vc to Uc until |Uc| = k or there does not exist any v ∈ Vc to make div(Uc ∪ {v}) ≥ d′ (Lines 5-9).
+At this point, each Uc is a partial group-specific solution, and the union V ′ = �
+c Uc of partial solutions is the
+coreset. Next, using a similar procedure to FMMD-E, it builds a graph G on V ′ with diversity threshold d′
+2 and
+solves the ILP of Eq. 3.1–3.5 on G to obtain a solution S (Lines 11-12). Finally, if |S| = k, we have got a solution
+S ∈ F with div(S) ≥ d′
+2 and S will be returned as the final solution; otherwise, d′ is decreased by a factor of
+1 − ε, where ε ∈ (0, 1) is an error parameter, and the above procedure is executed again for the smaller d′ until a
+feasible solution S is found (Lines 13-16).
+Theoretical Analysis. Next, we give the complexity and approximation factor of FMMD-S.
+Theorem 3.2. FMMD-S is a 1−ε
+5 -approximation algorithm for FMMD running in O
+�
+Ckn + Ck log 1
+ε
+�
+time.
+Proof. If there is any set S′ ⊆ V ′ s.t. S′ ∈ F and div(S′) ≥ d′
+2 , then FMMD-S identifies such S′ (Line 12) from the
+exact solution of the ILP in Eq. 3.1–3.5. In addition, since the greedy algorithm (Lines 1-3) is 1
+2-approximate [26],
+the initial value of d′ is at least 2 · OPT
+2 = OPT > 2
+5 · OPT. Therefore, to prove the approximation factor, it suffices
+to show that there exists some S′ ⊆ V ′ s.t. S′ ∈ F and div(S′) ≥ d′
+2 when d′ ∈ [ 2(1−ε)
+5
+· OPT, 2
+5 · OPT].
+Towards this end, we next construct such a set S′ from V ′. Let S∗ be the optimal solution for FMMD on V ,
+and S∗
+c = S∗ ∩ Vc be its subset from group c. First, we initialize S′ = ∅. Then, we consider two cases for different
+groups. We consider first the groups of Case #1 in arbitrary order, then those of Case #2 in arbitrary order, and
+select |S∗
+c | items from each group c into S′.
+Case #1 (|Uc| < k): Let f : Vc → Uc map each item v ∈ Vc to its nearest neighbor f(v) in Uc. Note that
+the condition in Line 7 ensures that d(v, u) < d′ for any v ∈ Vc and u ∈ Uc. For each item sc,i ∈ S∗
+c , we add
+item f(sc,i) into S′. We now show that the added items are distinct. Indeed, if f(sc,i) ≡ f(sc,j) for i ̸= j, then
+the triangle inequality would give d(sc,i, sc,j) ≤ d(sc,i, f(sc,i))+d(sc,j, f(sc,j)) = d(sc,i, f(sc,i))+d(sc,j, f(sc,i)) <
+2·d′ ≤ 4
+5 ·OPT < OPT; however, at the same time we have d(sc,i, sc,j) ≥ OPT because sc,i, sc,j ∈ S∗, which leads to a
+contradiction. Moreover, because we have identified for each sc,i ∈ S∗
+c one distinct item in Uc, we have |Uc| ≥ |S∗
+c |.
+After processing all the groups in Case #1, we have d(f(s∗
+i ), f(s∗
+j)) ≥ d(s∗
+i , s∗
+j) − d(s∗
+i , f(s∗
+i )) − d(s∗
+j, f(s∗
+j)) >
+OPT − 2d′ ≥ OPT
+5
+for any f(s∗
+i ), f(s∗
+j) ∈ S′ and thus div(S′) > OPT
+5 .
+Case #2 (|Uc| = k): Let g : Uc → S′ map each item u ∈ Uc to its nearest neighbor g(u) in the current
+instance of S′. We remove from Uc every u ∈ Uc with d(u, g(u)) < d′
+2 . Because the condition of Line 7 ensures
+d(uc,i, uc,j) ≥ d′ for any i ̸= j, there is at most one item removed for each item in S′ – otherwise, the triangle
+
+Table 1: Statistics of datasets used in our experiments
+Dataset
+Group
+C
+n
+dim
+Distance Metric
+Adult
+Sex
+2
+48,842
+6
+l2-distance
+Race
+5
+S+R
+10
+CelebA
+Sex
+2
+202,599
+25,088
+l1-distance
+Age
+2
+S+A
+4
+Census
+Sex
+2
+2,426,116
+25
+l1-distance
+Age
+7
+S+A
+14
+Twitter
+Sex
+3
+18,836
+1,024
+Angular distance
+inequality would give d(uc,i, uc,j) < d′, thus leading to a contradiction. Therefore, at least k − |S′| items remain
+in Uc. Moreover, |S∗
+c | ≤ k − |S′| because S′ always contains the same number of items from each considered
+group as S∗ throughout the construction process. We pick |S∗
+c | items from the remaining ones and add them
+to S′. After this operation, we still have div(S′) ≥ d′
+2 ≥ 1−ε
+5
+· OPT since our earlier removal of items from Uc
+ensured d(uc,i, g(uc,i)) ≥ d′
+2 for each added uc,i. Finally, after processing all groups in Case #2, we get a set
+S′ that contains the same number of items from each group c ∈ [C] as S∗, which implies that S′ ∈ F, and
+div(S′) ≥ 1−ε
+5
+· OPT. Therefore, we conclude that FMMD-S is a 1−ε
+5 -approximation algorithm for FMMD.
+Since it takes O(nk) time to compute U as well as Uc for each c ∈ [C], the total time to compute V ′ is O(Ckn)
+and |V ′| ≤ Ck. Then, the time to solve the ILP in Eq. 3.1–3.5 for FMMD-S is O(Ck) because there are at most
+�Ck
+k
+�
+= O(Ck) possible solutions to consider. Moreover, the number of iterations for d′ is O(log 1
+ε) since the ratio
+between the first and last values of d′ is O(1). Thus, the time complexity of FMMD-S is O
+�
+Ckn + Ck log 1
+ε
+�
+.
+When k = o(log n) and C = O(1), its time complexity is reduced to O
+�
+n(k + log 1
+ε)
+�
+. Additionally, its space
+complexity is O(n + C2k2) since the number of edges in G is O(C2k2).
+4
+Experimental Evaluation
+4.1
+Experimental Setup In this section, we conduct extensive experiments to evaluate the performance of our
+proposed algorithms, i.e., FMMD-E and FMMD-S. We compare them with the state-of-the-art FMMD algorithms,
+including FairSwap, FairFlow, and FairGMM in [23], FairGreedyFlow in [1], and SFDM1 and SFDM2 in [30]. We
+implemented all the above algorithms in Python 3 using the NetworkX library for building and manipulating
+graphs and the Gurobi optimizer for solving ILPs. All the experiments were carried out on a desktop with an
+Intel Core i5-9500 3.0GHz processor and 32GB RAM running Ubuntu 20.04.3 LTS. Each algorithm was run on a
+single thread. All data and code are publicly available at https://osf.io/te34m/.
+We use four public real-world datasets listed in Table 1, where dim is the dimensionality of the feature vector.
+The detailed information and preprocessing procedures on each dataset are described in Appendix A. The fairness
+constraints in our experiments are defined according to the proportional representation [7, 17]: For each group
+c ∈ [C], we set lc = max(1, (1 − α)k · |Vc|
+n ) and hc = (1 + α)k · |Vc|
+n
+with α = 0.2 in FMMD-E and FMMD-S and
+kc = ⌈k · |Vc|
+n ⌉ or ⌊k · |Vc|
+n ⌋ in all other algorithms. All the algorithms were executed ten times in each experiment.
+We report the average running time and average diversity value of the solutions provided by each algorithm. We
+use ‘N/A’ to indicate that an algorithm either does not find a solution within one day or does not work when
+C > 2 (i.e., FairSwap and SFDM1). In the preliminary experiments (see Appendix B), we find that the solution
+quality of FMMD-S hardly improves when ε is decreased below 0.05 and so we fix ε = 0.05 for FMMD-S in all
+the remaining experiments.
+4.2
+Experimental Results Table 2 shows the diversity achieved by different algorithms on “small” datasets,
+i.e., datasets that were obtained by sampling 1,000 items uniformly at random from each full dataset. Figures 3–
+4 illustrate the performance of different algorithms on small datasets with varying k. Note that FairSwap and
+SFDM1 are specific for the case of C = 2 and FairGMM fails to finish within one day when C > 3 or k > 10 since
+it has to enumerate
+�kC
+k
+�
+sets for solution computation. They are ignored in subsequent tables and figures when
+they cannot provide valid solutions.
+
+Table 2: Diversity values of the solutions returned by different algorithms on small datasets for solution size
+k = 10. The optimums OPT∗ without fairness constraints are reported to show “the price of fairness”.
+Dataset
+Group
+FairSwap
+FairFlow
+FairGMM
+FairGreedyFlow
+SFDM1
+SFDM2
+FMMD-E
+FMMD-S
+OPT∗
+Adult
+Sex
+4.51
+3.24
+4.81
+2.18
+4.03
+4.18
+5.30
+4.64
+5.30
+Race
+N/A
+1.73
+N/A
+1.33
+N/A
+2.83
+4.54
+4.01
+S+R
+N/A
+0.84
+N/A
+0.99
+N/A
+2.04
+3.12
+2.88
+CelebA
+Sex
+101457.3
+55540.7
+127354.6
+46266.9
+94873.3
+93216.6
+129818.2
+106959.3
+129871.5
+Age
+110098.1
+54649.6
+127871.2
+46312.3
+102762.9
+91578.2
+129871.5
+116701.6
+S+A
+N/A
+42412.7
+N/A
+39967.2
+N/A
+88026.7
+127974.6
+108055.3
+Census
+Sex
+28.4
+15.8
+29.8
+14.7
+27.2
+28.0
+34.0
+30.3
+35.0
+Age
+N/A
+7.6
+N/A
+9.3
+N/A
+15.7
+34.0
+30.3
+Twitter
+Sex
+N/A
+1.23
+1.44
+1.23
+N/A
+1.39
+1.51
+1.46
+1.51
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+FairFlow
+FairGMM
+FairSwap
+FairGreedyFlow
+SFDM1
+SFDM2
+FMMD-S
+FMMD-E
+10
+20
+30
+40
+50
+k
+0
+2
+4
+6
+8
+10
+Diversity
+(a) Adult (Sex)
+10
+20
+30
+40
+50
+k
+0
+1
+2
+3
+4
+5
+Diversity
+(b) Adult (Race)
+10
+20
+30
+40
+50
+k
+0
+1
+2
+3
+4
+Diversity
+(c) Adult (S+R)
+10
+20
+30
+40
+50
+k
+0.0
+0.5
+1.0
+1.5
+Diversity
+×105
+(d) CelebA (Sex)
+10
+20
+30
+40
+50
+k
+0.0
+0.5
+1.0
+1.5
+Diversity
+×105
+(e) CelebA (Age)
+10
+20
+30
+40
+50
+k
+0.0
+0.5
+1.0
+1.5
+Diversity
+×105
+(f) CelebA (S+A)
+10
+20
+30
+40
+50
+k
+0
+10
+20
+30
+40
+50
+Diversity
+(g) Census (Sex)
+10
+20
+30
+40
+50
+k
+0
+10
+20
+30
+40
+Diversity
+(h) Census (Age)
+20
+30
+40
+50
+k
+0
+5
+10
+15
+20
+25
+30
+Diversity
+(i) Census (S+A)
+10
+20
+30
+40
+50
+k
+0.9
+1.1
+1.3
+1.5
+1.7
+Diversity
+(j) Twitter (Sex)
+Figure 3: Diversity values of the solutions of different algorithms with varying solution size k on small datasets.
+In general, FMMD-E always provides optimal solutions for FMMD within the time limit (i.e., 24 hours)
+when n = 1, 000. The price of fairness, measured by the decrease in diversity due to the fairness constraints,
+is marginal on all datasets except Adult with C ≥ 5. FMMD-S shows much higher solution quality (up to 3.9×
+greater in diversity value) than all approximation algorithms except FairGMM. Although FairGMM sometimes
+provides slightly better solutions than FMMD-S, it runs more than two orders of magnitude slower. Moreover,
+the diversity values of all algorithms drop with k because div is a monotonically non-increasing function. The
+running time is independent of k for FMMD-E, grows exponentially with k for FairGMM, and increases linearly
+with k for FMMD-S and other algorithms, which all follow from their time complexities.
+Table 3 presents the diversity values and running time of different algorithms for solution size k = 50 on all
+the full datasets. The performance of different algorithms by varying the solution size k from 10 to 100 on full
+datasets (with all the items) is presented in Figures 5–6. In general, the running time of all algorithms increases
+substantially with n and dim. FairGMM and FMMD-E fail to finish within one day and thus are omitted from
+Table 3. FMMD-S provides better solutions (up to 7.2× higher in diversity value) than all the baselines in most
+cases. The only exception is that FMMD-S shows slightly lower solution quality than FairSwap and SFDM1 on
+CelebA when C = 2. This is because of the extremely high dimensionality of CelebA (d = 25, 088), where the
+distances between different pairs of points are less distinguishable. In such cases, the thresholding method in
+FMMD-S is inferior to the local search methods in FairSwap and SFDM1. Moreover, the time efficiency of FMMD-
+S is lower than the baselines, as solving ILPs is often time-consuming. Nevertheless, on Census, i.e., the largest
+dataset with more than two million items, FMMD-S still finishes the computation within 3 hours.
+To evaluate the scalability of different algorithms, we vary the number C of groups and the number n of
+points on synthetic datasets. In particular, each dataset consists of ten two-dimensional Gaussian isotropic blobs
+
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+FairFlow
+FairGMM
+FairSwap
+FairGreedyFlow
+SFDM1
+SFDM2
+FMMD-S
+FMMD-E
+10
+20
+30
+40
+50
+k
+10−1
+100
+101
+102
+103
+Time (s)
+(a) Adult (Sex)
+10
+20
+30
+40
+50
+k
+10−1
+100
+101
+102
+103
+Time (s)
+(b) Adult (Race)
+10
+20
+30
+40
+50
+k
+10−1
+100
+101
+102
+103
+Time (s)
+(c) Adult (S+R)
+10
+20
+30
+40
+50
+k
+10−1
+100
+101
+102
+103
+Time (s)
+(d) CelebA (Sex)
+10
+20
+30
+40
+50
+k
+10−1
+100
+101
+102
+103
+Time (s)
+(e) CelebA (Age)
+10
+20
+30
+40
+50
+k
+10−1
+100
+101
+102
+103
+Time (s)
+(f) CelebA (S+A)
+10
+20
+30
+40
+50
+k
+10−2
+10−1
+100
+101
+102
+103
+Time (s)
+(g) Census (Sex)
+10
+20
+30
+40
+50
+k
+10−1
+100
+101
+102
+103
+Time (s)
+(h) Census (Age)
+20
+30
+40
+50
+k
+10−1
+100
+101
+102
+103
+Time (s)
+(i) Census (S+A)
+10
+20
+30
+40
+50
+k
+10−1
+100
+101
+102
+103
+104
+Time (s)
+(j) Twitter (Sex)
+Figure 4: Running time of different algorithms with varying solution size k on small datasets.
+Table 3: Results of different algorithms on full datasets for solution size k = 50. FairGMM and FMMD-E are
+omitted because they cannot provide any solution within the time limit (i.e., 24 hours).
+Dataset Group
+FairSwap
+FairFlow
+FairGreedyFlow
+SFDM1
+SFDM2
+FMMD-S
+diversity time(s) diversity time(s) diversity time(s) diversity time(s) diversity time(s) diversity time(s)
+Adult
+Sex
+2.55
+28.45
+2.10
+26.47
+1.46
+99.61
+2.86
+8.17
+3.22
+13.71
+3.56
+46.38
+Race
+N/A
+1.43
+28.47
+0.78
+91.24
+N/A
+2.86
+15.81
+3.56
+48.10
+S+R
+N/A
+0.99
+32.64
+0.50
+148.12
+N/A
+2.55
+23.35
+3.61
+45.51
+CelebA
+Sex
+125865.8
+3093.9
+101303.9
+2624.1
+57696.8
+3558.0 128450.2
+1473.4
+117342.5
+1354.4
+123639.0
+4536.4
+Age
+112387.6
+4141.2
+74679.4
+2598.8
+55470.0
+2260.8 126446.9
+1024.0
+121056.9
+1222.3
+113798.5
+4626.6
+S+A
+N/A
+29278.4
+2589.5
+34066.4
+3871.3
+N/A
+118598.7
+1141.7 129866.3
+4752.5
+Census
+Sex
+22.8
+1533.3
+16.8
+1250.4
+N/A
+22.3
+328.7
+24.4
+459.14
+28.6
+3268.8
+Age
+N/A
+5.2
+1461.5
+N/A
+N/A
+12.5
+662.76
+16.0
+2728.2
+S+A
+N/A
+3.4
+1723.5
+N/A
+N/A
+11.1
+170.98
+15.0
+9224.8
+Twitter
+Sex
+N/A
+1.08
+120.85
+1.04
+783.7
+N/A
+1.33
+85.56
+1.38
+208.52
+with random centers in [−10, 10]2 and identity covariance matrices. Each point is assigned to one of the C groups
+uniformly at random. The Euclidean distance is used as the distance metric. For fixed C = 2 or 10, we obtain
+six datasets with n = 102, 103, . . . , 107; and for fixed n = 1, 000, ten datasets with C = 2, 4, . . . , 20.
+The performance of different algorithms by varying n and C on synthetic datasets for solution size k = 20 is
+presented in Figure 7. In terms of solution quality, the diversity values are steady for FMMD-E and FMMD-S but
+significantly drop for all other algorithms when C increases. In terms of efficiency, the running time of FMMD-E
+is hardly affected by C. Other algorithms run slower when C is larger. Nevertheless, FMMD-S runs faster than
+any other algorithm when C ≥ 8, and its advantages in time efficiency become more significant with increasing C.
+Finally, all algorithms’ diversity values and running time grow with the dataset size n. FMMD-E cannot scale to
+large datasets due to its exponential time complexity. All in all, FMMD-S outperforms all the other approximation
+algorithms in terms of solution quality for different C or n, and its advantages become more apparent when C or
+n is larger. These results confirm the scalability of FMMD-S concerning the group size C and dataset size n.
+5
+Conclusion
+We investigated the problem of max-min diversification with fairness constraints (FMMD) in this paper. We
+proposed an exact ILP-based algorithm for this problem on small datasets. We further designed a scalable 1−ε
+5 -
+approximation algorithm, where ε ∈ (0, 1), on massive datasets based on our exact algorithm and the notion of
+coresets. Extensive experimental results on four real-world datasets confirmed the effectiveness, efficiency, and
+scalability of our proposed algorithms.
+
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+FairFlow
+FairSwap
+FairGreedyFlow
+SFDM1
+SFDM2
+FMMD-S
+20
+40
+60
+80
+100
+k
+0
+2
+4
+6
+8
+Diversity
+(a) Adult (Sex)
+20
+40
+60
+80
+100
+k
+0
+2
+4
+6
+8
+Diversity
+(b) Adult (Race)
+20
+40
+60
+80
+100
+k
+0
+2
+4
+6
+Diversity
+(c) Adult (S+R)
+20
+40
+60
+80
+100
+k
+0.0
+0.5
+1.0
+1.5
+2.0
+Diversity
+×105
+(d) CelebA (Sex)
+20
+40
+60
+80
+100
+k
+0.0
+0.5
+1.0
+1.5
+2.0
+Diversity
+×105
+(e) CelebA (Age)
+20
+40
+60
+80
+100
+k
+0.0
+0.5
+1.0
+1.5
+2.0
+Diversity
+×105
+(f) CelebA (S+A)
+20
+40
+60
+80
+100
+k
+10
+20
+30
+40
+Diversity
+(g) Census (Sex)
+20
+40
+60
+80
+100
+k
+0
+10
+20
+30
+40
+Diversity
+(h) Census (Age)
+20
+40
+60
+80
+100
+k
+0
+10
+20
+30
+40
+Diversity
+(i) Census (S+A)
+20
+40
+60
+80
+100
+k
+0.0
+0.5
+1.0
+1.5
+2.0
+Diversity
+(j) Twitter (Sex)
+Figure 5: Diversity values of the solutions of different algorithms with varying solution size k on full datasets.
+While a step forward in both theoretical and experimental aspects of the algorithms for the fair variant
+of diversity maximization, our work leaves many open problems for future exploration. A natural question is
+whether there is any polynomial-time O(1)-approximation algorithm for FMMD, since our FMMD-S algorithm
+has an approximation factor of 1−ε
+5
+but runs in polynomial time only when C = O(1) and k = o(log n), whereas the
+best-known polynomial-time algorithm in [1] only achieves an approximation factor of
+1
+C+1. Moreover, it would
+also be interesting to study the fairness-aware variants of other diversity measures (e.g., the ones in [4,18]).
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+
+0.0
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+FairFlow
+FairSwap
+FairGreedyFlow
+SFDM1
+SFDM2
+FMMD-S
+20
+40
+60
+80
+100
+k
+100
+101
+102
+103
+Time (s)
+(a) Adult (Sex)
+20
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+100
+101
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+103
+Time (s)
+(b) Adult (Race)
+20
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+100
+101
+102
+103
+Time (s)
+(c) Adult (S+R)
+20
+40
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+k
+102
+103
+104
+Time (s)
+(d) CelebA (Sex)
+20
+40
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+80
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+k
+102
+103
+104
+Time (s)
+(e) CelebA (Age)
+20
+40
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+80
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+k
+102
+103
+104
+Time (s)
+(f) CelebA (S+A)
+20
+40
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+80
+100
+k
+102
+103
+104
+Time (s)
+(g) Census (Sex)
+20
+40
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+80
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+k
+102
+103
+104
+Time (s)
+(h) Census (Age)
+20
+40
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+80
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+k
+101
+102
+103
+104
+105
+Time (s)
+(i) Census (S+A)
+20
+40
+60
+80
+100
+k
+101
+102
+103
+104
+Time (s)
+(j) Twitter (Sex)
+Figure 6: Running time of different algorithms with varying solution size k on full datasets.
+0.0
+0.2
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+0.6
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+0.0
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+FairFlow
+FairGMM
+FairSwap
+FairGreedyFlow
+SFDM1
+SFDM2
+FMMD-S
+FMMD-E
+4
+8
+12
+16
+20
+C
+0
+1
+2
+3
+4
+5
+Diversity
+4
+8
+12
+16
+20
+C
+10−1
+10−1
+100
+101
+102
+Time (s)
+(a) Synthetic (n = 1, 000)
+102
+103
+104
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+106
+107
+n
+0
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+102 103 104 105 106 107
+n
+10−2
+10−1
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+101
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+103
+104
+Time (s)
+(b) Synthetic (C = 2)
+102
+103
+104
+105
+106
+107
+n
+0
+1
+2
+3
+4
+5
+6
+Diversity
+102 103 104 105 106 107
+n
+10−2
+10−1
+100
+101
+102
+103
+104
+Time (s)
+(c) Synthetic (C = 10)
+Figure 7: Results of different algorithms by varying the number C of groups and dataset size n for solution size
+k = 20. Since FMMD-E does not provide any solution within the time limit on datasets with n > 1, 000, FairGMM
+does not provide any solution within the time limit for k = 20, and FairSwap and SFDM1 cannot work with C > 2,
+they are not plotted in these cases.
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+A
+Data Preparation
+Detailed information about the four real-world datasets we use and the preprocessing procedures for them are
+presented as follows.
+• Adult is retrieved from the UCI Machine Learning Repository3. It is a collection of 48,842 records from
+the 1994 US Census database. We select six numeric attributes as features and normalize each to have zero
+mean and unit standard deviation. The l2-distance (Euclidean distance) is used as the distance metric. The
+groups are generated from two demographic attributes: sex and race. By using them individually and in
+combination, there are 2 (sex), 5 (race), and 10 (sex + race) groups, respectively.
+• CelebA is provided by Liu et al. [22] on the website4. It is a set of 202,599 images of human faces. We
+get a 25,088-dimensional (512 × 7 × 7) feature vector for each image from the pre-trained VGG16 model
+in Keras5. The l1-distance (Manhattan distance) between feature vectors is used as the distance metric.
+We generate 2 groups from a human-annotated class label “sex” {‘female’, ‘male’}, 2 groups from another
+human-annotated class label “age” {‘young’, ‘not young’}, and 4 groups from both of them, respectively.
+• Census is also retrieved from the UCI Machine Learning Repository6. It is a set of 2,426,116 records from
+the 1990 US Census data. We take 25 (normalized) numeric attributes as features and use the l1-distance
+(Manhattan distance) as the distance metric.
+We generate 2, 7, and 14 groups from two demographic
+attributes sex, age, and both of them, respectively.
+• Twitter is retrieved from Kaggle7. It is a collection of 18,836 tweets with user profiles. We transform
+each tweet into a 1,024-dimensional feature vector using the sentence-transformer model8 [27] (“bert-large-
+nli-stsb-mean-tokens”). The angular distance is used as the distance metric. We generate 3 groups from the
+attribute “sex” {‘female’, ‘male’, ‘non-human’} in user profiles.
+The proportion of each group on each dataset is provided in Table 4.
+B
+Parameter Tuning for FMMD-S
+Figure 8 illustrates the performance of FMMD-S by varying the parameter ε from 0.001 to 0.5 for solution size
+k = 10 on small datasets (n = 1, 000). Regarding solution quality, the diversity values of the solutions of FMMD-S
+remain approximately constant when ε takes small values. However, they decrease and become less stable as ε
+becomes larger. Especially, when ε = 0.5, FMMD-S cannot provide stable and high-quality solutions in most cases.
+In terms of efficiency, the running time increases significantly and becomes less stable for smaller ε, particularly
+when ε ≤ 0.01. These results conform to our theoretical analyses of FMMD-S. The above results show that the
+most appropriate value of ε lies in the range [0.01, 0.1] in most cases. Therefore, we decide to set ε = 0.05 in all
+other experiments throughout this paper.
+3archive.ics.uci.edu/ml/datasets/adult
+4mmlab.ie.cuhk.edu.hk/projects/CelebA.html
+5https://keras.io/api/applications/vgg/
+6archive.ics.uci.edu/ml/datasets/US+Census+Data+(1990)
+7www.kaggle.com/crowdflower/twitter-user-gender-classification
+8huggingface.co/models?library=sentence-transformers
+
+Table 4: Proportion of each group on each dataset.
+Dataset
+Group
+Proportion of Each Group
+Adult
+Sex
+‘Female’: 33.2%, ‘Male’: 66.8%
+Race
+‘Amer-Indian-Eskimo’: 1.0%, ‘Asian-Pac-Islander’: 3.1%, ‘Black’: 9.6%,
+‘Others’: 0.8%, ‘White’: 85.5%
+Sex+Race
+‘AIE+F’: 0.4%, ‘API+F’: 1.1%, ‘B+F’: 4.7%, ‘O+F’: 0.3%, ‘W+F’: 26.7%,
+‘AIE+M’: 0.6%, ‘API+M’: 2.0%, ‘B+M’: 4.9%, ‘O+M’: 0.5%, ‘W+M’: 58.8%
+CelebA
+Sex
+‘Female’: 58.3%, ‘Male’: 41.7%
+Age
+‘Not Young’: 22.6%, ‘Young’: 77.4%
+Sex+Age
+‘NY+F’: 7.3%, ‘Y+F’: 51.0%, ‘NY+M’: 15.3%, ‘Y+M’: 26.4%
+Census
+Sex
+‘Female’ : 51.6%, ‘Male’ : 48.4%
+Age
+‘A1 (12-)’: 18.2%, ‘A2 (13-19)’: 10.0%, ‘A3 (20-29)’: 15.3%,
+‘A4 (30-39)’: 16.7%, ‘A5 (40-49)’: 12.9%, ‘A6 (50-64)’: 13.6%,
+‘A7 (65+)’: 13.3%
+Sex+Age
+‘A1+F’: 8.9%, ‘A2+F’: 4.9%, ‘A3+F’: 7.7%, ‘A4+F’: 8.5%, ‘A5+F’: 6.6%,
+‘A6+F’: 7.2%, ‘A7+F’: 7.9%, ‘A1+M’: 9.3%, ‘A2+M’: 5.1%, ‘A3+M’: 7.6%,
+‘A4+M’: 8.2%, ‘A5+M’: 6.3%, ‘A6+M’: 6.5%, ‘A7+M’: 5.4%
+Twitter
+Sex
+‘Female’: 35.6%, ‘Male’: 32.9%, ‘Non-Human’: 31.5%
+0.001 0.005 0.01
+0.05
+0.1
+0.5
+ϵ
+4.0
+4.5
+5.0
+5.5
+Diversity
+0.001 0.005 0.01
+0.05
+0.1
+0.5
+ϵ
+0.17
+0.18
+0.19
+0.20
+0.21
+Time (s)
+(a) Adult (Sex)
+0.001 0.005
+0.01
+0.05
+0.1
+0.5
+ϵ
+2
+3
+4
+5
+Diversity
+0.001 0.005 0.01
+0.05
+0.1
+0.5
+ϵ
+10−1
+100
+101
+102
+Time (s)
+(b) Adult (Race)
+0.001 0.005 0.01
+0.05
+0.1
+0.5
+ϵ
+2.0
+2.5
+3.0
+3.5
+Diversity
+0.001 0.005 0.01
+0.05
+0.1
+0.5
+ϵ
+10−1
+100
+101
+102
+Time (s)
+(c) Adult (S+R)
+0.001 0.005 0.01
+0.05
+0.1
+0.5
+ϵ
+0.5
+1.0
+1.5
+Diversity
+×105
+0.001 0.005 0.01
+0.05
+0.1
+0.5
+ϵ
+100
+101
+102
+Time (s)
+(d) CelebA (Sex)
+0.001 0.005 0.01
+0.05
+0.1
+0.5
+ϵ
+0.5
+1.0
+1.5
+Diversity
+×105
+0.001 0.005 0.01
+0.05
+0.1
+0.5
+ϵ
+100
+101
+102
+103
+Time (s)
+(e) CelebA (Age)
+0.001 0.005 0.01
+0.05
+0.1
+0.5
+ϵ
+0.5
+1.0
+1.5
+Diversity
+×105
+0.001 0.005 0.01
+0.05
+0.1
+0.5
+ϵ
+100
+101
+102
+103
+Time (s)
+(f) CelebA (S+A)
+0.001 0.005 0.01
+0.05
+0.1
+0.5
+ϵ
+15
+20
+25
+30
+35
+Diversity
+0.001 0.005 0.01
+0.05
+0.1
+0.5
+ϵ
+10−1
+100
+101
+Time (s)
+(g) Census (Sex)
+0.001 0.005 0.01
+0.05
+0.1
+0.5
+ϵ
+15
+20
+25
+30
+35
+Diversity
+0.001 0.005 0.01
+0.05
+0.1
+0.5
+ϵ
+10−1
+100
+101
+Time (s)
+(h) Census (Age)
+0.001 0.005 0.01
+0.05
+0.1
+0.5
+ϵ
+1.3
+1.4
+1.5
+1.6
+Diversity
+0.001 0.005 0.01
+0.05
+0.1
+0.5
+ϵ
+100
+101
+Time (s)
+(i) Twitter (Sex)
+Figure 8: Results of FMMD-S with varying the parameter ε for solution size k = 10. The error bars in each plot
+indicate the maximum and minimum of the diversity value and running time over 10 runs, respectively. In all
+plots for diversity values, the blue horizontal lines denote the diversity values of the optimal solutions provided
+by FMMD-E, which measure the gaps between the approximate solutions of FMMD-S and the optimal ones.
+
diff --git a/c9A0T4oBgHgl3EQfGv_2/content/tmp_files/load_file.txt b/c9A0T4oBgHgl3EQfGv_2/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..926d0370404c9c688c025df347c06b018f6ba38e
--- /dev/null
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@@ -0,0 +1,1491 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf,len=1490
+page_content='Max-Min Diversification with Fairness Constraints:  Exact and Approximation Algorithms  Yanhao Wang∗  Michael Mathioudakis†  Jia Li‡  Francesco Fabbri§  Abstract  Diversity maximization aims to select a diverse and representative subset of items from a large dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' It is  a fundamental optimization task that finds applications in data summarization, feature selection, web search,  recommender systems, and elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' However, in a setting where data items are associated with different  groups according to sensitive attributes like sex or race, it is possible that algorithmic solutions for this task,  if left unchecked, will under- or over-represent some of the groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Therefore, we are motivated to address  the problem of max-min diversification with fairness constraints, aiming to select k items to maximize the  minimum distance between any pair of selected items while ensuring that the number of items selected from  each group falls within predefined lower and upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' In this work, we propose an exact algorithm based  on integer linear programming that is suitable for small datasets as well as a  1−ε  5 -approximation algorithm  for any ε ∈ (0, 1) that scales to large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Extensive experiments on real-world datasets demonstrate the  superior performance of our proposed algorithms over existing ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Keywords: max-min diversification, algorithmic fairness  1  Introduction  In recent years, algorithms have been increasingly used for data-driven automated decision-making in many  domains of everyday life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' This has raised concerns about the possibility that algorithms may produce unfair and  discriminatory decisions for specific population groups, particularly in sensitive socio-computational domains such  as voting, hiring, banking, education, and criminal justice [12, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' To alleviate such concerns, there has been  a lot of research devoted to incorporating fairness into the algorithms for automated decision tasks, including  classification [14], clustering [10], ranking [24,32], matching [28], and data summarization [8,20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  This paper considers the diversity maximization problem and addresses its fairness-aware variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  The  problem consists in selecting a diverse subset of items from a given dataset and is encountered in data  summarization [8, 23], web search [2], recommendation [21], feature selection [31], and elsewhere [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Existing  literature on the problem of diversity maximization primarily focuses on two objectives, namely max-min  diversification (MMD), which aims to maximize the minimum distance between any pair of selected items, and  max-sum diversification (MSD), which seeks to maximize the sum of pairwise distances between selected items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  As shown in Figure 1, MMD tends to cover the data range uniformly, while MSD tends to pick “outliers” and may  include highly similar items in the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Since the notion of diversity captured by MMD better represents the  property that data summarization, feature selection, and many other tasks target with their solutions, we will  only consider MMD in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' To be precise, given a set V of n items in a metric space and a positive integer  k ≤ n, MMD asks for a size-k subset S of V to maximize the minimum pairwise distance within S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  In particular, we study the fair max-min diversification (FMMD) problem, a variant of MMD that aims not  only to maximize the diversity measure defined above but also to guarantee the satisfaction of group fairness  constraints as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Let all the items in V be divided into C disjoint groups V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' , VC by a sensitive  attribute such as sex or race.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' To ensure a fair representation, the number of items selected from each group Vc,  where c ∈ [1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' , C], is limited to be between lower and upper bounds specified as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' This definition of group  fairness constraints captures and generalizes several existing notions of fairness for groups, including proportional  representation [7, 17], equal representation [19, 20], and statistical parity [14, 33], and has been widely used in  optimization problems such as top-k ranking [9], submodular maximization [17], and multiwinner voting [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  ∗East China Normal University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' yhwang@dase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='ecnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='cn  †University of Helsinki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' michael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='mathioudakis@helsinki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='fi  ‡East China Normal University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' jiali@stu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='ecnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='cn  §Spotify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' francescof@spotify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='com  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='02053v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='DS]  5 Jan 2023  (a) MMD  (b) MSD  Figure 1: An illustration of max-min diversification (MMD) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' max-sum diversification (MSD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' The items in the  solutions are marked in blue color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='1  Related Work Erkut [15] proved that the MMD problem is NP-hard in metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Ravi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' [26]  proposed a  1  2-approximation greedy algorithm [16] for MMD and proved that no polynomial algorithm could  achieve a better approximation factor unless P=NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Recently, many different algorithms have been proposed for  MMD in various settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Indyk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' [18] proposed a 1  3-approximation distributed algorithm for MMD based  on the notion of coresets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Drosou and Pitoura [13] designed a b−1  2b2 -approximation cover tree-based algorithm for  MMD on dynamic data, where b is the base of the cover tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Ceccarello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' [6] proposed ( 1  2 − ε)-approximate  MapReduce and streaming algorithms for MMD in metric spaces of bounded doubling dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Borassi et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' [5] proposed a sliding-window algorithm for MMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Nevertheless, none of the above algorithms are applicable  to FMMD because they cannot guarantee the fulfillment of fairness constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Moumoulidou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' [23] first proposed approximation algorithms for the fair variant of MMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Addanki et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' [1] improved the approximation ratios of the algorithms in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' [30] proposed two streaming  algorithms for the fair variant of MMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' However, these algorithms work for exact-size group fairness constraints,  a special case of our bounded-size group fairness constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Moreover, as shown empirically, these algorithms  provide lower-quality solutions than ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Besides MMD, many other optimization problems have similar group fairness-aware variants – e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=',  determinantal point processes [8], k-centers [11, 19, 20], top-k ranking [9], submodular maximization [17, 29],  and multiwinner voting [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' However, since their objectives differ from MMD, the algorithms proposed for their  fair variants are not directly applicable to FMMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='2  Our Results The main results of this paper are two novel algorithms for the fair max-min diversification  (FMMD) problem, which selects a size-k subset S from a dataset V that maximizes the diversity value while  satisfying group-fairness constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  We first propose FMMD-E, an exact algorithm that is suitable for solving FMMD on small datasets, despite  the NP-hardness of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  This algorithm exploits the connection between the MMD and maximum  independent set (MIS) problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' It formulates FMMD as the problem of finding an independent set of vertices  with group fairness constraints on an undirected graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Then, the optimal solution of FMMD can be obtained  in O(nk log n) time by solving the reduced problem via integer-linear programming (ILP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Since FMMD-E cannot scale to large datasets, we propose FMMD-S, a more scalable approximation algorithm  for FMMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Specifically, for any ε ∈ (0, 1), FMMD-S provides  1−ε  5 -approximate solutions for FMMD in  O  �  Ckn + Ck log 1  ε  �  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Under the assumptions that k = o(log n) and C = O(1), the time complexity of  FMMD-S is reduced to O  �  n(k + log 1  ε)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' The basic idea of FMMD-S is, in the first step, to limit the computation  to a considerably smaller subset of the original dataset (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=', coreset) by running the greedy algorithm [16, 26]  and, in the second step, to use an ILP-based method similar to FMMD-E to obtain an approximate solution to  FMMD from the subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Finally, we compare the performance of our algorithms with the state-of-the-art algorithms in [1, 23, 30]  for the FMMD problem on real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' The results show that i) FMMD-E provides exact solutions in  reasonable time on small datasets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=', n = 1, 000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' ii) FMMD-S returns solutions of higher quality than existing  approximation algorithms in comparable time while scaling to large datasets with millions of items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  2  Preliminaries  In this section, we first formally define the FMMD problem, a fairness-aware variant of max-min diversification  (MMD), and then provide its hardness result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Max-Min Diversification (MMD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Let V be a set of n items and d : V × V → R≥0 be a distance  metric that captures the dissimilarities between items in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' We remind that, by definition, d(·, ·) satisfies the  following properties for any u, v, w ∈ V : i) d(u, v) = 0 ⇔ u = v (identity of indiscernibles);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' ii) d(u, v) = d(v, u)  (symmetry);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' iii) d(u, v) + d(v, w) ≥ d(u, w) (triangle inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  For MMD, the diversity value div(S) of a  subset S ⊆ V is defined as the minimum among all pairwise distances between distinct items in S – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=',  div(S) = minu,v∈S : u̸=v d(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Given a set V , a distance function d(·, ·), and a positive integer k ≤ n, the  MMD problem asks for a size-k subset S of V such that div(S) is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Fair Max-Min Diversification (FMMD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Let the set V be divided into C disjoint groups V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' , VC by  a sensitive categorical attribute, such as sex or race.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' For FMMD, the fairness-aware variant of MMD, the group  fairness constraints restrict the selection of items from each group Vc for c ∈ [C] so that the number of items  selected from Vc lies within a range of values from lc to hc (both inclusive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Meanwhile, it also requires that the  total number of selected items is k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Formally, the collection F of all feasible solutions for FMMD is  F = {S ⊆ V : |S| = k ∧ lc ≤ |S ∩ Vc| ≤ hc, ∀c ∈ [C]}  and, to discard from consideration trivially empty sets F, we will further assume that lc ≤ hc ≤ |Vc| and  �C  c=1 lc ≤ k ≤ �C  c=1 hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  The FMMD problem asks for a subset S of V so that S satisfies the group  fairness constraints (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=', S ∈ F) and div(S) is maximized, or formally, S∗ = arg maxS∈F div(S), where S∗  and OPT = div(S∗) denote the optimal solution of FMMD and its diversity value, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Hardness of FMMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' By using a reduction from the Clique problem, MMD is proven to be NP-hard  for general metric spaces and cannot be approximated within any factor greater than 1  2 unless P=NP [15, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Nevertheless, a greedy algorithm provides the best possible 1  2-approximate solution in O(kn) time [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Although  the greedy algorithm does not work for FMMD directly, as it may provide solutions that do not fall within F (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=',  are not “fair”), it will be used as a subroutine for our FMMD-S algorithm in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='2 for data reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Since  MMD is a special case of FMMD when C = 1 and l1 ≤ k ≤ h1, the hardness result for MMD can be generalized  to FMMD as follows:  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' FMMD is NP-hard and cannot be approximated by a factor of 1  2 + ε for any parameter ε > 0  unless P=NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  3  Algorithms  In this section, we describe our proposed algorithms for FMMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' First, we propose FMMD-E, an exact algorithm  that runs in O(nk log n) time (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Second, we propose FMMD-S, a 1−ε  5 -approximation algorithm that  runs in O  �  Ckn + Ck log 1  ε  �  time for any error parameter ε ∈ (0, 1) (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='1  An Exact ILP-Based Algorithm To build an exact algorithm for FMMD, we use ideas similar to [3] for  the reduction from MMD to maximum independent set (MIS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Given the set of feasible solutions F and a positive  real number δ, the decision version of FMMD asks whether there is a set S ⊆ V such that S ∈ F and div(S) ≥ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Given an instance of the FMMD decision problem, we build an undirected graph G = (V, E) as follows: the set  of vertices in G is identical to V and there is an edge between two vertices u, v ∈ V if and only if d(u, v) < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' We  remind that a vertex set S is called an independent set iff no two vertices in S are adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Moreover, we define  the Fair Independent Set (FIS) problem that determines whether there exists an independent vertex set S ∈ F  on the graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Based on the above definitions, the lemma below asserts the equivalence between FMMD and  FIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' FMMD is equivalent to FIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' In the one direction, assume that the answer to FMMD is ‘yes’ – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=', there is a subset S ∈ F of V with  div(S) ≥ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Then, we have d(u, v) ≥ δ for any u, v ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Thus, by construction, there is no edge (u, v) ∈ E, and S  is an independent vertex set of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Therefore, the answer to FIS is ‘yes’ as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' In the opposite direction, assume  that the answer to FIS is ‘yes’ – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=', S ∈ F is an independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' By definition, there is no edge between any of  𝛿  (a) FMMD  (b) FIS  Figure 2: Example for the reduction from Fair Max-Min Diversification (FMMD) to Fair Independent Set (FIS)  on a dataset with n = 10 points and C = 2 groups in blue and red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' An FMMD instance with k = 5, lc = 2  and hc = 3 for c = 1, 2 is reduced to an FIS instance where a fair independent set of vertices is represented by  triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  its vertices in G, which by construction means that d(u, v) ≥ δ for any u, v ∈ S and, therefore, we have div(S) ≥ δ  for the given S ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Therefore, the answer to FMMD is also ‘yes’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' We thus prove that the answer to FMMD is  ‘yes’ if and only if the answer to FIS is ‘yes’, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Additionally, we have two observations for FMMD, which are easy to verify from its definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Fact 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' (Monotonicity) If there exists a set S ∈ F with div(S) ≥ δ, then such a set will exist for any δ′ ≤ δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  If there does not exist any set S ∈ F with div(S) ≥ δ, then such a set will not exist for any δ′ ≥ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' (Discontinuity) The optimal diversity value OPT for FMMD is always equal to the distance d(u, v)  between some pair of items u, v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  From all the above results, the following theorem asserts that FMMD is reducible to FIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' An exact solution of FMMD is obtained by solving O(log n) FIS instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Let us consider the following algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' First, compute and sort the distances between all pairs of items  in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Then, use a binary search on the sorted array of pairwise distances to find the largest d∗ such that the  answer to its corresponding FIS instance is ‘yes’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' The binary search finds d∗ in O(log n) steps, as the number  of pairwise distances is O(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' And it holds that div(S∗) ≥ d∗ from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Observation 1 guarantees that  there does not exist any S ∈ F with div(S) > d∗ due to the maximality of d∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Observation 2 ensures that d∗ is  exactly equal to OPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Thus, the above procedure identifies the exact solution to FMMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  The ILP Formulation of FMMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' In light of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='1, what remains to obtain an exact algorithm  for FMMD is to design an exact algorithm for FIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' We note that FIS without fairness constraints is equivalent to  the maximum independent set (MIS) problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' We thus adapt the edge-based integer-linear programming (ILP)  formulation of MIS by adding fairness constraints to define an FIS instance, as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  max  z =  n  �  i=1  xi  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='1)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  xi + xj ≤ 1, ∀(vi, vj) ∈ E  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='2)  n  �  i=1  xi ≤ k  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='3)  lc ≤  �  vi∈Vc  xi ≤ hc, ∀c ∈ [C]  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='4)  xi ∈ {0, 1}, ∀i ∈ [n]  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='5)  Algorithm 1 FMMD-E  Input: Dataset V = �C  c=1 Vc with n = |V |;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' lower and upper bounds lc, hc ∈ Z+ for c ∈ [C];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' size constraint k ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Output: A set S∗ ⊆ V such that S∗ ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  1: Compute the distances of all pairs of items in V and sort them ascendingly as D[1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' , n(n−1)  2  ]  2: Let L ← 1, H ← n(n−1)  2  , cur ← L+H  2  , S∗ ← ∅  3: while H > L do  4:  Build an undirected graph G = (V, E) where E = {(u, v) ∈ V × V | d(u, v) < D[cur]}  5:  Compute the solution x of the ILP in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='5  6:  Find a set S = {vi ∈ V : xi = 1, i ∈ [n]} based on x  7:  if |S| = k then  8:  If div(S) > div(S∗) or S∗ = ∅, then S∗ ← S  9:  Let L ← cur + 1 and cur ← L+H  2  10:  else  11:  Let H ← cur − 1 and cur ← L+H  2  12: return S∗  where xi is a binary variable to indicate whether vi ∈ V is included in the solution or not, the objective function  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='1 and the first constraint in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='2 are the same as the edge-based ILP formulation of MIS, the second  constraint in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='3 limits the solution size to at most k, and the third constraint in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='4 is on the upper and  lower bounds of the number of items chosen from each group Vc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' By solving the ILP in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='5 optimally, we  will either find a fair independent set S = {vi ∈ V : xi = 1, i ∈ [n]} of G if z = k or confirm that there does not  exist such a set if z < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Algorithm Description and Complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' By combining the constructive proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='1 and the  ILP formulation of FMMD, we obtain FMMD-E, an exact algorithm for FMMD, as presented in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  First, it computes the distances of all n(n−1)  2  pairs of distinct items in V in O(n2) steps and sorts them in ascending  order in an array D[1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' , n(n−1)  2  ] in O(n2 log n) steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Then, a binary search is performed on D to find OPT in  O(log n) steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' For each guess D[cur] of OPT, it builds an undirected graph G in O(n2) steps and finds a set S  by solving the ILP in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='5 in  �n  k  �  = O(nk) steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' If |S| = k, then S ∈ F and div(S) ≥ D[cur].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' In this  case, the search space is narrowed to the upper half to check whether there is a better solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Otherwise, or if  |S| < k, then OPT < D[cur] and the search space is narrowed to the lower half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Finally, when the binary search  is terminated, the algorithm has found the exact solution S∗ to FMMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' The time complexity of FMMD-E is  O(nk log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Moreover, since |D| = |E| = O(n2), its space complexity is O(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='2  A More Scalable Approximation Algorithm The main drawback of FMMD-E is that it cannot handle  large datasets due to exponential complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Standard optimization libraries, such as CPLEX1 and Gurobi2,  can only solve ILPs with up to several thousand variables optimally in a reasonable time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' A natural approach to  addressing this challenge is to identify a “coreset”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=', a small subset of the original dataset on which the exact  algorithm is run to look for approximate solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Formally, a subset V ′ ⊆ V is called an α-coreset (0 ≤ α ≤ 1)  of V for FMMD if OPT[V ′] ≥ α · OPT, where OPT[V ′] is the optimal diversity value for FMMD on V ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  It now remains to answer i) how such a coreset is built and ii) what approximation factor is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' For  i), we are inspired by the notion of composable coresets [18,31] for MMD in streaming and distributed settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  The basic idea is first to partition the dataset and run the greedy algorithm of [16] on each partition to obtain a  partial solution and then compute a final solution from the union of partial solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' In the context of FMMD,  the dataset is naturally divided into C groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Thus, we first find a solution from each group, then consider the  union of all group-specific solutions as our coreset, and finally use FMMD-E to obtain a solution from the coreset,  which is feasible since the coreset size is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' We refer to the resulting algorithm as FMMD-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' For ii), we prove  that the obtained solution offers an approximation factor of 1−ε  5  for any ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Algorithm Description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' FMMD-S is described in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Initially, it invokes the greedy algorithm on  V without fairness constraints to compute an initial solution U (Lines 1-3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Note that the greedy algorithm is 1  2-  approximate for MMD, and any feasible solution of FMMD must also be feasible for MMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Therefore, the optimal  1www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='com/products/ilog-cplex-optimization-studio  2www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='gurobi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='com/products/gurobi-optimizer/  Algorithm 2 FMMD-S  Input: Dataset V = �C  c=1 Vc with n = |V |;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' lower and upper bounds lc, hc ∈ Z+ for c ∈ [C];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' size constraint k ∈ Z+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' error  parameter ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Output: A set S ⊆ V such that S ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  1: Pick an arbitrary item u1 from V and set U = {u1}  2: for i ← 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' , k do  3:  ui ← arg maxv∈V minu∈U d(u, v) and U ← U ∪ {ui}  4: Set Uc ← Vc ∩ U, d′ ← 2 · div(U)  5: repeat  6:  for c ← 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' , C do  7:  while |Uc| < k and there exists some item v ∈ Vc such that div(Uc ∪ {v}) ≥ d′ do  8:  u′  c ← arg maxv∈Vc minu∈Uc d(u, v)  9:  Uc ← Uc ∪ {u′  c}  10:  Build an undirected graph G = (V ′, E), where V ′ = �  c Uc and E = {(u, v) ∈ V ′ × V ′ | d(u, v) < d′  2 }  11:  Compute the solution x of the ILP in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='5  12:  Find a set S = {vi ∈ V : xi = 1, i ∈ [n]} based on x  13:  if |S| < k then  14:  S ← ∅ and d′ ← (1 − ε) · d′  15: until S ̸= ∅  16: return S  diversity OPT of FMMD is bounded by 2·div(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Subsequently, the algorithm divides U by group into U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' , UC  and guesses OPT as its upper bound d′ = 2 · div(U) (Line 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' For each c ∈ [C], it runs the greedy algorithm to add  new items from Vc to Uc until |Uc| = k or there does not exist any v ∈ Vc to make div(Uc ∪ {v}) ≥ d′ (Lines 5-9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  At this point, each Uc is a partial group-specific solution, and the union V ′ = �  c Uc of partial solutions is the  coreset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Next, using a similar procedure to FMMD-E, it builds a graph G on V ′ with diversity threshold d′  2 and  solves the ILP of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='5 on G to obtain a solution S (Lines 11-12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Finally, if |S| = k, we have got a solution  S ∈ F with div(S) ≥ d′  2 and S will be returned as the final solution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' otherwise, d′ is decreased by a factor of  1 − ε, where ε ∈ (0, 1) is an error parameter, and the above procedure is executed again for the smaller d′ until a  feasible solution S is found (Lines 13-16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Theoretical Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Next, we give the complexity and approximation factor of FMMD-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' FMMD-S is a 1−ε  5 -approximation algorithm for FMMD running in O  �  Ckn + Ck log 1  ε  �  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' If there is any set S′ ⊆ V ′ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' S′ ∈ F and div(S′) ≥ d′  2 , then FMMD-S identifies such S′ (Line 12) from the  exact solution of the ILP in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' In addition, since the greedy algorithm (Lines 1-3) is 1  2-approximate [26],  the initial value of d′ is at least 2 · OPT  2 = OPT > 2  5 · OPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Therefore, to prove the approximation factor, it suffices  to show that there exists some S′ ⊆ V ′ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' S′ ∈ F and div(S′) ≥ d′  2 when d′ ∈ [ 2(1−ε)  5  OPT, 2  5 · OPT].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Towards this end, we next construct such a set S′ from V ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Let S∗ be the optimal solution for FMMD on V ,  and S∗  c = S∗ ∩ Vc be its subset from group c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' First, we initialize S′ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Then, we consider two cases for different  groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' We consider first the groups of Case #1 in arbitrary order, then those of Case #2 in arbitrary order, and  select |S∗  c | items from each group c into S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Case #1 (|Uc| < k): Let f : Vc → Uc map each item v ∈ Vc to its nearest neighbor f(v) in Uc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Note that  the condition in Line 7 ensures that d(v, u) < d′ for any v ∈ Vc and u ∈ Uc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' For each item sc,i ∈ S∗  c , we add  item f(sc,i) into S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' We now show that the added items are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Indeed, if f(sc,i) ≡ f(sc,j) for i ̸= j, then  the triangle inequality would give d(sc,i, sc,j) ≤ d(sc,i, f(sc,i))+d(sc,j, f(sc,j)) = d(sc,i, f(sc,i))+d(sc,j, f(sc,i)) <  2·d′ ≤ 4  5 ·OPT < OPT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' however, at the same time we have d(sc,i, sc,j) ≥ OPT because sc,i, sc,j ∈ S∗, which leads to a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Moreover, because we have identified for each sc,i ∈ S∗  c one distinct item in Uc, we have |Uc| ≥ |S∗  c |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  After processing all the groups in Case #1, we have d(f(s∗  i ), f(s∗  j)) ≥ d(s∗  i , s∗  j) − d(s∗  i , f(s∗  i )) − d(s∗  j, f(s∗  j)) >  OPT − 2d′ ≥ OPT  5  for any f(s∗  i ), f(s∗  j) ∈ S′ and thus div(S′) > OPT  5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Case #2 (|Uc| = k): Let g : Uc → S′ map each item u ∈ Uc to its nearest neighbor g(u) in the current  instance of S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' We remove from Uc every u ∈ Uc with d(u, g(u)) < d′  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Because the condition of Line 7 ensures  d(uc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' uc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='j) ≥ d′ for any i ̸= j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' there is at most one item removed for each item in S′ – otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' the triangle  Table 1: Statistics of datasets used in our experiments  Dataset  Group  C  n  dim  Distance Metric  Adult  Sex  2  48,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='842  6  l2-distance  Race  5  S+R  10  CelebA  Sex  2  202,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='599  25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='088  l1-distance  Age  2  S+A  4  Census  Sex  2  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='426,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='116  25  l1-distance  Age  7  S+A  14  Twitter  Sex  3  18,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='836  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='024  Angular distance  inequality would give d(uc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' uc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='j) < d′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' thus leading to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Therefore, at least k − |S′| items remain  in Uc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Moreover, |S∗  c | ≤ k − |S′| because S′ always contains the same number of items from each considered  group as S∗ throughout the construction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' We pick |S∗  c | items from the remaining ones and add them  to S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' After this operation, we still have div(S′) ≥ d′  2 ≥ 1−ε  5  OPT since our earlier removal of items from Uc  ensured d(uc,i, g(uc,i)) ≥ d′  2 for each added uc,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Finally, after processing all groups in Case #2, we get a set  S′ that contains the same number of items from each group c ∈ [C] as S∗, which implies that S′ ∈ F, and  div(S′) ≥ 1−ε  5  OPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Therefore, we conclude that FMMD-S is a 1−ε  5 -approximation algorithm for FMMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Since it takes O(nk) time to compute U as well as Uc for each c ∈ [C], the total time to compute V ′ is O(Ckn)  and |V ′| ≤ Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Then, the time to solve the ILP in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='5 for FMMD-S is O(Ck) because there are at most  �Ck  k  �  = O(Ck) possible solutions to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Moreover, the number of iterations for d′ is O(log 1  ε) since the ratio  between the first and last values of d′ is O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Thus, the time complexity of FMMD-S is O  �  Ckn + Ck log 1  ε  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  When k = o(log n) and C = O(1), its time complexity is reduced to O  �  n(k + log 1  ε)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Additionally, its space  complexity is O(n + C2k2) since the number of edges in G is O(C2k2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  4  Experimental Evaluation  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='1  Experimental Setup In this section, we conduct extensive experiments to evaluate the performance of our  proposed algorithms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=', FMMD-E and FMMD-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' We compare them with the state-of-the-art FMMD algorithms,  including FairSwap, FairFlow, and FairGMM in [23], FairGreedyFlow in [1], and SFDM1 and SFDM2 in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' We  implemented all the above algorithms in Python 3 using the NetworkX library for building and manipulating  graphs and the Gurobi optimizer for solving ILPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' All the experiments were carried out on a desktop with an  Intel Core i5-9500 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='0GHz processor and 32GB RAM running Ubuntu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='3 LTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Each algorithm was run on a  single thread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' All data and code are publicly available at https://osf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='io/te34m/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  We use four public real-world datasets listed in Table 1, where dim is the dimensionality of the feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  The detailed information and preprocessing procedures on each dataset are described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' The fairness  constraints in our experiments are defined according to the proportional representation [7, 17]: For each group  c ∈ [C], we set lc = max(1, (1 − α)k · |Vc|  n ) and hc = (1 + α)k · |Vc|  n  with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='2 in FMMD-E and FMMD-S and  kc = ⌈k · |Vc|  n ⌉ or ⌊k · |Vc|  n ⌋ in all other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' All the algorithms were executed ten times in each experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  We report the average running time and average diversity value of the solutions provided by each algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' We  use ‘N/A’ to indicate that an algorithm either does not find a solution within one day or does not work when  C > 2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=', FairSwap and SFDM1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' In the preliminary experiments (see Appendix B), we find that the solution  quality of FMMD-S hardly improves when ε is decreased below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='05 and so we fix ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='05 for FMMD-S in all  the remaining experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='2  Experimental Results Table 2 shows the diversity achieved by different algorithms on “small” datasets,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=', datasets that were obtained by sampling 1,000 items uniformly at random from each full dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Figures 3–  4 illustrate the performance of different algorithms on small datasets with varying k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Note that FairSwap and  SFDM1 are specific for the case of C = 2 and FairGMM fails to finish within one day when C > 3 or k > 10 since  it has to enumerate  �kC  k  �  sets for solution computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' They are ignored in subsequent tables and figures when  they cannot provide valid solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Table 2: Diversity values of the solutions returned by different algorithms on small datasets for solution size  k = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' The optimums OPT∗ without fairness constraints are reported to show “the price of fairness”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Dataset  Group  FairSwap  FairFlow  FairGMM  FairGreedyFlow  SFDM1  SFDM2  FMMD-E  FMMD-S  OPT∗  Adult  Sex  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='51  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='24  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='81  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='18  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='03  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='18  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='30  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='64  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='30  Race  N/A  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='73  N/A  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='33  N/A  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='83  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='54  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='01  S+R  N/A  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='84  N/A  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='99  N/A  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='04  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='12  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='88  CelebA  Sex  101457.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='3  55540.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
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+page_content='5  Age  110098.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='1  54649.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='6  127871.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='2  46312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
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+page_content='9  91578.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='2  129871.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='5  116701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='6  S+A  N/A  42412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='7  N/A  39967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='2  N/A  88026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='7  127974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
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+page_content='3  Census  Sex  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
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+page_content='0  FairFlow  FairGMM  FairSwap  FairGreedyFlow  SFDM1  SFDM2  FMMD-S  FMMD-E  10  20  30  40  50  k  0  2  4  6  8  10  Diversity  (a) Adult (Sex)  10  20  30  40  50  k  0  1  2  3  4  5  Diversity  (b) Adult (Race)  10  20  30  40  50  k  0  1  2  3  4  Diversity  (c) Adult (S+R)  10  20  30  40  50  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
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+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='5  Diversity  ×105  (d) CelebA (Sex)  10  20  30  40  50  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
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+page_content='5  Diversity  ×105  (e) CelebA (Age)  10  20  30  40  50  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
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+page_content='5  Diversity  ×105  (f) CelebA (S+A)  10  20  30  40  50  k  0  10  20  30  40  50  Diversity  (g) Census (Sex)  10  20  30  40  50  k  0  10  20  30  40  Diversity  (h) Census (Age)  20  30  40  50  k  0  5  10  15  20  25  30  Diversity  (i) Census (S+A)  10  20  30  40  50  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='7  Diversity  (j) Twitter (Sex)  Figure 3: Diversity values of the solutions of different algorithms with varying solution size k on small datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  In general, FMMD-E always provides optimal solutions for FMMD within the time limit (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=', 24 hours)  when n = 1, 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' The price of fairness, measured by the decrease in diversity due to the fairness constraints,  is marginal on all datasets except Adult with C ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' FMMD-S shows much higher solution quality (up to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='9×  greater in diversity value) than all approximation algorithms except FairGMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Although FairGMM sometimes  provides slightly better solutions than FMMD-S, it runs more than two orders of magnitude slower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Moreover,  the diversity values of all algorithms drop with k because div is a monotonically non-increasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' The  running time is independent of k for FMMD-E, grows exponentially with k for FairGMM, and increases linearly  with k for FMMD-S and other algorithms, which all follow from their time complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Table 3 presents the diversity values and running time of different algorithms for solution size k = 50 on all  the full datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' The performance of different algorithms by varying the solution size k from 10 to 100 on full  datasets (with all the items) is presented in Figures 5–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' In general, the running time of all algorithms increases  substantially with n and dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' FairGMM and FMMD-E fail to finish within one day and thus are omitted from  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' FMMD-S provides better solutions (up to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='2× higher in diversity value) than all the baselines in most  cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' The only exception is that FMMD-S shows slightly lower solution quality than FairSwap and SFDM1 on  CelebA when C = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' This is because of the extremely high dimensionality of CelebA (d = 25, 088), where the  distances between different pairs of points are less distinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' In such cases, the thresholding method in  FMMD-S is inferior to the local search methods in FairSwap and SFDM1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Moreover, the time efficiency of FMMD-  S is lower than the baselines, as solving ILPs is often time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Nevertheless, on Census, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=', the largest  dataset with more than two million items, FMMD-S still finishes the computation within 3 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  To evaluate the scalability of different algorithms, we vary the number C of groups and the number n of  points on synthetic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' In particular, each dataset consists of ten two-dimensional Gaussian isotropic blobs  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='FairFlow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='FairGMM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='FairSwap  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='FairGreedyFlow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='SFDM1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='SFDM2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='FMMD-S  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='FMMD-E  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='k  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='10−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='101  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='102  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='103  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='Time (s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='(a) Adult (Sex)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
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+page_content='52  with random centers in [−10, 10]2 and identity covariance matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Each point is assigned to one of the C groups  uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' The Euclidean distance is used as the distance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' For fixed C = 2 or 10, we obtain  six datasets with n = 102, 103, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' , 107;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' and for fixed n = 1, 000, ten datasets with C = 2, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' , 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  The performance of different algorithms by varying n and C on synthetic datasets for solution size k = 20 is  presented in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' In terms of solution quality, the diversity values are steady for FMMD-E and FMMD-S but  significantly drop for all other algorithms when C increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' In terms of efficiency, the running time of FMMD-E  is hardly affected by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Other algorithms run slower when C is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Nevertheless, FMMD-S runs faster than  any other algorithm when C ≥ 8, and its advantages in time efficiency become more significant with increasing C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Finally, all algorithms’ diversity values and running time grow with the dataset size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' FMMD-E cannot scale to  large datasets due to its exponential time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' All in all, FMMD-S outperforms all the other approximation  algorithms in terms of solution quality for different C or n, and its advantages become more apparent when C or  n is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' These results confirm the scalability of FMMD-S concerning the group size C and dataset size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  5  Conclusion  We investigated the problem of max-min diversification with fairness constraints (FMMD) in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' We  proposed an exact ILP-based algorithm for this problem on small datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' We further designed a scalable 1−ε  5 -  approximation algorithm, where ε ∈ (0, 1), on massive datasets based on our exact algorithm and the notion of  coresets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Extensive experimental results on four real-world datasets confirmed the effectiveness, efficiency, and  scalability of our proposed algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
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+page_content='0  FairFlow  FairSwap  FairGreedyFlow  SFDM1  SFDM2  FMMD-S  20  40  60  80  100  k  0  2  4  6  8  Diversity  (a) Adult (Sex)  20  40  60  80  100  k  0  2  4  6  8  Diversity  (b) Adult (Race)  20  40  60  80  100  k  0  2  4  6  Diversity  (c) Adult (S+R)  20  40  60  80  100  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
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+page_content='0  Diversity  ×105  (d) CelebA (Sex)  20  40  60  80  100  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
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+page_content='0  Diversity  ×105  (e) CelebA (Age)  20  40  60  80  100  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
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+page_content='0  Diversity  ×105  (f) CelebA (S+A)  20  40  60  80  100  k  10  20  30  40  Diversity  (g) Census (Sex)  20  40  60  80  100  k  0  10  20  30  40  Diversity  (h) Census (Age)  20  40  60  80  100  k  0  10  20  30  40  Diversity  (i) Census (S+A)  20  40  60  80  100  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
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+page_content='0  Diversity  (j) Twitter (Sex)  Figure 5: Diversity values of the solutions of different algorithms with varying solution size k on full datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  While a step forward in both theoretical and experimental aspects of the algorithms for the fair variant  of diversity maximization, our work leaves many open problems for future exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' A natural question is  whether there is any polynomial-time O(1)-approximation algorithm for FMMD, since our FMMD-S algorithm  has an approximation factor of 1−ε  5  but runs in polynomial time only when C = O(1) and k = o(log n), whereas the  best-known polynomial-time algorithm in [1] only achieves an approximation factor of  1  C+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Moreover, it would  also be interesting to study the fairness-aware variants of other diversity measures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=', the ones in [4,18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  References  [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
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+page_content=' Celis, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Keswani, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Straszak, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Deshpande, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Kathuria, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Vishnoi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Fair and diverse DPP-based  data summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' In ICML, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' 715–724, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  [9] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Celis, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Straszak, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Vishnoi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
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+page_content=' In ICALP, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' 28:1–28:15, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
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+page_content=' Chierichetti, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Kumar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Lattanzi, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Vassilvitskii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
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+page_content=' In NIPS, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' 5029–5037,  2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Chiplunkar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Kale, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Ramamoorthy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' How to solve fair k-center in massive data models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
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+page_content=' 1877–1886, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
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+page_content=' Chouldechova and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Roth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' A snapshot of the frontiers of fairness in machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' ACM, 63:5  (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
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+page_content='  [13] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Drosou and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Pitoura.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Diverse set selection over dynamic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Knowl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Data Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
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+page_content=' Pitassi, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Reingold, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Zemel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Fairness through awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' In ITCS, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' 214–226,  2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  [15] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Erkut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
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+page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
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+page_content='Figure 6: Running time of different algorithms with varying solution size k on full datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
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+page_content='0  FairFlow  FairGMM  FairSwap  FairGreedyFlow  SFDM1  SFDM2  FMMD-S  FMMD-E  4  8  12  16  20  C  0  1  2  3  4  5  Diversity  4  8  12  16  20  C  10−1  10−1  100  101  102  Time (s)  (a) Synthetic (n = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' 000)  102  103  104  105  106  107  n  0  1  2  3  4  5  6  Diversity  102 103 104 105 106 107  n  10−2  10−1  100  101  102  103  104  Time (s)  (b) Synthetic (C = 2)  102  103  104  105  106  107  n  0  1  2  3  4  5  6  Diversity  102 103 104 105 106 107  n  10−2  10−1  100  101  102  103  104  Time (s)  (c) Synthetic (C = 10)  Figure 7: Results of different algorithms by varying the number C of groups and dataset size n for solution size  k = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Since FMMD-E does not provide any solution within the time limit on datasets with n > 1, 000, FairGMM  does not provide any solution within the time limit for k = 20, and FairSwap and SFDM1 cannot work with C > 2,  they are not plotted in these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  [16] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Gonzalez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Clustering to minimize the maximum intercluster distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
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+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=', 38 (1985), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' 293–  306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Halabi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Mitrovi´c, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Norouzi-Fard, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Tardos, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Tarnawski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
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+page_content=' 2876–2883, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  [32] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Zehlike, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Bonchi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Castillo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Hajian, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Megahed, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Baeza-Yates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Fa*ir: A fair top-k ranking algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  In CIKM, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' 1569–1578, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  [33] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Zemel, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Wu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Swersky, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Pitassi, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Dwork.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Learning fair representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' In ICML, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' 325–333,  2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  [34] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Pan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Chang, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Overcoming multi-model forgetting in one-shot NAS with diversity  maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' In CVPR, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' 7806–7815, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  A  Data Preparation  Detailed information about the four real-world datasets we use and the preprocessing procedures for them are  presented as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Adult is retrieved from the UCI Machine Learning Repository3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' It is a collection of 48,842 records from  the 1994 US Census database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' We select six numeric attributes as features and normalize each to have zero  mean and unit standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' The l2-distance (Euclidean distance) is used as the distance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' The  groups are generated from two demographic attributes: sex and race.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' By using them individually and in  combination, there are 2 (sex), 5 (race), and 10 (sex + race) groups, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  CelebA is provided by Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' [22] on the website4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' It is a set of 202,599 images of human faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' We  get a 25,088-dimensional (512 × 7 × 7) feature vector for each image from the pre-trained VGG16 model  in Keras5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' The l1-distance (Manhattan distance) between feature vectors is used as the distance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  We generate 2 groups from a human-annotated class label “sex” {‘female’, ‘male’}, 2 groups from another  human-annotated class label “age” {‘young’, ‘not young’}, and 4 groups from both of them, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Census is also retrieved from the UCI Machine Learning Repository6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' It is a set of 2,426,116 records from  the 1990 US Census data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' We take 25 (normalized) numeric attributes as features and use the l1-distance  (Manhattan distance) as the distance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  We generate 2, 7, and 14 groups from two demographic  attributes sex, age, and both of them, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Twitter is retrieved from Kaggle7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' It is a collection of 18,836 tweets with user profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' We transform  each tweet into a 1,024-dimensional feature vector using the sentence-transformer model8 [27] (“bert-large-  nli-stsb-mean-tokens”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' The angular distance is used as the distance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' We generate 3 groups from the  attribute “sex” {‘female’, ‘male’, ‘non-human’} in user profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  The proportion of each group on each dataset is provided in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  B  Parameter Tuning for FMMD-S  Figure 8 illustrates the performance of FMMD-S by varying the parameter ε from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='001 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='5 for solution size  k = 10 on small datasets (n = 1, 000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Regarding solution quality, the diversity values of the solutions of FMMD-S  remain approximately constant when ε takes small values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' However, they decrease and become less stable as ε  becomes larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Especially, when ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='5, FMMD-S cannot provide stable and high-quality solutions in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  In terms of efficiency, the running time increases significantly and becomes less stable for smaller ε, particularly  when ε ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' These results conform to our theoretical analyses of FMMD-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' The above results show that the  most appropriate value of ε lies in the range [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='1] in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' Therefore, we decide to set ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='05 in all  other experiments throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  3archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
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+page_content='edu/ml/datasets/US+Census+Data+(1990)  7www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='com/crowdflower/twitter-user-gender-classification  8huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='co/models?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='library=sentence-transformers  Table 4: Proportion of each group on each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='  Dataset  Group  Proportion of Each Group  Adult  Sex  ‘Female’: 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='2%, ‘Male’: 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='8%  Race  ‘Amer-Indian-Eskimo’: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='0%, ‘Asian-Pac-Islander’: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='1%, ‘Black’: 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='6%,  ‘Others’: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='8%, ‘White’: 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='5%  Sex+Race  ‘AIE+F’: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='4%, ‘API+F’: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='1%, ‘B+F’: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='7%, ‘O+F’: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='3%, ‘W+F’: 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='7%,  ‘AIE+M’: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='6%, ‘API+M’: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='0%, ‘B+M’: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='9%, ‘O+M’: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='5%, ‘W+M’: 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='8%  CelebA  Sex  ‘Female’: 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='3%, ‘Male’: 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content='7%  Age  ‘Not Young’: 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
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+page_content='5  ϵ  100  101  Time (s)  (i) Twitter (Sex)  Figure 8: Results of FMMD-S with varying the parameter ε for solution size k = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' The error bars in each plot  indicate the maximum and minimum of the diversity value and running time over 10 runs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
+page_content=' In all  plots for diversity values, the blue horizontal lines denote the diversity values of the optimal solutions provided  by FMMD-E, which measure the gaps between the approximate solutions of FMMD-S and the optimal ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9A0T4oBgHgl3EQfGv_2/content/2301.02053v1.pdf'}
diff --git a/cdE_T4oBgHgl3EQf0Bzf/content/tmp_files/2301.08327v1.pdf.txt b/cdE_T4oBgHgl3EQf0Bzf/content/tmp_files/2301.08327v1.pdf.txt
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+1
+Blind as a bat: audible echolocation on small robots
+Frederike D¨umbgen
+Adrien Hoffet
+Mihailo Kolundˇzija
+Adam Scholefield
+Martin Vetterli
+Abstract—For safe and efficient operation, mobile robots need
+to perceive their environment, and in particular, perform tasks
+such as obstacle detection, localization, and mapping. Although
+robots are often equipped with microphones and speakers, the
+audio modality is rarely used for these tasks. Compared to the
+localization of sound sources, for which many practical solutions
+exist, algorithms for active echolocation are less developed and
+often rely on hardware requirements that are out of reach for
+small robots.
+We propose an end-to-end pipeline for sound-based localiza-
+tion and mapping that is targeted at, but not limited to, robots
+equipped with only simple buzzers and low-end microphones.
+The method is model-based, runs in real time, and requires no
+prior calibration or training. We successfully test the algorithm
+on the e-puck robot with its integrated audio hardware, and on
+the Crazyflie drone, for which we design a reproducible audio
+extension deck. We achieve centimeter-level wall localization on
+both platforms when the robots are static during the measure-
+ment process. Even in the more challenging setting of a flying
+drone, we can successfully localize walls, which we demonstrate
+in a proof-of-concept multi-wall localization and mapping demo.
+Index Terms—Range Sensing, Robot Audition, SLAM, Aerial
+Systems: Perception and Autonomy
+I. INTRODUCTION
+A
+bat’s ability to “see” in the dark using sound has fasci-
+nated scientists for centuries [1]. Bats emit ultrasonic
+chirps, and based on the timing and form of the echoes,
+localize objects of interest such as food [2], obstacles [3], or
+water resources [4].
+Similarly, most mobile robots need to localize targets and
+obstacles in their surroundings in order to perform meaningful
+operations such as search and rescue, exploration, and delivery.
+Most robots that are currently deployed in the real world rely
+on rich sensors for localization and mapping — sensors that
+provide a plethora of information — such as cameras, lidar [5]
+or radar [6].
+However, just like bats have relatively poor eyesight, not all
+robots can be equipped with rich sensing equipment. Consider
+the Crazyflie, a developer-friendly nano drone [7]. This drone
+is very limited in terms of admissible payload, which rules out
+heavy sensors such as lidar or radar. A single camera can be
+attached through the AI-deck; however, this does not provide
+full coverage for obstacle avoidance. More coverage can be
+obtained with the multi-ranger deck, designed specifically for
+obstacle detection using four infrared sensors.
+We propose to instead use audible sound for obstacle detec-
+tion and mapping. Microphones come in lighter and smaller
+This work was supported in part by “SNF-SESAM – Sensing and Sampling:
+Theory and Algorithms.” n◦ 200021 181978/1.
+All authors are with the School of Computer and Communication Sci-
+ences, EPFL, 1015 Lausanne, Switzerland. Corresponding author: Frederike
+D¨umbgen, frederike.duembgen@gmail.com.
+13 cm
+7 cm
+microphone
+microphone
+buzzer
+Fig. 1: The robotic platforms used for echolocation in this paper.
+Depicted on the left is the e-puck robot [9], a wheeled robot developed
+for education. On the right, we show the Crazyflie drone [7] with our
+custom audio extension deck, equipped with four microphones and a
+piezoelectric buzzer. The proposed echolocation pipeline can be used
+on any platform with at least one low-key microphone and sound
+source, respectively.
+form factors than rich sensors and demand less memory for
+processing. Compared to proximity sensors such as infrared
+and ultrasound, they are less directional, allowing fewer sen-
+sors to achieve full spatial coverage. Moreover, many robots
+that are designed to communicate with users come equipped
+with microphones, and thus using audio for navigation tasks
+may require little additional payload.
+In this paper, we develop a novel interference-based echolo-
+cation algorithm, which includes a simultaneous localization
+and mapping (SLAM) framework, integrating a particle filter
+and a factor graph [8], which accounts for the nonlinear, multi-
+modal nature of echolocation measurements. For experimental
+evaluation, we develop and provide an audio extension deck
+for the Crazyflie drone. It consists of a small piezoelectric
+buzzer and four microphones embedded in a custom PCB,
+including its own microcontroller for acquisition and prepro-
+cessing. The methods developed in this paper are however
+applicable to any robot equipped with at least one microphone
+and a speaker. To underline this point, we also provide
+experimental validation on the ground-based e-puck robot [9].
+Both platforms are shown in Figure 1. To summarize, our main
+contributions are:
+• a
+model-based,
+probabilistic
+echolocation
+algorithm
+based on audio interference measurements,
+• a combined particle filtering and SLAM pipeline suited
+for the non-Gaussian nature of echolocation measure-
+ments, and
+• a Crazyflie audio extension deck with associated software
+for research on audio-based navigation on drones.
+The paper is structured as follows. After reviewing related
+work in Section II, we describe our method for going from
+low-level audio signals to wall and robot location estimates
+arXiv:2301.08327v1  [cs.RO]  19 Jan 2023
+
+40602
+in Section III. In Section IV, we provide implementation
+details for the two used hardware platforms. In Section V, we
+evaluate the method on these platforms – with finely controlled
+motion in Section V-A, on the flying drone with one wall in
+Section V-B, and in a multi-wall demo in Section V-C. We
+conclude with a discussion of the results in Section VI.
+II. RELATED WORK
+We first give an overview of existing solutions for echolo-
+cation from raw audio signals, also called the front-end in the
+SLAM literature [5], before going over relevant work in the
+back-end state estimation procedure.
+a) Sound-based front end: Existing methods for sound-
+based sensing can be loosely categorized into passive so-
+lutions, which estimate the direction of arrival (DOA) of
+external sound sources, and active solutions, which probe the
+environment with an onboard sound source. While DOA for
+robotics is a rather active research field [10], sound-based
+active solutions, the focus of this paper, are considerably less
+explored.
+Amongst the most popular active solutions are methods
+estimating the time of arrival (TOA) of the echoes of reflecting
+objects. In rooms, this information is contained in the so-
+called room impulse response (RIR), consisting of shifted,
+attenuated peaks for the early echoes, and a more noise-like
+reverberation tail. It can be estimated in frequency domain
+through the deconvolution of a known wide-band source signal
+and its response. When the locations of microphones and
+speakers are known, the first peaks of the RIR can be used to
+approximately recover a room’s geometry [11]. Adapting this
+method to a moving setup, the authors of [12] map out the
+room, one wall at a time, using a small loudspeaker and high-
+quality microphones mounted on a wheeled robot. The authors
+of [13] use a similar platform but recover the peaks of the RIR
+through nonlinear least-squares optimization on the wide-band
+frequency response. In follow-up work [14], this method is
+extended with a DOA estimator. Although promising, these
+methods use loudspeakers and microphones with high signal-
+to-noise ratio (SNR) and flat frequency responses, which is
+not available on the platforms studied in this paper.
+The TOA of reflectors can also be determined by emitting
+and cross-correlating short, frequency-modulated pulses, a
+concept known as pulse compression. Using this technique,
+room geometry is estimated, using a static omnidirectional
+speaker and a tablet’s microphones, in [15]. Moving to a
+fully mobile setup, the authors of [16] perform room mapping
+on a smartphone. Pulse compression is also commonly used
+for underwater acoustic ranging [17], [18]. Note that these
+methods require fine control of the emitted audio pulses.
+This last point is overcome by methods exploiting sound
+interference. The idea is to detect close-by reflectors by mea-
+suring the change in magnitude caused by interference from
+its echoes. In the first work bringing this idea to robotics, it
+was shown that a reflector can be detected based on the change
+in magnitude of the propeller’s ego-noise [19]. The reported
+results are obtained using static measurement microphones,
+and an offline calibration phase of the propeller’s ego noise.
+The results highlight one advantage of interference-based
+methods: since they do not require emission of specifically
+designed short pulses, they can be used on noise inherent to
+the system.
+In this paper, we take an interference-based approach. Our
+system is lighter compared to RIR-based methods, both in
+terms of hardware requirements and computing power: it runs
+on computationally limited platforms with simplistic speakers
+and low-key microphones. Compared to pulse-compression
+techniques, the sound signal is simpler and does not need to be
+exactly known — this can be an advantage when the available
+buzzer does not allow for fine-grained control, or when sound
+sources inherent to the system are used. As opposed to [19],
+our system does not require prior calibration, uses on-board
+low-key microphones and tightly integrates the motion of the
+device, as discussed next.
+b) Back-end state estimation: For a practical solution,
+the measurements taken at subsequent times and poses need
+to be fused into a coherent map. This can be posed as a factor
+graph inference problem, where all unknown states (poses
+and walls) are linked by measurement factors [8]. Inference
+on factor graphs can become prohibitively expensive as the
+number of unknown states increases over time, unless certain
+Gaussian and sparsity assumptions are made, which is ex-
+ploited by modern solvers [20], [21], [22]. While the Gaussian
+assumption may hold for many commonly used sensors, it does
+not for the sound-based front end considered in this paper:
+sound is known to lead to multi-modal distributions [23],
+[24] and thus requires different inference approaches [25],
+[26]. Our state estimation pipeline is most similar in nature
+to the work by [17], where a particle filter is used as a
+pre-filtering step to render the bearing and range pseudo-
+distributions approximately Gaussian. In our case, the particle
+filter helps not only reduce ambiguities, but also allows us
+to convert the raw path difference measurements to distance
+and angle estimates, which can be fed directly to the SLAM
+algorithm.
+III. METHODS
+In this section, we describe the estimation pipeline, going
+from raw audio measurements to pose and plane estimates.
+We first derive the model of the microphone measurements as
+a function of the robot pose and plane in Section III-A. We
+derive our measurement model, which can be used to estimate
+the path difference, in Section III-B, and infer distance and
+angle distributions using discrete filtering in Section III-C.
+We conclude with a standard SLAM framework to jointly
+estimate poses and planes in Section III-D. An overview of
+the estimation pipeline is given in Figure 2.
+A. Derivation of signal model
+We first derive the model of the individual microphone
+responses. A sketch of a sample setup is shown Figure 3. We
+place the robot’s body frame (also called the local frame) at the
+sound source location s ∈ R2, and denote the Nm microphone
+locations in this frame by xm ∈ R2. The pose, consisting of
+the robot’s translation xn ∈ R2 and orientation φn ∈ [0, 2π)
+
+3
+particle filter
+Section III.B
+Section III.C
+Section III.D
+Fig. 2: Overview of estimation pipeline: we depict two microphones and their measurements in green and blue, respectively, at two subsequent
+poses. The raw measurements y(m)
+nl (f), shown on the left, are used to estimate the location of nearby walls p(πl) and the robot’s poses
+over time p(ξn), shown on the right.
+at time tn, is denoted by ξn, expressed in the fixed inertial
+frame (also called global frame). Considering the studied
+experimental platforms (a ground-based robot and a drone
+hovering at constant height) we operate in two dimensions,
+but this is not a hard requirement for the proposed solution.
+We denote the Nl planes by πl, and we parametrize them
+with their global distance dl and angle θl from the origin.
+The recorded signal at each microphone is the sum of the
+direct sound and the reflected sound. Accounting for free-space
+attenuation and energy dissipation according to [27], it takes
+the form:
+z(m)
+nl (t) := z(m)(t, ξn, πl)
+(1)
+= g(m)
+�
+1
+4πℓ(m) s
+�
+t − ℓ(m)
+c
+�
++ 1 − ρ
+4πr(m)
+nl
+s
+�
+t − r(m)
+nl
+c
+��
+,
+where c is the speed of sound, ρ the wall absorption coef-
+ficient, and g(m) the unknown, frequency-dependent, micro-
+phone gain. The lengths of the direct and reflected paths are
+respectively given by ℓ(m) := ∥x(m) − s∥, and
+r(m)
+nl
+=
+��
+ℓ(m)�2 + 4(dnl)2 − 4dnlℓ(m) cos(θ(m) − θnl),
+(2)
+where θ(m) = ∠(x(m) − s) denotes the bearing of the m-th
+microphone, and θnl, dnl denote the distance and angle of
+plane l as seen from pose ξn:
+dnl = dl − x⊤
+n n
+�
+θl
+�
+and
+θnl = θl − θn,
+(3)
+Fig. 3:
+Sketch of the experimental setup of echolocation with one
+wall, two microphones and one source. We can measure the difference
+between the reflected path lengths r(m)
+nl
+for mics m = 0, 1, and
+the direct path lengths ℓ(m), respectively, which are related to the
+distance dnl and angle θnl of the wall πl at pose ξn.
+where n(θ) is a normal vector with direction θ. Solving (2)
+for the distance dnl given a fixed angle, we have:
+dnl := h
+�
+r(m)
+nl , θl
+�
+= 1
+2
+�
+ℓ(m) cos (θ(m) − θnl)
+(4)
++
+��
+ℓ(m)�2 �
+cos2 (θ(m) − θnl) − 1
+�
++
+�
+r(m)
+nl
+�2�
+,
+which, for the limit case of a co-located phone and speaker
+simplifies to the widely used approximation dnl ≈ r(m)
+nl /2.
+B. Interference: from audio to path differences
+Next, we derive our measurement function. In the frequency
+domain, the squared magnitude of (1) becomes:
+y(m)
+nl (f) = g(m)(f)2|ˆs(f)|2
+(4π)2
+�
+1
+ℓ(m)2 + (1 − ρ)2
+�
+r(m)
+nl
+�2
++ 2
+1 − ρ
+ℓ(m)r(m)
+nl
+cos
+�
+2πf(r(m)
+nl
+− ℓ(m))/c
+�
+�
+��
+�
+:=˜y(m)
+nl (f)
+�
+,
+(5)
+where we introduce ∆(m)
+nl
+:= r(m)
+nl
+− ℓ(m) for the path
+difference. In theory, y(m)
+nl (f) could be used as the mea-
+surement model: evaluating it at Nf different frequencies
+fk, the robot state ξn and wall location πl can be inferred
+using nonlinear optimization. However, the function is highly
+nonlinear in the unknowns and, in particular, non-bijective,
+making it prone to yield ambiguous results. To make things
+worse, the unknown parameters g(m)(f), |ˆs(f)| and ρ in (5)
+render the optimization problem more costly, should we try
+to either estimate them (thus adding more dimensions to the
+state vector), or marginalize them out.
+Thankfully, since the “frequency”1 of the cosine in (5)
+depends only on the unknown state vectors and sound fre-
+quency, we can infer probability distributions of the path
+difference ∆(m) by studying its frequency response. We in-
+troduce ˜y(m)
+nl (f) for the cosine, as shown in (5). It can be
+estimated from y(m)
+nl (f) through online calibration, which we
+will further develop in Section IV-C.2
+1This “frequency” has the unit of seconds, and is not to be confused with
+the “real” frequencies fk (expressed in Hz) at which we measure.
+2An interesting result states that the FFT-based solution we propose is in
+fact equivalent to least-squares optimization with amplitude and phase of the
+cosine marginalized out [28].
+
+4
+Using a classical result from Bayesian theory, we know that
+the probability of a signal to be a sinusoid of frequency f is
+directly related to the signal’s periodogram [29]. Applying this
+result here, the probability p(m)
+n
+(∆) of microphone m being
+at a path difference ∆ at time n is given by:
+p(m)
+n
+(∆) = η
+�
+1 −
+2P (m)
+˜y
+(∆/c)
+�Nf
+k=1
+�
+˜y(m)
+nl (f)
+�2
+� 2−Nf
+2
+,
+(6)
+with P (m)
+˜y
+(f) :=
+1
+Nf
+������
+Nf
+�
+k=1
+˜y(m)
+nl (f)ej2πffk
+������
+2
+,
+(7)
+where η is a normalization factor. By choosing a roughly
+uniform spacing of frequencies, we can use the FFT of ˜y to
+calculate P (m)
+˜y
+(f). For visualization purposes, we sketch an
+example of ˜y(m)
+nl (f), for different frequencies and distances,
+in Figure 4. Intuitively speaking, we sample an approximately
+vertical slice of this function, and the periodicity of the slice
+is related to the path difference of the closest wall.
+Next, we fuse multiple path difference distributions to obtain
+distance and angle distributions that can be fed to our SLAM
+framework.
+Fig. 4: Sketch of interference patterns over path differences and
+frequencies, with the samples recorded by a drone (left) and a ground-
+based robot (right). For different distances, we measure different
+periodicities in the interference. Note that the ground-based robot
+can measure clean vertical slices, as opposed to the drone.
+C. Particle filter: from path differences to planes
+Because of the non-linear measurement model and the poor
+signal quality that common light-weight microphones and
+buzzers provide, the path difference distributions are subject
+to high noise levels and ambiguities. We propose a particle
+filtering approach for converting them to more unimodal
+distance and angle distributions. The proposed filter implicitly
+assumes that, for a small time window, the pose estimates are
+subject to only little drift, and can thus be used to resolve the
+ambiguities in the distributions. Mathematically speaking, we
+want to convert our potentially ambiguous path distributions
+p(m)
+n
+(∆), obtained from (6), to unimodal distance- and angle
+distributions p(dl) and p(θl).
+a) Initialization: We initialize Np particles, containing
+the (local) angle and distance candidates (d0,k, θ0,k), with k =
+0 . . . Np − 1. The distances and angles are chosen uniformly
+from an area of size [0, dmax] × [0, 2π). We denote by dmax
+the maximal distance that we aim to estimate, given by the
+maximally resolvable path difference.3 Because of low SNR
+at high distances, we clip this maximal distance at 80 cm. We
+initialize all particles with equal weights wk = 1/Np.
+b) Prediction: At each timestep, we use the current
+movement estimates to transform the particles to the new local
+coordinates:
+dn,k = dn−1,k − n(θn−1,k + φn)⊤(xn − xn−1)
+θn,k = θn−1,k − (φn − φn−1),
+(8)
+where φn and xn are relative pose estimates. We add a fixed
+proportion (10%) of uniformly distributed particles, in order
+to prevent the filter from converging to local minima.
+c) Update: Next, we update the particle weights using
+the newest measurements p(m)
+n
+(∆). Using each particle’s
+(dn,k, θn,k) values, we calculate the corresponding path dif-
+ference ∆(m)
+k
+with (2) and evaluate the distributions using
+linear interpolation. Assuming the microphone measurements
+are independent, we set the value wk to the product of the Nm
+probabilities thus obtained.
+d) Resampling: After each update step, we resample
+the particles according to their weights, using the stratified
+resampling algorithm [30].
+D. Planar simultaneous localization and mapping
+In the final step, we aim to feed the filtered plane measure-
+ments, along with potential pose measurements, to a SLAM
+framework. This ensures long-term consistency, and enables
+crucial tasks such as data association and path planning.
+In the factor graph representation of our problem, the
+nodes correspond to poses and planes. The pose estimates
+˜ξn ∈ SE(d) are incorporated in unary factors of the form:
+cn =
+���ξn ⊖ ˜ξn
+���
+Σn,
+(9)
+where ⊖ is the difference operator defined in the Lie algebra
+of SE(d), and Σn is the diagonal pose covariance matrix,
+fixed heuristically according to the pose estimation accuracy
+of the state estimator (σx = σy = 1 cm, σφ = 5 deg).
+For the plane measurements, we first extract distance and
+angle estimates ˜dnl, ˜θnl and standard deviations σθ
+nl, σd
+nl from
+the particle filter, using the weighted particle mean and
+standard deviation. Approximating the distance and angle
+distributions as Gaussians, we use the binary plane factors
+of the form [31]:
+cnl =
+����
+�R⊤
+n n(θl)
+x⊤
+n n(θl)
+�
+−
+�n(˜θnl)
+˜dnl
+�����
+Σnl
+,
+(10)
+where Rn is the rotation matrix corresponding to φn. Σnl is
+the noise covariance matrix, calculated from σθ
+nl, σd
+nl, with the
+upper-left block given by:
+(σθ
+nl)2 ∂n
+∂θ
+���˜θnl
+∂n
+∂θ
+���
+⊤
+˜θnl
+.
+(11)
+3By the Nyquist criterion we have dmax = h(∆max + ℓ) = h(c/2δf +
+ℓ) ≈ c/4δf, with δf the (average) difference between the sampled frequen-
+cies.
+
+5
+request
+wall estimate
+SPI
+Bluetooth
+CRTP
+buzzer control
+feature extraction
+audio deck
+audio data
+audio data
+pose data
+commands
+ground station
+crazyflie_crtp/gateway
+crazyflie_demo/wall_detector
+audio_gtsam/wall_mapper
+send commands
+distributions
+build 
+factor 
+graph
+solve 
+factor 
+graph
+run state 
+machine
+audio driver
+conversion of 
+CRTP    SPI
+conversion of 
+ROS    CRTP
+generate 
+generate 
+commands
+audio data
+pose data
+conversion of 
+ROS    USB
+epuck_com/serial
+buzzer control
+feature extraction
+epuck main
+USB
+audio data
+pose data
+commands
+server/
+client
+publisher/
+subscriber
+Fig. 5: Processing pipeline from firmware to the ground station’s ROS processing pipeline. The audio data and pose data is sent from either
+the Crazyflie drone (top row) or the e-puck robot (bottom row) to the ground station, where they are processed in parallel by a series of
+ROS nodes. Note that service/client connections (dotted lines) are only triggered upon request, while listener/publisher connections (solid
+black lines) are running continuously.
+To determine whether a new measurement corresponds to a
+new plane or to a revisited plane, we introduce the following
+data association loss between planes πi and πj:
+e(πi, πj) = ∥n(θi)di − n(θj)dj∥.
+(12)
+This loss measures how far the normal points of the two planes
+are from each other, thus incorporating both angle as well as
+distance discrepancies. We associate a new plane measurement
+to a previously seen plane if (12) falls below a given threshold.
+We found that a threshold of 30 cm performed well throughout
+our experiments. We solve the factor graph using the iSAM2
+solver implemented in gtsam [21].
+IV. EXPERIMENTAL PLATFORMS
+To demonstrate the described algorithm’s performance, the
+main experimental platform studied is the Crazyflie drone,
+a micro-drone with relatively poor computing resources (mi-
+crocontroller STM32F405) and strict payload constraints. To
+fit these requirements, we design an audio-deck optimized to
+add little weight, and equipped with its own microcontroller
+(STM32F446) to offload the low-level audio pre-processing.
+The deck conforms with the standard format of Crazyflie
+extension decks, thus adhering to its plug-and-play philosophy.
+The deck contains four MEMS microphones attached to its
+extremities, a positioning chosen for its good trade-off between
+minimal propeller noise and off-centre weight. A piezoelectric
+buzzer, which has a reduced PCB footprint and energy con-
+sumption compared to a speaker, is placed in the centre of the
+deck. Note that one microphone is enough for the described
+algorithms, but having more microphones both increases the
+SNR and enables other applications such as DOA estimation.
+Relative pose estimates are obtained using the drone’s optical
+flow deck. A photo of the Crazyflie equipped with the audio
+deck is shown in Figure 1, and all hardware design files are
+made publicly available.4
+We also test our algorithms on the wheeled e-puck robot
+(version 2) [9]. Just like the custom audio deck, the robot
+possesses four MEMS microphones, but it has a more versatile
+and louder electromagnetic speaker. Being a ground-based
+robot, the e-puck allows for the evaluation of the methods in
+more controlled settings, eliminating both position jitter and
+propeller noise. Relative pose estimates are obtained through
+wheel odometry.
+4The code base including hardware design files and ROS pipeline is
+available at https://github.com/LCAV/audioROS.
+A. Low-level audio processing
+The full processing pipeline is visualized in Figure 5. In
+the firmware (left part of the plot), the signals recorded by the
+four microphones are first stored in buffers of size N = 2048
+at a sampling rate of Fs = 48 kHz. With these values, the
+resulting frequency resolution, using the real-valued FFT, is
+∆f = Fs/N ≈ 23 Hz.
+The piecewise constant frequency sweep of the buzzer is
+generated through PWM signals. For the drone, to increase
+SNR, the frequency range is chosen as to be minimally
+affected by propeller noise under average drone hovering
+thrusts, while ensuring that the played frequencies match the
+available frequency analysis bins. Each note is played until
+the microphones have recorded one full buffer, which takes
+N/Fs ≈ 40 ms. For a sweep of Nf = 20 discrete frequencies,
+this results in circa 1 s per sweep. After each sweep, the
+complex response is calculated through a windowed FFT,
+using the flattop window, which yields faithful magnitude
+measurements even under small frequency errors [32]. The
+response at the current buzzer frequency is extracted and added
+to an outgoing buffer.
+After completion of each sweep, the next sweep is started
+only once the data was successfully sent to the ground station.
+For the Crazyflie drone, the buffer is passed first via SPI to
+the main processor and then through the custom Bluetooth
+protocol (CRTP) to the ground station. The data throughput
+of both SPI and CRTP being significantly higher than the
+playback rate of the sweeps Nf × 2 × 32 = 1280 bps, we can
+ensure smooth transmission. For the e-puck robot, we omit
+the intermediate SPI and CRTP communication and send the
+audio data directly over USB to the ground station.
+B. Ground station processing
+On the ground station (right half of Figure 5), the received
+audio data is passed through a modular processing pipeline
+implementing the steps described in Sections III-B to III-D
+through custom nodes and interfaces, written in python in the
+Robot Operating Systems (ROS) framework, version Galactic.
+The full software suite, including a simulation framework
+and implementations of DOA methods, are made publicly
+available.
+
+6
+20
+40
+distance [cm]
+3000
+3500
+4000
+frequency [Hz]
+measured
+20
+40
+distance [cm]
+theoretical
+Crazyflie drone
+20
+40
+60
+distance [cm]
+2000
+3000
+4000
+5000
+frequency [Hz]
+measured
+20
+40
+60
+distance [cm]
+theoretical
+e-puck robot
+Fig. 6: Interference patterns measured at one microphone, using the drone’s stepper-motor setup and the e-puck setup. Photos of the
+experimental setups are shown to the left. Note that the measured values have the same patterns as predicted by theory, using equation (5).
+C. Online joint calibration
+In our model (5), there are two unwanted and unknown,
+frequency-dependent gains: the highly non-linear frequency
+response of the piezo buzzer |ˆs(f)|, and the gains of the
+different microphones g(m)(f). If we remove these two fac-
+tors, we can use the measurement model ˜y(m)
+nl (f), and path
+difference estimation is reduced to a simple frequency anal-
+ysis, as described in III-B. Since the two unknown gains are
+entangled, we choose to estimate them jointly. The intuition
+of the calibration is the following: while the robot is moving,
+it measures a portion of the distance-frequency function, as
+visualized in Figure 4. Even for small movements, i.e. small
+variations of wall distance d, we are likely to sample both
+destructive and constructive interference regimes. We can thus
+expect that taking a moving average over a sufficiently long
+time window yields a good estimate of the joint speaker-
+microphone gain. To reduce the computational cost of the
+calibration, we implement the average using a simple infinite
+impulse response (IIR) low-pass filter:
+˜g(m)
+n
+(f) = (1 − λ)˜g(m)
+n−1(f) + λy(m)
+nl (f),
+(13)
+where we have introduced ˜g(m)
+n
+(f), the estimate of the joint
+gain function g(m)(f)2|ˆs(f)|2 at time index n. The parameter
+λ controls the effective window width, and we use a value of
+λ = 0.3 throughout all experiments.
+V. RESULTS
+We evaluate our systems in three increasingly difficult
+stages. First, we obtain fine-grained measurements of the
+wall’s interference patterns by moving the drone and e-puck
+robot in small steps towards the wall. For this purpose, the
+drone is mounted on a linear stepper motor. In a second stage,
+we evaluate the drone’s performance in localizing a single wall
+during flight, before concluding with a multi-wall demo.
+A. Stepper motor and ground-based robot
+Moving the drone and e-puck robot in centimeter-steps
+perpendicular to a glass wall, we create the measured inter-
+ference matrices shown in Figure 6. The frequency sweeps
+consist of Nf = 32 uniformly spaced frequencies, where the
+range for the drone is chosen so as to avoid the propeller
+noise harmonics. We note that, although noisy, we can clearly
+−40
+−20
+0
+error [cm]
+error evolution
+20
+40
+60
+distance [cm]
+−100
+0
+100
+error [deg]
+0
+10
+20
+30
+absolute error [cm]
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+cdf [-]
+distance estimation
+0
+100
+200
+absolute error [deg]
+angle estimation
+used data
+drone
+e-puck
+random
+fixed
+Fig. 7: Performance of distance and angle estimation. Left: distance
+and angle error as a function of wall distance. Right: cumulative
+distribution functions (cdf) of absolute errors. Shown are the exper-
+imental results from the drone on the stepper motor and the e-puck
+robot. For comparison, we also include the results using simulated
+data. The error using random guesses or one fixed guess serve as
+worse-case baselines.
+discern the interference patterns predicted by theory. The
+pattern is particularly pronounced closer to the wall. We also
+note a strong frequency-dependent gain, particularly for the
+e-puck robot, which we remove through online calibration as
+described in Section IV-C.
+Next, we apply our estimation pipeline to these measure-
+ments. Treating each frequency sweep (each column in Fig-
+ure 6) separately, we evaluate the performance of both distance
+and angle estimation. We found that Np = 400 particles
+gave consistently good results throughout all experiments, with
+which one full estimation (including update, prediction and
+resampling) takes less than 75 ms. We first investigate the
+estimation error as we increase the distance to the wall (left
+plots in Figure 7). Distance estimation is reliable up to 40 cm
+from the wall, with the maximum error staying below 2 cm.
+Angle estimation is less reliable, in particular for the drone.
+However, up to ca. 30 cm, the maximum error is smaller than
+20 deg. We take a closer look at the cumulative probability
+distributions (cdfs) in the right plots of Figure 7. As worst-
+case baselines, we plot the result of choosing random angles or
+distances, or using a fixed value at the mid-point. As best-case
+baseline, we run the proposed solution on simulated data. We
+observe that the median error for distance estimation for both
+e-puck and drone experiments is less than 3 cm. The e-puck
+robot also performs particularly well at angle estimation with
+a median error below 10 deg.
+
+7
+0
+10
+20
+30
+absolute error [cm]
+0.0
+0.5
+1.0
+cdf [-]
+distance estimation
+0
+100
+200
+absolute error [deg]
+angle estimation
+used mics
+0
+1
+2
+3
+0-1
+0-2
+0-3
+1-2
+1-3
+2-3
+0-1-2
+0-1-3
+0-2-3
+1-2-3
+0-1-2-3
+Fig. 8: Ablation study for the number of microphones used for
+distance and angle estimation on the drone (stepper-motor setup).
+While performance degrades with fewer microphones, it is still
+satisfactory even for a single microphone.
+Finally, we perform an ablation study on the number of
+microphones used for estimation. Focusing on the drone results
+for brevity, Figure 8 shows that the performance goes down
+as we exclude more microphones. However, even for a single
+microphone, we still achieve median errors of less than 5 cm
+and 90 deg, which is sufficient for robust wall detection.
+B. Results on flying drone
+The results thus far suggest that the proposed algorithm
+works well on robots with precise motion. For the remainder of
+this paper, we move to the more challenging setup of a flying
+drone. We fly the drone at constant speed towards a flat surface
+(glass or whiteboard), of which we estimate the location. This
+setup adds to the above studies the nuisance factors of 1) lat-
+eral movement during measurements and 2) varying propeller
+noise. To reduce both, we pick a shorter frequency sweep of
+Nf = 20 and omit the softer middle frequencies. The results
+are more variable, as shown in Figure 9. The drone however
+still localizes the wall with a median distance error of less than
+8 cm for almost all test runs. The angle estimates, on the other
+hand, lose in accuracy, with performances ranging from30 deg
+up to 120 deg median error. One possible explanation is that
+the movement between two consecutive sweeps is relatively
+small, making angle estimation very sensitive to noise.
+0
+10
+20
+30
+absolute error [cm]
+0.00
+0.25
+0.50
+0.75
+1.00
+cdf [-]
+distance estimation
+0
+100
+200
+absolute error [deg]
+angle estimation
+used data
+dataset
+glass wall
+dataset
+whiteboard
+random
+fixed
+Fig. 9: Distance and angle estimation by the flying drone approaching
+the wall. Top: cumulative distribution functions of the absolute errors,
+where each line corresponds to a different experiment. The same
+baselines as for Figure 7 are used. Bottom: merged consecutive
+frames of two example experiments (bird view).
+C. Multi-wall demo
+Finally, we implement a multi-wall detection demo. The
+drone flies back and forth between two whiteboards using only
+sound to detect, avoid, and map them, as shown in Figure 10.
+We control the drone manually, but the processing runs in real
+time on the recorded bag files.
+We assume there is a wall ahead of the drone when the
+distance estimate is below a threshold of 20 cm. Once a wall
+is detected, we perform data association by (12). Because of
+the low accuracy of angle estimates observed in the flying
+experiments, we use the simplification that a detected wall is
+always in front of the drone with respect to its movement.
+The average processing time of the non-optimized python
+implementation of the factor-graph inference, is less than 1 ms,
+even when running on all nodes (up to 200 for the second
+demo). Together with the inference time from the particle filter,
+the state estimation can thus run at ca. 10 Hz.
+We qualitatively evaluate the performance of the full
+pipeline in Figure 10, where we show the drone’s pose
+estimates, wall estimates and ground truth wall locations. The
+drone successfully maps and avoids the walls, and the labelling
+works reliably. In particular, in the second experiment (right
+plot in Figure 10), new measurements are correctly associated
+with π0 when revisiting the first wall.
+Fig. 10: Live wall avoidance experiments. Top: Estimated poses
+and planes after factor-graph inference, for two experiments. The
+positions over time are plotted in yellow to dark-blue. Wall estimates
+are plotted with decreasing transparency over time. Bottom: merged
+consecutive frames of first experiment (side view).
+VI. CONCLUSION AND FUTURE WORK
+We have proposed an end-to-end system for wall localiza-
+tion based on audio measurements. Using our custom audio
+deck for the Crazyflie drone and the off-the-shelf e-puck robot,
+we show how an approximate interference model can be used
+to determine the location of a reflector. This algorithm is suited
+for light-weight and cheap hardware, and runs entirely in real
+time without prior calibration.
+For ground-based robots or drones with a controlled move-
+ment, we achieve a precision of less than 2 cm and 30 deg
+
+-8
+median error over a distance of 50 cm. For the more challeng-
+ing setting of a flying drone, the accuracy lowers to ca. 8 cm,
+but the method is still able to perform wall detection reliably,
+which we demonstrate in a multi-wall detection demo. For the
+latter, the estimates are fed into a factor-graph SLAM pipeline
+which also performs data association.
+The performance drop when moving from ground-based to
+flying platforms suggests that the system would benefit from
+more sophisticated algorithms for handling propeller noise.
+For example, varying propeller noise and sound interference
+(e.g. background noise like voices or music) may affect the
+performance. Adaptive frequency sweep selection based on
+surrounding noise bears the potential to greatly increase the
+generalizability and robustness of the system. Compared to
+distance estimation, angle estimation was found to be more
+sensitive to noise. In future work, we plan to add other
+direction indicators such as sound level or phase differences
+as input to the angle estimation. Furthermore, an implicit
+assumption of the proposed method is that the robot is static
+during each frequency sweep. While this is easily achieved for
+ground-based robots, a method more specifically targeted to
+drones could account for its noisy motion. Finally, to overcome
+the potential nuisance of audible signals, a promising direction
+of future research consists of replacing the buzzer signal with
+a system’s inherent sound sources such as propeller noise.
+ACKNOWLEDGMENTS
+We would like to thank Daniel Burnier for providing the
+e-puck robot and technical support, the students Isaac, Kaour-
+intin and Emilia for their valuable help, and Karen, Eric and
+Mia for their feedback on the manuscript.
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diff --git a/cdE_T4oBgHgl3EQf0Bzf/content/tmp_files/load_file.txt b/cdE_T4oBgHgl3EQf0Bzf/content/tmp_files/load_file.txt
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@@ -0,0 +1,631 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf,len=630
+page_content='1  Blind as a bat: audible echolocation on small robots  Frederike D¨umbgen  Adrien Hoffet  Mihailo Kolundˇzija  Adam Scholefield  Martin Vetterli  Abstract—For safe and efficient operation, mobile robots need  to perceive their environment, and in particular, perform tasks  such as obstacle detection, localization, and mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Although  robots are often equipped with microphones and speakers, the  audio modality is rarely used for these tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Compared to the  localization of sound sources, for which many practical solutions  exist, algorithms for active echolocation are less developed and  often rely on hardware requirements that are out of reach for  small robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  We propose an end-to-end pipeline for sound-based localiza-  tion and mapping that is targeted at, but not limited to, robots  equipped with only simple buzzers and low-end microphones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  The method is model-based, runs in real time, and requires no  prior calibration or training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' We successfully test the algorithm  on the e-puck robot with its integrated audio hardware, and on  the Crazyflie drone, for which we design a reproducible audio  extension deck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' We achieve centimeter-level wall localization on  both platforms when the robots are static during the measure-  ment process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Even in the more challenging setting of a flying  drone, we can successfully localize walls, which we demonstrate  in a proof-of-concept multi-wall localization and mapping demo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  Index Terms—Range Sensing, Robot Audition, SLAM, Aerial  Systems: Perception and Autonomy  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' INTRODUCTION  A  bat’s ability to “see” in the dark using sound has fasci-  nated scientists for centuries [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Bats emit ultrasonic  chirps, and based on the timing and form of the echoes,  localize objects of interest such as food [2], obstacles [3], or  water resources [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  Similarly, most mobile robots need to localize targets and  obstacles in their surroundings in order to perform meaningful  operations such as search and rescue, exploration, and delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  Most robots that are currently deployed in the real world rely  on rich sensors for localization and mapping — sensors that  provide a plethora of information — such as cameras, lidar [5]  or radar [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  However, just like bats have relatively poor eyesight, not all  robots can be equipped with rich sensing equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Consider  the Crazyflie, a developer-friendly nano drone [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' This drone  is very limited in terms of admissible payload, which rules out  heavy sensors such as lidar or radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' A single camera can be  attached through the AI-deck;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' however, this does not provide  full coverage for obstacle avoidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' More coverage can be  obtained with the multi-ranger deck, designed specifically for  obstacle detection using four infrared sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  We propose to instead use audible sound for obstacle detec-  tion and mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Microphones come in lighter and smaller  This work was supported in part by “SNF-SESAM – Sensing and Sampling:  Theory and Algorithms.” n◦ 200021 181978/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  All authors are with the School of Computer and Communication Sci-  ences, EPFL, 1015 Lausanne, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Corresponding author: Frederike  D¨umbgen, frederike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='duembgen@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  13 cm  7 cm  microphone  microphone  buzzer  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' 1: The robotic platforms used for echolocation in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  Depicted on the left is the e-puck robot [9], a wheeled robot developed  for education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' On the right, we show the Crazyflie drone [7] with our  custom audio extension deck, equipped with four microphones and a  piezoelectric buzzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' The proposed echolocation pipeline can be used  on any platform with at least one low-key microphone and sound  source, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  form factors than rich sensors and demand less memory for  processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Compared to proximity sensors such as infrared  and ultrasound, they are less directional, allowing fewer sen-  sors to achieve full spatial coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Moreover, many robots  that are designed to communicate with users come equipped  with microphones, and thus using audio for navigation tasks  may require little additional payload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  In this paper, we develop a novel interference-based echolo-  cation algorithm, which includes a simultaneous localization  and mapping (SLAM) framework, integrating a particle filter  and a factor graph [8], which accounts for the nonlinear, multi-  modal nature of echolocation measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' For experimental  evaluation, we develop and provide an audio extension deck  for the Crazyflie drone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' It consists of a small piezoelectric  buzzer and four microphones embedded in a custom PCB,  including its own microcontroller for acquisition and prepro-  cessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' The methods developed in this paper are however  applicable to any robot equipped with at least one microphone  and a speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' To underline this point, we also provide  experimental validation on the ground-based e-puck robot [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  Both platforms are shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' To summarize, our main  contributions are:  a  model-based,  probabilistic  echolocation  algorithm  based on audio interference measurements,  a combined particle filtering and SLAM pipeline suited  for the non-Gaussian nature of echolocation measure-  ments, and  a Crazyflie audio extension deck with associated software  for research on audio-based navigation on drones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' After reviewing related  work in Section II, we describe our method for going from  low-level audio signals to wall and robot location estimates  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='08327v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='RO]  19 Jan 2023  40602  in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' In Section IV, we provide implementation  details for the two used hardware platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' In Section V, we  evaluate the method on these platforms – with finely controlled  motion in Section V-A, on the flying drone with one wall in  Section V-B, and in a multi-wall demo in Section V-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' We  conclude with a discussion of the results in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' RELATED WORK  We first give an overview of existing solutions for echolo-  cation from raw audio signals, also called the front-end in the  SLAM literature [5], before going over relevant work in the  back-end state estimation procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  a) Sound-based front end: Existing methods for sound-  based sensing can be loosely categorized into passive so-  lutions, which estimate the direction of arrival (DOA) of  external sound sources, and active solutions, which probe the  environment with an onboard sound source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' While DOA for  robotics is a rather active research field [10], sound-based  active solutions, the focus of this paper, are considerably less  explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  Amongst the most popular active solutions are methods  estimating the time of arrival (TOA) of the echoes of reflecting  objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' In rooms, this information is contained in the so-  called room impulse response (RIR), consisting of shifted,  attenuated peaks for the early echoes, and a more noise-like  reverberation tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' It can be estimated in frequency domain  through the deconvolution of a known wide-band source signal  and its response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' When the locations of microphones and  speakers are known, the first peaks of the RIR can be used to  approximately recover a room’s geometry [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Adapting this  method to a moving setup, the authors of [12] map out the  room, one wall at a time, using a small loudspeaker and high-  quality microphones mounted on a wheeled robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' The authors  of [13] use a similar platform but recover the peaks of the RIR  through nonlinear least-squares optimization on the wide-band  frequency response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' In follow-up work [14], this method is  extended with a DOA estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Although promising, these  methods use loudspeakers and microphones with high signal-  to-noise ratio (SNR) and flat frequency responses, which is  not available on the platforms studied in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  The TOA of reflectors can also be determined by emitting  and cross-correlating short, frequency-modulated pulses, a  concept known as pulse compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Using this technique,  room geometry is estimated, using a static omnidirectional  speaker and a tablet’s microphones, in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Moving to a  fully mobile setup, the authors of [16] perform room mapping  on a smartphone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Pulse compression is also commonly used  for underwater acoustic ranging [17], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Note that these  methods require fine control of the emitted audio pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  This last point is overcome by methods exploiting sound  interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' The idea is to detect close-by reflectors by mea-  suring the change in magnitude caused by interference from  its echoes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' In the first work bringing this idea to robotics, it  was shown that a reflector can be detected based on the change  in magnitude of the propeller’s ego-noise [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' The reported  results are obtained using static measurement microphones,  and an offline calibration phase of the propeller’s ego noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  The results highlight one advantage of interference-based  methods: since they do not require emission of specifically  designed short pulses, they can be used on noise inherent to  the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  In this paper, we take an interference-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Our  system is lighter compared to RIR-based methods, both in  terms of hardware requirements and computing power: it runs  on computationally limited platforms with simplistic speakers  and low-key microphones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Compared to pulse-compression  techniques, the sound signal is simpler and does not need to be  exactly known — this can be an advantage when the available  buzzer does not allow for fine-grained control, or when sound  sources inherent to the system are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' As opposed to [19],  our system does not require prior calibration, uses on-board  low-key microphones and tightly integrates the motion of the  device, as discussed next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  b) Back-end state estimation: For a practical solution,  the measurements taken at subsequent times and poses need  to be fused into a coherent map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' This can be posed as a factor  graph inference problem, where all unknown states (poses  and walls) are linked by measurement factors [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Inference  on factor graphs can become prohibitively expensive as the  number of unknown states increases over time, unless certain  Gaussian and sparsity assumptions are made, which is ex-  ploited by modern solvers [20], [21], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' While the Gaussian  assumption may hold for many commonly used sensors, it does  not for the sound-based front end considered in this paper:  sound is known to lead to multi-modal distributions [23],  [24] and thus requires different inference approaches [25],  [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Our state estimation pipeline is most similar in nature  to the work by [17], where a particle filter is used as a  pre-filtering step to render the bearing and range pseudo-  distributions approximately Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' In our case, the particle  filter helps not only reduce ambiguities, but also allows us  to convert the raw path difference measurements to distance  and angle estimates, which can be fed directly to the SLAM  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' METHODS  In this section, we describe the estimation pipeline, going  from raw audio measurements to pose and plane estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  We first derive the model of the microphone measurements as  a function of the robot pose and plane in Section III-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' We  derive our measurement model, which can be used to estimate  the path difference, in Section III-B, and infer distance and  angle distributions using discrete filtering in Section III-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  We conclude with a standard SLAM framework to jointly  estimate poses and planes in Section III-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' An overview of  the estimation pipeline is given in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Derivation of signal model  We first derive the model of the individual microphone  responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' A sketch of a sample setup is shown Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' We  place the robot’s body frame (also called the local frame) at the  sound source location s ∈ R2, and denote the Nm microphone  locations in this frame by xm ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' The pose, consisting of  the robot’s translation xn ∈ R2 and orientation φn ∈ [0, 2π)  3  particle filter  Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='B  Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='C  Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='D  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' 2: Overview of estimation pipeline: we depict two microphones and their measurements in green and blue, respectively, at two subsequent  poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' The raw measurements y(m)  nl (f), shown on the left, are used to estimate the location of nearby walls p(πl) and the robot’s poses  over time p(ξn), shown on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  at time tn, is denoted by ξn, expressed in the fixed inertial  frame (also called global frame).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Considering the studied  experimental platforms (a ground-based robot and a drone  hovering at constant height) we operate in two dimensions,  but this is not a hard requirement for the proposed solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  We denote the Nl planes by πl, and we parametrize them  with their global distance dl and angle θl from the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  The recorded signal at each microphone is the sum of the  direct sound and the reflected sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Accounting for free-space  attenuation and energy dissipation according to [27], it takes  the form:  z(m)  nl (t) := z(m)(t, ξn, πl)  (1)  = g(m)  �  1  4πℓ(m) s  �  t − ℓ(m)  c  �  + 1 − ρ  4πr(m)  nl  s  �  t − r(m)  nl  c  ��  ,  where c is the speed of sound, ρ the wall absorption coef-  ficient, and g(m) the unknown, frequency-dependent, micro-  phone gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' The lengths of the direct and reflected paths are  respectively given by ℓ(m) := ∥x(m) − s∥, and  r(m)  nl  =  ��  ℓ(m)�2 + 4(dnl)2 − 4dnlℓ(m) cos(θ(m) − θnl),  (2)  where θ(m) = ∠(x(m) − s) denotes the bearing of the m-th  microphone, and θnl, dnl denote the distance and angle of  plane l as seen from pose ξn:  dnl = dl − x⊤  n n  �  θl  �  and  θnl = θl − θn,  (3)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' 3:  Sketch of the experimental setup of echolocation with one  wall, two microphones and one source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' We can measure the difference  between the reflected path lengths r(m)  nl  for mics m = 0, 1, and  the direct path lengths ℓ(m), respectively, which are related to the  distance dnl and angle θnl of the wall πl at pose ξn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  where n(θ) is a normal vector with direction θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Solving (2)  for the distance dnl given a fixed angle, we have:  dnl := h  �  r(m)  nl , θl  �  = 1  2  �  ℓ(m) cos (θ(m) − θnl)  (4)  +  ��  ℓ(m)�2 �  cos2 (θ(m) − θnl) − 1  �  +  �  r(m)  nl  �2�  ,  which, for the limit case of a co-located phone and speaker  simplifies to the widely used approximation dnl ≈ r(m)  nl /2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Interference: from audio to path differences  Next, we derive our measurement function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' In the frequency  domain, the squared magnitude of (1) becomes:  y(m)  nl (f) = g(m)(f)2|ˆs(f)|2  (4π)2  �  1  ℓ(m)2 + (1 − ρ)2  �  r(m)  nl  �2  + 2  1 − ρ  ℓ(m)r(m)  nl  cos  �  2πf(r(m)  nl  − ℓ(m))/c  �  �  ��  �  :=˜y(m)  nl (f)  �  ,  (5)  where we introduce ∆(m)  nl  := r(m)  nl  − ℓ(m) for the path  difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' In theory, y(m)  nl (f) could be used as the mea-  surement model: evaluating it at Nf different frequencies  fk, the robot state ξn and wall location πl can be inferred  using nonlinear optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' However, the function is highly  nonlinear in the unknowns and, in particular, non-bijective,  making it prone to yield ambiguous results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' To make things  worse, the unknown parameters g(m)(f), |ˆs(f)| and ρ in (5)  render the optimization problem more costly, should we try  to either estimate them (thus adding more dimensions to the  state vector), or marginalize them out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  Thankfully, since the “frequency”1 of the cosine in (5)  depends only on the unknown state vectors and sound fre-  quency, we can infer probability distributions of the path  difference ∆(m) by studying its frequency response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' We in-  troduce ˜y(m)  nl (f) for the cosine, as shown in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' It can be  estimated from y(m)  nl (f) through online calibration, which we  will further develop in Section IV-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='2  1This “frequency” has the unit of seconds, and is not to be confused with  the “real” frequencies fk (expressed in Hz) at which we measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  2An interesting result states that the FFT-based solution we propose is in  fact equivalent to least-squares optimization with amplitude and phase of the  cosine marginalized out [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  4  Using a classical result from Bayesian theory, we know that  the probability of a signal to be a sinusoid of frequency f is  directly related to the signal’s periodogram [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Applying this  result here, the probability p(m)  n  (∆) of microphone m being  at a path difference ∆ at time n is given by:  p(m)  n  (∆) = η  �  1 −  2P (m)  ˜y  (∆/c)  �Nf  k=1  �  ˜y(m)  nl (f)  �2  � 2−Nf  2  ,  (6)  with P (m)  ˜y  (f) :=  1  Nf  ������  Nf  �  k=1  ˜y(m)  nl (f)ej2πffk  ������  2  ,  (7)  where η is a normalization factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' By choosing a roughly  uniform spacing of frequencies, we can use the FFT of ˜y to  calculate P (m)  ˜y  (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' For visualization purposes, we sketch an  example of ˜y(m)  nl (f), for different frequencies and distances,  in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Intuitively speaking, we sample an approximately  vertical slice of this function, and the periodicity of the slice  is related to the path difference of the closest wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  Next, we fuse multiple path difference distributions to obtain  distance and angle distributions that can be fed to our SLAM  framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' 4: Sketch of interference patterns over path differences and  frequencies, with the samples recorded by a drone (left) and a ground-  based robot (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' For different distances, we measure different  periodicities in the interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Note that the ground-based robot  can measure clean vertical slices, as opposed to the drone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Particle filter: from path differences to planes  Because of the non-linear measurement model and the poor  signal quality that common light-weight microphones and  buzzers provide, the path difference distributions are subject  to high noise levels and ambiguities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' We propose a particle  filtering approach for converting them to more unimodal  distance and angle distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' The proposed filter implicitly  assumes that, for a small time window, the pose estimates are  subject to only little drift, and can thus be used to resolve the  ambiguities in the distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Mathematically speaking, we  want to convert our potentially ambiguous path distributions  p(m)  n  (∆), obtained from (6), to unimodal distance- and angle  distributions p(dl) and p(θl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  a) Initialization: We initialize Np particles, containing  the (local) angle and distance candidates (d0,k, θ0,k), with k =  0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Np − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' The distances and angles are chosen uniformly  from an area of size [0, dmax] × [0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' We denote by dmax  the maximal distance that we aim to estimate, given by the  maximally resolvable path difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='3 Because of low SNR  at high distances, we clip this maximal distance at 80 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' We  initialize all particles with equal weights wk = 1/Np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  b) Prediction: At each timestep, we use the current  movement estimates to transform the particles to the new local  coordinates:  dn,k = dn−1,k − n(θn−1,k + φn)⊤(xn − xn−1)  θn,k = θn−1,k − (φn − φn−1),  (8)  where φn and xn are relative pose estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' We add a fixed  proportion (10%) of uniformly distributed particles, in order  to prevent the filter from converging to local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  c) Update: Next, we update the particle weights using  the newest measurements p(m)  n  (∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Using each particle’s  (dn,k, θn,k) values, we calculate the corresponding path dif-  ference ∆(m)  k  with (2) and evaluate the distributions using  linear interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Assuming the microphone measurements  are independent, we set the value wk to the product of the Nm  probabilities thus obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  d) Resampling: After each update step, we resample  the particles according to their weights, using the stratified  resampling algorithm [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Planar simultaneous localization and mapping  In the final step, we aim to feed the filtered plane measure-  ments, along with potential pose measurements, to a SLAM  framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' This ensures long-term consistency, and enables  crucial tasks such as data association and path planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  In the factor graph representation of our problem, the  nodes correspond to poses and planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' The pose estimates  ˜ξn ∈ SE(d) are incorporated in unary factors of the form:  cn =  ���ξn ⊖ ˜ξn  ���  Σn,  (9)  where ⊖ is the difference operator defined in the Lie algebra  of SE(d), and Σn is the diagonal pose covariance matrix,  fixed heuristically according to the pose estimation accuracy  of the state estimator (σx = σy = 1 cm, σφ = 5 deg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  For the plane measurements, we first extract distance and  angle estimates ˜dnl, ˜θnl and standard deviations σθ  nl, σd  nl from  the particle filter, using the weighted particle mean and  standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Approximating the distance and angle  distributions as Gaussians, we use the binary plane factors  of the form [31]:  cnl =  ����  �R⊤  n n(θl)  x⊤  n n(θl)  �  −  �n(˜θnl)  ˜dnl  �����  Σnl  ,  (10)  where Rn is the rotation matrix corresponding to φn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Σnl is  the noise covariance matrix, calculated from σθ  nl, σd  nl, with the  upper-left block given by:  (σθ  nl)2 ∂n  ∂θ  ���˜θnl  ∂n  ∂θ  ���  ⊤  ˜θnl  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  (11)  3By the Nyquist criterion we have dmax = h(∆max + ℓ) = h(c/2δf +  ℓ) ≈ c/4δf, with δf the (average) difference between the sampled frequen-  cies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='request  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='wall estimate  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='SPI  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='Bluetooth  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='CRTP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='buzzer control  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='feature extraction  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='audio deck  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='audio data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='audio data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='pose data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='commands  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='ground station  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='crazyflie_crtp/gateway  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='crazyflie_demo/wall_detector  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='audio_gtsam/wall_mapper  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='send commands  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='distributions  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='build  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='factor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='graph  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='solve  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='factor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='graph  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='run state  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='machine  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='audio driver  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='conversion of  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='CRTP    ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='SPI  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='conversion of  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='ROS    ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='CRTP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='generate  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='generate  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='commands  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='audio data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='pose data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='conversion of  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='ROS    ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='USB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='epuck_com/serial  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='buzzer control  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='feature extraction  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='epuck main  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='USB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='audio data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='pose data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='commands  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='server/  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='client  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='publisher/  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='subscriber  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' 5: Processing pipeline from firmware to the ground station’s ROS processing pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' The audio data and pose data is sent from either  the Crazyflie drone (top row) or the e-puck robot (bottom row) to the ground station, where they are processed in parallel by a series of  ROS nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Note that service/client connections (dotted lines) are only triggered upon request, while listener/publisher connections (solid  black lines) are running continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  To determine whether a new measurement corresponds to a  new plane or to a revisited plane, we introduce the following  data association loss between planes πi and πj:  e(πi, πj) = ∥n(θi)di − n(θj)dj∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  (12)  This loss measures how far the normal points of the two planes  are from each other, thus incorporating both angle as well as  distance discrepancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' We associate a new plane measurement  to a previously seen plane if (12) falls below a given threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  We found that a threshold of 30 cm performed well throughout  our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' We solve the factor graph using the iSAM2  solver implemented in gtsam [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' EXPERIMENTAL PLATFORMS  To demonstrate the described algorithm’s performance, the  main experimental platform studied is the Crazyflie drone,  a micro-drone with relatively poor computing resources (mi-  crocontroller STM32F405) and strict payload constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' To  fit these requirements, we design an audio-deck optimized to  add little weight, and equipped with its own microcontroller  (STM32F446) to offload the low-level audio pre-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  The deck conforms with the standard format of Crazyflie  extension decks, thus adhering to its plug-and-play philosophy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  The deck contains four MEMS microphones attached to its  extremities, a positioning chosen for its good trade-off between  minimal propeller noise and off-centre weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' A piezoelectric  buzzer, which has a reduced PCB footprint and energy con-  sumption compared to a speaker, is placed in the centre of the  deck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Note that one microphone is enough for the described  algorithms, but having more microphones both increases the  SNR and enables other applications such as DOA estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  Relative pose estimates are obtained using the drone’s optical  flow deck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' A photo of the Crazyflie equipped with the audio  deck is shown in Figure 1, and all hardware design files are  made publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='4  We also test our algorithms on the wheeled e-puck robot  (version 2) [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Just like the custom audio deck, the robot  possesses four MEMS microphones, but it has a more versatile  and louder electromagnetic speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Being a ground-based  robot, the e-puck allows for the evaluation of the methods in  more controlled settings, eliminating both position jitter and  propeller noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Relative pose estimates are obtained through  wheel odometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  4The code base including hardware design files and ROS pipeline is  available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='com/LCAV/audioROS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Low-level audio processing  The full processing pipeline is visualized in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' In  the firmware (left part of the plot), the signals recorded by the  four microphones are first stored in buffers of size N = 2048  at a sampling rate of Fs = 48 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' With these values, the  resulting frequency resolution, using the real-valued FFT, is  ∆f = Fs/N ≈ 23 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  The piecewise constant frequency sweep of the buzzer is  generated through PWM signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' For the drone, to increase  SNR, the frequency range is chosen as to be minimally  affected by propeller noise under average drone hovering  thrusts, while ensuring that the played frequencies match the  available frequency analysis bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Each note is played until  the microphones have recorded one full buffer, which takes  N/Fs ≈ 40 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' For a sweep of Nf = 20 discrete frequencies,  this results in circa 1 s per sweep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' After each sweep, the  complex response is calculated through a windowed FFT,  using the flattop window, which yields faithful magnitude  measurements even under small frequency errors [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' The  response at the current buzzer frequency is extracted and added  to an outgoing buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  After completion of each sweep, the next sweep is started  only once the data was successfully sent to the ground station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  For the Crazyflie drone, the buffer is passed first via SPI to  the main processor and then through the custom Bluetooth  protocol (CRTP) to the ground station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' The data throughput  of both SPI and CRTP being significantly higher than the  playback rate of the sweeps Nf × 2 × 32 = 1280 bps, we can  ensure smooth transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' For the e-puck robot, we omit  the intermediate SPI and CRTP communication and send the  audio data directly over USB to the ground station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Ground station processing  On the ground station (right half of Figure 5), the received  audio data is passed through a modular processing pipeline  implementing the steps described in Sections III-B to III-D  through custom nodes and interfaces, written in python in the  Robot Operating Systems (ROS) framework, version Galactic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  The full software suite, including a simulation framework  and implementations of DOA methods, are made publicly  available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  6  20  40  distance [cm]  3000  3500  4000  frequency [Hz]  measured  20  40  distance [cm]  theoretical  Crazyflie drone  20  40  60  distance [cm]  2000  3000  4000  5000  frequency [Hz]  measured  20  40  60  distance [cm]  theoretical  e-puck robot  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' 6: Interference patterns measured at one microphone, using the drone’s stepper-motor setup and the e-puck setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Photos of the  experimental setups are shown to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Note that the measured values have the same patterns as predicted by theory, using equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Online joint calibration  In our model (5), there are two unwanted and unknown,  frequency-dependent gains: the highly non-linear frequency  response of the piezo buzzer |ˆs(f)|, and the gains of the  different microphones g(m)(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' If we remove these two fac-  tors, we can use the measurement model ˜y(m)  nl (f), and path  difference estimation is reduced to a simple frequency anal-  ysis, as described in III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Since the two unknown gains are  entangled, we choose to estimate them jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' The intuition  of the calibration is the following: while the robot is moving,  it measures a portion of the distance-frequency function, as  visualized in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Even for small movements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' small  variations of wall distance d, we are likely to sample both  destructive and constructive interference regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' We can thus  expect that taking a moving average over a sufficiently long  time window yields a good estimate of the joint speaker-  microphone gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' To reduce the computational cost of the  calibration, we implement the average using a simple infinite  impulse response (IIR) low-pass filter:  ˜g(m)  n  (f) = (1 − λ)˜g(m)  n−1(f) + λy(m)  nl (f),  (13)  where we have introduced ˜g(m)  n  (f), the estimate of the joint  gain function g(m)(f)2|ˆs(f)|2 at time index n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' The parameter  λ controls the effective window width, and we use a value of  λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='3 throughout all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' RESULTS  We evaluate our systems in three increasingly difficult  stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' First, we obtain fine-grained measurements of the  wall’s interference patterns by moving the drone and e-puck  robot in small steps towards the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' For this purpose, the  drone is mounted on a linear stepper motor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' In a second stage,  we evaluate the drone’s performance in localizing a single wall  during flight, before concluding with a multi-wall demo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Stepper motor and ground-based robot  Moving the drone and e-puck robot in centimeter-steps  perpendicular to a glass wall, we create the measured inter-  ference matrices shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' The frequency sweeps  consist of Nf = 32 uniformly spaced frequencies, where the  range for the drone is chosen so as to avoid the propeller  noise harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' We note that, although noisy, we can clearly  −40  −20  0  error [cm]  error evolution  20  40  60  distance [cm]  −100  0  100  error [deg]  0  10  20  30  absolute error [cm]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='0  cdf [-]  distance estimation  0  100  200  absolute error [deg]  angle estimation  used data  drone  e-puck  random  fixed  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' 7: Performance of distance and angle estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Left: distance  and angle error as a function of wall distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Right: cumulative  distribution functions (cdf) of absolute errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Shown are the exper-  imental results from the drone on the stepper motor and the e-puck  robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' For comparison, we also include the results using simulated  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' The error using random guesses or one fixed guess serve as  worse-case baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  discern the interference patterns predicted by theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' The  pattern is particularly pronounced closer to the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' We also  note a strong frequency-dependent gain, particularly for the  e-puck robot, which we remove through online calibration as  described in Section IV-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  Next, we apply our estimation pipeline to these measure-  ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Treating each frequency sweep (each column in Fig-  ure 6) separately, we evaluate the performance of both distance  and angle estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' We found that Np = 400 particles  gave consistently good results throughout all experiments, with  which one full estimation (including update, prediction and  resampling) takes less than 75 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' We first investigate the  estimation error as we increase the distance to the wall (left  plots in Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Distance estimation is reliable up to 40 cm  from the wall, with the maximum error staying below 2 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  Angle estimation is less reliable, in particular for the drone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  However, up to ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' 30 cm, the maximum error is smaller than  20 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' We take a closer look at the cumulative probability  distributions (cdfs) in the right plots of Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' As worst-  case baselines, we plot the result of choosing random angles or  distances, or using a fixed value at the mid-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' As best-case  baseline, we run the proposed solution on simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' We  observe that the median error for distance estimation for both  e-puck and drone experiments is less than 3 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' The e-puck  robot also performs particularly well at angle estimation with  a median error below 10 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  7  0  10  20  30  absolute error [cm]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='0  cdf [-]  distance estimation  0  100  200  absolute error [deg]  angle estimation  used mics  0  1  2  3  0-1  0-2  0-3  1-2  1-3  2-3  0-1-2  0-1-3  0-2-3  1-2-3  0-1-2-3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' 8: Ablation study for the number of microphones used for  distance and angle estimation on the drone (stepper-motor setup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  While performance degrades with fewer microphones, it is still  satisfactory even for a single microphone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  Finally, we perform an ablation study on the number of  microphones used for estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Focusing on the drone results  for brevity, Figure 8 shows that the performance goes down  as we exclude more microphones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' However, even for a single  microphone, we still achieve median errors of less than 5 cm  and 90 deg, which is sufficient for robust wall detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Results on flying drone  The results thus far suggest that the proposed algorithm  works well on robots with precise motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' For the remainder of  this paper, we move to the more challenging setup of a flying  drone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' We fly the drone at constant speed towards a flat surface  (glass or whiteboard), of which we estimate the location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' This  setup adds to the above studies the nuisance factors of 1) lat-  eral movement during measurements and 2) varying propeller  noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' To reduce both, we pick a shorter frequency sweep of  Nf = 20 and omit the softer middle frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' The results  are more variable, as shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' The drone however  still localizes the wall with a median distance error of less than  8 cm for almost all test runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' The angle estimates, on the other  hand, lose in accuracy, with performances ranging from30 deg  up to 120 deg median error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' One possible explanation is that  the movement between two consecutive sweeps is relatively  small, making angle estimation very sensitive to noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  0  10  20  30  absolute error [cm]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='00  cdf [-]  distance estimation  0  100  200  absolute error [deg]  angle estimation  used data  dataset  glass wall  dataset  whiteboard  random  fixed  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' 9: Distance and angle estimation by the flying drone approaching  the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Top: cumulative distribution functions of the absolute errors,  where each line corresponds to a different experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' The same  baselines as for Figure 7 are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Bottom: merged consecutive  frames of two example experiments (bird view).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Multi-wall demo  Finally, we implement a multi-wall detection demo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' The  drone flies back and forth between two whiteboards using only  sound to detect, avoid, and map them, as shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  We control the drone manually, but the processing runs in real  time on the recorded bag files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  We assume there is a wall ahead of the drone when the  distance estimate is below a threshold of 20 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Once a wall  is detected, we perform data association by (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Because of  the low accuracy of angle estimates observed in the flying  experiments, we use the simplification that a detected wall is  always in front of the drone with respect to its movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  The average processing time of the non-optimized python  implementation of the factor-graph inference, is less than 1 ms,  even when running on all nodes (up to 200 for the second  demo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Together with the inference time from the particle filter,  the state estimation can thus run at ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' 10 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  We qualitatively evaluate the performance of the full  pipeline in Figure 10, where we show the drone’s pose  estimates, wall estimates and ground truth wall locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' The  drone successfully maps and avoids the walls, and the labelling  works reliably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' In particular, in the second experiment (right  plot in Figure 10), new measurements are correctly associated  with π0 when revisiting the first wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' 10: Live wall avoidance experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Top: Estimated poses  and planes after factor-graph inference, for two experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' The  positions over time are plotted in yellow to dark-blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Wall estimates  are plotted with decreasing transparency over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Bottom: merged  consecutive frames of first experiment (side view).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' CONCLUSION AND FUTURE WORK  We have proposed an end-to-end system for wall localiza-  tion based on audio measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Using our custom audio  deck for the Crazyflie drone and the off-the-shelf e-puck robot,  we show how an approximate interference model can be used  to determine the location of a reflector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' This algorithm is suited  for light-weight and cheap hardware, and runs entirely in real  time without prior calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  For ground-based robots or drones with a controlled move-  ment, we achieve a precision of less than 2 cm and 30 deg  8  median error over a distance of 50 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' For the more challeng-  ing setting of a flying drone, the accuracy lowers to ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' 8 cm,  but the method is still able to perform wall detection reliably,  which we demonstrate in a multi-wall detection demo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' For the  latter, the estimates are fed into a factor-graph SLAM pipeline  which also performs data association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  The performance drop when moving from ground-based to  flying platforms suggests that the system would benefit from  more sophisticated algorithms for handling propeller noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  For example, varying propeller noise and sound interference  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' background noise like voices or music) may affect the  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Adaptive frequency sweep selection based on  surrounding noise bears the potential to greatly increase the  generalizability and robustness of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Compared to  distance estimation, angle estimation was found to be more  sensitive to noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' In future work, we plan to add other  direction indicators such as sound level or phase differences  as input to the angle estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Furthermore, an implicit  assumption of the proposed method is that the robot is static  during each frequency sweep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' While this is easily achieved for  ground-based robots, a method more specifically targeted to  drones could account for its noisy motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content=' Finally, to overcome  the potential nuisance of audible signals, a promising direction  of future research consists of replacing the buzzer signal with  a system’s inherent sound sources such as propeller noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  We would like to thank Daniel Burnier for providing the  e-puck robot and technical support, the students Isaac, Kaour-  intin and Emilia for their valuable help, and Karen, Eric and  Mia for their feedback on the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE_T4oBgHgl3EQf0Bzf/content/2301.08327v1.pdf'}
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diff --git a/ctAyT4oBgHgl3EQfwfma/content/tmp_files/2301.00650v1.pdf.txt b/ctAyT4oBgHgl3EQfwfma/content/tmp_files/2301.00650v1.pdf.txt
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+ 
+ 
+ 
+HYBRID DEEP REINFORCEMENT LEARNING AND PLANNING FOR 
+SAFE  AND  COMFORTABLE  AUTOMATED  DRIVING 
+ 
+ 
+ 
+A PREPRINT 
+ 
+ 
+Dikshant Gupta 
+Saarland University, Computer Science Department 
+Saarbruecken, Germany dikshant2210@gmail.com 
+ 
+Matthias Klusch 
+German Research Center for Artificial Intelligence 
+Saarbruecken, Germany matthias.klusch@dfki.de 
+ 
+ 
+ 
+ABSTRACT 
+We present a novel hybrid learning method, HyLEAR, for solving the collision-free navigation 
+problem for self-driving cars in POMDPs. HyLEAR leverages interposed learning to embed knowl- 
+edge of a hybrid planner into a deep reinforcement learner to faster determine safe and comfortable 
+driving policies. In particular, the hybrid planner combines pedestrian path prediction and risk-aware 
+path planning with driving-behavior rule-based reasoning such that the driving policies also take 
+into account, whenever possible, the ride comfort and a given set of driving-behavior rules. Our 
+experimental performance analysis over the CARLA-CTS1 benchmark of critical traffic scenarios 
+revealed that HyLEAR can significantly outperform the selected baselines in terms of safety and ride 
+comfort. 
+ 
+ 
+1 Introduction 
+ 
+Problem. We consider the basic problem of collision-free navigation (CFN) of a self-driving car as, in short, to navigate 
+on a driveable path to a given goal in minimal time and collisions with objects such as other cars or pedestrians in a 
+partially observable traffic environment. The CFN problem can be modelled as a partially observable Markov decision 
+process (POMDP) to be solved by the car online and subject to the given car and pedestrian model. In this work, 
+we adopt the POMDP definition and models for the CFN problem from [17] and address the additional challenge to 
+compute driving policies that are not only safe but, whenever possible, also passenger comfortable. 
+Related work. Current CFN methods for self-driving cars can leverage either a deep reinforcement learner (DRL) 
+[10, 6], or an approximate POMDP planner (APPL) [15, 1], or a hybrid combination of thereof [17]. While a hybrid 
+system of DRL-supported online planning for CFN can outperform its individual components for CFN in terms of 
+safety [17], it may suffer from long training and online planning time. However, it is not known, whether some 
+alternative hybrid system of planning-supported DRL for CFN can be more efficient and safe. The rule-interposed 
+learning framework in [21] was shown to enable faster training of vanilla DRL but has not been applied to hybrid 
+DRL for CFN yet. On the other hand, many works and user studies investigated the effect of various road and load 
+disturbance factors including human-perceived risk [18, 9, 11] on passenger or ride comfort [19, 3, 7]. In this work, we 
+measure ride comfort based on the load disturbance factors smoothness and human- perceived risk of the ride by the 
+car. However, the potential of hybrid CFN methods taking ride comfort into account without compromising safety in 
+critical scenarios remains unclear. Besides, self-driving cars are expected to navigate whenever possible in compliance 
+with a given set of default driving-behavior or traffic rules, such as not to drive on sidewalks, or to keep the lane. In 
+some critical situations, however, safe driving may require the violation of certain rules as experienced human drivers 
+might do without making it a habit and still being able to explain their decision for the exceptional trajectory. In [5], a 
+hierarchical rulebook is used for explainable (production rule) reasoning for filtering rule-compliant safe trajectories but 
+without integrated hybrid CFN learning and ride comfort. 
+Contributions. To this end, we developed HyLEAR, a novel hybrid planning-assisted DRL method for the purpose 
+of safe navigation of self-driving cars in POMDPs considering, whenever possible, risk-aware ride comfort 
+
+HyLEAR  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+and driving-behavior rule compliance. For our experiments, we created the CARLA-CTS benchmark of critical traffic 
+scenarios based on the GIDAS accident study [2] and simulated in the driving simulator CARLA. 
+ 
+2 Method HyLEAR 
+ 
+Overview. HyLEAR consists of a soft actor-critic deep reinforcement learner, named NavSAC, that is assisted by a 
+hybrid planner. The hybrid planner leverages functional modules for (a) risk-sensitive planning of alternative safe and 
+short paths, (b) driving behavior rule-based reasoning for the selection of one path with minimal risk and rule violations, 
+and (c) online POMDP planning of an optimal speed action for the next step on this path. In particular, at each time step 
+of driving simulation during training and testing of HyLEAR in CARLA, the risk-aware path planner uses an anytime 
+weighted hybrid A* k-path planner and a pedestrian intention estimator M2P3 [16] to determine three alternative safe 
+and short paths together with their estimated human-perceived risk values. These risk values are computed based on the 
+driver risk fields for the paths as in [11] together with the respective cost-maps for planning. While the first path is 
+planned with given problem related cost-map, the planning of the other two relies on modified cost-maps with lower 
+costs of driving on free sidewalks, respectively, additional predicted pedestrian positions. The rule-based reasoner 
+performs hierarchical rule reasoning on a given rulebook [5] of initially four priority-ordered default driving-behavior 
+rules (avoid driving on sidewalks, minimize risk (blockage), keep the lane, take shortest path) to select the safe path with 
+minimal human-perceived risk and rule violations. For the next full car control action, the steering angle is extracted 
+from this selected path and the optimal speed action is determined during training by the POMDP planner IS-DESPOT 
+[15, 1] and during testing by the trained learner NavSAC of HyLEAR. 
+ 
+ 
+ 
+Figure 1: HyLEAR training architecture. 
+ 
+Training. The training of HyLEAR with the architecture shown in Figure 1 follows the general interposed learning 
+framework [21]: At each time step t, a full car control action at is generated by combining steering angle from the 
+selected path and speed action sampled from categorical distribution initialized with the hybrid planner speed policy. 
+Action at gets executed in the CARLA driving simulation environment and the respective POMDP belief state transition 
+tuple (ot, at, Rt+1, ot+1) is stored in the DRL memory buffer D. The deep reinforcement learner NavSAC takes as 
+input (a) a small RGB segmentation image as snippet from the planning cost-map that includes observed environmental 
+information of the actual scene together with past and future path of the car and pedestrian, as well as (b) the current 
+reward, speed, and previous action to learn to output the optimal speed action policy. For this purpose, it processes 
+
+NavSAC
+X4/2X2
+Learner
+Sampleactionfrom
+PPR
++
+Hybrid
+Planner
+Hierarchy of
+Rules
+Velocity Planner
+No Sidewalk
+IS-DESPOT
+Paths with
+Blockage
+Best Path in
+perceived Risk
+accordanceto
+rules
+Lane Keeping
+Belief(b)
+Belief Tracker
+Path Cength
+(Particle Belief)
+Risk Aware Hybrid A* Path Planner
+Paths with
+PedestrianPath
+PedestrianM2P3-Pedestrian Path
+perceived Risk
+Updated Cost Map
+Path
+Prediction
+DriverRisk Field
+Evaluation
+Driver Risk Field
+k-shortest paths(steering angle)
+k-shortest path Hybrid
+Cost Map
+4.HyLEAR  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+the image with a convolutional neural network, and samples a set of transitions from D for its off-policy soft actor-critic 
+learning with the respective loss functions 
+ 
+ 
+ 
+ 
+where observation ot and action at are sampled from D, V ψ is state value function, Qθ is Q-function, πϕ (speed) 
+actor policy function, and ψ, θ and ϕ are the NavSAC network weights to be trained. This hybrid planning-assisted 
+informed state-action space sampling may reduce the training time of NavSAC by avoiding unnecessary explorations. 
+Testing. The testing architecture of HyLEAR is the same as for training except that it does not include the computation- 
+ally expensive planner IS-DESPOT to determine an optimal speed actions for given situation and path. This capability 
+is now embedded in and performed much faster by the trained learner NavSAC. During testing, the hybrid planner of 
+HyLEAR generates the shortest safe path with minimal human-perceived risk and rule violations as input for the trained 
+NavSAC such that the extracted steering angle together with the optimal speed action determined now by the trained 
+NavSAC instead of the planner IS-DESPOT is then executed as a full control action by the car in the driving simulator 
+CARLA. 
+ 
+ 
+3 Evaluation 
+ 
+Experimental setting. Our experimental comparative performance evaluation of HyLEAR has been conducted over 
+the synthetic benchmark CARLA-CTS1, which consists of twelve parameterized scenarios with about thirty thousand 
+scenes in total simulated in the driving simulator CARLA. Most of the traffic scenarios are taken from the GIDAS 
+accident study [2] where the car is confronted with street crossing pedestrians, possibly occluded by some parking car, 
+an incoming car and intersections. The scenes per scenario are generated with varying speed and crossing distance 
+of pedestrians from the car. The selected baselines are (a) the individual CFN action planning and learning methods 
+IS-DESPOT-p and NavSAC-p each of which guided by a hybrid A* path planner, as well as the socially-aware DRL 
+method A2C-CADRL [8] and the hybrid learning-assisted planning method HyLEAP [17] for CFN. The performance 
+of each method is measured in terms of (a) overall safety index (SI) defined as total number of scenarios in which 
+the method is below given percentages of crashes and near-misses (5, 10); (b) crash and near-miss rates, and time 
+to goal (TTG); (c) ride comfort defined as being inversely proportional to the equally weighted sum of jerks and 
+human-perceived risk of planned trajectory, risk normalized to [0,1] with risk threshold set to 0.1 [11]; and (d) training 
+and execution time. HyLEAR is implemented in Python and PyTorch framework; all methods were trained and tested 
+on the DL supercomputer NVIDIA DGX-1 at DFKI Kaiserslautern. 
+Results. The overall results of our experiments, averaged across all scenarios, are shown in Table 1. 
+ 
+ 
+Method 
+Safety (SI) 
+Crash (%) 
+Near-miss (%) 
+TTG (s) 
+Comfort 
+Training (d) 
+Exec (ms) 
+NavSAC-p 
+1 
+21.44 
+9.24 
+17.57 
+0.262 
+10 
+60.29 
+IS-DESPOT-p 
+1 
+21.01 
+6.05 
+16.20 
+0.689 
+N/A 
+259.56 
+A2C-CADRL-p 
+0 
+25.14 
+11.47 
+14.26 
+1.010 
+4 
+58.57 
+HyLEAP 
+4 
+19.26 
+7.41 
+16.16 
+0.803 
+5 
+215.80 
+HyLEAR 
+5 
+19.88 
+8.49 
+15.86 
+1.064 
+3 
+71.50 
+Table 1: Overview of results on the CARLA-CTS1 benchmark 
+ 
+In general, HyLEAR provided a safer and more comfortable ride than the other selected baselines, following com- 
+paratively the fastest training and with a relatively acceptable time to goal and execution time. In some cases, taking 
+the safe and more comfortable but not shortest route by HyLEAR may come at the expense of minimal time to goal 
+compared to the fastest method A2C-CADRL. The latter, however, performed worse on safety due to local minima of 
+always accelerating to reach the goal, which resulted in second best comfort due to zero jerks. On average, the n-step 
+look-ahead action planning with IS-DESPOT-p was as safe as the learner NavSAC-p and with more ride comfort, in 
+particular in scenarios with temporarily occluded pedestrians but required extremely more execution time due to online 
+planning than both DRL methods. The interposed learning allowed HyLEAR to learn the optimal speed action for 
+given situation and safe path faster than the other DRL methods, while due to its hybrid planning HyLEAR performed 
+best in terms of safety and ride comfort, only driving on safe paths through free sidewalks if there are no alternatives 
+with acceptable human-perceived risk. While both hybrid methods, HyLEAP and HyLEAR, are by far more safe than 
+the tested individual planning and DRL methods, the hybrid planning-assisted learning of HyLEAR outperformed the 
+DRL-assisted online planning of HyLEAP in safety, comfort, time to goal, training and execution time. 
+ 
+ 
+
+JQ(0)=E(ot,at)~D[(Q°(ot,at)-Q(ot,at))2];
+J(Φ)=Eot~D log πP(atlot)-Q°(ot,at)HyLEAR  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+4 Conclusion 
+ 
+We presented HyLEAR, the first hybrid planning-assisted deep reinforcement learning method for collision-free driving 
+policies for self-driving cars that also take into account, whenever possible, the ride comfort and a given set of driving- 
+behavior rules. The experimental results over the CARLA-CTS benchmark revealed that HyLEAR can outperform the 
+selected baselines in safety and ride comfort with faster training and acceptable execution time. 
+Acknowledgments. This work has been funded by the German Ministry for Education and Research (BMB+F) in the 
+project MOMENTUM. 
+ 
+References 
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+European Conference on Computer Vision (ECCV). 
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+driving. Proc. IEEE Intelligent Vehicles Symposium (IV). IEEE. 
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+[16] Poibrenski, A. et al. (2020): Multimodal multi-pedestrian path prediction for autonomous cars. Applied Computing 
+Review, 20(4), ACM. 
+[17]  Pusse, F. & Klusch, M. (2019):  Hybrid online POMDP planning and deep reinforcement learning for safer 
+self-driving cars. Proc. IEEE Intelligent Vehicles Symposium (IV). IEEE. 
+[18] Skoglund, C. (2021): Risk-aware autonomous driving using POMDPs and responsibility-sensitive safety. Master 
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+learning. arXiv preprint arXiv:1910.09986 
+
diff --git a/ctAyT4oBgHgl3EQfwfma/content/tmp_files/load_file.txt b/ctAyT4oBgHgl3EQfwfma/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf,len=250
+page_content='HYBRID DEEP REINFORCEMENT LEARNING AND PLANNING FOR SAFE  AND  COMFORTABLE  AUTOMATED  DRIVING  A PREPRINT  Dikshant Gupta Saarland University, Computer Science Department Saarbruecken, Germany dikshant2210@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='com  Matthias Klusch German Research Center for Artificial Intelligence Saarbruecken, Germany matthias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='klusch@dfki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='de  ABSTRACT We present a novel hybrid learning method, HyLEAR, for solving the collision-free navigation problem for self-driving cars in POMDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' HyLEAR leverages interposed learning to embed knowl- edge of a hybrid planner into a deep reinforcement learner to faster determine safe and comfortable driving policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' In particular, the hybrid planner combines pedestrian path prediction and risk-aware path planning with driving-behavior rule-based reasoning such that the driving policies also take into account, whenever possible, the ride comfort and a given set of driving-behavior rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' Our experimental performance analysis over the CARLA-CTS1 benchmark of critical traffic scenarios revealed that HyLEAR can significantly outperform the selected baselines in terms of safety and ride comfort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='  1 Introduction  Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' We consider the basic problem of collision-free navigation (CFN) of a self-driving car as, in short, to navigate on a driveable path to a given goal in minimal time and collisions with objects such as other cars or pedestrians in a partially observable traffic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' The CFN problem can be modelled as a partially observable Markov decision process (POMDP) to be solved by the car online and subject to the given car and pedestrian model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' In this work, we adopt the POMDP definition and models for the CFN problem from [17] and address the additional challenge to compute driving policies that are not only safe but, whenever possible, also passenger comfortable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' Related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' Current CFN methods for self-driving cars can leverage either a deep reinforcement learner (DRL) [10, 6], or an approximate POMDP planner (APPL) [15, 1], or a hybrid combination of thereof [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' While a hybrid system of DRL-supported online planning for CFN can outperform its individual components for CFN in terms of safety [17], it may suffer from long training and online planning time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' However, it is not known, whether some alternative hybrid system of planning-supported DRL for CFN can be more efficient and safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' The rule-interposed learning framework in [21] was shown to enable faster training of vanilla DRL but has not been applied to hybrid DRL for CFN yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='  On the other hand, many works and user studies investigated the effect of various road and load disturbance factors including human-perceived risk [18, 9, 11] on passenger or ride comfort [19, 3, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' In this work, we measure ride comfort based on the load disturbance factors smoothness and human- perceived risk of the ride by the car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' However, the potential of hybrid CFN methods taking ride comfort into account without compromising safety in critical scenarios remains unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' Besides, self-driving cars are expected to navigate whenever possible in compliance with a given set of default driving-behavior or traffic rules, such as not to drive on sidewalks, or to keep the lane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' In some critical situations, however, safe driving may require the violation of certain rules as experienced human drivers might do without making it a habit and still being able to explain their decision for the exceptional trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' In [5], a hierarchical rulebook is used for explainable (production rule) reasoning for filtering rule-compliant safe trajectories but without integrated hybrid CFN learning and ride comfort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' To this end, we developed HyLEAR, a novel hybrid planning-assisted DRL method for the purpose of safe navigation of self-driving cars in POMDPs considering, whenever possible, risk-aware ride comfort  HyLEAR  and driving-behavior rule compliance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' For our experiments, we created the CARLA-CTS benchmark of critical traffic scenarios based on the GIDAS accident study [2] and simulated in the driving simulator CARLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='  2 Method HyLEAR  Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' HyLEAR consists of a soft actor-critic deep reinforcement learner, named NavSAC, that is assisted by a hybrid planner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' The hybrid planner leverages functional modules for (a) risk-sensitive planning of alternative safe and short paths, (b) driving behavior rule-based reasoning for the selection of one path with minimal risk and rule violations, and (c) online POMDP planning of an optimal speed action for the next step on this path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' In particular, at each time step of driving simulation during training and testing of HyLEAR in CARLA, the risk-aware path planner uses an anytime weighted hybrid A* k-path planner and a pedestrian intention estimator M2P3 [16] to determine three alternative safe and short paths together with their estimated human-perceived risk values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' These risk values are computed based on the driver risk fields for the paths as in [11] together with the respective cost-maps for planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' While the first path is planned with given problem related cost-map, the planning of the other two relies on modified cost-maps with lower costs of driving on free sidewalks, respectively, additional predicted pedestrian positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' The rule-based reasoner performs hierarchical rule reasoning on a given rulebook [5] of initially four priority-ordered default driving-behavior rules (avoid driving on sidewalks, minimize risk (blockage), keep the lane, take shortest path) to select the safe path with minimal human-perceived risk and rule violations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='  For the next full car control action, the steering angle is extracted from this selected path and the optimal speed action is determined during training by the POMDP planner IS-DESPOT [15, 1] and during testing by the trained learner NavSAC of HyLEAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='  Figure 1: HyLEAR training architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='  Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' The training of HyLEAR with the architecture shown in Figure 1 follows the general interposed learning framework [21]: At each time step t, a full car control action at is generated by combining steering angle from the selected path and speed action sampled from categorical distribution initialized with the hybrid planner speed policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' Action at gets executed in the CARLA driving simulation environment and the respective POMDP belief state transition tuple (ot, at, Rt+1, ot+1) is stored in the DRL memory buffer D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' The deep reinforcement learner NavSAC takes as input (a) a small RGB segmentation image as snippet from the planning cost-map that includes observed environmental information of the actual scene together with past and future path of the car and pedestrian, as well as (b) the current reward, speed, and previous action to learn to output the optimal speed action policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' For this purpose,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' it processes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='NavSAC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='X4/2X2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='Learner  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='Sampleactionfrom  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='PPR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='Hybrid  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='Planner  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='Hierarchy of  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='Rules  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='Velocity Planner  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='No Sidewalk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='IS DESPOT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='Paths with  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='Blockage  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='Best Path in  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='perceived Risk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='accordanceto  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='rules  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='Lane Keeping  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='Belief(b)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='Belief Tracker  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='Path Cength  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='(Particle Belief)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='Risk Aware Hybrid A Path Planner  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='Paths with  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='PedestrianPath  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='PedestrianM2P3 Pedestrian Path  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='perceived Risk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='Updated Cost Map  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='Path  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='Prediction  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='DriverRisk Field  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='Evaluation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='Driver Risk Field  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='k shortest paths(steering angle)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='k shortest path Hybrid  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='Cost Map  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='HyLEAR  the image with a convolutional neural network, and samples a set of transitions from D for its off-policy soft actor-critic learning with the respective loss functions  where observation ot and action at are sampled from D, V ψ is state value function, Qθ is Q-function, πϕ (speed) actor policy function, and ψ, θ and ϕ are the NavSAC network weights to be trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' This hybrid planning-assisted informed state-action space sampling may reduce the training time of NavSAC by avoiding unnecessary explorations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' Testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' The testing architecture of HyLEAR is the same as for training except that it does not include the computation- ally expensive planner IS-DESPOT to determine an optimal speed actions for given situation and path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' This capability is now embedded in and performed much faster by the trained learner NavSAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' During testing, the hybrid planner of HyLEAR generates the shortest safe path with minimal human-perceived risk and rule violations as input for the trained NavSAC such that the extracted steering angle together with the optimal speed action determined now by the trained NavSAC instead of the planner IS-DESPOT is then executed as a full control action by the car in the driving simulator CARLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='  3 Evaluation  Experimental setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' Our experimental comparative performance evaluation of HyLEAR has been conducted over the synthetic benchmark CARLA-CTS1, which consists of twelve parameterized scenarios with about thirty thousand scenes in total simulated in the driving simulator CARLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' Most of the traffic scenarios are taken from the GIDAS accident study [2] where the car is confronted with street crossing pedestrians, possibly occluded by some parking car, an incoming car and intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' The scenes per scenario are generated with varying speed and crossing distance of pedestrians from the car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' The selected baselines are (a) the individual CFN action planning and learning methods IS-DESPOT-p and NavSAC-p each of which guided by a hybrid A* path planner, as well as the socially-aware DRL method A2C-CADRL [8] and the hybrid learning-assisted planning method HyLEAP [17] for CFN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' The performance of each method is measured in terms of (a) overall safety index (SI) defined as total number of scenarios in which the method is below given percentages of crashes and near-misses (5, 10);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' (b) crash and near-miss rates, and time to goal (TTG);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' (c) ride comfort defined as being inversely proportional to the equally weighted sum of jerks and human-perceived risk of planned trajectory, risk normalized to [0,1] with risk threshold set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='1 [11];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' and (d) training and execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='  HyLEAR is implemented in Python and PyTorch framework;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' all methods were trained and tested on the DL supercomputer NVIDIA DGX-1 at DFKI Kaiserslautern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' The overall results of our experiments, averaged across all scenarios, are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='  Method Safety (SI) Crash (%) Near-miss (%) TTG (s) Comfort Training (d) Exec (ms) NavSAC-p 1 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='44 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='24 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='262 10 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='29 IS-DESPOT-p 1 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='01 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='05 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='689 N/A 259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='56 A2C-CADRL-p 0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='14 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='47 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='26 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='010 4 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='57 HyLEAP 4 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='26 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='41 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='803 5 215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='80 HyLEAR 5 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='88 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='49 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='064 3 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='50 Table 1: Overview of results on the CARLA-CTS1 benchmark  In general, HyLEAR provided a safer and more comfortable ride than the other selected baselines, following com- paratively the fastest training and with a relatively acceptable time to goal and execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' In some cases, taking the safe and more comfortable but not shortest route by HyLEAR may come at the expense of minimal time to goal compared to the fastest method A2C-CADRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' The latter, however, performed worse on safety due to local minima of always accelerating to reach the goal, which resulted in second best comfort due to zero jerks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' On average, the n-step look-ahead action planning with IS-DESPOT-p was as safe as the learner NavSAC-p and with more ride comfort, in particular in scenarios with temporarily occluded pedestrians but required extremely more execution time due to online planning than both DRL methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' The interposed learning allowed HyLEAR to learn the optimal speed action for given situation and safe path faster than the other DRL methods, while due to its hybrid planning HyLEAR performed best in terms of safety and ride comfort, only driving on safe paths through free sidewalks if there are no alternatives with acceptable human-perceived risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' While both hybrid methods, HyLEAP and HyLEAR, are by far more safe than the tested individual planning and DRL methods, the hybrid planning-assisted learning of HyLEAR outperformed the DRL-assisted online planning of HyLEAP in safety, comfort, time to goal, training and execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='  JQ(0)=E(ot,at)~D[(Q°(ot,at) Q(ot,at))2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content='  J(Φ)=Eot~D log πP(atlot) Q°(ot,at)HyLEAR  4 Conclusion  We presented HyLEAR, the first hybrid planning-assisted deep reinforcement learning method for collision-free driving policies for self-driving cars that also take into account, whenever possible, the ride comfort and a given set of driving- behavior rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' The experimental results over the CARLA-CTS benchmark revealed that HyLEAR can outperform the selected baselines in safety and ride comfort with faster training and acceptable execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
+page_content=' This work has been funded by the German Ministry for Education and Research (BMB+F) in the project MOMENTUM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfwfma/content/2301.00650v1.pdf'}
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+Recursive Neural Networks with Bottlenecks Diagnose
+(Non-)Compositionality
+Verna Dankers1 and Ivan Titov1,2
+1ILCC, University of Edinburgh
+2ILLC, University of Amsterdam
+vernadankers@gmail.com, ititov@inf.ed.ac.uk
+Abstract
+A recent line of work in NLP focuses on
+the (dis)ability of models to generalise com-
+positionally for artificial languages. However,
+when considering natural language tasks, the
+data involved is not strictly, or locally, compo-
+sitional. Quantifying the compositionality of
+data is a challenging task, which has been in-
+vestigated primarily for short utterances. We
+use recursive neural models (Tree-LSTMs)
+with bottlenecks that limit the transfer of infor-
+mation between nodes. We illustrate that com-
+paring data’s representations in models with
+and without the bottleneck can be used to pro-
+duce a compositionality metric. The procedure
+is applied to the evaluation of arithmetic ex-
+pressions using synthetic data, and sentiment
+classification using natural language data. We
+demonstrate that compression through a bot-
+tleneck impacts non-compositional examples
+disproportionately and then use the bottleneck
+compositionality metric (BCM) to distinguish
+compositional from non-compositional sam-
+ples, yielding a compositionality ranking over
+a dataset.
+1
+Introduction
+Compositional generalisation in contemporary
+NLP research investigates models’ ability to com-
+pose the meanings of expressions from their parts
+and is often investigated with artificial languages
+(e.g. Lake and Baroni, 2018; Hupkes et al., 2020) or
+highly-structured natural language data (e.g. Key-
+sers et al., 2019). For such tasks, the local compo-
+sitionality definition of Szabó (2012, p. 10) illus-
+trates how meaning can be algebraically composed:
+“The meaning of a complex expression
+is determined by the meanings its con-
+stituents have individually and the way
+those constituents are combined.”
+In natural language, there are fragments whose
+meaning can be composed as with arithmetic (e.g.
+removing
+the
+ruler
+from
+the classroom
+removing
+the
+pencil
+the classroom
+locally compositional processing
+recursive processing
+from
+unambiguous example:
+interpretation is the same
+ambiguous example:
+interpretation changes
+Figure 1: When processing this phrase, “the ruler” is
+interpreted differently when comparing recursive pro-
+cessing with local processing. We enforce local pro-
+cessing by equipping models with bottlenecks, and our
+bottleneck compositionality metric (BCM) then com-
+pares inputs’ representations before and after compres-
+sion through the bottleneck.
+“the cat is in the house”), while others carry con-
+textual dependencies (e.g. “the kiwi grows on the
+farm”). Can we characterise whether an input’s
+meaning arises from strictly local compositions?
+Existing work in that direction mostly focuses
+on providing a ‘compositionality rating’1 for figura-
+tive utterances since figurative language is assumed
+to be less compositional (Ramisch et al., 2016; Nan-
+dakumar et al., 2019; Reddy et al., 2011). Andreas
+(2018) suggests a general-purpose formulation for
+measuring the compositionality of examples using
+their numerical representations, through the Tree
+Reconstruction Error (TRE), expressing the dis-
+tance between a model’s representation of an input
+and a strictly compositional reconstruction of that
+representation. Determining how to compute that
+reconstruction is far from trivial.
+Inspired by TRE, we use recursive neural net-
+works, Tree-LSTMs (Tai et al., 2015), to process
+inputs according to their syntactic structure. We
+augment Tree-LSTMs with bottlenecks to compute
+1We colloquially refer to the ‘compositionality ratings’ of
+phrases, but a more appropriate way to express the same would
+be to refer to ‘the extent to which the meaning of a phrase
+arises from a compositional syntax and semantics’. After all,
+compositionality is a property of a language, not of a phrase.
+arXiv:2301.13714v1  [cs.CL]  31 Jan 2023
+
+the task-specific meaning of an input in a more lo-
+cally compositional manner. We use these models
+to distinguish more compositional examples from
+less compositional ones in a bottleneck compo-
+sitionality metric (BCM). Figure 1 provides an
+intuition for how a bottleneck can provide a met-
+ric. For fragments that violate the assumption that
+meanings of subexpressions can be computed lo-
+cally (on the left side), one could end up with dif-
+ferent interpretations when comparing a contex-
+tualised interpretation (in blue) with one locally
+computed (in green): disambiguating “ruler” re-
+quires postponed meaning computation, and thus
+local processing is likely to lead to different results
+from regular processing. For fragments that are
+non-ambiguous (on the right side) the two types
+of processing can yield the same interpretation be-
+cause the interpretation of “pencil” is likely to be
+the same, with or without the context. The bottle-
+neck hinders the model in postponing computations
+and more strongly impacts non-compositional sam-
+ples compared to compositional ones, thus acting
+as a metric.
+In the remainder of the paper, we firstly discuss
+the related work in §2. §3 elaborates on the models
+used that either apply a deep variational informa-
+tion bottleneck (DVIB) (Alemi et al., 2017) or com-
+press representations through increased dropout
+or smaller hidden dimensionalities. In §4, we pro-
+vide a proof-of-concept in a controlled environment
+where non-compositional examples are manually
+introduced, after which §5 elaborates on the natural
+language example of sentiment analysis. For both
+tasks, we (1) demonstrate that compression through
+a bottleneck encourages local processing and (2)
+show that the bottleneck can act as a metric dis-
+tinguishing compositional from less compositional
+examples.
+2
+Related Work
+Multi-word expressions
+The majority of the re-
+lated work in the past two decades has discussed
+the compositionality of phrases in the context of
+figurative language, such as phrasal verbs (“to eat
+up”) (McCarthy et al., 2003), noun compounds
+(“cloud nine” vs “swimming pool”) (Reddy et al.,
+2011; Yazdani et al., 2015; Ramisch et al., 2016;
+Nandakumar et al., 2019), verb-noun collocations
+(“take place” vs “take a gift”) (Venkatapathy and
+Joshi, 2005; McCarthy et al., 2007), and adjective-
+noun pairs (“nice house”) (Guevara, 2010). Compo-
+sitionality judgements were obtained from humans,
+who indicated to what extent the meaning of the
+compound is that of the words when combined lit-
+erally, and various computational methods were
+applied to learn that mapping. Those methods were
+initially thesaurus-based (McCarthy et al., 2003),
+relied on word vectors from co-occurrence matri-
+ces later on (Reddy et al., 2011), or employed deep
+neural networks (Nandakumar et al., 2019).
+Compositionality by reconstruction
+TRE (An-
+dreas, 2018) is a task-agnostic metric that evalu-
+ates the compositionality of data representations:
+TRE(x) = δ(f(x), ˆfη(d)). It is the distance be-
+tween the representation of x constructed by f
+and the compositionally reconstructed variant ˆfη(d)
+based on the derivation of x (d). When employ-
+ing the metric, one should define an appropriate
+distance function (δ) and define ˆfη parametrised
+by η. Andreas illustrates the TRE’s versatility by
+instantiating it for three scenarios: to investigate
+whether image representations are similar to com-
+posed image attributes, whether phrase embeddings
+are similar to the vector addition of their compo-
+nents, and whether generalisation accuracy in a
+reference game positively correlates with TRE.
+Bhathena et al. (2020) present two methods
+based on TRE to obtain compositionality ratings
+for sentiment trees, referred to as tree impurity and
+weighted node switching that express the differ-
+ence between the sentiment label of the root and
+the other nodes in the tree. Zheng and Jiang (2022)
+ranked examples of sentiment analysis based on
+the extent to which neural models should memorise
+examples in order to capture their target correctly.
+While different from TRE, memorisation could be
+related to non-compositionality in the sense that
+non-compositional examples require more memori-
+sation, akin to formulaic language requiring mem-
+orisation in humans (Wray and Perkins, 2000).
+Other instantiations of the TRE are from litera-
+ture on language emergence in signalling games,
+where the degree of compositionality of that lan-
+guage is measured. Korbak et al. (2020) contrast
+TRE and six other compositionality metrics for
+signalling games where the colour and shape of
+an object are communicated. Examples of such
+metrics are topographic similarity, positional disen-
+tanglement and context independence. These are
+not directly related to our work, considering that
+they aim to provide a metric for a language rather
+than single utterances. Appendix B.2 elaborates on
+topographic similarity and the metrics of Bhathena
+
+et al. (2020) and Zheng and Jiang (2022), compar-
+ing them to our metric for sentiment analysis.
+Compositional data splits
+Recent work on com-
+positional generalisation using artificial languages
+or highly-structured natural language data focuses
+on creating data splits that have systematic sepa-
+ration of input combinations in train and test data.
+The aim is to create test sets that should not be hard
+when computing meaning compositionally, but, in
+practice, are very challenging. An example compo-
+sitionality metric for semantic parsing is maximum
+compound divergence (Keysers et al., 2019; Shaw
+et al., 2021), that minimises train-test differences
+in word distributions while maximising the differ-
+ences in compound usage. This only applies to a
+data split as a whole, and – differently from the
+work at hand – does not rate individual samples.
+More recently, Bogin et al. (2022) discussed a
+diagnostic metric for semantic parsing, that pre-
+dicts model success on examples based on their
+local structure. Because models struggle with sys-
+tematically assigning the same meaning to subex-
+pressions when they re-appear in new syntactic
+structures, such structural deviation diagnoses gen-
+eralisation failures. Notice that the aim of our work
+is different, namely identifying examples that are
+not compositional, rather than investigating gener-
+alisation failure for compositional examples.
+3
+Model
+The model we employ is the Tree-LSTM (Tai et al.,
+2015), which is a generalisation of LSTMs to tree-
+structured network topologies. The LSTM com-
+putes symbols’ representations by incorporating
+previous time steps, visiting symbols in linear or-
+der. A sentence representation is simply the final
+time step. A Tree-LSTM, instead, uses a tree’s root
+node representation as the sentence representation,
+and computes the representation of a non-terminal
+node using the node’s children.
+Equations 1 and 2 illustrate the difference be-
+tween the LSTM and an N-ary Tree-LSTM for
+the input gate. The LSTM computes the gate’s ac-
+tivation for time step t using input vector xt and
+previous hidden state ht−1. The Tree-LSTM does
+so for node j using the input vector xj and the
+hidden states of up to N children of node j.
+it = σ(W (i)xt + U (i)ht−1 + b(i))
+(1)
+ij = σ(W (i)xj +
+N
+�
+ℓ=1
+U (i)
+ℓ hjℓ + b(i))
+(2)
+In addition to the input gate, the Tree-LSTM’s
+specification for non-terminal j (with its kth child
+indicated as hjk) involves an output gate oj (equa-
+tion analogous to 2), a forget gate fjk (Equation 3),
+cell input activation vector uj (equation analogous
+to 2, with the σ function replaced by tanh), and
+memory cell state cj (Equation 4). Finally, cj feeds
+into the computation of hidden state hj (Equa-
+tion 5).
+fjk = σ(W (f)xj +
+N
+�
+ℓ=1
+U (f)
+kℓ hjℓ + b(f))
+(3)
+cj = ij ⊙ uj +
+N
+�
+ℓ=1
+fjℓ ⊙ cjℓ
+(4)
+hj = oj ⊙ tanh(cj)
+(5)
+We apply a binary Tree-LSTM to compute hidden
+state hj and memory cell state cj, that thus uses
+separate parameters in the gates for the left and
+right child.
+Tree-LSTMs process inputs according to their
+syntactic structure, which has been associated with
+more compositional processing (Socher et al., 2013;
+Tai et al., 2015). Yet, although the topology encour-
+ages compositional processing, there is no mecha-
+nism to explicitly regulate how much information
+is passed from children to parent nodes – e.g. given
+enough capacity, the hidden representations could
+store every input encountered and postpone pro-
+cessing until the very end. We add such a mecha-
+nism by introducing a bottleneck.
+1.
+Deep Variational Information Bottleneck
+The information bottleneck of Alemi et al. (2017)
+assumes random variables X and Y for the input
+and output, and emits a compressed representation
+Z that preserves information about Y , by minimis-
+ing the loss LIB in Equation 6. This loss is in-
+tractable, which motivates the variational estimate
+LV IB provided in Equation 7 (Alemi et al., 2017)
+that we use to train the deep variational informa-
+tion bottleneck (DVIB) version of our model.
+LIB = βI(X, Z) − I(Z, Y )
+(6)
+LV IB = βE
+x[KL[pθ(z|x), r(z)]]
+�
+��
+�
+information loss
++
+E
+z∼pθ(z|x)[−logqφ(y|z)]
+�
+��
+�
+task loss
+(7)
+
+In the information loss, r(z) and pθ(z|x) estimate
+the prior and posterior probability over z, respec-
+tively. In the task loss, qφ(y|z) is a parametric
+approximation of p(y|z).
+In order to allow an
+analytic computation of the KL-divergence, we
+consider Gaussian distributions r(z) and pθ(z|x),
+namely r(z) = N(z|µ0, Σ0) and pθ(z|x) =
+N(z|µ(x), Σ(x)), where µ(x) and µ0 are mean
+vectors, and Σ(x) and Σ0 are diagonal covariance
+matrices. The reparameterisation trick is used to
+estimate the gradients: z = µ(x)+Σ(x)⊙ϵ, where
+ϵ ∼ N(0, I).
+We sample z once per non-terminal node, and
+average the KL terms of all non-terminal nodes,
+where x is the hidden state hj or the cell state cj
+(that have separate bottlenecks), and µ(x) and Σ(x)
+are computed by feeding x to two linear layers. β
+regulates the impact of the DVIB, and is gradually
+increased during training. During inference, we
+use z = µ(x).
+2. Dropout bottleneck
+Binary dropout (Srivas-
+tava et al., 2014) is commonly applied when train-
+ing neural models, to prevent overfitting. With
+a probability p hidden units are set to zero, and
+during the evaluation all units are kept, but the acti-
+vations are scaled down. Dropout encourages dis-
+tributing the most salient information over multiple
+neurons, which comes at the cost of idiosyncratic
+patterns that networks may memorise otherwise.
+We hypothesise that this hurts non-compositional
+examples most. We apply dropout to the Tree-
+LSTM’s hidden states (hj) and memory cell states
+(cj).
+3.
+Hidden dimensionality bottleneck
+Simi-
+larly, decreasing the number of hidden units is
+expected to act as a bottleneck. We decrease the
+number of hidden units in the Tree-LSTM, keep-
+ing the embedding and task classifier dimensions
+stable, where possible.
+The different bottlenecks have different merits:
+whereas the hidden dimensionality and dropout
+bottlenecks shine through simplicity, they are rigid
+in how they affect the model and apply in the same
+way at every node. The DVIB allows for more
+flexibility in how compression is achieved through
+learnt Σ(x) and by requiring an overall reduction
+in the information loss term, without enforcing the
+same bottleneck at every node in the tree.
+From bottleneck to compositionality metric
+BCM compares Tree-LSTMs with and without a
+bottleneck. We experiment with two methods, in-
+spired by TRE (Andreas, 2018). TRE aims to find
+η such that δ(f(x), ˆfη(d)) is minimised, for inputs
+x, their derivations d, distance function δ, a model
+f and its compositional approximation ˆfη.
+• In the TRE training (BCM-TT) setup, we
+include the distance (δ) between the hidden
+representations of f and ˆfη in the loss when
+training ˆfη. When training ˆfη with TRE train-
+ing, f is frozen, and f and ˆfη share the final
+linear layer of the classification module. In the
+arithmetic task, δ is the mean-squared error
+(MSE) (i.e. the squared Euclidean distance).
+In sentiment analysis, δ is the Cosine distance
+function.
+• In the post-processing (BCM-PP) setup, we
+train the two models separately, extract hid-
+den representations and apply canonical cor-
+relation analysis (CCA) (Hotelling, 1936)
+to minimise the distance between the sets
+of hidden representations. Assume matrices
+A ∈ RdA×N and B ∈ RdB×N represent-
+ing N inputs with dimensionalities dA and
+dB. CCA linearly transforms these subspaces
+A′ = WA, B′ = V B to maximise the cor-
+relations {ρ1, . . . , ρmin(dA,dB)} of the trans-
+formed subspaces. We treat the number of
+CCA dimensions to use as a hyperparameter.
+4
+Proof-of-concept: Arithmetic
+Given a task, we assign ratings to inputs that ex-
+press to what extent their task-dependent meaning
+arises in a locally compositional manner. To inves-
+tigate the impact of our metric on compositional
+and non-compositional examples in a controlled
+environment, we first use perfectly compositional
+arithmetic expressions and introduce exceptions to
+that compositionality manually.
+4.1
+Data and model training
+Math problems have previously been used to exam-
+ine neural models’ compositional reasoning (e.g.
+Saxton et al., 2018; Hupkes et al., 2018; Russin
+et al., 2021). Arithmetic expressions are suited for
+our application, in particular since they can be rep-
+resented as trees. We use expressions containing
+brackets, integers -10 to 10, and + and - operators –
+e.g. “( 10 - ( 5 + 3 ))” (using data from Hupkes
+et al., 2018). The output is an integer. This is mod-
+elled as a regression problem with the MSE loss.
+The ‘meaning’ (the numerical value) of a subex-
+pression can be locally computed at each node in
+
+0
+2
+-4
++
+-
+0
+-4
++
+-
+2
+0 in left subtree? 
+interpret 0 as 0
+0 in right subtree? treat
+0 as the 1st leaf node
+Figure 2: Illustration of the ‘exceptions’ in the arith-
+metic task: the value of “0” depends on its position and
+on the value of the leftmost leaf node in the tree.
+the tree: there are no contextual dependencies.
+In this controlled environment, we introduce ex-
+ceptions by making “0” ambiguous. When located
+in the subtree headed by the root node’s left child,
+it takes on its regular value, but when located in
+the right subtree, it takes on the value of the left-
+most leaf node of the entire tree (see Figure 2).
+The model is thus encouraged to perform non-
+compositional processing to keep track of all occur-
+rences of “0” and store the first leaf node’s value
+throughout the tree. 88% of the training data are the
+original arithmetic expressions, and 12% are such
+exceptions. We can thus track what happens to the
+two categories when we introduce the bottleneck.
+The training data consist of 14903 expressions with
+1 to 9 numbers. We test on expressions with lengths
+5 to 9, using 5000 examples per length. The Tree-
+LSTMs trained on this dataset have embeddings
+and hidden states of sizes 150 and are trained for
+50 epochs with learning rate 2e−4 with AdamW
+and a batch size of 32. The base Tree-LSTMs in
+all setups use the same architecture, namely the
+Tree-LSTM architecture required for the DVIB,
+but with β = 0. All results are averaged over mod-
+els trained using ten different random seeds. In the
+Tree-LSTM, the numbers are leaf nodes and the
+labels of non-terminal nodes are the operators.2
+4.2
+Task performance: Hierarchy without
+compositionality?
+Figures 3a and 3b visualise the performance for
+the regular examples and exceptions, respectively,
+when increasing β for the DVIB. The DVIB dis-
+proportionately harms the exceptions; when β is
+too high the model cannot capture the non-local
+dependencies. Appendix A.1 shows how the hid-
+den dimensionality and dropout bottlenecks have a
+similar effect. Figure 4 and Appendix A.2 provide
+insights in the training dynamics of the models: ini-
+tially, all models will treat “0” as a regular number,
+independent of the bottleneck. Close to conver-
+2Appendix C further elaborates on the experimental setup.
+5
+6
+7
+8
+9
+number of leaf nodes
+0
+1
+2
+3
+MSE
+.0
+.01
+.025
+.05
+.1
+.25
+.5
+1.0
+(a) Regular examples
+5
+6
+7
+8
+9
+number of leaf nodes
+0
+20
+40
+60
+MSE
+.0
+.01
+.025
+.05
+.1
+.25
+.5
+1.0
+(b) Exceptions
+Figure 3: Performance (MSE) on the arithmetic task
+for the Tree-LSTM with the DVIB (darker colours cor-
+respond to higher β). Exceptions have a contextual de-
+pendency and cannot be computed bottom up.
+20
+40
+epochs
+0
+2
+4
+6
+8
+10
+MSE
+0
+.01
+.025
+.05
+.1
+.25
+.5
+1
+(a) Compositional targets
+20
+40
+epochs
+0
+2
+4
+6
+8
+10
+MSE
+0
+.01
+.025
+.05
+.1
+.25
+.5
+1
+(b) Adapted targets
+Figure 4: Training dynamics for the Tree-LSTM with
+the DVIB: for all test examples we compute the MSE
+over the course of training on the validation set using
+(a) the compositional targets, or (b) the targets from the
+adapted dataset, of which a subset is not compositional.
+gence, models trained with a low β have learnt to
+capture the ambiguities, whereas models trained
+with a higher β will remain in a more locally com-
+positional state.3
+Bottlenecks restrict information passed through-
+out the tree. To process an arithmetic subexpres-
+sion, all that is needed is to pass on its outcome,
+not the subexpression itself – e.g. in Figure 2, one
+could simply store the value 6 instead of the subex-
+pression “2 - -4”. The former represents local
+processing and is more efficient (i.e. it requires
+storing less information), while the latter leads to
+information from the leaf nodes being passed to
+non-terminal nodes higher up the tree. Storing in-
+formation about the leaf nodes would be required
+to cope with the exceptions in the data. That the
+model could get close to accurate predictions for
+these exceptions in the first place suggests Tree-
+LSTMs can process inputs according to the hier-
+archical structure without being locally composi-
+tional. Increasing compression using bottlenecks
+enforces local processing.
+3Comparing ‘early’ and ‘late’ models may yield similar
+results as comparing base and bottleneck models. Yet, without
+labels of which examples are compositional, it is hard to know
+when the model transitions from the early to the late stage.
+
+PP, DVIB
+PP, dropout
+PP, dim.
+TT, DVIB
+TT, dropout
+TT, dim.
+bottleneck
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+relative position
+(a) All BCM variants
+0
+10000
+20000
+position in ranking
+0
+200
+400
+600
+800
+1000
+1200
+#samples
+regular
+exceptions
+(b) Ranking for β = 0.25
+Figure 5: Rankings of arithmetic examples. (a) shows
+the relative position of regular examples and exceptions
+in the rankings of all setups, where 0 corresponds to the
+start of the ranking and 1 to the end. (b) illustrates the
+result of BCM-TT with the DVIB, β = 0.25.
+4.3
+The Bottleneck Compositionality Metric
+The bottleneck Tree-LSTM harms the exceptions
+disproportionately in terms of task performance,
+and through BCM we can exploit the difference be-
+tween the base and bottleneck model to distinguish
+compositional from non-compositional examples.
+As laid out in §3, we use the TT or PP method
+to compare pairs of Tree-LSTMs: the base Tree-
+LSTM with β = 0, no dropout and a hidden dimen-
+sion of 150 (base model) is paired up with Tree-
+LSTMs with the same architecture, but a different
+β, a different dropout probability p or a different
+hidden dimensionality d (bottleneck model). All
+Tree-LSTMs have a classification module that con-
+sists of two linear layers, where the first layer maps
+the Tree-LSTM’s hidden representation to a vector
+of 100 units, and the second layer emits the pre-
+dicted value. The 100-dimensional vector is used
+to apply the BCM:
+• In BCM-PP the vector feeds into the CCA
+computation, that compares the hidden repre-
+sentation of the base model and the bottleneck
+model using their Cosine distance. We rank
+examples according to that distance, and use
+all CCA directions.
+• In BCM-TT, the vector feeds into the TRE
+loss component. We train the base model,
+freeze that network, and add the MSE of
+the hidden representations of the bottleneck
+model and the base model to the loss. After
+training, the remaining MSE is used to rank
+examples.
+Both setups have the same output, namely a com-
+positionality ranking of examples in a dataset. A
+successful ranking would put the exceptions last.
+0
+2.5e-05
+.000625
+.00125
+.0025
+.00625
+.0125
+.025
+0.3
+0.4
+0.5
+performance
+(a) DVIB
+0
+.1
+.25
+.5
+.65
+.75
+.85
+.9
+dropout
+sentiment-only baseline
+(b) Dropout
+150
+125
+100
+75
+50
+25
+10
+5
+size
+BCM-PP
+BCM-TT
+(c) Hidden dim.
+Figure 6: The accuracy (solid) and macro-averaged F1-
+scores (dashed) for the SST test set, for base models,
+bottleneck models and a sentiment-only baseline.
+Figure 5a illustrates the relative position of regular
+examples and exceptions for all bottlenecks and
+BCM variants, for β = 0.25, p = 0.5 and d = 25.
+The change in MSE observed in §4.2 is reflected
+in the quality of the ranking, but the success does
+not depend on the specific selection of β, p or d, as
+long as they are large (β, p) or small enough (d).
+Figure 5b illustrates one of the rankings.
+Summarising, we illustrated that recursive models
+can employ strategies that do not locally compose
+the meaning of arithmetic subexpressions but carry
+tokens’ identities throughout the tree. We can make
+a model more locally compositional using bottle-
+necks and can use a model’s hidden states to infer
+which examples required non-local processing af-
+terwards, acting as our compositionality metric.
+5
+Sentiment analysis
+We apply the metric to the task of sentiment anal-
+ysis, for which Moilanen and Pulman (2007, p. 1)
+suggest the following notion of compositionality:
+“For if the meaning of a sentence is a
+function of the meanings of its parts then
+the global polarity of a sentence is a func-
+tion of the polarities of its parts.”
+Sentiment is quasi-compositional: even though the
+sentiment of an expression is often a straightfor-
+ward function of the sentiment of its parts, there are
+exceptions – e.g. consider cases of sarcasm, such
+as “I love it when people yell at me first thing in
+the morning” (Barnes et al., 2019), which makes it
+a suitable testing bed.
+5.1
+Data and model training
+We use the SST-5 subtask from the Stanford Sen-
+timent Treebank (SST) (Socher et al., 2013), that
+contains sentences from movie reviews collected
+
+++
+++
+++
+++
++
++
+positive
+-
+-
+-
++
++
++
+--
+--
+--
+--
+-
+-
+negative
+~
+-
+--
+++
++
+--
++
+-
+--
+-
+~
++
+-
++
+++
+~
++
+++
++
+~
+~
+~
+~
+-
+neutral
+~
+~
+~
+in-between
+continuity
+++
+~
+-
+symmetric
+symmetric
+symmetric
+symmetric
+symmetric
+symmetric
+symmetric
+symmetric
+-
+--
+--
+Figure 7: Illustration of the predictions of a sentiment-
+only baseline model. We indicate the predicted senti-
+ment given two inputs. The labels range from very neg-
+ative (‘- -’) to neutral (‘∼’) to very positive (‘++’).
+by Pang and Lee (2005). The SST-5 subtask re-
+quires classifying sentences into one of five classes
+ranging from very negative to very positive. The
+standard train, development and test subsets have
+8544, 1101 and 2210 examples, respectively. The
+sentences were originally parsed with the Stanford
+Parser (Klein and Manning, 2003), and the dataset
+includes sentiment labels for all nodes of those
+parse trees. Typically, labels for all phrases are in-
+cluded in training, but the evaluation is conducted
+for the root nodes of the test set, only.
+Following Tai et al. (2015), we use GloVe word
+embeddings (Pennington et al., 2014), that we
+freeze across models.4 The Tree-LSTM has 150
+hidden units and is trained for 10 epochs with a
+learning rate of 2e−4 and the AdamW optimiser.
+During each training update, the loss is computed
+over all subexpressions of 4 trees. Training is re-
+peated with 10 random seeds.5 Figure 6 provides
+the performance on the test set for the base and bot-
+tleneck models, using the accuracy and the macro-
+averaged F1-score. Tai et al. (2015) obtained an
+accuracy of 0.497 using frozen embeddings.
+In sentiment analysis,
+as in our pseudo-
+arithmetic task, a successful model would often
+have to deviate from local processing. After all,
+the correct interpretation of a leaf node is often
+unknown without access to the context – e.g. in
+the case of ambiguous words like “sick” which is
+likely to refer to being ill, but could also mean
+“awesome”. Being successful at the task thus re-
+quires a recursive model to keep track of informa-
+4The notion of local compositionality, relied on in this
+work, assumes that tokens are not disambiguated, which is
+why we refrain from using state-of-the-art contextualised rep-
+resentations. The focus of this work is on developing a compo-
+sitionality metric rather than on improving sentiment analysis.
+5Appendix C further elaborates on the experimental setup.
+2.5e-05
+.000625
+.00125
+.0025
+.00625
+.0125
+.025
+0.55
+0.60
+0.65
+pearson's r
+(a) DVIB
+.1
+.25
+.5
+.65
+.75
+.85
+.9
+dropout
+(b) Dropout
+125
+100
+75
+50
+25
+10
+5
+hidden dim.
+BCM-PP
+BCM-TT
+(c) Hidden dim.
+2.5e-05
+.000625
+.00125
+.0025
+.00625
+.0125
+.025
+0.0
+0.2
+0.4
+0.6
+spearman's 
+(d) DVIB
+.1
+.25
+.5
+.65
+.75
+.85
+.9
+dropout
+(e) Dropout
+125
+100
+75
+50
+25
+10
+5
+hidden dim.
+25
+50
+75
+100
+(f) Hidden dim.
+Figure 8: Pearson’s r for the predictions of sentiment-
+only baselines and bottleneck models (a-c) and Spear-
+man’s ρ for the SST validation set compositionality
+ranking of the baselines and bottleneck models (d-f),
+when varying the number of CCA dimensions used.
+tion about (leaf) nodes while recursively processing
+the input, and more so for non-compositional ex-
+amples than for compositional examples. As with
+the arithmetic task, local processing – enforced in
+the bottleneck models – should disproportionately
+hinder processing of non-compositional examples.
+5.2
+A sentiment-only baseline
+To assert that the bottlenecks make the models
+more compositional, we create a sentiment-only
+baseline that is given as input not words but their
+sentiment, and has a hidden dimensionality d = 25.
+Non-compositional patterns that arise from the
+composition of certain words rather than the senti-
+ment of those words (e.g. “drop dead gorgeous”)
+could hardly be captured by that model. As such,
+the model exemplifies how sentiment can be com-
+puted more compositionally. Figure 7 illustrates the
+default sentiment this model predicts for various
+input combinations. Its predictions can be charac-
+terised by i) predicting positive for positive inputs,
+ii) predicting negative for negative inputs, iii) pre-
+dicting neutral if one input is neutral, iv) predict-
+ing a class in between the input classes or as v)
+predicting the same class as its inputs (continuity).
+The performance of the model is included in Fig-
+ure 6, and Figure 8 (a-c) indicates the Pearson’s
+r between the sentiment predictions of bottleneck
+models and baseline models. Generally, a higher
+β or dropout probability, or a lower hidden dimen-
+
+0.40
+0.45
+0.50
+0.55
+0.60
+0.65
+in_between
+continuity (neutral)
+neutral
+positive
+amplification
+continuity (non-neutral)
+neutral<->polarised
+switch
+attenuation
+negative
+mixed
+idioms
+difficult-vocab
+world-knowledge
+amplified
+comparative
+strong
+desirable-element
+no-sentiment
+negated
+sarcasm/irony
+morphology
+Figure 9: Categories of SST examples and their average position on the compositionality ranking visualised for the
+BCM-PP with the hidden dimensionality bottleneck and d = 25. Categories in black are assigned by us; categories
+in gray are from Barnes et al. (2019). The categories are further explained in the main text and Appendix B.1.
+Jittering was applied to better visualise overlapping categories.
+sionality, leads to predictions that are more similar
+to this sentiment-only model, unless the amount
+of regularisation is too extreme, debilitating the
+model (e.g. for dropout with probability 0.9).
+5.3
+The Bottleneck Compositionality Metric
+Now we use BCM to obtain a ranking over the SST
+dataset. We separate the dataset into four folds,
+train on those folds for the base and bottleneck
+model, and compute the cosine distances for the
+hidden representations of examples in the test sets
+(using BCM-PP or BCM-TT). We merge the co-
+sine distances for the different folds, averaged over
+models trained with 10 random seeds, and order ex-
+amples based on the resulting distance. We select
+the values for β, dropout and the hidden dimension-
+ality, as well as the number of CCA directions to
+use, based on rankings computed over the SST val-
+idation data. Figure 8 (d-f) illustrates how the rank-
+ings of bottleneck models correlate with rankings
+constructed using the sentiment-only baseline. We
+select 25 directions, β = 0.0025, dropout p = 0.65
+and a hidden dimensionality of 25 to compute the
+full ranking. Contrary to the arithmetic task, BCM-
+TT underperforms compared to BCM-PP.
+Different from the arithmetic task, it is unclear
+which examples should be at the start or end of
+the ranking. Therefore, we examine the relative
+position of categories of examples in Figure 9 for
+the BCM-PP with the hidden dimensionality bot-
+tleneck, and in Appendix B.3 for the remaining
+rankings. The categories include the previously in-
+troduced ones, augmented with the following four:
+• amplification: the root is even more posi-
+tive/negative than its top two children;
+• attenuation: the root is less positive/negative
+than its top two children;
+• switch: the children are positive/negative, but
+the root node flips that sentiment;
+• neutral↔polarised: the inputs are sentiment-
+laden, but the root is neutral, or vice versa.
+We also include characterisations from Barnes et al.
+(2019), who label examples from the SST test set,
+for which state-of-the-art sentiment models can-
+not seem to predict the correct label, including, for
+example, cases where a sentence contains mixed
+sentiment, or sentences with idioms, irony or sar-
+casm. Appendix B.1 elaborates on the meaning of
+the categories. Figure 9 illustrates the relative posi-
+tions of our sentiment characterisations and those
+of Barnes et al. on that ranking. Patterns associ-
+ated with more compositional sentiment process-
+ing, such as ‘positive’, ‘negative’ and ‘in between’
+lead to hidden representations that are more simi-
+lar between the base model and bottleneck models
+than the dataset average (the mid point, 0.5). Atyp-
+ical patterns like ‘switch’ and ‘neutral↔polarised’,
+on the other hand, along with the characterisations
+by Barnes et al. lead to less similar hidden repre-
+sentations. Appendix B.3 presents the same results
+for all six rankings considered, along with example
+sentences from across the ranking, to illustrate the
+types of sentences encountered among the most
+compositional and the least compositional ones.
+5.4
+Example use cases
+Compositionality rankings can be used in multiple
+manners, of which we illustrate two below.
+When data is scarce: use compositional exam-
+ples
+Assuming that most of natural language is
+compositional, one would expect that when limit-
+ing the training data, selecting compositional exam-
+ples yields the best test performance. To investigate
+this, we train models on various training dataset
+sizes, and evaluate with the regular test set. The
+training data is taken from the start of the ranking
+for the ‘compositional’ setup, and from the end
+of the ranking for the ‘non-compositional’ setup
+
+0.01
+0.05
+0.1
+0.2
+0.3
+0.4
+0.5
+training ratio
+0.2
+0.3
+0.4
+0.5
+0.6
+performance
+(a) LSTM
+0.01
+0.05
+0.1
+0.2
+0.3
+0.4
+0.5
+training ratio
+0.2
+0.3
+0.4
+0.5
+0.6
+performance
+comp.
+non-comp.
+(b) Roberta
+Figure 10: Change in SST test set accuracy (solid) and
+macro-averaged F1-score (dashed) as the training set
+size increases, for LSTM and Roberta models. The ex-
+amples are from the most (in blue) or the least composi-
+tional (in green) portion of the ranking from the BCM-
+PP metric with the hidden dimensionality bottleneck.
+Model
+Comp.
+Non-comp.
+Random
+Acc.
+F1
+Acc.
+F1
+Acc.
+F1
+Roberta
+.546
+.535
+.516
+.487
+.565
+.549
+LSTM
+.505
+.485
+.394
+.310
+.478
+.447
+Table 1: Performance on the new SST compositionality
+splits, generated using the ranking from the BCM-PP
+metric with the hidden dimensionality bottleneck.
+(excluding test data), while ensuring equal distri-
+butions over input lengths and output classes. We
+train a two-layer bidirectional LSTM with 300 hid-
+den units, and Roberta-base (Liu et al., 2019), us-
+ing batch size 4. The models are trained for 10 and
+5 epochs, respectively, with learning rates 2e − 4
+and 5e − 6. Because the ranking is computed over
+full sentences, and not subexpressions, we train the
+models on the sentiment labels for the root nodes.
+Figure 10 presents the results, consolidating that
+when data is scarce, using compositional examples
+is beneficial.
+Non-compositional examples are challenging
+For the same models, Table 1 indicates how perfor-
+mance changes if we redistribute train and test data
+such that the test set contains the most composi-
+tional examples, or the least compositional ones
+(keeping length and class distributions similar).
+The non-compositional test setup is more challeng-
+ing, with an 11 percentage point drop in accuracy
+for the LSTM, and a 3 point decrease for Roberta.
+In conclusion, applying the BCM to the sentiment
+data has confirmed findings previously observed
+for the arithmetic toy task. While it is harder to
+understand whether the method actually filters out
+non-compositional examples, both comparisons to
+a sentiment-only baseline, and the average position
+of cases for which the composition of sentiment is
+known to be challenging (e.g. for ‘mixed’ senti-
+ment, ‘comparative’ sentiment or ‘sarcasm’), sug-
+gest that compression acts as a compositionality
+metric. We also illustrated two ways in which the
+resulting ranking can be used.
+6
+Conclusion
+This work presents the Bottleneck Compositional-
+ity Metric, a TRE-based metric (Andreas, 2018)
+that is task-independent and can be applied to in-
+puts of varying lengths: we pair up Tree-LSTMs
+where one of them has more compressed repre-
+sentations due to a bottleneck (the DVIB, hidden
+dimensionality bottleneck or dropout bottleneck),
+and use the distance between their hidden represen-
+tations as a per-datum metric. The method was ap-
+plied to rank examples in datasets from most com-
+positional to least compositional, which is of inter-
+est due to the growing relevance of compositional
+generalisation research in NLP, which assumes the
+compositionality of natural language, and encour-
+ages models to compose meanings of expressions
+rather than to memorise phrases as chunks. We
+provided a proof-of-concept using an arithmetic
+task but also applied the metric to the much more
+noisy domain of sentiment analysis.
+The different bottlenecks lead to qualitatively
+similar results. This suggests that, while DVIB
+might be better motivated (it directly optimises an
+estimate of the Shannon information passed across
+the network), its alternatives may be preferable in
+practice due to their simplicity.
+Because natural language itself is not fully com-
+positional, graded metrics like the ones we pre-
+sented can support future research, such as i) learn-
+ing from data according to a compositionality-
+based curriculum to improve task performance, ii)
+filtering datasets to improve compositional gener-
+alisation, or iii) developing more and less compo-
+sitional models depending on the desiderata for a
+task – e.g. to perform well on sentences with id-
+ioms, one may desire a more non-compositional
+model. In addition, the formulation of the metric
+was general enough to be expanded upon in the
+future: one could pair up other models, such as an
+LSTM and a Tree-LSTM, or a Transformer and its
+recursive variant, as long as one keeps in mind that
+the compositional reconstruction itself should not
+be too powerful. After all, even Tree-LSTMs could
+capture the exceptions in the arithmetic dataset de-
+spite their hierarchical inductive bias.
+
+Limitations
+We identify three types of limitations for the work
+presented:
+• A conceptual limitation is that we work
+from a very strict definition of composition-
+ality (local compositionality), which essen-
+tially equates language with arithmetic. While
+overly restrictive, current datasets testing com-
+positional generalisation follow this notion.
+The framework might be extensible to more
+relaxed notions by allowing for token disam-
+biguation by using contextualised token em-
+beddings and only enforcing a bottleneck on
+the amount of further contextual integration
+within the model added on top of the token
+embeddings.
+• In terms of methodological limitations, the
+use of only Tree-LSTMs – although well-
+motivated from the perspective of composi-
+tional processing – is a major limitation. Tree-
+LSTMs are most suited for sentence classifica-
+tion tasks, limiting the approach’s applicabil-
+ity to sequence-to-sequence tasks. Nonethe-
+less, the bottlenecks can be integrated in other
+types of architectures that process inputs in
+a hierarchical manner, such as sequence-to-
+sequence models inducing latent source and
+target trees (Kim, 2021) to yield an alternative
+implementation of the BCM.
+Our work also assumes that an input’s tree
+structure is known, which might not always
+be the case. Therefore, the compositionality
+ranking obtained using BCM always depends
+on the trees used: what is non-compositional
+given one (potentially inadequate) structure
+might be more compositional given another
+(improved) structure.
+• Lastly, the evaluation of our approach is lim-
+ited in the natural domain through the absence
+of gold labels of the compositionality of ex-
+amples in the sentiment analysis task, but for
+other tasks that could have been considered,
+the same limitation would have applied.
+Acknowledgements
+We thank Chris Lucas for his contributions to this
+project when it was still in an early stage, Kenny
+Smith for his comments on the first draft of this
+paper, and Matthias Lindemann for excellent sug-
+gestions for the camera-ready version. VD is sup-
+ported by the UKRI Centre for Doctoral Training
+in Natural Language Processing, funded by the
+UKRI (grant EP/S022481/1) and the University
+of Edinburgh, School of Informatics and School
+of Philosophy, Psychology & Language Sciences.
+IT acknowledges the support of the European Re-
+search Council (ERC StG BroadSem 678254) and
+the Dutch National Science Foundation (NWO Vidi
+639.022.518).
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+
+A
+Bottlenecks for arithmetic
+A.1
+Performance for other bottlenecks
+Figures 11 and 12 display the MSE for models with
+the hidden dimensionality bottleneck and dropout
+bottleneck.
+5
+6
+7
+8
+9
+number of leaf nodes
+0
+1
+2
+3
+MSE
+dim
+5
+10
+25
+50
+75
+100
+125
+150
+(a) Regular examples
+5
+6
+7
+8
+9
+number of leaf nodes
+0
+20
+40
+60
+MSE
+dim
+5
+10
+25
+50
+75
+100
+125
+150
+(b) Exceptions
+Figure 11: Performance on the arithmetic task for the
+Tree-LSTM with varying hidden dimensionalities.
+5
+6
+7
+8
+9
+number of leaf nodes
+0
+2
+4
+6
+8
+10
+MSE
+dropout
+.0
+.1
+.25
+.5
+.65
+.75
+.85
+.9
+(a) Regular examples
+5
+6
+7
+8
+9
+number of leaf nodes
+0
+20
+40
+60
+MSE
+dropout
+.0
+.1
+.25
+.5
+.65
+.75
+.85
+.9
+(b) Exceptions
+Figure 12: Performance on the arithmetic task for the
+Tree-LSTM with varying dropout probabilities.
+A.2
+Bottleneck training dynamics
+The exceptions in the arithmetic task had both a
+compositional and non-compositional interpreta-
+tion, essentially giving us two sets of targets for
+measuring the models’ performance during train-
+ing using the validation data. Figures 4 (in the
+main paper), 13, and 14 illustrate how early on
+during training, the MSE is lowest for the composi-
+tional targets for all hyperparameters used for the
+bottlenecks: the models overgeneralise the regu-
+lar interpretation of “0”, applying it to all inputs.
+Later on, the models that have a high β, a high
+dropout probability or a small dimension stay in
+that ‘compositional state’, whereas the remaining
+models learn to capture the ambiguity.
+B
+Sentiment analysis
+B.1
+Categories of Barnes et al. (2019)
+Barnes et al. collect examples from sentiment anal-
+ysis datasets that three state-of-the-art classifiers
+struggle with, and annotate them for the presence
+of 18 (para-)linguistic phenomena. We include the
+20
+40
+epochs
+0
+2
+4
+6
+8
+10
+MSE
+dim
+150
+125
+100
+75
+50
+25
+10
+5
+(a) Compositional targets
+20
+40
+epochs
+0
+2
+4
+6
+8
+10
+MSE
+dim
+150
+125
+100
+75
+50
+25
+10
+5
+(b) Non-comp. targets
+Figure 13: Training dynamics for the hidden dimen-
+sionality bottleneck.
+20
+40
+epochs
+0
+2
+4
+6
+8
+10
+MSE
+dropout
+0
+.1
+.25
+.5
+.65
+.75
+.85
+.9
+(a) Compositional targets
+20
+40
+epochs
+0
+2
+4
+6
+8
+10
+MSE
+dropout
+0
+.1
+.25
+.5
+.65
+.75
+.85
+.9
+(b) Non-comp. targets
+Figure 14: Training dynamics for the dropout bottle-
+neck.
+following categories from the SST test set in the
+visualisation of the SST ranking:
+1. Negated:
+phrases or sentences that are
+negated, where Barnes et al. identify a pattern
+of irrelevant negation throwing models off.
+2. Amplified: cases where neutral modifiers act
+as strong contextual valence shifters.
+3. Strong: very positive or very negative cases.
+4. Desirable element: sentiment dominated by
+one ‘desirable element’, such as “pool”.
+5. Comparative: cases that express sentiment by
+comparison (e.g. using “better than”).
+6. Idioms: cases containing idioms, for which
+the mapping from the words to the sentiment
+is often not straightforward.
+7. Mixed: mixed positive & negative sentiment.
+8. Difficult-vocab: e.g. “engrossing and psycho-
+logically resonant suspenser”.
+9. World-knowledge: for instance comparisons
+between entities, where the entities imply a
+certain sentiment.
+10. Sarcasm/irony: sarcasm is often present in
+negative examples, where the speaker is in-
+tending the opposite of what is said.
+11. No-sentiment: neutral labelled examples.
+12. Morphology: examples with morphological
+features that affect sentiment very positively
+or very negatively.
+B.2
+Alternative metrics
+Tree impurity score (TIS)
+Bhathena et al.
+
+PP
+dim.
+PP
+dropout
+PP
+DVIB
+TT
+dim.
+TT
+dropout
+TT
+DVIB
+`topographic'
+TIS
+memorisation
+0.12
+0.15
+0.09
+0.26
+0.33
+0.31
+-0.049
+-0.021
+-0.014
+-0.079
+-0.099
+-0.097
+0.41
+0.37
+0.33
+0.39
+0.45
+0.46
+0.00
+0.25
+0.50
+0.75
+spearman's 
+Figure 15: Illustration of how BCM rankings over the
+SST data correlate with baseline metrics that ought
+to be correlated: topographic similarity, tree impurity
+score and memorisation as per Spearman’s ρ.
+(2020) propose basic compositionality metrics that
+rely on the difference between the label of the root
+node, and labels of subexpressions. We report their
+tree impurity score (TIS): a simple metric that mea-
+sures the absolute difference between the label of
+the root node and the average of all labels in a tree.
+‘Topographic’ similarity
+A compositionality
+metric from language emergence literature is to-
+pographic similarity (Brighton and Kirby, 2006),
+that given a set of objects, their meanings and
+the associated signals computes the correlation of
+the distances between corresponding pairs of sig-
+nals and meanings. For instance, Lazaridou et al.
+(2018) compare symbolic signals in a referential
+game using the Levenshtein edit distance, and com-
+pare vector meaning representations through cosine
+distance. Topographic similarity assumes that in
+a compositional language, similar signals should
+yield similar meaning representations. However,
+the use of edit distance to directly compare sen-
+tences does not readily transfer to sentiment analy-
+sis, where one word changed in the input space can
+yield a large change in the predicted sentiment.
+To approximate topographic similarity of mean-
+ings and signals, we instead manipulate the input
+in ways that should yield a similar prediction, and
+then rate examples based on the change in the pre-
+dictions observed. We replace nouns, verbs, adjec-
+tives or adverbs with a word that in SST has the
+same POS tag and sentiment label. If a change
+is observed, the example is more likely to be a
+non-compositional example.
+We apply this to the base models from the main
+paper, by randomly replacing one token that is a
+verb, noun, adjective or adverb with a different
+token of the same POS tag and sentiment label.
+We make 50 such modifications per sentence per
+model seed. We record the average change in the
+predicted sentiment, and use this as a metric akin
+to topographic similarity.
+PP
+dim.
+PP
+dropout
+PP
+DVIB
+TT
+dim.
+TT
+dropout
+PP
+dropout
+PP
+DVIB
+TT
+dim.
+TT
+dropout
+TT
+DVIB
+0.64
+0.67
+0.68
+0.31
+0.35
+0.32
+0.35
+0.41
+0.35
+0.65
+0.33
+0.41
+0.38
+0.67
+0.8
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+spearman's 
+Figure 16: Illustration of how the BCM rankings over
+the SST dataset correlate as per Spearman’s ρ.
+Memorisation score
+Zheng and Jiang (2022)
+empirically evaluate the long tail theory posed by
+Feldman and Zhang (2020) (who validate their hy-
+potheses using computer vision tasks), that states
+that for data distributions with a long tail, memo-
+risation of training examples is required for near-
+optimal performance on the test data. Zheng and
+Jiang put this theory to the test for sentiment classi-
+fication. The metric of Zheng and Jiang expresses
+how the likelihood of the target changes when an
+example is down-weighted during training. That
+change should be larger for memorised examples.
+Intuitively, non-compositionality and memorisa-
+tion are related: non-compositional patterns in data
+require memorising the atypical interpretation of
+words in specific contexts. The metric is expected
+to be different from our metric – ours is measured
+using test data, while memorisation occurs during
+training – but a positive correlation is expected be-
+tween the two, nonetheless. Zheng and Jiang use
+the binary SST subtask and report their scores on
+preprocessed versions of SST sentences. We report
+the correlation for the examples for which we could
+find matching surface forms only.
+Figure 15 indicates how the different BCM rank-
+ings correlate with these three alternative metrics.
+Surprisingly, TIS negatively correlates with our
+rankings, but only weakly. This may be due to
+the simplicity of that metric, that ignores the tree
+structure and simply averages all sentiment of all
+nodes. The devised ‘topographic’ similarity posi-
+tively correlates with our rankings, but only up to
+ρ = 0.33. The memorisation score has a moderate
+correlation with our rankings, of up to ρ = 0.46 for
+the BCM-TT. Figure 16 illustrates how the rank-
+ings correlate with each other. The figure suggests
+that TRE training yields rankings that are still quite
+different from the post-processing ones.
+
+B.3
+Additional results for SST rankings
+Here, we present further results for the BCM rankings over the SST datasets: firstly, Figure 17 provides
+the rankings for the six different BCM setups. They have commonalities, but also differences, e.g. in the
+BCM-TT setups the ‘neutral’ sentiment is closer to the end of the ranking.
+0.40
+0.45
+0.50
+0.55
+0.60
+0.65
+in_between
+continuity (neutral)
+neutral
+positive
+amplification
+continuity (non-neutral)
+neutral<->polarised
+switch
+attenuation
+negative
+mixed
+idioms
+difficult-vocab
+world-knowledge
+amplified
+comparative
+strong
+desirable-element
+no-sentiment
+negated
+sarcasm/irony
+morphology
+(a) BCM-PP, Hidden dimensionality bottleneck
+0.35
+0.40
+0.45
+0.50
+0.55
+0.60
+0.65
+in_between
+continuity (neutral)
+neutral
+positive
+amplification
+continuity (non-neutral)
+neutral<->polarised
+switch
+attenuation
+negative
+mixed
+idioms
+difficult-vocab
+world-knowledge
+amplified
+comparative
+strong
+desirable-element
+no-sentiment
+negated
+sarcasm/irony
+morphology
+(b) BCM-PP, DVIB bottleneck
+0.45
+0.50
+0.55
+0.60
+in_between
+continuity (neutral)
+neutral
+positive
+amplification
+continuity (non-neutral)
+neutral<->polarised
+switch
+attenuation
+negative
+mixed
+idioms
+difficult-vocab
+world-knowledge
+amplified
+comparative
+strong
+desirable-element
+no-sentiment
+negated
+sarcasm/irony
+morphology
+(c) BCM-PP, Dropout bottleneck
+0.35
+0.40
+0.45
+0.50
+0.55
+0.60
+in_between
+continuity (neutral)
+neutral
+positive
+amplification
+continuity (non-neutral)
+neutral<->polarised
+switch
+attenuation
+negative
+mixed
+idioms
+difficult-vocab
+world-knowledge
+amplified
+comparative
+strong
+desirable-element
+no-sentiment
+negated
+sarcasm/irony
+morphology
+(d) BCM-TT, Hidden dimensionality bottleneck
+0.35
+0.40
+0.45
+0.50
+0.55
+0.60
+in_between
+continuity (neutral)
+neutral
+positive
+amplification
+continuity (non-neutral)
+neutral<->polarised
+switch
+attenuation
+negative
+mixed
+idioms
+difficult-vocab
+world-knowledge
+amplified
+comparative
+strong
+desirable-element
+no-sentiment
+negated
+sarcasm/irony
+morphology
+(e) BCM-TT, DVIB bottleneck
+0.35
+0.40
+0.45
+0.50
+0.55
+0.60
+0.65
+in_between
+continuity (neutral)
+neutral
+positive
+amplification
+continuity (non-neutral)
+neutral<->polarised
+switch
+attenuation
+negative
+mixed
+idioms
+difficult-vocab
+world-knowledge
+amplified
+comparative
+strong
+desirable-element
+no-sentiment
+negated
+sarcasm/irony
+morphology
+(f) BCM-TT, Dropout bottleneck
+Figure 17: Example categories for SST test set examples, and their average position on the compositionality
+ranking for the hidden dimensionality bottleneck with d = 25, the dropout bottleneck with p = 0.65 or the DVIB
+with β = 0.0025. The categories in black are assigned by us, and the categories in gray are from Barnes et al.
+(2019). Jittering was applied to better visualise overlapping categories.
+
+Secondly, Table 2 provides example sentences from different parts of the rankings, randomly sampled. 0
+indicates the most compositional examples, and 1 the least compositional ones.
+Relative position
+Target
+Sentence
+- BCM-PP, Hidden dim. bottleneck
+0.07
+3
+tsai convincingly paints a specifically urban sense of disassociation here .
+0.17
+0
+the film desperately sinks further and further into comedy futility .
+0.21
+3
+a cop story that understands the medium amazingly well .
+0.40
+3
+one scarcely needs the subtitles to enjoy this colorful action farce .
+0.42
+1
+the entire movie is about a boring , sad man being boring and sad .
+0.53
+2
+not everyone will play the dark , challenging tune taught by the piano teacher .
+0.66
+3
+this is the stuff that disney movies are made of .
+0.75
+3
+daughter from danang sticks with its subjects a little longer and tells a deeper story
+0.86
+2
+not kids , who do n’t need the lesson in repugnance .
+0.98
+0
+lacks heart , depth and , most of all , purpose .
+- BCM-PP, Dropout bottleneck
+0.09
+4
+it ’s a cool event for the whole family .
+0.14
+0
+done in mostly by a weak script that ca n’t support the epic treatment .
+0.20
+3
+some movies are like a tasty hors-d’oeuvre ; this one is a feast .
+0.37
+4
+my oh my , is this an invigorating , electric movie .
+0.41
+3
+its director ’s most substantial feature for some time .
+0.51
+4
+the modern master of the chase sequence returns with a chase to end all chases
+0.62
+3
+like its bizarre heroine , it irrigates our souls .
+0.78
+1
+sadly , ‘ garth ’ has n’t progressed as nicely as ‘ wayne . ’
+0.86
+2
+you ’re too conscious of the effort it takes to be this spontaneous .
+0.95
+0
+a film of empty , fetishistic violence in which murder is casual and fun .
+- BCM-PP, DVIB
+0.07
+4
+highly recommended viewing for its courage , ideas , technical proficiency and great acting .
+0.18
+3
+it manages to squeeze by on angelina jolie ’s surprising flair for self-deprecating comedy .
+0.25
+1
+goes on and on to the point of nausea .
+0.38
+3
+the best part about “ gangs ” was daniel day-lewis .
+0.44
+2
+a dopey movie clothed in excess layers of hipness .
+0.51
+2
+a perplexing example of promise unfulfilled , despite many charming moments .
+0.64
+0
+the entire film is one big excuse to play one lewd scene after another .
+0.79
+1
+the problematic characters and overly convenient plot twists foul up shum ’s good intentions .
+0.82
+3
+the jabs it employs are short , carefully placed and dead-center .
+0.92
+2
+then nadia ’s birthday might not have been such a bad day after all .
+- BCM-TT, Hidden dim. bottleneck
+0.00
+3
+the story is smart and entirely charming in intent and execution .
+0.02
+2
+for single digits kidlets stuart little 2 is still a no brainer .
+0.17
+1
+outer-space buffs might love this film , but others will find its pleasures intermittent .
+0.29
+3
+although shot with little style , skins is heartfelt and achingly real .
+0.37
+3
+a knowing look at female friendship , spiked with raw urban humor .
+0.50
+3
+there ’s an energy to y tu mamá también .
+0.57
+2
+a piquant meditation on the things that prevent people from reaching happiness .
+0.63
+3
+too daft by half ... but supremely good natured .
+0.71
+1
+the feature-length stretch ... strains the show ’s concept .
+0.88
+1
+a thriller without thrills and a mystery devoid of urgent questions .
+0.97
+1
+the movie does n’t generate a lot of energy .
+- BCM-TT, Dropout bottleneck
+0.03
+3
+an extremely funny , ultimately heartbreaking look at life in contemporary china .
+0.14
+1
+the pretensions – and disposable story – sink the movie .
+0.22
+4
+that rara avis : the intelligent romantic comedy with actual ideas on its mind .
+0.39
+2
+admirable , certainly , but not much fun to watch .
+0.50
+1
+the end result is a film that ’s neither .
+0.59
+2
+throwing it all away for the fleeting joys of love ’s brief moment .
+0.63
+4
+kids should have a stirring time at this beautifully drawn movie .
+0.77
+3
+the obnoxious title character provides the drama that gives added clout to this doc .
+0.87
+0
+stitch is a bad mannered , ugly and destructive little **** .
+0.98
+0
+there is no pleasure in watching a child suffer .
+- BCM-TT, DVIB
+0.03
+4
+quite simply , a joy to watch and – especially – to listen to .
+0.14
+1
+a fake street drama that keeps telling you things instead of showing them .
+0.22
+1
+it tries too hard , and overreaches the logic of its own world .
+0.34
+1
+plays less like a coming-of-age romance than an infomercial .
+0.48
+4
+it ’s one of the most honest films ever made about hollywood .
+0.52
+3
+a low-key labor of love that strikes a very resonant chord .
+0.69
+3
+it ’s never laugh-out-loud funny , but it is frequently amusing .
+0.71
+3
+a meditation on faith and madness , frailty is blood-curdling stuff .
+0.81
+1
+a beautifully shot but dull and ankle-deep ‘ epic . ’
+0.91
+2
+it uses the pain and violence of war as background material for color .
+Table 2: Examples from across the six rankings, randomly sampled.
+
+Thirdly, Figure 18 illustrates how training on different subsets of the ranking leads to different
+performance on the test set. In general, it is better to train on the compositional examples.
+0.01
+0.05
+0.1
+0.2
+0.3
+0.4
+0.5
+training ratio
+0.2
+0.3
+0.4
+0.5
+0.6
+performance
+(a) LSTM, Hidden dim., BCM-PP
+0.01
+0.05
+0.1
+0.2
+0.3
+0.4
+0.5
+training ratio
+0.2
+0.3
+0.4
+0.5
+0.6
+performance
+(b) LSTM, dropout, BCM-PP
+0.01
+0.05
+0.1
+0.2
+0.3
+0.4
+0.5
+training ratio
+0.2
+0.3
+0.4
+0.5
+0.6
+performance
+(c) LSTM, DVIB, BCM-PP
+0.01
+0.05
+0.1
+0.2
+0.3
+0.4
+0.5
+training ratio
+0.2
+0.3
+0.4
+0.5
+0.6
+performance
+(d) LSTM, Hidden dim., BCM-TT
+0.01
+0.05
+0.1
+0.2
+0.3
+0.4
+0.5
+training ratio
+0.2
+0.3
+0.4
+0.5
+0.6
+performance
+(e) LSTM, dropout, BCM-TT
+0.01
+0.05
+0.1
+0.2
+0.3
+0.4
+0.5
+training ratio
+0.2
+0.3
+0.4
+0.5
+0.6
+performance
+(f) LSTM, DVIB, BCM-TT
+0.01
+0.05
+0.1
+0.2
+0.3
+0.4
+0.5
+training ratio
+0.2
+0.3
+0.4
+0.5
+0.6
+performance
+comp.
+non-comp.
+(g) Roberta, Hidden dim., BCM-PP
+0.01
+0.05
+0.1
+0.2
+0.3
+0.4
+0.5
+training ratio
+0.2
+0.3
+0.4
+0.5
+0.6
+performance
+comp.
+non-comp.
+(h) Roberta, dropout, BCM-PP
+0.01
+0.05
+0.1
+0.2
+0.3
+0.4
+0.5
+training ratio
+0.2
+0.3
+0.4
+0.5
+0.6
+performance
+comp.
+non-comp.
+(i) Roberta, DVIB, BCM-PP
+0.01
+0.05
+0.1
+0.2
+0.3
+0.4
+0.5
+training ratio
+0.2
+0.3
+0.4
+0.5
+0.6
+performance
+comp.
+non-comp.
+(j) Roberta, hidden dim., BCM-TT
+0.01
+0.05
+0.1
+0.2
+0.3
+0.4
+0.5
+training ratio
+0.2
+0.3
+0.4
+0.5
+0.6
+performance
+comp.
+non-comp.
+(k) Roberta, dropout, BCM-TT
+0.01
+0.05
+0.1
+0.2
+0.3
+0.4
+0.5
+training ratio
+0.2
+0.3
+0.4
+0.5
+0.6
+performance
+comp.
+non-comp.
+(l) Roberta, DVIB, BCM-TT
+Figure 18: Change in SST test set accuracy (solid) and macro-averaged F1-score (dashed) as the training set size
+increases, for LSTM and Roberta models.
+
+C
+Reproducibility details
+Visit https://github.com/vernadankers/bottleneck_
+compositionality_metric for the code and data. Be-
+low we collect an overview of the settings involved
+in the model training and model evaluation:
+• Model specifications: Following Tai et al.
+(2015) we use 150 hidden dimensions, and
+300-dimensional word embeddings for the
+sentiment analysis task. For consistency, we
+adopt the 150 dimensions in the arithmetic
+task, as well, but reduce the size of the word
+embeddings to 150. The arithmetic base mod-
+els have 426k trainable parameters, and the
+sentiment base models have 514k trainable
+parameters (excluding the frozen word em-
+beddings).
+• Training procedure:
+The sentiment and
+arithmetic models are trained for 10 and 50
+epochs, respectively, based on model conver-
+gence. These numbers are fixed across mod-
+els, to ensure that when multiple models are
+combined in the compositionality metric, they
+have been trained for the same amount of time.
+For sentiment, a batch size of 4 was used, and
+batch sizes {1, 4, 8} were experimented with.
+For arithmetic, batch sizes {16, 32, 64} were
+experimented with across 5 seeds, where 32
+was selected. Selection is based on validation
+performance across five seeds and training
+speed (e.g. while 1 slightly outperformed 4
+for sentiment analysis, we opted for 4 for com-
+putational efficiency). For both tasks, learning
+rates {0.001, 2e-4, 1e-4} were experimented
+with, and 2e-4 was selected based on model
+performance on the validation set across 5
+seeds. In these hyperparameter trials, the ac-
+curacy was used for sentiment analysis and
+the MSE was used for the arithmetic task.
+• Hidden dimensionality bottleneck:
+To
+increase compression, the dimensionality
+should clearly become smaller, so we man-
+ually select 8 values to run ranging from 150
+(the standard dimension) to 5.
+• Dropout bottleneck: For dropout, the closer
+to 1, the more compression there is. We manu-
+ally select 8 values to run from 0 (the standard
+amount) to 0.9.
+• DVIB: For DVIB, preliminary experiments
+indicated that β > 1 for arithmetic or β > 0.1
+for sentiment debilitates the model. We man-
+ually selected 8 values to run for β based
+on that information. We use the implemen-
+tation of Li and Eisner (2019) for the DVIB,
+available at: https://github.com/XiangLi1999/
+syntactic-VIB.
+• Evaluation metrics: The evaluation metric
+used for arithmetic is the MSE. The evaluation
+metrics for sentiment analysis are the accuracy
+of the predicted sentiment class, and the F1-
+score, macro-averaged.
+• Number of runs & run time: The results for
+both tasks are averaged over ten seeds. Train-
+ing one model with one seed on CPU lasts
+up to 40 minutes for the sentiment analysis
+models, and up to 30 minutes for the arith-
+metic task. The TRE-training setup typically
+takes twice as long. We utilise CPUs from the
+icelake partition of the CSD3 cluster.
+For the datasets used, the following are relevant
+details in terms of their size and preprocessing:
+• Arithmetic task: The arithmetic task was
+generated using the implementation of Hup-
+kes et al. (2018), available at https://github.
+com/dieuwkehupkes/processing_arithmetics.
+We augment the data with the ambiguous
+examples ourselves. The training data consist
+of 14903 expressions with 1 to 9 numbers.
+We test on expressions with lengths 5 to
+9, using 5000 examples per length.
+2100
+additional examples are used as validation
+data, to track the model’s behaviour during
+training.
+• Stanford Sentiment Treebank:
+We col-
+lect the SST data from the pytreebank
+package (https://github.com/JonathanRaiman/
+pytreebank). For the (Tree-)LSTM models, we
+further preprocess the inputs by lowercasing
+the sentences. We use SST-5, that classifies
+sentences using five classes ranging from very
+negative to very positive. The standard train,
+validation and test subsets have 8544, 1101
+and 2210 examples, respectively. Each node
+in the input trees has its own sentiment label.
+For the experiments contained in Section 5.4,
+we did not run an extensive hyperparameter search.
+Those results are averaged over five seeds, and the
+models were trained on one NVIDIA A100-SXM-
+80GB, where training one seed lasted up to 15
+minutes.
+
diff --git a/dNFST4oBgHgl3EQfEjgF/content/tmp_files/load_file.txt b/dNFST4oBgHgl3EQfEjgF/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,1218 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf,len=1217
+page_content='Recursive Neural Networks with Bottlenecks Diagnose  (Non-)Compositionality  Verna Dankers1 and Ivan Titov1,2  1ILCC, University of Edinburgh  2ILLC, University of Amsterdam  vernadankers@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='com, ititov@inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='uk  Abstract  A recent line of work in NLP focuses on  the (dis)ability of models to generalise com-  positionally for artificial languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' However,  when considering natural language tasks, the  data involved is not strictly, or locally, compo-  sitional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Quantifying the compositionality of  data is a challenging task, which has been in-  vestigated primarily for short utterances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We  use recursive neural models (Tree-LSTMs)  with bottlenecks that limit the transfer of infor-  mation between nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We illustrate that com-  paring data’s representations in models with  and without the bottleneck can be used to pro-  duce a compositionality metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The procedure  is applied to the evaluation of arithmetic ex-  pressions using synthetic data, and sentiment  classification using natural language data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We  demonstrate that compression through a bot-  tleneck impacts non-compositional examples  disproportionately and then use the bottleneck  compositionality metric (BCM) to distinguish  compositional from non-compositional sam-  ples, yielding a compositionality ranking over  a dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  1  Introduction  Compositional generalisation in contemporary  NLP research investigates models’ ability to com-  pose the meanings of expressions from their parts  and is often investigated with artificial languages  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Lake and Baroni, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Hupkes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=', 2020) or  highly-structured natural language data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Key-  sers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' For such tasks, the local compo-  sitionality definition of Szabó (2012, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 10) illus-  trates how meaning can be algebraically composed:  “The meaning of a complex expression  is determined by the meanings its con-  stituents have individually and the way  those constituents are combined.”  In natural language, there are fragments whose  meaning can be composed as with arithmetic (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  removing  the  ruler  from  the classroom  removing  the  pencil  the classroom  locally compositional processing  recursive processing  from  unambiguous example:  interpretation is the same  ambiguous example:  interpretation changes  Figure 1: When processing this phrase, “the ruler” is  interpreted differently when comparing recursive pro-  cessing with local processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We enforce local pro-  cessing by equipping models with bottlenecks, and our  bottleneck compositionality metric (BCM) then com-  pares inputs’ representations before and after compres-  sion through the bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  “the cat is in the house”), while others carry con-  textual dependencies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' “the kiwi grows on the  farm”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Can we characterise whether an input’s  meaning arises from strictly local compositions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Existing work in that direction mostly focuses  on providing a ‘compositionality rating’1 for figura-  tive utterances since figurative language is assumed  to be less compositional (Ramisch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Nan-  dakumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Reddy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Andreas  (2018) suggests a general-purpose formulation for  measuring the compositionality of examples using  their numerical representations, through the Tree  Reconstruction Error (TRE), expressing the dis-  tance between a model’s representation of an input  and a strictly compositional reconstruction of that  representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Determining how to compute that  reconstruction is far from trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Inspired by TRE, we use recursive neural net-  works, Tree-LSTMs (Tai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=', 2015), to process  inputs according to their syntactic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We  augment Tree-LSTMs with bottlenecks to compute  1We colloquially refer to the ‘compositionality ratings’ of  phrases, but a more appropriate way to express the same would  be to refer to ‘the extent to which the meaning of a phrase  arises from a compositional syntax and semantics’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' After all,  compositionality is a property of a language, not of a phrase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='13714v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='CL]  31 Jan 2023  the task-specific meaning of an input in a more lo-  cally compositional manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We use these models  to distinguish more compositional examples from  less compositional ones in a bottleneck compo-  sitionality metric (BCM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Figure 1 provides an  intuition for how a bottleneck can provide a met-  ric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' For fragments that violate the assumption that  meanings of subexpressions can be computed lo-  cally (on the left side), one could end up with dif-  ferent interpretations when comparing a contex-  tualised interpretation (in blue) with one locally  computed (in green): disambiguating “ruler” re-  quires postponed meaning computation, and thus  local processing is likely to lead to different results  from regular processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' For fragments that are  non-ambiguous (on the right side) the two types  of processing can yield the same interpretation be-  cause the interpretation of “pencil” is likely to be  the same, with or without the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The bottle-  neck hinders the model in postponing computations  and more strongly impacts non-compositional sam-  ples compared to compositional ones, thus acting  as a metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  In the remainder of the paper, we firstly discuss  the related work in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' §3 elaborates on the models  used that either apply a deep variational informa-  tion bottleneck (DVIB) (Alemi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=', 2017) or com-  press representations through increased dropout  or smaller hidden dimensionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' In §4, we pro-  vide a proof-of-concept in a controlled environment  where non-compositional examples are manually  introduced, after which §5 elaborates on the natural  language example of sentiment analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' For both  tasks, we (1) demonstrate that compression through  a bottleneck encourages local processing and (2)  show that the bottleneck can act as a metric dis-  tinguishing compositional from less compositional  examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  2  Related Work  Multi-word expressions  The majority of the re-  lated work in the past two decades has discussed  the compositionality of phrases in the context of  figurative language, such as phrasal verbs (“to eat  up”) (McCarthy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=', 2003), noun compounds  (“cloud nine” vs “swimming pool”) (Reddy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=',  2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Yazdani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Ramisch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Nandakumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=', 2019), verb-noun collocations  (“take place” vs “take a gift”) (Venkatapathy and  Joshi, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' McCarthy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=', 2007), and adjective-  noun pairs (“nice house”) (Guevara, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Compo-  sitionality judgements were obtained from humans,  who indicated to what extent the meaning of the  compound is that of the words when combined lit-  erally, and various computational methods were  applied to learn that mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Those methods were  initially thesaurus-based (McCarthy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=', 2003),  relied on word vectors from co-occurrence matri-  ces later on (Reddy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=', 2011), or employed deep  neural networks (Nandakumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Compositionality by reconstruction  TRE (An-  dreas, 2018) is a task-agnostic metric that evalu-  ates the compositionality of data representations:  TRE(x) = δ(f(x), ˆfη(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' It is the distance be-  tween the representation of x constructed by f  and the compositionally reconstructed variant ˆfη(d)  based on the derivation of x (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' When employ-  ing the metric, one should define an appropriate  distance function (δ) and define ˆfη parametrised  by η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Andreas illustrates the TRE’s versatility by  instantiating it for three scenarios: to investigate  whether image representations are similar to com-  posed image attributes, whether phrase embeddings  are similar to the vector addition of their compo-  nents, and whether generalisation accuracy in a  reference game positively correlates with TRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Bhathena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' (2020) present two methods  based on TRE to obtain compositionality ratings  for sentiment trees, referred to as tree impurity and  weighted node switching that express the differ-  ence between the sentiment label of the root and  the other nodes in the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Zheng and Jiang (2022)  ranked examples of sentiment analysis based on  the extent to which neural models should memorise  examples in order to capture their target correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  While different from TRE, memorisation could be  related to non-compositionality in the sense that  non-compositional examples require more memori-  sation, akin to formulaic language requiring mem-  orisation in humans (Wray and Perkins, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Other instantiations of the TRE are from litera-  ture on language emergence in signalling games,  where the degree of compositionality of that lan-  guage is measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Korbak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' (2020) contrast  TRE and six other compositionality metrics for  signalling games where the colour and shape of  an object are communicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Examples of such  metrics are topographic similarity, positional disen-  tanglement and context independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' These are  not directly related to our work, considering that  they aim to provide a metric for a language rather  than single utterances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='2 elaborates on  topographic similarity and the metrics of Bhathena  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' (2020) and Zheng and Jiang (2022), compar-  ing them to our metric for sentiment analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Compositional data splits  Recent work on com-  positional generalisation using artificial languages  or highly-structured natural language data focuses  on creating data splits that have systematic sepa-  ration of input combinations in train and test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  The aim is to create test sets that should not be hard  when computing meaning compositionally, but, in  practice, are very challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' An example compo-  sitionality metric for semantic parsing is maximum  compound divergence (Keysers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Shaw  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=', 2021), that minimises train-test differences  in word distributions while maximising the differ-  ences in compound usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' This only applies to a  data split as a whole, and – differently from the  work at hand – does not rate individual samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  More recently, Bogin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' (2022) discussed a  diagnostic metric for semantic parsing, that pre-  dicts model success on examples based on their  local structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Because models struggle with sys-  tematically assigning the same meaning to subex-  pressions when they re-appear in new syntactic  structures, such structural deviation diagnoses gen-  eralisation failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Notice that the aim of our work  is different, namely identifying examples that are  not compositional, rather than investigating gener-  alisation failure for compositional examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  3  Model  The model we employ is the Tree-LSTM (Tai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=',  2015), which is a generalisation of LSTMs to tree-  structured network topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The LSTM com-  putes symbols’ representations by incorporating  previous time steps, visiting symbols in linear or-  der.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' A sentence representation is simply the final  time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' A Tree-LSTM, instead, uses a tree’s root  node representation as the sentence representation,  and computes the representation of a non-terminal  node using the node’s children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Equations 1 and 2 illustrate the difference be-  tween the LSTM and an N-ary Tree-LSTM for  the input gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The LSTM computes the gate’s ac-  tivation for time step t using input vector xt and  previous hidden state ht−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The Tree-LSTM does  so for node j using the input vector xj and the  hidden states of up to N children of node j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  it = σ(W (i)xt + U (i)ht−1 + b(i))  (1)  ij = σ(W (i)xj +  N  �  ℓ=1  U (i)  ℓ hjℓ + b(i))  (2)  In addition to the input gate, the Tree-LSTM’s  specification for non-terminal j (with its kth child  indicated as hjk) involves an output gate oj (equa-  tion analogous to 2), a forget gate fjk (Equation 3),  cell input activation vector uj (equation analogous  to 2, with the σ function replaced by tanh), and  memory cell state cj (Equation 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Finally, cj feeds  into the computation of hidden state hj (Equa-  tion 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  fjk = σ(W (f)xj +  N  �  ℓ=1  U (f)  kℓ hjℓ + b(f))  (3)  cj = ij ⊙ uj +  N  �  ℓ=1  fjℓ ⊙ cjℓ  (4)  hj = oj ⊙ tanh(cj)  (5)  We apply a binary Tree-LSTM to compute hidden  state hj and memory cell state cj, that thus uses  separate parameters in the gates for the left and  right child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Tree-LSTMs process inputs according to their  syntactic structure, which has been associated with  more compositional processing (Socher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Tai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Yet, although the topology encour-  ages compositional processing, there is no mecha-  nism to explicitly regulate how much information  is passed from children to parent nodes – e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' given  enough capacity, the hidden representations could  store every input encountered and postpone pro-  cessing until the very end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We add such a mecha-  nism by introducing a bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Deep Variational Information Bottleneck  The information bottleneck of Alemi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' (2017)  assumes random variables X and Y for the input  and output, and emits a compressed representation  Z that preserves information about Y , by minimis-  ing the loss LIB in Equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' This loss is in-  tractable, which motivates the variational estimate  LV IB provided in Equation 7 (Alemi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=', 2017)  that we use to train the deep variational informa-  tion bottleneck (DVIB) version of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  LIB = βI(X, Z) − I(Z, Y )  (6)  LV IB = βE  x[KL[pθ(z|x), r(z)]]  �  ��  �  information loss  +  E  z∼pθ(z|x)[−logqφ(y|z)]  �  ��  �  task loss  (7)  In the information loss, r(z) and pθ(z|x) estimate  the prior and posterior probability over z, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' In the task loss, qφ(y|z) is a parametric  approximation of p(y|z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  In order to allow an  analytic computation of the KL-divergence, we  consider Gaussian distributions r(z) and pθ(z|x),  namely r(z) = N(z|µ0, Σ0) and pθ(z|x) =  N(z|µ(x), Σ(x)), where µ(x) and µ0 are mean  vectors, and Σ(x) and Σ0 are diagonal covariance  matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The reparameterisation trick is used to  estimate the gradients: z = µ(x)+Σ(x)⊙ϵ, where  ϵ ∼ N(0, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  We sample z once per non-terminal node, and  average the KL terms of all non-terminal nodes,  where x is the hidden state hj or the cell state cj  (that have separate bottlenecks), and µ(x) and Σ(x)  are computed by feeding x to two linear layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' β  regulates the impact of the DVIB, and is gradually  increased during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' During inference, we  use z = µ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Dropout bottleneck  Binary dropout (Srivas-  tava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=', 2014) is commonly applied when train-  ing neural models, to prevent overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' With  a probability p hidden units are set to zero, and  during the evaluation all units are kept, but the acti-  vations are scaled down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Dropout encourages dis-  tributing the most salient information over multiple  neurons, which comes at the cost of idiosyncratic  patterns that networks may memorise otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  We hypothesise that this hurts non-compositional  examples most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We apply dropout to the Tree-  LSTM’s hidden states (hj) and memory cell states  (cj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Hidden dimensionality bottleneck  Simi-  larly, decreasing the number of hidden units is  expected to act as a bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We decrease the  number of hidden units in the Tree-LSTM, keep-  ing the embedding and task classifier dimensions  stable, where possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  The different bottlenecks have different merits:  whereas the hidden dimensionality and dropout  bottlenecks shine through simplicity, they are rigid  in how they affect the model and apply in the same  way at every node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The DVIB allows for more  flexibility in how compression is achieved through  learnt Σ(x) and by requiring an overall reduction  in the information loss term, without enforcing the  same bottleneck at every node in the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  From bottleneck to compositionality metric  BCM compares Tree-LSTMs with and without a  bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We experiment with two methods, in-  spired by TRE (Andreas, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' TRE aims to find  η such that δ(f(x), ˆfη(d)) is minimised, for inputs  x, their derivations d, distance function δ, a model  f and its compositional approximation ˆfη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  In the TRE training (BCM-TT) setup, we  include the distance (δ) between the hidden  representations of f and ˆfη in the loss when  training ˆfη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' When training ˆfη with TRE train-  ing, f is frozen, and f and ˆfη share the final  linear layer of the classification module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' In the  arithmetic task, δ is the mean-squared error  (MSE) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' the squared Euclidean distance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  In sentiment analysis, δ is the Cosine distance  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  In the post-processing (BCM-PP) setup, we  train the two models separately, extract hid-  den representations and apply canonical cor-  relation analysis (CCA) (Hotelling, 1936)  to minimise the distance between the sets  of hidden representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Assume matrices  A ∈ RdA×N and B ∈ RdB×N represent-  ing N inputs with dimensionalities dA and  dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' CCA linearly transforms these subspaces  A′ = WA, B′ = V B to maximise the cor-  relations {ρ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' , ρmin(dA,dB)} of the trans-  formed subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We treat the number of  CCA dimensions to use as a hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  4  Proof-of-concept: Arithmetic  Given a task, we assign ratings to inputs that ex-  press to what extent their task-dependent meaning  arises in a locally compositional manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' To inves-  tigate the impact of our metric on compositional  and non-compositional examples in a controlled  environment, we first use perfectly compositional  arithmetic expressions and introduce exceptions to  that compositionality manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='1  Data and model training  Math problems have previously been used to exam-  ine neural models’ compositional reasoning (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Saxton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Hupkes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Russin  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Arithmetic expressions are suited for  our application, in particular since they can be rep-  resented as trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We use expressions containing  brackets, integers -10 to 10, and + and - operators –  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' “( 10 - ( 5 + 3 ))” (using data from Hupkes  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The output is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' This is mod-  elled as a regression problem with the MSE loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  The ‘meaning’ (the numerical value) of a subex-  pression can be locally computed at each node in  0  2  4  + 0  4  + 2  0 in left subtree?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  interpret 0 as 0  0 in right subtree?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' treat  0 as the 1st leaf node  Figure 2: Illustration of the ‘exceptions’ in the arith-  metic task: the value of “0” depends on its position and  on the value of the leftmost leaf node in the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  the tree: there are no contextual dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  In this controlled environment, we introduce ex-  ceptions by making “0” ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' When located  in the subtree headed by the root node’s left child,  it takes on its regular value, but when located in  the right subtree, it takes on the value of the left-  most leaf node of the entire tree (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  The model is thus encouraged to perform non-  compositional processing to keep track of all occur-  rences of “0” and store the first leaf node’s value  throughout the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 88% of the training data are the  original arithmetic expressions, and 12% are such  exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We can thus track what happens to the  two categories when we introduce the bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  The training data consist of 14903 expressions with  1 to 9 numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We test on expressions with lengths  5 to 9, using 5000 examples per length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The Tree-  LSTMs trained on this dataset have embeddings  and hidden states of sizes 150 and are trained for  50 epochs with learning rate 2e−4 with AdamW  and a batch size of 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The base Tree-LSTMs in  all setups use the same architecture, namely the  Tree-LSTM architecture required for the DVIB,  but with β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' All results are averaged over mod-  els trained using ten different random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' In the  Tree-LSTM, the numbers are leaf nodes and the  labels of non-terminal nodes are the operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='2  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='2  Task performance: Hierarchy without  compositionality?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Figures 3a and 3b visualise the performance for  the regular examples and exceptions, respectively,  when increasing β for the DVIB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The DVIB dis-  proportionately harms the exceptions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' when β is  too high the model cannot capture the non-local  dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='1 shows how the hid-  den dimensionality and dropout bottlenecks have a  similar effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Figure 4 and Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='2 provide  insights in the training dynamics of the models: ini-  tially, all models will treat “0” as a regular number,  independent of the bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Close to conver-  2Appendix C further elaborates on the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  5  6  7  8  9  number of leaf nodes  0  1  2  3  MSE  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='01  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='025  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='05  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='25  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='0  (a) Regular examples  5  6  7  8  9  number of leaf nodes  0  20  40  60  MSE  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='01  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='025  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='05  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='25  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='0  (b) Exceptions  Figure 3: Performance (MSE) on the arithmetic task  for the Tree-LSTM with the DVIB (darker colours cor-  respond to higher β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Exceptions have a contextual de-  pendency and cannot be computed bottom up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  20  40  epochs  0  2  4  6  8  10  MSE  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='01  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='025  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='05  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='25  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='5  1  (a) Compositional targets  20  40  epochs  0  2  4  6  8  10  MSE  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='01  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='025  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='05  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='25  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='5  1  (b) Adapted targets  Figure 4: Training dynamics for the Tree-LSTM with  the DVIB: for all test examples we compute the MSE  over the course of training on the validation set using  (a) the compositional targets, or (b) the targets from the  adapted dataset, of which a subset is not compositional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  gence, models trained with a low β have learnt to  capture the ambiguities, whereas models trained  with a higher β will remain in a more locally com-  positional state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='3  Bottlenecks restrict information passed through-  out the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' To process an arithmetic subexpres-  sion, all that is needed is to pass on its outcome,  not the subexpression itself – e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' in Figure 2, one  could simply store the value 6 instead of the subex-  pression “2 - -4”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The former represents local  processing and is more efficient (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' it requires  storing less information), while the latter leads to  information from the leaf nodes being passed to  non-terminal nodes higher up the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Storing in-  formation about the leaf nodes would be required  to cope with the exceptions in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' That the  model could get close to accurate predictions for  these exceptions in the first place suggests Tree-  LSTMs can process inputs according to the hier-  archical structure without being locally composi-  tional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Increasing compression using bottlenecks  enforces local processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  3Comparing ‘early’ and ‘late’ models may yield similar  results as comparing base and bottleneck models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Yet, without  labels of which examples are compositional, it is hard to know  when the model transitions from the early to the late stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  PP, DVIB  PP, dropout  PP, dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  TT, DVIB  TT, dropout  TT, dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  bottleneck  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='0  relative position  (a) All BCM variants  0  10000  20000  position in ranking  0  200  400  600  800  1000  1200  #samples  regular  exceptions  (b) Ranking for β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='25  Figure 5: Rankings of arithmetic examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' (a) shows  the relative position of regular examples and exceptions  in the rankings of all setups, where 0 corresponds to the  start of the ranking and 1 to the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' (b) illustrates the  result of BCM-TT with the DVIB, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='3  The Bottleneck Compositionality Metric  The bottleneck Tree-LSTM harms the exceptions  disproportionately in terms of task performance,  and through BCM we can exploit the difference be-  tween the base and bottleneck model to distinguish  compositional from non-compositional examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  As laid out in §3, we use the TT or PP method  to compare pairs of Tree-LSTMs: the base Tree-  LSTM with β = 0, no dropout and a hidden dimen-  sion of 150 (base model) is paired up with Tree-  LSTMs with the same architecture, but a different  β, a different dropout probability p or a different  hidden dimensionality d (bottleneck model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' All  Tree-LSTMs have a classification module that con-  sists of two linear layers, where the first layer maps  the Tree-LSTM’s hidden representation to a vector  of 100 units, and the second layer emits the pre-  dicted value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The 100-dimensional vector is used  to apply the BCM:  In BCM-PP the vector feeds into the CCA  computation, that compares the hidden repre-  sentation of the base model and the bottleneck  model using their Cosine distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We rank  examples according to that distance, and use  all CCA directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  In BCM-TT, the vector feeds into the TRE  loss component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We train the base model,  freeze that network, and add the MSE of  the hidden representations of the bottleneck  model and the base model to the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' After  training, the remaining MSE is used to rank  examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Both setups have the same output, namely a com-  positionality ranking of examples in a dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' A  successful ranking would put the exceptions last.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='5e-05  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='000625  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='00125  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='0025  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='00625  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='0125  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='5  performance  (a) DVIB  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='25  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='5  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='65  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='75  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='85  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='9  dropout  sentiment-only baseline  (b) Dropout  150  125  100  75  50  25  10  5  size  BCM-PP  BCM-TT  (c) Hidden dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Figure 6: The accuracy (solid) and macro-averaged F1-  scores (dashed) for the SST test set, for base models,  bottleneck models and a sentiment-only baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Figure 5a illustrates the relative position of regular  examples and exceptions for all bottlenecks and  BCM variants, for β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='25, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='5 and d = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  The change in MSE observed in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='2 is reflected  in the quality of the ranking, but the success does  not depend on the specific selection of β, p or d, as  long as they are large (β, p) or small enough (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Figure 5b illustrates one of the rankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Summarising, we illustrated that recursive models  can employ strategies that do not locally compose  the meaning of arithmetic subexpressions but carry  tokens’ identities throughout the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We can make  a model more locally compositional using bottle-  necks and can use a model’s hidden states to infer  which examples required non-local processing af-  terwards, acting as our compositionality metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  5  Sentiment analysis  We apply the metric to the task of sentiment anal-  ysis, for which Moilanen and Pulman (2007, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 1)  suggest the following notion of compositionality:  “For if the meaning of a sentence is a  function of the meanings of its parts then  the global polarity of a sentence is a func-  tion of the polarities of its parts.”  Sentiment is quasi-compositional: even though the  sentiment of an expression is often a straightfor-  ward function of the sentiment of its parts, there are  exceptions – e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' consider cases of sarcasm, such  as “I love it when people yell at me first thing in  the morning” (Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=', 2019), which makes it  a suitable testing bed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='1  Data and model training  We use the SST-5 subtask from the Stanford Sen-  timent Treebank (SST) (Socher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=', 2013), that  contains sentences from movie reviews collected  ++  ++  ++  ++  +  +  positive +  +  +  --  --  --  -- negative  ~ --  ++  +  --  + -- ~  + +  ++  ~  +  ++  +  ~  ~  ~  ~ neutral  ~  ~  ~  in-between  continuity  ++  ~ symmetric  symmetric  symmetric  symmetric  symmetric  symmetric  symmetric  symmetric --  --  Figure 7: Illustration of the predictions of a sentiment-  only baseline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We indicate the predicted senti-  ment given two inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The labels range from very neg-  ative (‘- -’) to neutral (‘∼’) to very positive (‘++’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  by Pang and Lee (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The SST-5 subtask re-  quires classifying sentences into one of five classes  ranging from very negative to very positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The  standard train, development and test subsets have  8544, 1101 and 2210 examples, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The  sentences were originally parsed with the Stanford  Parser (Klein and Manning, 2003), and the dataset  includes sentiment labels for all nodes of those  parse trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Typically, labels for all phrases are in-  cluded in training, but the evaluation is conducted  for the root nodes of the test set, only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Following Tai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' (2015), we use GloVe word  embeddings (Pennington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=', 2014), that we  freeze across models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='4 The Tree-LSTM has 150  hidden units and is trained for 10 epochs with a  learning rate of 2e−4 and the AdamW optimiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  During each training update, the loss is computed  over all subexpressions of 4 trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Training is re-  peated with 10 random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='5 Figure 6 provides  the performance on the test set for the base and bot-  tleneck models, using the accuracy and the macro-  averaged F1-score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Tai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' (2015) obtained an  accuracy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='497 using frozen embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  In sentiment analysis,  as in our pseudo-  arithmetic task, a successful model would often  have to deviate from local processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' After all,  the correct interpretation of a leaf node is often  unknown without access to the context – e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' in  the case of ambiguous words like “sick” which is  likely to refer to being ill, but could also mean  “awesome”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Being successful at the task thus re-  quires a recursive model to keep track of informa-  4The notion of local compositionality, relied on in this  work, assumes that tokens are not disambiguated, which is  why we refrain from using state-of-the-art contextualised rep-  resentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The focus of this work is on developing a compo-  sitionality metric rather than on improving sentiment analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  5Appendix C further elaborates on the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='5e-05  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='000625  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
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+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content="65  pearson's r  (a) DVIB  ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
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+page_content='75  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='85  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='9  dropout  (b) Dropout  125  100  75  50  25  10  5  hidden dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  BCM-PP  BCM-TT  (c) Hidden dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='5e-05  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='000625  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
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+page_content='0025  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='00625  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='0125  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
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+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content="6  spearman's  (d) DVIB  ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='25  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='5  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='65  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='75  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='85  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='9  dropout  (e) Dropout  125  100  75  50  25  10  5  hidden dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  25  50  75  100  (f) Hidden dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Figure 8: Pearson’s r for the predictions of sentiment-  only baselines and bottleneck models (a-c) and Spear-  man’s ρ for the SST validation set compositionality  ranking of the baselines and bottleneck models (d-f),  when varying the number of CCA dimensions used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  tion about (leaf) nodes while recursively processing  the input, and more so for non-compositional ex-  amples than for compositional examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' As with  the arithmetic task, local processing – enforced in  the bottleneck models – should disproportionately  hinder processing of non-compositional examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='2  A sentiment-only baseline  To assert that the bottlenecks make the models  more compositional, we create a sentiment-only  baseline that is given as input not words but their  sentiment, and has a hidden dimensionality d = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Non-compositional patterns that arise from the  composition of certain words rather than the senti-  ment of those words (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' “drop dead gorgeous”)  could hardly be captured by that model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' As such,  the model exemplifies how sentiment can be com-  puted more compositionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Figure 7 illustrates the  default sentiment this model predicts for various  input combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Its predictions can be charac-  terised by i) predicting positive for positive inputs,  ii) predicting negative for negative inputs, iii) pre-  dicting neutral if one input is neutral, iv) predict-  ing a class in between the input classes or as v)  predicting the same class as its inputs (continuity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  The performance of the model is included in Fig-  ure 6, and Figure 8 (a-c) indicates the Pearson’s  r between the sentiment predictions of bottleneck  models and baseline models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Generally, a higher  β or dropout probability, or a lower hidden dimen-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='65  in_between  continuity (neutral)  neutral  positive  amplification  continuity (non-neutral)  neutral<->polarised  switch  attenuation  negative  mixed  idioms  difficult-vocab  world-knowledge  amplified  comparative  strong  desirable-element  no-sentiment  negated  sarcasm/irony  morphology  Figure 9: Categories of SST examples and their average position on the compositionality ranking visualised for the  BCM-PP with the hidden dimensionality bottleneck and d = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Categories in black are assigned by us;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' categories  in gray are from Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The categories are further explained in the main text and Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Jittering was applied to better visualise overlapping categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  sionality, leads to predictions that are more similar  to this sentiment-only model, unless the amount  of regularisation is too extreme, debilitating the  model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' for dropout with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='3  The Bottleneck Compositionality Metric  Now we use BCM to obtain a ranking over the SST  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We separate the dataset into four folds,  train on those folds for the base and bottleneck  model, and compute the cosine distances for the  hidden representations of examples in the test sets  (using BCM-PP or BCM-TT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We merge the co-  sine distances for the different folds, averaged over  models trained with 10 random seeds, and order ex-  amples based on the resulting distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We select  the values for β, dropout and the hidden dimension-  ality, as well as the number of CCA directions to  use, based on rankings computed over the SST val-  idation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Figure 8 (d-f) illustrates how the rank-  ings of bottleneck models correlate with rankings  constructed using the sentiment-only baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We  select 25 directions, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='0025, dropout p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='65  and a hidden dimensionality of 25 to compute the  full ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Contrary to the arithmetic task, BCM-  TT underperforms compared to BCM-PP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Different from the arithmetic task, it is unclear  which examples should be at the start or end of  the ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Therefore, we examine the relative  position of categories of examples in Figure 9 for  the BCM-PP with the hidden dimensionality bot-  tleneck, and in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='3 for the remaining  rankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The categories include the previously in-  troduced ones, augmented with the following four:  amplification: the root is even more posi-  tive/negative than its top two children;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  attenuation: the root is less positive/negative  than its top two children;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  switch: the children are positive/negative, but  the root node flips that sentiment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  neutral↔polarised: the inputs are sentiment-  laden, but the root is neutral, or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  We also include characterisations from Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  (2019), who label examples from the SST test set,  for which state-of-the-art sentiment models can-  not seem to predict the correct label, including, for  example, cases where a sentence contains mixed  sentiment, or sentences with idioms, irony or sar-  casm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='1 elaborates on the meaning of  the categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Figure 9 illustrates the relative posi-  tions of our sentiment characterisations and those  of Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' on that ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Patterns associ-  ated with more compositional sentiment process-  ing, such as ‘positive’, ‘negative’ and ‘in between’  lead to hidden representations that are more simi-  lar between the base model and bottleneck models  than the dataset average (the mid point, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Atyp-  ical patterns like ‘switch’ and ‘neutral↔polarised’,  on the other hand, along with the characterisations  by Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' lead to less similar hidden repre-  sentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='3 presents the same results  for all six rankings considered, along with example  sentences from across the ranking, to illustrate the  types of sentences encountered among the most  compositional and the least compositional ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='4  Example use cases  Compositionality rankings can be used in multiple  manners, of which we illustrate two below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  When data is scarce: use compositional exam-  ples  Assuming that most of natural language is  compositional, one would expect that when limit-  ing the training data, selecting compositional exam-  ples yields the best test performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' To investigate  this, we train models on various training dataset  sizes, and evaluate with the regular test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The  training data is taken from the start of the ranking  for the ‘compositional’ setup, and from the end  of the ranking for the ‘non-compositional’ setup  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
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+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='5  training ratio  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
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+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='6  performance  (a) LSTM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='5  training ratio  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='6  performance  comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  non-comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  (b) Roberta  Figure 10: Change in SST test set accuracy (solid) and  macro-averaged F1-score (dashed) as the training set  size increases, for LSTM and Roberta models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The ex-  amples are from the most (in blue) or the least composi-  tional (in green) portion of the ranking from the BCM-  PP metric with the hidden dimensionality bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Model  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Non-comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Random  Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  F1  Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  F1  Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  F1  Roberta  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='546  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='535  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='516  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='487  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='565  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='549  LSTM  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='505  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='485  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='394  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='310  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='478  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='447  Table 1: Performance on the new SST compositionality  splits, generated using the ranking from the BCM-PP  metric with the hidden dimensionality bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  (excluding test data), while ensuring equal distri-  butions over input lengths and output classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We  train a two-layer bidirectional LSTM with 300 hid-  den units, and Roberta-base (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=', 2019), us-  ing batch size 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The models are trained for 10 and  5 epochs, respectively, with learning rates 2e − 4  and 5e − 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Because the ranking is computed over  full sentences, and not subexpressions, we train the  models on the sentiment labels for the root nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Figure 10 presents the results, consolidating that  when data is scarce, using compositional examples  is beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Non-compositional examples are challenging  For the same models, Table 1 indicates how perfor-  mance changes if we redistribute train and test data  such that the test set contains the most composi-  tional examples, or the least compositional ones  (keeping length and class distributions similar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  The non-compositional test setup is more challeng-  ing, with an 11 percentage point drop in accuracy  for the LSTM, and a 3 point decrease for Roberta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  In conclusion, applying the BCM to the sentiment  data has confirmed findings previously observed  for the arithmetic toy task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' While it is harder to  understand whether the method actually filters out  non-compositional examples, both comparisons to  a sentiment-only baseline, and the average position  of cases for which the composition of sentiment is  known to be challenging (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' for ‘mixed’ senti-  ment, ‘comparative’ sentiment or ‘sarcasm’), sug-  gest that compression acts as a compositionality  metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We also illustrated two ways in which the  resulting ranking can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  6  Conclusion  This work presents the Bottleneck Compositional-  ity Metric, a TRE-based metric (Andreas, 2018)  that is task-independent and can be applied to in-  puts of varying lengths: we pair up Tree-LSTMs  where one of them has more compressed repre-  sentations due to a bottleneck (the DVIB, hidden  dimensionality bottleneck or dropout bottleneck),  and use the distance between their hidden represen-  tations as a per-datum metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The method was ap-  plied to rank examples in datasets from most com-  positional to least compositional, which is of inter-  est due to the growing relevance of compositional  generalisation research in NLP, which assumes the  compositionality of natural language, and encour-  ages models to compose meanings of expressions  rather than to memorise phrases as chunks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We  provided a proof-of-concept using an arithmetic  task but also applied the metric to the much more  noisy domain of sentiment analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  The different bottlenecks lead to qualitatively  similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' This suggests that, while DVIB  might be better motivated (it directly optimises an  estimate of the Shannon information passed across  the network), its alternatives may be preferable in  practice due to their simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Because natural language itself is not fully com-  positional, graded metrics like the ones we pre-  sented can support future research, such as i) learn-  ing from data according to a compositionality-  based curriculum to improve task performance, ii)  filtering datasets to improve compositional gener-  alisation, or iii) developing more and less compo-  sitional models depending on the desiderata for a  task – e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' to perform well on sentences with id-  ioms, one may desire a more non-compositional  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' In addition, the formulation of the metric  was general enough to be expanded upon in the  future: one could pair up other models, such as an  LSTM and a Tree-LSTM, or a Transformer and its  recursive variant, as long as one keeps in mind that  the compositional reconstruction itself should not  be too powerful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' After all, even Tree-LSTMs could  capture the exceptions in the arithmetic dataset de-  spite their hierarchical inductive bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Limitations  We identify three types of limitations for the work  presented:  A conceptual limitation is that we work  from a very strict definition of composition-  ality (local compositionality), which essen-  tially equates language with arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' While  overly restrictive, current datasets testing com-  positional generalisation follow this notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  The framework might be extensible to more  relaxed notions by allowing for token disam-  biguation by using contextualised token em-  beddings and only enforcing a bottleneck on  the amount of further contextual integration  within the model added on top of the token  embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  In terms of methodological limitations, the  use of only Tree-LSTMs – although well-  motivated from the perspective of composi-  tional processing – is a major limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Tree-  LSTMs are most suited for sentence classifica-  tion tasks, limiting the approach’s applicabil-  ity to sequence-to-sequence tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Nonethe-  less, the bottlenecks can be integrated in other  types of architectures that process inputs in  a hierarchical manner, such as sequence-to-  sequence models inducing latent source and  target trees (Kim, 2021) to yield an alternative  implementation of the BCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Our work also assumes that an input’s tree  structure is known, which might not always  be the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Therefore, the compositionality  ranking obtained using BCM always depends  on the trees used: what is non-compositional  given one (potentially inadequate) structure  might be more compositional given another  (improved) structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Lastly, the evaluation of our approach is lim-  ited in the natural domain through the absence  of gold labels of the compositionality of ex-  amples in the sentiment analysis task, but for  other tasks that could have been considered,  the same limitation would have applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Acknowledgements  We thank Chris Lucas for his contributions to this  project when it was still in an early stage, Kenny  Smith for his comments on the first draft of this  paper, and Matthias Lindemann for excellent sug-  gestions for the camera-ready version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' VD is sup-  ported by the UKRI Centre for Doctoral Training  in Natural Language Processing, funded by the  UKRI (grant EP/S022481/1) and the University  of Edinburgh, School of Informatics and School  of Philosophy, Psychology & Language Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  IT acknowledges the support of the European Re-  search Council (ERC StG BroadSem 678254) and  the Dutch National Science Foundation (NWO Vidi  639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='518).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  References  Alexander A Alemi, Ian Fischer, Joshua V Dillon, and  Kevin Murphy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Deep variational information  bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' In ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Jacob Andreas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Measuring compositionality in  representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  In International Confer-  ence on Learning Representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Jeremy Barnes, Lilja Øvrelid, and Erik Velldal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Sentiment analysis is not solved!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Assessing and  probing sentiment classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' In Proceedings of  the 2019 ACL Workshop BlackboxNLP: Analyzing  and Interpreting Neural Networks for NLP, pages  12–23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Hanoz Bhathena, Angelica Willis, and Nathan Dass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Evaluating compositionality of sentence rep-  resentation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' In Proceedings of the 5th Work-  shop on Representation Learning for NLP, pages  185–193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Ben Bogin, Shivanshu Gupta, and Jonathan Berant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Unobserved local structures make com-  positional generalization hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  arXiv preprint  arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='05899.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Henry Brighton and Simon Kirby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Understand-  ing linguistic evolution by visualizing the emergence  of topographic mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Artificial life, 12(2):229–  242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Vitaly Feldman and Chiyuan Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' What neu-  ral networks memorize and why: Discovering the  long tail via influence estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Advances in Neu-  ral Information Processing Systems, 33:2881–2891.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Emiliano Guevara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  A regression model of  adjective-noun compositionality in distributional se-  mantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' In Proceedings of the 2010 Workshop on  GEometrical Models of Natural Language Seman-  tics, pages 33–37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Citeseer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Harold Hotelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 1936.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Relations between two sets of  variates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Biometrika, 28(3/4):321–377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Dieuwke Hupkes, Verna Dankers, Mathijs Mul, and  Elia Bruni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Compositionality decomposed:  How do neural networks generalise?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Journal of Ar-  tificial Intellgence Research, 67:757–795.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Dieuwke Hupkes, Sara Veldhoen, and Willem Zuidema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Visualisation and ‘diagnostic classifiers’ re-  veal how recurrent and recursive neural networks  process hierarchical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Journal of Artificial  Intelligence Research, 61:907–926.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Daniel Keysers, Nathanael Schärli, Nathan Scales,  Hylke Buisman, Daniel Furrer, Sergii Kashubin,  Nikola Momchev, Danila Sinopalnikov, Lukasz  Stafiniak, Tibor Tihon, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Measuring com-  positional generalization: A comprehensive method  on realistic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  In International Conference on  Learning Representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Yoon Kim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Sequence-to-sequence learning with  latent neural grammars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Advances in Neural Infor-  mation Processing Systems, 34:26302–26317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Dan Klein and Christopher D Manning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Accu-  rate unlexicalized parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  In Proceedings of the  41st annual meeting of the association for compu-  tational linguistics, pages 423–430.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Tomasz Korbak, Julian Zubek, and Joanna R ˛aczaszek-  Leonardi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Measuring non-trivial composition-  ality in emergent communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' In NeurIPS 2020  workshop on Emergent Communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Brenden Lake and Marco Baroni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Generalization  without systematicity: On the compositional skills  of sequence-to-sequence recurrent networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' In In-  ternational Conference on Machine Learning, pages  2873–2882.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Angeliki Lazaridou, Karl Moritz Hermann, Karl Tuyls,  and Stephen Clark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Emergence of linguis-  tic communication from referential games with sym-  bolic and pixel input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' In International Conference  on Learning Representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Xiang Lisa Li and Jason Eisner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Specializing  word embeddings (for parsing) by information bot-  tleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' In Proceedings of the 2019 Conference on  Empirical Methods in Natural Language Processing  and the 9th International Joint Conference on Natu-  ral Language Processing (EMNLP-IJCNLP), pages  2744–2754.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Man-  dar Joshi, Danqi Chen, Omer Levy, Mike Lewis,  Luke Zettlemoyer, and Veselin Stoyanov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Roberta: A robustly optimized BERT pretraining ap-  proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Diana McCarthy, Bill Keller, and John A Carroll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Detecting a continuum of compositionality  in phrasal verbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' In Proceedings of the ACL 2003  workshop on Multiword expressions: analysis, ac-  quisition and treatment, pages 73–80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Diana McCarthy, Sriram Venkatapathy, and Aravind  Joshi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Detecting compositionality of verb-  object combinations using selectional preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  In Proceedings of the 2007 Joint Conference on  Empirical Methods in Natural Language Process-  ing and Computational Natural Language Learning  (EMNLP-CoNLL), pages 369–379.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Karo Moilanen and Stephen Pulman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Senti-  ment composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  In Proceedings of the Recent  Advances in Natural Language Processing Interna-  tional Conference, pages 378–382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Association for  Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Navnita Nandakumar, Timothy Baldwin, and Bahar  Salehi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' How well do embedding models cap-  ture non-compositionality?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' A view from multiword  expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' In Proceedings of the 3rd Workshop on  Evaluating Vector Space Representations for NLP,  pages 27–34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Bo Pang and Lillian Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Seeing stars: Exploit-  ing class relationships for sentiment categorization  with respect to rating scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' In Proceedings of the  43rd Annual Meeting of the Association for Compu-  tational Linguistics (ACL’05), pages 115–124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Jeffrey Pennington, Richard Socher, and Christopher D  Manning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Glove: Global vectors for word rep-  resentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' In Proceedings of the 2014 conference  on empirical methods in natural language process-  ing (EMNLP), pages 1532–1543.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Carlos Ramisch, Silvio Cordeiro, Leonardo Zilio,  Marco Idiart, and Aline Villavicencio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' How  naked is the naked truth?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  A multilingual lexicon  of nominal compound compositionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' In Proceed-  ings of the 54th Annual Meeting of the Association  for Computational Linguistics (Volume 2: Short Pa-  pers), pages 156–161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Siva Reddy, Diana McCarthy, and Suresh Manandhar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  An empirical study on compositionality in  compound nouns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  In Proceedings of 5th Interna-  tional Joint Conference on Natural Language Pro-  cessing, pages 210–218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Jacob Russin, Roland Fernandez, Hamid Palangi, Eric  Rosen, Nebojsa Jojic, Paul Smolensky, and Jian-  feng Gao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Compositional processing emerges  in neural networks solving math problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' In An-  nual Conference of the Cognitive Science Society  (CogSci), volume 2021, page 1767.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  David Saxton, Edward Grefenstette, Felix Hill, and  Pushmeet Kohli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Analysing mathematical rea-  soning abilities of neural models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' In International  Conference on Learning Representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Peter Shaw, Ming-Wei Chang, Panupong Pasupat, and  Kristina Toutanova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Compositional general-  ization and natural language variation: Can a se-  mantic parsing approach handle both?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' In Proceed-  ings of the 59th Annual Meeting of the Association  for Computational Linguistics and the 11th Interna-  tional Joint Conference on Natural Language Pro-  cessing (Volume 1: Long Papers), pages 922–938.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Richard Socher, Alex Perelygin, Jean Wu, Jason  Chuang, Christopher D Manning, Andrew Y Ng,  and Christopher Potts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Recursive deep mod-  els for semantic compositionality over a sentiment  treebank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' In Proceedings of the 2013 conference on  empirical methods in natural language processing,  pages 1631–1642.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky,  Ilya Sutskever, and Ruslan Salakhutdinov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Dropout: a simple way to prevent neural networks  from overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The journal of machine learning  research, 15(1):1929–1958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Zoltan Szabó.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  The case for compositionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  The Oxford handbook of compositionality, 64:80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Kai Sheng Tai, Richard Socher, and Christopher D  Manning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Improved semantic representations  from tree-structured long short-term memory net-  works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  In Proceedings of the 53rd Annual Meet-  ing of the Association for Computational Linguistics  and the 7th International Joint Conference on Natu-  ral Language Processing (Volume 1: Long Papers),  pages 1556–1566.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Sriram Venkatapathy and Aravind Joshi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Measur-  ing the relative compositionality of verb-noun (vn)  collocations by integrating features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  In Proceed-  ings of Human Language Technology Conference  and Conference on Empirical Methods in Natural  Language Processing, pages 899–906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Alison Wray and Michael R Perkins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The func-  tions of formulaic language: An integrated model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Language & Communication, 20(1):1–28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Majid Yazdani, Meghdad Farahmand, and James Hen-  derson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Learning semantic composition to de-  tect non-compositionality of multiword expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  In Proceedings of the 2015 Conference on Empiri-  cal Methods in Natural Language Processing, pages  1733–1742.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Xiaosen Zheng and Jing Jiang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  An empirical  study of memorization in NLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  In Proceedings  of the 60th Annual Meeting of the Association for  Computational Linguistics (Volume 1: Long Papers),  pages 6265–6278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  A  Bottlenecks for arithmetic  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='1  Performance for other bottlenecks  Figures 11 and 12 display the MSE for models with  the hidden dimensionality bottleneck and dropout  bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  5  6  7  8  9  number of leaf nodes  0  1  2  3  MSE  dim  5  10  25  50  75  100  125  150  (a) Regular examples  5  6  7  8  9  number of leaf nodes  0  20  40  60  MSE  dim  5  10  25  50  75  100  125  150  (b) Exceptions  Figure 11: Performance on the arithmetic task for the  Tree-LSTM with varying hidden dimensionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  5  6  7  8  9  number of leaf nodes  0  2  4  6  8  10  MSE  dropout  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='25  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='5  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='65  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='75  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='85  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='9  (a) Regular examples  5  6  7  8  9  number of leaf nodes  0  20  40  60  MSE  dropout  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='25  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='5  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='65  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='75  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='85  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='9  (b) Exceptions  Figure 12: Performance on the arithmetic task for the  Tree-LSTM with varying dropout probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='2  Bottleneck training dynamics  The exceptions in the arithmetic task had both a  compositional and non-compositional interpreta-  tion, essentially giving us two sets of targets for  measuring the models’ performance during train-  ing using the validation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Figures 4 (in the  main paper), 13, and 14 illustrate how early on  during training, the MSE is lowest for the composi-  tional targets for all hyperparameters used for the  bottlenecks: the models overgeneralise the regu-  lar interpretation of “0”, applying it to all inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Later on, the models that have a high β, a high  dropout probability or a small dimension stay in  that ‘compositional state’, whereas the remaining  models learn to capture the ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  B  Sentiment analysis  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='1  Categories of Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' (2019)  Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' collect examples from sentiment anal-  ysis datasets that three state-of-the-art classifiers  struggle with, and annotate them for the presence  of 18 (para-)linguistic phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We include the  20  40  epochs  0  2  4  6  8  10  MSE  dim  150  125  100  75  50  25  10  5  (a) Compositional targets  20  40  epochs  0  2  4  6  8  10  MSE  dim  150  125  100  75  50  25  10  5  (b) Non-comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' targets  Figure 13: Training dynamics for the hidden dimen-  sionality bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  20  40  epochs  0  2  4  6  8  10  MSE  dropout  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='25  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='5  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='65  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='75  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='85  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='9  (a) Compositional targets  20  40  epochs  0  2  4  6  8  10  MSE  dropout  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='25  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='5  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='65  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='75  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='85  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='9  (b) Non-comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' targets  Figure 14: Training dynamics for the dropout bottle-  neck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  following categories from the SST test set in the  visualisation of the SST ranking:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Negated:  phrases or sentences that are  negated, where Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' identify a pattern  of irrelevant negation throwing models off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Amplified: cases where neutral modifiers act  as strong contextual valence shifters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Strong: very positive or very negative cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Desirable element: sentiment dominated by  one ‘desirable element’, such as “pool”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Comparative: cases that express sentiment by  comparison (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' using “better than”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Idioms: cases containing idioms, for which  the mapping from the words to the sentiment  is often not straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Mixed: mixed positive & negative sentiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Difficult-vocab: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' “engrossing and psycho-  logically resonant suspenser”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' World-knowledge: for instance comparisons  between entities, where the entities imply a  certain sentiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Sarcasm/irony: sarcasm is often present in  negative examples, where the speaker is in-  tending the opposite of what is said.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' No-sentiment: neutral labelled examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Morphology: examples with morphological  features that affect sentiment very positively  or very negatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='2  Alternative metrics  Tree impurity score (TIS)  Bhathena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  PP  dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  PP  dropout  PP  DVIB  TT  dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content="  TT  dropout  TT  DVIB  `topographic'  TIS  memorisation  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
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+page_content="75  spearman's  Figure 15: Illustration of how BCM rankings over the  SST data correlate with baseline metrics that ought  to be correlated: topographic similarity, tree impurity  score and memorisation as per Spearman’s ρ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  (2020) propose basic compositionality metrics that  rely on the difference between the label of the root  node, and labels of subexpressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We report their  tree impurity score (TIS): a simple metric that mea-  sures the absolute difference between the label of  the root node and the average of all labels in a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  ‘Topographic’ similarity  A compositionality  metric from language emergence literature is to-  pographic similarity (Brighton and Kirby, 2006),  that given a set of objects, their meanings and  the associated signals computes the correlation of  the distances between corresponding pairs of sig-  nals and meanings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' For instance, Lazaridou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  (2018) compare symbolic signals in a referential  game using the Levenshtein edit distance, and com-  pare vector meaning representations through cosine  distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Topographic similarity assumes that in  a compositional language, similar signals should  yield similar meaning representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' However,  the use of edit distance to directly compare sen-  tences does not readily transfer to sentiment analy-  sis, where one word changed in the input space can  yield a large change in the predicted sentiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  To approximate topographic similarity of mean-  ings and signals, we instead manipulate the input  in ways that should yield a similar prediction, and  then rate examples based on the change in the pre-  dictions observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We replace nouns, verbs, adjec-  tives or adverbs with a word that in SST has the  same POS tag and sentiment label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' If a change  is observed, the example is more likely to be a  non-compositional example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  We apply this to the base models from the main  paper, by randomly replacing one token that is a  verb, noun, adjective or adverb with a different  token of the same POS tag and sentiment label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  We make 50 such modifications per sentence per  model seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We record the average change in the  predicted sentiment, and use this as a metric akin  to topographic similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  PP  dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  PP  dropout  PP  DVIB  TT  dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  TT  dropout  PP  dropout  PP  DVIB  TT  dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
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+page_content="8  spearman's  Figure 16: Illustration of how the BCM rankings over  the SST dataset correlate as per Spearman’s ρ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Memorisation score  Zheng and Jiang (2022)  empirically evaluate the long tail theory posed by  Feldman and Zhang (2020) (who validate their hy-  potheses using computer vision tasks), that states  that for data distributions with a long tail, memo-  risation of training examples is required for near-  optimal performance on the test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Zheng and  Jiang put this theory to the test for sentiment classi-  fication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The metric of Zheng and Jiang expresses  how the likelihood of the target changes when an  example is down-weighted during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' That  change should be larger for memorised examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Intuitively, non-compositionality and memorisa-  tion are related: non-compositional patterns in data  require memorising the atypical interpretation of  words in specific contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The metric is expected  to be different from our metric – ours is measured  using test data, while memorisation occurs during  training – but a positive correlation is expected be-  tween the two, nonetheless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Zheng and Jiang use  the binary SST subtask and report their scores on  preprocessed versions of SST sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We report  the correlation for the examples for which we could  find matching surface forms only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Figure 15 indicates how the different BCM rank-  ings correlate with these three alternative metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Surprisingly, TIS negatively correlates with our  rankings, but only weakly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' This may be due to  the simplicity of that metric, that ignores the tree  structure and simply averages all sentiment of all  nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The devised ‘topographic’ similarity posi-  tively correlates with our rankings, but only up to  ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The memorisation score has a moderate  correlation with our rankings, of up to ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='46 for  the BCM-TT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Figure 16 illustrates how the rank-  ings correlate with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The figure suggests  that TRE training yields rankings that are still quite  different from the post-processing ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='3  Additional results for SST rankings  Here, we present further results for the BCM rankings over the SST datasets: firstly, Figure 17 provides  the rankings for the six different BCM setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' They have commonalities, but also differences, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' in the  BCM-TT setups the ‘neutral’ sentiment is closer to the end of the ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
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+page_content='65  in_between  continuity (neutral)  neutral  positive  amplification  continuity (non-neutral)  neutral<->polarised  switch  attenuation  negative  mixed  idioms  difficult-vocab  world-knowledge  amplified  comparative  strong  desirable-element  no-sentiment  negated  sarcasm/irony  morphology  (a) BCM-PP, Hidden dimensionality bottleneck  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
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+page_content='65  in_between  continuity (neutral)  neutral  positive  amplification  continuity (non-neutral)  neutral<->polarised  switch  attenuation  negative  mixed  idioms  difficult-vocab  world-knowledge  amplified  comparative  strong  desirable-element  no-sentiment  negated  sarcasm/irony  morphology  (b) BCM-PP, DVIB bottleneck  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
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+page_content='60  in_between  continuity (neutral)  neutral  positive  amplification  continuity (non-neutral)  neutral<->polarised  switch  attenuation  negative  mixed  idioms  difficult-vocab  world-knowledge  amplified  comparative  strong  desirable-element  no-sentiment  negated  sarcasm/irony  morphology  (c) BCM-PP, Dropout bottleneck  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
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+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='60  in_between  continuity (neutral)  neutral  positive  amplification  continuity (non-neutral)  neutral<->polarised  switch  attenuation  negative  mixed  idioms  difficult-vocab  world-knowledge  amplified  comparative  strong  desirable-element  no-sentiment  negated  sarcasm/irony  morphology  (d) BCM-TT, Hidden dimensionality bottleneck  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='60  in_between  continuity (neutral)  neutral  positive  amplification  continuity (non-neutral)  neutral<->polarised  switch  attenuation  negative  mixed  idioms  difficult-vocab  world-knowledge  amplified  comparative  strong  desirable-element  no-sentiment  negated  sarcasm/irony  morphology  (e) BCM-TT, DVIB bottleneck  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='65  in_between  continuity (neutral)  neutral  positive  amplification  continuity (non-neutral)  neutral<->polarised  switch  attenuation  negative  mixed  idioms  difficult-vocab  world-knowledge  amplified  comparative  strong  desirable-element  no-sentiment  negated  sarcasm/irony  morphology  (f) BCM-TT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Dropout bottleneck  Figure 17: Example categories for SST test set examples,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' and their average position on the compositionality  ranking for the hidden dimensionality bottleneck with d = 25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' the dropout bottleneck with p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='65 or the DVIB  with β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='0025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The categories in black are assigned by us, and the categories in gray are from Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Jittering was applied to better visualise overlapping categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Secondly, Table 2 provides example sentences from different parts of the rankings, randomly sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' 0  indicates the most compositional examples, and 1 the least compositional ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Relative position  Target  Sentence  BCM-PP, Hidden dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' bottleneck  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='07  3  tsai convincingly paints a specifically urban sense of disassociation here .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='17  0  the film desperately sinks further and further into comedy futility .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='21  3  a cop story that understands the medium amazingly well .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='40  3  one scarcely needs the subtitles to enjoy this colorful action farce .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='42  1  the entire movie is about a boring , sad man being boring and sad .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='53  2  not everyone will play the dark , challenging tune taught by the piano teacher .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='66  3  this is the stuff that disney movies are made of .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='75  3  daughter from danang sticks with its subjects a little longer and tells a deeper story  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='86  2  not kids , who do n’t need the lesson in repugnance .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='98  0  lacks heart , depth and , most of all , purpose .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  BCM-PP, Dropout bottleneck  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='09  4  it ’s a cool event for the whole family .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='14  0  done in mostly by a weak script that ca n’t support the epic treatment .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='20  3  some movies are like a tasty hors-d’oeuvre ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' this one is a feast .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='37  4  my oh my , is this an invigorating , electric movie .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='41  3  its director ’s most substantial feature for some time .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='51  4  the modern master of the chase sequence returns with a chase to end all chases  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='62  3  like its bizarre heroine , it irrigates our souls .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='78  1  sadly , ‘ garth ’ has n’t progressed as nicely as ‘ wayne .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' ’  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='86  2  you ’re too conscious of the effort it takes to be this spontaneous .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='95  0  a film of empty , fetishistic violence in which murder is casual and fun .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  BCM-PP, DVIB  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='07  4  highly recommended viewing for its courage , ideas , technical proficiency and great acting .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='18  3  it manages to squeeze by on angelina jolie ’s surprising flair for self-deprecating comedy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='25  1  goes on and on to the point of nausea .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='38  3  the best part about “ gangs ” was daniel day-lewis .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='44  2  a dopey movie clothed in excess layers of hipness .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='51  2  a perplexing example of promise unfulfilled , despite many charming moments .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='64  0  the entire film is one big excuse to play one lewd scene after another .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='79  1  the problematic characters and overly convenient plot twists foul up shum ’s good intentions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='82  3  the jabs it employs are short , carefully placed and dead-center .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='92  2  then nadia ’s birthday might not have been such a bad day after all .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  BCM-TT, Hidden dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' bottleneck  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='00  3  the story is smart and entirely charming in intent and execution .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='02  2  for single digits kidlets stuart little 2 is still a no brainer .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='17  1  outer-space buffs might love this film , but others will find its pleasures intermittent .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='29  3  although shot with little style , skins is heartfelt and achingly real .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='37  3  a knowing look at female friendship , spiked with raw urban humor .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='50  3  there ’s an energy to y tu mamá también .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='57  2  a piquant meditation on the things that prevent people from reaching happiness .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='63  3  too daft by half .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' but supremely good natured .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='71  1  the feature-length stretch .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' strains the show ’s concept .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='88  1  a thriller without thrills and a mystery devoid of urgent questions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='97  1  the movie does n’t generate a lot of energy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  BCM-TT, Dropout bottleneck  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='03  3  an extremely funny , ultimately heartbreaking look at life in contemporary china .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='14  1  the pretensions – and disposable story – sink the movie .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='22  4  that rara avis : the intelligent romantic comedy with actual ideas on its mind .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='39  2  admirable , certainly , but not much fun to watch .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
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+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='59  2  throwing it all away for the fleeting joys of love ’s brief moment .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='63  4  kids should have a stirring time at this beautifully drawn movie .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='77  3  the obnoxious title character provides the drama that gives added clout to this doc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='87  0  stitch is a bad mannered , ugly and destructive little **** .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='98  0  there is no pleasure in watching a child suffer .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  BCM-TT, DVIB  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='03  4  quite simply , a joy to watch and – especially – to listen to .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
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+page_content='14  1  a fake street drama that keeps telling you things instead of showing them .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
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+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
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+page_content='71  3  a meditation on faith and madness , frailty is blood-curdling stuff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='81  1  a beautifully shot but dull and ankle-deep ‘ epic .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' ’  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
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+page_content='  Table 2: Examples from across the six rankings, randomly sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Thirdly, Figure 18 illustrates how training on different subsets of the ranking leads to different  performance on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' In general, it is better to train on the compositional examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
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+page_content='  non-comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  (l) Roberta, DVIB, BCM-TT  Figure 18: Change in SST test set accuracy (solid) and macro-averaged F1-score (dashed) as the training set size  increases, for LSTM and Roberta models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  C  Reproducibility details  Visit https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='com/vernadankers/bottleneck_  compositionality_metric for the code and data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Be-  low we collect an overview of the settings involved  in the model training and model evaluation:  Model specifications: Following Tai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  (2015) we use 150 hidden dimensions, and  300-dimensional word embeddings for the  sentiment analysis task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' For consistency, we  adopt the 150 dimensions in the arithmetic  task, as well, but reduce the size of the word  embeddings to 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The arithmetic base mod-  els have 426k trainable parameters, and the  sentiment base models have 514k trainable  parameters (excluding the frozen word em-  beddings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Training procedure:  The sentiment and  arithmetic models are trained for 10 and 50  epochs, respectively, based on model conver-  gence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' These numbers are fixed across mod-  els, to ensure that when multiple models are  combined in the compositionality metric, they  have been trained for the same amount of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  For sentiment, a batch size of 4 was used, and  batch sizes {1, 4, 8} were experimented with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  For arithmetic, batch sizes {16, 32, 64} were  experimented with across 5 seeds, where 32  was selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Selection is based on validation  performance across five seeds and training  speed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' while 1 slightly outperformed 4  for sentiment analysis, we opted for 4 for com-  putational efficiency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' For both tasks, learning  rates {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='001, 2e-4, 1e-4} were experimented  with, and 2e-4 was selected based on model  performance on the validation set across 5  seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' In these hyperparameter trials, the ac-  curacy was used for sentiment analysis and  the MSE was used for the arithmetic task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Hidden dimensionality bottleneck:  To  increase compression, the dimensionality  should clearly become smaller, so we man-  ually select 8 values to run ranging from 150  (the standard dimension) to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Dropout bottleneck: For dropout, the closer  to 1, the more compression there is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We manu-  ally select 8 values to run from 0 (the standard  amount) to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  DVIB: For DVIB, preliminary experiments  indicated that β > 1 for arithmetic or β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='1  for sentiment debilitates the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We man-  ually selected 8 values to run for β based  on that information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We use the implemen-  tation of Li and Eisner (2019) for the DVIB,  available at: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='com/XiangLi1999/  syntactic-VIB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Evaluation metrics: The evaluation metric  used for arithmetic is the MSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The evaluation  metrics for sentiment analysis are the accuracy  of the predicted sentiment class, and the F1-  score, macro-averaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Number of runs & run time: The results for  both tasks are averaged over ten seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Train-  ing one model with one seed on CPU lasts  up to 40 minutes for the sentiment analysis  models, and up to 30 minutes for the arith-  metic task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The TRE-training setup typically  takes twice as long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We utilise CPUs from the  icelake partition of the CSD3 cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  For the datasets used, the following are relevant  details in terms of their size and preprocessing:  Arithmetic task: The arithmetic task was  generated using the implementation of Hup-  kes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' (2018), available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  com/dieuwkehupkes/processing_arithmetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  We augment the data with the ambiguous  examples ourselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The training data consist  of 14903 expressions with 1 to 9 numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  We test on expressions with lengths 5 to  9, using 5000 examples per length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  2100  additional examples are used as validation  data, to track the model’s behaviour during  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Stanford Sentiment Treebank:  We col-  lect the SST data from the pytreebank  package (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='com/JonathanRaiman/  pytreebank).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' For the (Tree-)LSTM models, we  further preprocess the inputs by lowercasing  the sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' We use SST-5, that classifies  sentences using five classes ranging from very  negative to very positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' The standard train,  validation and test subsets have 8544, 1101  and 2210 examples, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content=' Each node  in the input trees has its own sentiment label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  For the experiments contained in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='4,  we did not run an extensive hyperparameter search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
+page_content='  Those results are averaged over five seeds, and the  models were trained on one NVIDIA A100-SXM-  80GB, where training one seed lasted up to 15  minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFST4oBgHgl3EQfEjgF/content/2301.13714v1.pdf'}
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+Draft version January 5, 2023
+Typeset using LATEX twocolumn style in AASTeX631
+TESS Discovery of Twin Planets near 2:1 Resonance around Early M-Dwarf TOI 4342
+Evan Tey
+,1 Chelsea X. Huang
+,2 Michelle Kunimoto
+,1 Andrew Vanderburg
+,1 Avi Shporer
+,1
+Samuel N. Quinn
+,3 George Zhou
+,2 Karen A. Collins
+,3 Kevin I. Collins
+,4 Eric L. N. Jensen
+,5
+Richard P. Schwarz
+,3 Ramotholo Sefako,6 Tianjun Gan
+,7 Elise Furlan
+,8 Crystal L. Gnilka
+,9
+Steve B. Howell
+,9 Kathryn V. Lester
+,9 Carl Ziegler,10 C´esar Brice˜no,11 Nicholas Law,12
+Andrew W. Mann
+,12 George R. Ricker
+,1 Roland K. Vanderspek
+,1 David W. Latham
+,3
+S. Seager
+,1, 13, 14 Jon M. Jenkins
+,9 Joshua N. Winn,15 Douglas A. Caldwell
+,9, 16 David Charbonneau
+,3
+Christopher J. Burke
+,1 and Zahra Essack
+1, 13
+1Department of Physics and Kavli Institute for Astrophysics and Space Science, Massachusetts Institute of Technology, 77 Massachusetts
+Ave, Cambridge, MA, 02139, USA
+2University of Southern Queensland, West St, Darling Heights, Toowoomba, Queensland, 4350, Australia
+3Center for Astrophysics — Harvard & Smithsonian, 60 Garden St, Cambridge, MA, 02138, USA
+4George Mason University, 4400 University Drive, Fairfax, VA, 22030, USA
+5Department of Physics and Astronomy, Swarthmore College, 500 College Ave, Swarthmore, PA, 19081, USA
+6South African Astronomical Observatory, P.O. Box 9, Observatory, Cape Town, 7935, South Africa
+7Department of Astronomy and Tsinghua Centre for Astrophysics, Tsinghua University, Beijing, 100084, China
+8Caltech/IPAC-NExScI, NASA Exoplanet Science Institute, 1200 East California Boulevard, Mail Code 100-22, Pasadena, CA, 91125,
+USA
+9NASA Ames Research Center, Moffett Field, CA, 94035, USA
+10Department of Physics, Engineering and Astronomy, Stephen F. Austin State University, 1936 North St, Nacogdoches, TX, 75962, USA
+11Cerro-Tololo Inter-American Observatory, Casilla 603, La Serena, Chile
+12Department of Physics and Astronomy, The University of North Carolina at Chapel Hill, Chapel Hill, NC, 27599-3255, USA
+13Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge,
+MA, 02139, USA
+14Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139,
+USA
+15Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ, 08544, USA
+16SETI Institute, Moffett Field, Mountain View, CA, 94035, USA
+ABSTRACT
+With data from the Transiting Exoplanet Survey Satellite (TESS), we showcase improvements to the
+MIT Quick-Look Pipeline (QLP) through the discovery and validation of a multi-planet system around
+M-dwarf TOI 4342 (Tmag = 11.032, M⋆ = 0.63 M⊙, R⋆ = 0.60 R⊙, Teff = 3900 K, d=61.54 pc). With
+updates to QLP, including a new multi-planet search, as well as faster cadence data from TESS ’ First
+Extended Mission, we discovered two sub-Neptunes (Rb = 2.266+0.038
+−0.038 R⊕ and Rc = 2.415+0.043
+−0.040 R⊕;
+Pb = 5.538 days and Pc = 10.689 days) and validated them with ground-based photometry, spectra,
+and speckle imaging. Both planets notably have high transmission spectroscopy metrics (TSMs) of 36
+and 32, making TOI 4342 one of the best systems for comparative atmospheric studies. This system
+demonstrates how improvements to QLP, along with faster cadence Full-Frame Images (FFIs), can
+lead to the discovery of new multi-planet systems.
+Keywords: planetary systems, planets and satellites: detection, stars: individual (TOI 4342)
+1. INTRODUCTION
+The Transiting Exoplanet Survey Satellite (TESS,
+Ricker et al. 2015) launched on April 18, 2018 with a
+goal of discovering transiting exoplanets around bright
+stars across the entire sky. During every sector (∼ 27
+days), TESS observes a 24◦ × 96◦ swath of the sky dur-
+ing two elongated orbits around the Earth before shift-
+ing to the next sector. During its Primary Mission (2018
+Jul 25 – 2020 Jul 04), TESS collected photometry at a
+2 minute cadence for ∼ 20,000 targets each sector pre-
+selected from the Candidate Target List (CTL, Stassun
+et al. 2018), while Full-Frame Images (FFIs) were col-
+arXiv:2301.01370v1  [astro-ph.EP]  3 Jan 2023
+
+ID2
+Tey et al.
+lected at a 30 minute cadence. In these 26 sectors, TESS
+covered 70% of the sky and found 2241 transiting planet
+candidates (Guerrero et al. 2021).
+The MIT Quick Look Pipeline (QLP, Huang et al.
+2020a) has been an important contributor to detecting
+these planet candidates, expanding the search from the
+∼ 200, 000 CTL stars to millions of stars brighter than
+Tmag = 10.5 in the FFIs. Every sector, QLP searches
+for planet candidates around stars with Tmag < 10.5,
+making full use of all past sectors of data.
+Notably,
+∼ 1000 candidates from the Primary Mission were found
+around stars not on the CTL – a majority of which were
+detected by QLP.
+With its 1st Extended Mission (2020 Jul 04 – 2022 Sep
+01), TESS, and QLP in particular, was well-positioned
+to yield even more candidates, especially as the FFI
+recording cadence was reduced from 30 minutes to 10
+minutes. On 2022 Sep 01, TESS will have started its
+Second Extended Mission, and the FFI recording ca-
+dence will be reduced further to 200 seconds.
+These
+reductions allow the flux-time series from the FFIs to
+better resolve transit ingresses and egresses and there-
+fore improve planet detection efficiency. QLP, however,
+still has room for improvement. We are introducing here
+three major changes to the pipeline:
+• Systematic Multi-planet Search: In the Primary
+Mission, QLP only sent the most promising tran-
+sit candidate from each light curve through future
+stages of vetting. Additional candidates could be
+reported if the TOI vetters noticed them during
+visual inspection of the light curves, but no ex-
+plicit search was otherwise done for these addi-
+tional planets. Based on our knowledge of close-in
+exoplanet systems from the Kepler mission, a ma-
+jority of the small planets reside in multiple plane-
+tary systems (Winn & Fabrycky 2015). Adding an
+automatic search for multiple transit signals will
+potentially introduce new interesting systems for
+follow-up studies and allow future statistical stud-
+ies of system architecture.
+• Improved Difference Images:
+Difference images
+(Bryson et al. 2013) are an effective method of
+determining the source location of a transit signal
+in the sky. For the Primary Mission, QLP used a
+simplistic algorithm for creating difference images
+by directly subtracting the median stacked frames
+in the in-transit and out-of-transit time windows.
+By more intelligently selecting frames to avoid sys-
+tematics, transit ingress / egress, and additional
+planets in the system, we improve the robustness
+of the difference images. This reduces QLP’s false
+positive rate and allows us to reduce human-review
+times or alternatively expand our search to more
+stars.
+• Quaternion Detrending:
+Previously,
+QLP de-
+trended light curves by fitting and removing ba-
+sis splines to correct for long-timescale systemat-
+ics (Vanderburg & Johnson 2014). However, there
+are residual systematics on shorter timescales due
+to space craft jitter motions that can have signif-
+icant effects on light curves, especially for bright
+stars (Vanderburg et al. 2019; Huang et al. 2020b).
+We can measure and correct for this jitter with
+TESS quaternion data – 3-vector time series data
+describing spacecraft attitude every two seconds.1
+For transit signals with relatively small depth and
+short duration, correcting these systematics is im-
+portant to improving QLP’s detection sensitivity.
+Altogether, these improvements to QLP – along with
+extended light curves at faster cadence – will increase the
+scientific output of TESS as a whole. One population
+that will especially benefit from this are multi-planet
+systems around M-dwarfs.
+M-dwarfs are known to
+frequently host multi-planetary systems (Ballard 2019;
+Dressing & Charbonneau 2015). The duration of plane-
+tary transits around M-dwarfs are often relatively short,
+making these transits easily diluted in the 30 minute ca-
+dence TESS FFI light curves from the Primary Mission.
+M-dwarfs are also the most abundant type of star in our
+galaxy. So, even though many have been selected for 2
+minute cadence target pixel stamp observations, plenty
+are only monitored by the FFIs.
+The regular multi-
+sector processing approach of the QLP also enables us
+to take the most advantage of all the available data. Fi-
+nally, M-dwarfs are also ideal host stars for atmospheric
+characterization of their transiting planets. Determining
+atmospheric composition, especially for multiple planets
+in a single system, allows us to compare formation and
+evolutionary histories.
+As of July 2022, TESS has detected 373 candidate
+transiting planetary systems around nearby M-dwarfs
+2. 36 of these systems host multiple planets, including
+TOI 270 (G¨unther et al. 2019; Van Eylen et al. 2021),
+TOI 175 (Kostov et al. 2019; Cloutier et al. 2019), and
+TOI 700 (Gilbert et al. 2020; Rodriguez et al. 2020).
+Building off the original Kepler mission (Kepler Mission
+1 Quaternion data is available online at https://archive.stsci.edu/
+missions/tess/engineering/
+2 The TESS candidate list was downloaded from the NASA Exo-
+planet Archive on July 7, 2022.
+
+TESS Discovery of Twin Planets Around TOI 4342
+3
+2019), this doubles the number of discovered transiting
+M-dwarf multi-planet systems.
+The discovery of TOI 4342, an M-dwarf system with
+two transiting sub-Neptunes, showcases the power of
+the Extended Mission FFIs with the newly added im-
+provements to QLP for detecting multi-planet M-dwarf
+systems. With the longer baseline, we have more op-
+portunities to catch transits and, together with the
+faster cadence, we have better statistics to detect the
+shorter-duration transits that come with M-dwarf sys-
+tems.
+Adding short-timescale systematic corrections
+adds to this sensitivity, and the multi-planet search
+is what allows us to discover this multi-planet system
+where the previous version of QLP could not make this
+discovery.
+TOI 4342 (TIC 354944123; Tmag = 11.032; d =
+61.54 pc) is an M0V-type star with two transiting sub-
+Neptunes TOI 4342 b (2.27 R⊕) and TOI 4342 c (2.41
+R⊕). The planets are near a 2:1 mean-motion resonance
+with periods of 5.538 and 10.689 days. Both planets are
+good targets for atmospheric characterization & com-
+parison studies with transmission spectroscopy metrics
+≥ 30 (TSM, Kempton et al. 2018).
+Both planets were detected with TESS and followed
+up with ground-based photometry, reconnaissance spec-
+troscopy, and high resolution imaging. In Section 2 we
+describe these observations. In Section 3 we perform fits
+and validate both signals as planetary transits around
+an M0V host star. In Section 4 we re-emphasize the im-
+provements to QLP and describe the TOI 4342 system
+in the context of small planets, multi-planet systems,
+and planets orbiting M-dwarfs.
+2. OBSERVATIONS AND DATA ANALYSIS
+2.1. TESS
+TOI 4342 was first observed by TESS in Sector 13
+(Primary Mission, 2019 Jun 19 – Jul 18) then observed
+again in Sector 27 (1st Extended Mission, 2020 Jul 4 –
+30, 2020) as a 2-min target due to its brightness and
+small radius (Stassun et al. 2019), as well as in FFIs.
+The 2-min data were reduced by the Science Pro-
+cessing Operations Center (SPOC) pipeline at NASA
+(Jenkins et al. 2016). The planets’ transit signals were
+detected during a SPOC multisector search of Sec-
+tors 13 and 27 on 26 May 2021 with an adaptive,
+noise-compensating matched filter (Jenkins 2002; Jenk-
+ins et al. 2010, 2020).
+In the multisector search, the
+5-day signal was detected with SNR 10.8 and multi-
+ple event statistic (MES) of 8.5, and the 10-day signal
+was detected with SNR 9.0 and MES 8.2. Both tran-
+sit signatures passed all the diagnostic tests reported in
+the SPOC data validation reports (Twicken et al. 2018;
+Li et al. 2019), including the difference image centroid
+tests, which located the source of the transits to within
+8.7 +/- 5.5 arcsec and 3.4 +/- 3.6 arc sec of the target
+star image for TOI 4342 b and c, respectively. Both can-
+didates were classified as high quality candidates by the
+TESS-ExoClass classifier 3.
+TESS-ExoClass applies a
+series of tests based on the Kepler Robovetter (Coughlin
+et al. 2016; Thompson et al. 2018) to pass the best can-
+didates on to the manual TOI vetting process (Guerrero
+et al. 2021). These exoplanet signatures were alerted as
+TOI 4342.01 and .02 on 28 July 2021 (Guerrero et al.
+2021).
+The FFIs meanwhile were reduced by QLP, but TOI
+4342’s TESS band brightness is below the threshold to
+be released as a QLP TOI (10.5) (Guerrero et al. 2021).
+While developing the QLP improvements listed in Sec-
+tion 1, we conducted an independent search for multi-
+planet systems in QLP light curves and found two planet
+candidates in TOI 4342. For both sectors, our analysis
+started with raw QLP light curves, which uses calibrated
+FFIs from the MIT TESS image calibration software
+(TICA, Fausnaugh et al. 2020) 4. Figure 1 shows TOI
+4342 and the surrounding field.
+This multi-planet search consisted of iterative applica-
+tions of the Box Least Squares algorithm (BLS, Kov´acs
+et al. 2002) as implemented in VARTOOLS (Hartman &
+Bakos 2016), masking out transits for previously found
+signals when searching for new planets. We then also
+performed an additional step of light curve detrending
+to remove short-timescale systematics. This was done
+by decorrelating spacecraft motion from the light curve
+using leading statistical moments (mean, standard devi-
+ation, skewness) of and covariances between the quater-
+nion time series components Q1, Q2, and Q3 calculated
+within each exposure (Vanderburg et al. 2019). Together
+with basis splines to remove long-timescale trends (Van-
+derburg & Johnson 2014), we perform iterative fits to
+the light curve, removing 3-σ outliers until the fit con-
+verged. The resulting combined trend was subtracted
+out to produce a final corrected QLP light curve.
+With these adjustments, we found two signals. The
+first signal has a period of 5.538 days with signal-to-
+noise ratio (SNR) of 13.126 (5.14 per transit), and the
+second has a period of 10.688 days with SNR 11.506
+(6.74 per transit). To highlight the impact of the first
+Extended Mission and QLP improvements, we specif-
+ically searched for planets in Sector 13 (Primary Mis-
+3 https://github.com/christopherburke/TESS-ExoClass
+4 TICA FFIs are available as High Level Science Products at the
+Mikulski Archive for Space Telescopes (MAST): https://archive.
+stsci.edu/hlsp/tica
+
+4
+Tey et al.
+Figure 1. Images of the field surrounding TOI 4342 in 1976 (left), 1992 (top right), and 1995 (bottom right). In all images,
+the orange circle shows the 2021.0 location of TOI 4342 and the blue circle shows the location on J2000. Both circles have radii
+of 2′′. Nearby stars in the TIC are shown in green with marker size corresponding to brightness. In red is a 10′′ scale (∼ half a
+TESS pixel).
+sion) and Sector 27 (first Extended Mission) separately.
+Corrected for the number of transits seen in each sec-
+tor, we saw average SNR per transit of 3.70 and 4.83 in
+Sector 13 for each signal respectively. In Sector 27, this
+improved to 5.52 and 5.99. We also compared the SNRs
+with the original QLP detrending method (only correct-
+ing for long-timescale systematics) and found worse per-
+formance without the quaternion correction (SNRs per
+transit of 5.07 and 5.32).
+For the remainder of the system modeling in this work,
+we use the Simple Aperture Photometry (SAP) light
+curve from the SPOC pipeline (Twicken et al. 2010;
+Morris et al. 2020) with the following pre-processing.
+First we removed contamination from nearby stars by
+scaling the light curves by the SPOC-provided CROWDSAP
+values. We also ignored data with nonzero SPOC quality
+flags. This included an anomalous event during Sector
+13 (TJD 1665.2983 to 1665.3501) where the spacecraft
+fell out of fine pointing. The resulting light curve can
+be seen in Figure 3 with its Lomb-Scargle periodogram.
+In the periodogram, we see a peak signifying stellar
+variability at a period of about 13 days.
+To remove
+this variability and other instrumental systematics, we
+conducted our own correction similar to our new QLP
+correction. First, we excluded data from QLP-predicted
+transit times, then we split the light curve into indi-
+vidual spacecraft orbits (with two orbits per sector) to
+be corrected separately. We again iteratively removed
+short-timescale systematics with quaternion time se-
+ries statistics and long-timescale systematics with basis
+spline fits. This final fit was then divided out to produce
+our corrected light curve. Figure 2 shows the detrended
+2-minute cadence light curve with transits highlighted.
+2.2. Ground-based Photometry
+We obtained seeing-limited ground-based follow-up
+observations from the TESS Follow-up Observing Pro-
+gram Sub Group 1 (TFOP SG1; Collins 2019).
+The
+ground-based observations have much higher spatial res-
+olution than TESS and can help confirm the source
+location of a TESS transit signal and can provide ad-
+ditional transit observations to refine ephemerides for
+predicting future transits. We used the TESS Transit
+Finder, which is a customized version of the Tapir soft-
+ware package (Jensen 2013), to schedule our transit ob-
+servations.
+
+SERC-J Survey (1976-08-29)
+AAO-SES Survey (1992-08-21)
+Blue Photographic Plate
+Red Photographic Plate
+TESS Magnitude
+40"
+20
+18
+17
+45"
+-77°58'20"
+Dec
+50"
+-77°58'55'
+Dec
+40"
+SERC-I Survey (1995-09-03)
+IR Photographic, Plate
+40"
+45"
+Dec
+59'00"
+50"
+2000
+2021
+-77°58'55'
+21h37m42s
+36s
+30s
+21h37m36s
+34s
+32s
+RA
+RATESS Discovery of Twin Planets Around TOI 4342
+5
+Figure 2. Top: Detrended TESS light curve in gray, binned values in purple, with transits highlighted for each TOI 4342 b
+(blue) and TOI 4342 c (orange). Middle: TESS and LCOGT light curves folded on the best fit period and epoch found in
+Section 3.2.1. Different background-subtracted observations are shown in different colors for each transit with the best fit model
+in black. Bottom: Residual flux between the fully combined light curve and the model.
+Between UT 2021-05-30 and 2021-09-14, 4 transits of
+TOI 4342 b and 5 transits of TOI 4342 c were observed in
+the Sloan i′ band using the Las Cumbres Observatory
+Global Telescope (LCOGT; Brown et al. 2013) 1.0 m
+network.
+The observations were taken at the Siding
+Spring Observatory (SSO), South Africa Astronomical
+Observatory (SAAO) and Cerro Tololo Inter-American
+Observatory (CTIO) nodes of the LCOGT network and
+are summarized in Table 1. We use 7 of the 9 transits
+as 2 were cut short for bad weather conditions. The 1 m
+telescopes are equipped with 4096 × 4096 SINISTRO
+cameras having an image scale of 0.′′389 per pixel, re-
+sulting in a 26′ × 26′ field of view. The images were cal-
+ibrated by the standard LCOGT BANZAI pipeline (Mc-
+Cully et al. 2018). Differential photometric data were
+extracted with AstroImageJ (Collins et al. 2017a) using
+circular photometric apertures with radii 6.′′0. Thus, the
+TOI 4342 aperture excludes flux from the nearest known
+Gaia DR3 and TICv8 star (TIC 2025922721) 16′′ east
+of TOI 4342. As shown in Section 3.2, the transit sig-
+nals are detected on-target relative to known Gaia DR3
+stars.
+We also checked the light curves of Gaia DR3
+sources within 2.′5 of TOI 4342 and found no evidence
+of nearby eclipsing binary systems that could be causing
+the TESS detection. All SG1 data can be found online
+at ExoFOP (2019).
+2.3. Reconnaissance Spectroscopy
+We obtained 9 spectra of TOI 4342 over two seasons
+in slicer mode with the fiber-fed high resolution echelle
+spectrograph CHIRON (Tokovinin et al. 2013).
+CHI-
+
+3000
+2000
+: Flux (ppm)
+1000
+0.
+Relative I
+-1000
+-2000
+-3000
+1655
+1660
+1665
+1670
+1675
+1680
+2040
+2045
+2050
+2055
+2060
+BJD-2457000 [Days]
+BJD-2457000 [Days]
+TOI 4342 b
+TOI 4342 c
+3000
+3000
+TESS
+TESS
+2000
+SAAO 2021-06-19 UT
+2000
+CTIOa 2021-08-13 UT
+(ppm)
+CTIO 2021-07-12 UT
+CTIOb 2021-08-13 UT
+SSO 2021-08-14 UT
+CTIO 2021-09-14 UT
+1000
+1000
+: Flux
+CTIO 2021-08-31 UT
+0
+0:
+Relative
+-1000
+-1000
+2000
+-2000
+-3000
+-3000
+-4 
+-2
+0
+2
+4
+-4
+-2
+0
+2
+4
+Hours frorm mid-transit
+Hours from mid-transit
+3000
+3000
+Combined
+2000
+2000
+(wdd)
+1000
+1000
+Residual flux (
+0
+0.
+-1000
+-1000
+2000
+-2000
+Combined
+-3000
+-3000
+0
+2
+-2
+4
+-4
+0
+2
+Hours frorm mid-transit
+Hours from mid-transit6
+Tey et al.
+Table 1.
+SG1 Follow-Up Observations
+Target
+Instrument
+Date (UT)
+Filter
+Aperture
+Observing Notes
+TOI 4342 c
+LCO-CTIO 1.0m
+2021-09-14
+i′
+5.1′′
+TOI 4342 b
+LCO-CTIO 1.0m
+2021-08-31
+i′
+6.2′′
+TOI 4342 b
+LCO-SSO 1.0m
+2021-08-14
+i′
+5.1′′
+TOI 4342 c
+LCO-CTIO 1.0m a
+2021-08-13
+i′
+5.9′′
+Simultaneous observation a
+TOI 4342 c
+LCO-CTIO 1.0m b
+2021-08-13
+i′
+5.9′′
+Simultaneous observation a
+TOI 4342 b
+LCO-CTIO 1.0m
+2021-07-12
+i′
+6.2′′
+TOI 4342 c
+LCO-SAAO 1.0m
+2021-07-11
+i′
+7.0′′
+Noisy, cut short by weather
+TOI 4342 b
+LCO-SAAO 1.0m
+2021-06-19
+i′
+8.6′′
+TOI 4342 c
+LCO-CTIO 1.0m
+2021-05-30
+i′
+7.8”
+Partial, cut short by weather
+aTransit observed simultaneously by two distinct telescopes at CTIO.
+Figure 3. Top: SPOC SAP light curve with our correction trend (described in Section 2.1) in blue. This trend, formed from
+quaternion data and basis splines to remove systematics and long timescale variability, is divided out of the raw light curve
+to create our final flattened light curve. Bottom: Lomb-Scargle periodograms of the SAP SPOC light curve for each sector of
+observation. We see peaks around 14 days marking stellar variability, but we cannot pin down a specific period due to the large
+gap between observations.
+RON is mounted on the 1.5 m SMARTS telescope, lo-
+cated at the CTIO, Chile, and has a spectral resolving
+power of 80,000. The spectra were taken using 1 hour
+long exposures and were extracted by the standard CH-
+IRON pipeline (Paredes et al. 2021).
+We derived the radial velocities using a cross corre-
+lation against a median combined template spectrum.
+The template spectrum is composed of a median combi-
+nation of all CHIRON spectra, each shifted to rest after
+an approximate velocity measurement via a cross corre-
+lation against a synthetic template. The measured ve-
+locity of each spectrum is that of the mean velocity from
+each spectral order, weighted by their cross correlation
+function heights. The velocity uncertainties were esti-
+mated from the scatter of the per-order velocities. We
+find a mean internal uncertainty of ∼ 15 m s−1, with an
+RMS of ∼ 22 m s−1 between the 9 measurements. The
+full dataset can be found in Table 2.
+2.4. High-Resolution Speckle Imaging
+“Third-light” flux contamination from a close stellar
+companion can lead to an underestimated planetary ra-
+dius if not accounted for in the transit model (Ciardi
+et al. 2015) and even cause non-detections of small plan-
+ets residing within the same exoplanetary system (Lester
+
+1.02
+0.98
+1654
+1658
+1662
+1666
+1669
+1673
+1677
+1681
+2037
+2041
+2045
+2049
+2050
+2054
+2058
+2062
+BJD - 2457000 [Days]
+BJD - 2457000 [Days]
+BJD - 2457000 [Days]
+BJD - 2457000 [Days]
+0.8
+S13
+S27
+0.4
+0.2 -
+0.0 -
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+22
+24
+26
+DaysTESS Discovery of Twin Planets Around TOI 4342
+7
+Table 2.
+Radial Velocities of TOI 4342
+BJD
+RV
+σRV
+Instrument
+(km s−1)
+(km s−1)
+2459359.91837
+−5.726
+0.013
+CHIRON
+2459360.85475
+−5.760
+0.018
+CHIRON
+2459379.77461
+−5.795
+0.020
+CHIRON
+2459384.82649
+−5.770
+0.011
+CHIRON
+2459409.75314
+−5.788
+0.012
+CHIRON
+2459700.86421
+−5.734
+0.018
+CHIRON
+2459725.87319
+−5.747
+0.016
+CHIRON
+2459740.83584
+−5.737
+0.016
+CHIRON
+2459742.79459
+−5.761
+0.012
+CHIRON
+et al. 2021). The discovery of close, bound companion
+stars, which exist in nearly one-half of FGK type stars
+(Matson et al. 2018) and less so for M class stars, pro-
+vides crucial information toward our understanding of
+exoplanetary formation, dynamics and evolution (How-
+ell et al. 2021).
+Thus, to search for close-in bound
+companions unresolved by TESS, Gaia, or ground-based
+seeing-limited follow-up observations, we obtained high-
+resolution imaging speckle observations of TOI 4342.
+TOI 4342 was observed at the 4.1-m Southern Astro-
+physical Research (SOAR) telescope (Tokovinin 2018)
+on 1 October 2021 UT, in Cousins I-band, a similar vis-
+ible bandpass as TESS. This observation was sensitive to
+a 5-magnitude-fainter star at an angular distance of 1′′
+from the target. More details of the observations within
+this survey are available in Ziegler et al. (2020). The 5σ
+detection sensitivity and speckle auto-correlation func-
+tions from the observations are shown in Figure 4. No
+nearby stars were detected within 3′′ of TOI 4342 in the
+SOAR observations.
+TOI 4342 was also observed on July 23 2021 UT using
+the Zorro speckle instrument on the Gemini South 8-m
+telescope5 (Scott et al. 2021). Zorro provides simultane-
+ous speckle imaging in two bands (562 nm and 832 nm)
+with output data products including a reconstructed im-
+age with robust contrast limits on companion detections
+(e.g., Howell et al. 2016). During this observing run, the
+blue channel was inoperable, thus only 832 nm observa-
+tions were obtained. Thirteen sets of 1000 X 0.06 sec
+exposures were collected and subjected to Fourier anal-
+ysis in the standard reduction pipeline (see Howell et al.
+2011). Figure 5 shows our final contrast curve and the
+5 https://www.gemini.edu/sciops/instruments/alopeke-zorro/
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+∆arcsec
+0
+1
+2
+3
+4
+5
+6
+∆magnitude (I-band)
+2
+0
+-2
+∆α [arcsec]
+-2
+0
+2
+∆δ [arcsec]
+SOAR Speckle ACF
+TIC354944123
+Figure 4.
+SOAR speckle auto-correlation function (inset
+plot) and its 5 sigma detection sensitivity curve (main plot)
+in the Cousins-I band.
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+angular separation (arcsec)
+0
+1
+2
+3
+4
+5
+6
+7
+m
+832 nm
+TOI 4342
+1"
+832 nm
+Figure 5. Gemini South Zorro Speckle Imaging 5 sigma con-
+trast curve (full plot) and reconstructed images (inset plot)
+in the 832 nm band. The diffraction limit of the instrument
+is 20 mas.
+832 nm reconstructed speckle image. We find that TOI
+4342 is a single star with no companion brighter than
+5-6 magnitudes below that of the target star from very
+close in (0.1′′) out to 1.2′′. At the distance of TOI 4342
+(61.54 pc) these angular distances correspond to spatial
+distances of 6.2 to 74 au.
+3. ANALYSIS
+3.1. Stellar parameters
+We re-derive stellar parameters according to the fol-
+lowing empirical relations. First we take the observed
+Ks magnitude from 2MASS and the Gaia Data Release
+3 (DR3) (Gaia Collaboration et al. 2022) parallax to find
+
+8
+Tey et al.
+the absolute Ks magnitude of TOI 4342. Then, using
+the relation from Benedict et al. (2016), we get a stellar
+mass of 0.6296 ± 0.0086 M⊙. With this mass, we use
+the mass-radius relation from Boyajian et al. (2012) to
+find a stellar radius of 0.599 ± 0.013 R⊙. We check this
+with the absolute Ks magnitude-radius relation from
+Mann et al. (2015) where we find a consistent value of
+0.598 ± 0.018 R⊙.
+From Mann et al. (2015), we also
+calculate a bolometric luminosity of 0.0746 ± 0.0053 L⊙
+via the observed V and J magnitudes and the resulting
+bolometric correction. Using the Stefan-Boltzmann law,
+we find an effective temperature of 3901 ± 69 K, giving
+TOI 4342 a spectral type around M0V. These values are
+consistent with the stellar parameters listed in TIC 8.2
+of 0.587±0.020 M⊙, 0.598±0.018 R⊙, 0.073±0.016 L⊙,
+and 3880 ± 160 K, which were derived using the older
+Gaia DR2.
+3.2. Light curve modeling
+3.2.1. Best Fit Model
+We simultaneously fit the detrended TESS light curve
+and all 7 of the 9 SG1 light curves unaffected by weather
+(see Table 1) using exoplanet (Foreman-Mackey et al.
+2021).
+For TESS light curves, we approximate the per point
+measured uncertainty as 1.4826 times the median abso-
+lute deviation (MAD) of the flux within each TESS or-
+bit. For each SG1 light curve, we use the reported flux
+uncertainties, and on top of that, fit for a jitter term
+(added in quadrature) to capture additional errors in
+the observation’s error budget. We also simultaneously
+fit a second order polynomial to the SG1 light curves
+to account for nightly trends. To model the stellar limb
+darkening we use quadratic limb darkening models with
+uninformative priors following Kipping (2013a) for each
+observation band.
+For both orbits, we assume an eccentricity of 0.
+6
+Periods and epochs for each planet were given uniform
+priors centered on the values found via BLS search to
+±10% of the period. The ratios of planet to stellar radii
+had uniform priors from 0 to 1, and impact parameters
+had uniform priors from −(1+Rp/R∗) to +(1+Rp/R∗)
+(though when reported, we take the absolute value). Fi-
+nally, the stellar mass and radius were given normal pri-
+ors using the parameters derived in Section 3.1.
+exoplanet uses PyMC3 (Salvatier et al. 2016) to per-
+form No U-turn Sampling (Hoffman & Gelman 2011)
+6 Based on priors from Kipping (2013b) and Eylen et al. (2019), we
+expect the eccentricities to be low. We also performed a separate
+fit with an uninformed prior on the eccentricity and argument of
+periapsis, and did not see significant changes in the results.
+from the posterior distribution. We sampled five inde-
+pendent chains with 5000 tuning steps and 5000 draws.
+All parameters converged with Gelman-Rubin statistic
+≤ 1.01 (Gelman & Rubin 1992).
+Table 3 shows the
+median sampled and derived values with 1 sigma confi-
+dence intervals, and Figure 2 shows the median posterior
+model with residuals.
+3.2.2. Transit Shape Model
+Separately, using exoplanet’s SimpleTransitOrbit
+we also performed a TESS-only fit to best constrain each
+candidate’s transit shape – the ratio between the dura-
+tion of the flat part of the transit (tF ) to total transit
+duration (tT ).
+Again, we assume a Gaussian noise model on top of a
+constant baseline. Priors on period and epoch were set
+to uniform distributions centered around the BLS peri-
+ods and epochs with bounds of ±10% of the period. Du-
+rations were given uniform priors from 0 to 2 times the
+BLS durations, Rp/R⋆ were given uniform priors from 0
+to 1, and impact parameters were given uniform priors
+from −(1 + Rp/R∗) to +(1 + Rp/R∗). Since we want
+to find the best characterization of the transit shape for
+each planet independently from the host star, we use two
+independent star models with loose stellar radius priors
+from 0 to 2 R⊙ and quadratic limb darkening with Kip-
+ping (2013a)’s priors.
+Following the same fitting configuration as Section
+3.2.1, we can calculate the transit shape following Equa-
+tion 15 from (Seager & Mall´en-Ornelas 2003).
+�tF
+tT
+�2
+= (1 − Rp/R∗)2 − b2
+(1 + Rp/R∗)2 − b2
+(1)
+where Rp is planet radius, R∗ is stellar radius, and b
+is impact parameter. For TOI 4342 b and TOI 4342 c
+respectively, we find median and one sigma confidence
+intervals of 0.912+0.013
+−0.026 and 0.907+0.016
+−0.039.
+3.3. Radial Velocity Modeling
+Using the CHIRON data (Table 2), we can place mass
+upper limits on both planets by fitting for the ampli-
+tudes of simple sinusoids assuming circular orbits. This
+puts an upper bound on the radial velocity variations,
+meaning we can constrain any transiting companion
+masses to be planetary rather than stellar.
+We define our model with three unknowns: the base-
+line radial velocity, Kb, and Kc, where the K’s are
+semi-amplitudes of sinusoids set to the median periods
+and epochs from Section 3. After fitting, we find semi-
+amplitudes of 19.4+5.1
+−5.0 m s−1 and 14.4+3.6
+−3.6 m s−1 for each
+signal respectively. These posteriors give us 3-σ mass
+upper limits of 0.321 MJup and 0.289 MJup both much
+smaller than stellar masses.
+
+TESS Discovery of Twin Planets Around TOI 4342
+9
+3.4. Photocenter motion
+Photocenter motion analysis can help determine if the
+location of a transit signal matches the location of the
+target star on the sky.
+An effective method of determining both of these loca-
+tions is with the difference imaging technique (Bryson
+et al. 2013), designed initially for the Kepler Mission
+and inherited by the SPOC pipeline, whereby the dif-
+ference of averaged in- and out-of-transit pixel images is
+found. Assuming stellar variability and/or instrumental
+systematics are negligible on transit timescales, the dif-
+ference image should appear star-like at the location of
+the transit signal source. Meanwhile, the out-of-transit
+image should represent a direct image of the field sur-
+rounding the target star. If the field is relatively un-
+crowded and the target star is indeed the source of the
+transit signal, the difference and direct images should
+appear similar.
+To produce difference and direct images, we used
+TESS-plots7, a publicly available Python package for
+robust pixel-level analysis of TESS FFIs. For planets
+in multi-planet systems such as TOI 4342, TESS-plots
+masks out all cadences corresponding to other planet
+transits. This ensures that the primary source of vari-
+ability in the difference image is due to the planet of
+interest. TESS-plots also puts more care in choosing
+which in- and out-of-transit frames are used compared
+to the original difference images in QLP. It ignores the
+first and last 5% of the transit duration to better avoid
+ingress / egress. It also places a buffer between in- and
+out-of-transit frames to handle underestimated transit
+durations. Finally, “bad” transits (transits with lots of
+missing or poor quality datapoints) are discarded to pre-
+vent contamination of the overall difference image. Al-
+together, with these improvements, TESS-plots marks
+an important update over the Primary Mission QLP dif-
+ference images.
+Figure 6 shows difference images for TOI 4342 b and
+TOI 4342 c next to the direct image for TOI 4342, us-
+ing Sector 27 FFIs. The difference images confidently
+rule out the transit signals as coming from other nearby
+sources listed in the TIC, consistent with the centroid
+analysis derived by the SPOC pipeline. For TOI 4342
+b, the SPOC-derived photo centroid is only 2.5 σ away
+from the location of TIC 2025922721 (T=19.805 mag).
+However, this star is too faint to produce the transit
+depths observed on TOI 4342.
+3.5. Possible False Positive Scenarios
+7 https://github.com/mkunimoto/TESS-plots
+In this section we rule out possible false positive sce-
+narios where the signals are not coming from a multi-
+planet system.
+3.5.1. TOI 4342 is an eclipsing star system
+One possible source of false positives could be signals
+from eclipsing stellar companions rather than planetary
+companions. We can rule out this scenario by consider-
+ing our radial velocity model in Section 3.3. We found
+3-σ upper limits on semi-amplitude magnitudes of 50
+m s−1 and 36 m s−1 which translate to mass limits of
+0.321 MJup and 0.289 MJup for TOI 4342 b and TOI
+4342 c respectively. In other words, if our transit sig-
+nals are caused by gravitationally bound companions
+that block out light from TOI 4342, those companions
+must have sub-Jupiter masses and cannot be from stars.
+3.5.2. Contamination from a Nearby Eclipsing Binary
+Another major source of false positives is the signal of
+an eclipsing binary (EB) in the field near our target of
+interest. Because photometers measure all light within a
+specific aperture, eclipses from a nearby EB (NEB) can
+contaminate the target aperture and cause transit-like
+events in the light curve. These false positives account
+for as much as 40% of transit-like signals at the lowest
+Galactic latitudes in the Kepler field (Morton & Johnson
+2011; Bryson et al. 2013).
+We can start ruling out NEBs by restricting the sig-
+nal source to be near TOI 4342. Our photocenter mo-
+tion analysis in Section 3.4 rules out signals from known
+TICv8 stars, constraining the signal to be within ∼ 21′′
+(one TESS pixel) of TOI 4342. In Section 2.2, SG1 ob-
+servations rule out signals from the nearest Gaia DR3
+stars. In DR3 (Gaia Collaboration et al. 2022; Fabricius
+et al. 2021), Gaia has a resolution down to ∼ 0.7′′, so
+the signal must be on-target or from an NEB within an
+arcsecond of TOI 4342.
+Next, we can rule out potential NEBs by showing they
+must be brighter than certain magnitudes to cause either
+transit signal. First, we note the observed transit depth
+δobs is
+δobs = δEB ·
+f
+1 + f
+(2)
+where δEB the “true” depth (the NEB’s primary
+eclipse depth if TOI 4342 were not present), and f is
+the flux ratio between the NEB and TOI 4342.
+Then, we can place an upper bound on δEB by assum-
+ing b = 0 as in Eq. 21 from Seager & Mall´en-Ornelas
+(2003):
+δEB ≤
+(1 − tF
+tT )2
+(1 + tF
+tT )2
+(3)
+
+10
+Tey et al.
+Figure 6. Difference images for TOI 4342 b and TOI 4342 c, made using 20×20 pixel cutouts of the Sector 27 TESS FFIs (top).
+A close-up of the central 5×5 pixels is also shown (bottom). The third column shows the average of out-of-transit images, which
+should represent a direct image of the field near the target star. The target TIC-354944123 is indicated with a pink star, while
+nearby stars down to ∆T = 4 mag are plotted as white circles with sizes scaled by brightness. The difference images for both
+planet candidates indicate the transit sources are co-located with the target star. The color bars are all in units of electrons per
+second.
+Together with Equation 2, this places a lower bound
+on f purely as a function of δobs and transit shape ( tF
+tT ):
+δobs/δEB
+1 − δobs/δEB
+≤ f
+(4)
+which we can rearrange to an upper bound on NEB
+magnitude (mEB − m∗ = −2.5 log10 f where m∗ is the
+11.032, the Tmag of TOI 4342).
+Using the result of the transit shape model in Section
+3.2.2, we can derive 3-σ lower bounds on transit shape
+for each signal. These correspond to NEB magnitude
+upper bounds of 15.38 and 16.70 (∆T = 4.35 and 5.67
+mag). In other words, for an NEB to cause the TOI
+4342 b signal, it is likely within 4.35 mag of TOI 4342.
+In Figure 4, we see no stars within 5-6 mag of TOI
+4342 at separations of 1′′ – 3.0′′. This supports the rul-
+ing out of Gaia sources down to around 1′′. In Figure 5,
+no neighbors within 5-6 mag of TOI 4342 were detected
+from 0.1′′ – 1.2′′. So, altogether we can rule out NEBs
+greater than 0.1′′away from TOI 4342 as sources of the
+transit signals.
+Lastly, taking advantage of the relatively high proper-
+motion of TOI 4342, we can use archival images to rule
+out NEBs within 0.1′′of TOI 4342’s current location.
+Figure 1 shows the field surrounding TOI 4342 in 1976
+along with all known nearby TIC stars for magnitude
+reference. We see that in 1976, the 2021 location of TOI
+4342 is clear of any stars with Tmag ≲ 17 (∆T ≃ 6 mag).
+This rules out NEBs within 0.1′′of TOI 4342.
+Altogether, using our photocenter motion analysis,
+high-resolution speckle imaging, and archival field im-
+ages, we are able to rule out contamination from an
+NEB as the source of either transit signal.
+3.5.3. TOI 4342 is a hierarchical triple
+The final scenario we consider is an EB gravitationally
+bound to TOI 4342 (i.e. a hierarchical triple system).
+This EB would cause the same aperture contamination
+described in Section 3.5.2 (with the same magnitude lim-
+its), but could have evaded detection in Figure 1 because
+it would stay close to TOI 4342.
+In this scenario, if we assume the EB’s orbit is within
+0.1 arcsec of TOI 4342, its semimajor axis would be at
+most ∼ 6.2 AU (at a distance of 61.54 pc via Gaia DR3
+parallax). With Kepler’s third law and the stellar mass
+from Section 3.1, this gives the NEB an orbital period
+of ∼ 19.2 years around TOI 4342.
+From Section 3.5.2, we saw that NEBs should be
+brighter than ∆T ≃ 5 mag to cause the transit signals.
+
+Direct Image
+Difference Image
+Difference Image
+1.0
+3.0
+1500
+775
+775
+775
+0.5
+2.5
+Pixel
+1000
+770
+0.0
+770
+770
+2.0
+Row
+-0.5 765
+765
+F1.5
+765
+500
+C
+TOI-4342 b
+-1.0
+TO1-4342 
+1.0
+TO1-4342
+760
+760
+760
+595
+600
+595
+600
+605
+610
+595
+600
+605
+605
+610
+610
+772
+1.0
+772
+772
+3.0
+1500
+771
+771
+771
+0.5
+2.5
+770
+770
+770
+Pix
+1000
+-0.0
+F2.0
+☆
+Row
+769
+769
+769
+1.5
+-0.5
+500
+768
+768
+768
+TOI-4342 b
+TOI-4342 C
+TOI-4342
+1.0
+767
+-1.0 767
+767
+602
+604
+606
+602
+604
+606
+602
+604
+606
+Column Pixel
+Column Pixel
+Column PixelTESS Discovery of Twin Planets Around TOI 4342
+11
+This translates to a minimum luminosity of 7.5×10−4L⊙
+and, using L ∝ M 3.5, a minimum mass of 0.13M⊙.
+For a gravitationally bound NEB with a mass of at
+least 0.13M⊙, we would expect an edge-on radial veloc-
+ity semiamplitude of ∼ 1.8 km s−1. Gaia DR3 observed
+TOI 4342 over a 34 month baseline. In this timeframe,
+if there were a 0.13 M⊙ companion at 6.2 AU, we would
+expect an RV shift of ∼ 1 km s−1. The reported mean
+RV error though is only ∼ 0.36 km s−1. Similarly, from
+Section 3.3, our RV data spans more than a year. Based
+on this observation timeline, we expect an RV scatter
+from a companion NEB at a distance of 6.2 AU with
+0.13M⊙ would be larger than our observed 22 m s−1 at
+least 95% of the time.
+Together, since the actual RV error is lower than ex-
+pected from a companion EB for both our RV data
+and Gaia data, we conclude that these scenarios of EBs
+graviationally bound to TOI 4342 are unlikely causes of
+the transit signals.
+3.5.4. TRICERATOPS & Summary
+Using two sectors of TESS data along with additional
+photometric and spectroscopic observations, we were
+able to rule out most astrophysical false positives as
+sources of our transit signals. We considered the pos-
+sibilities that our signals are caused directly by an EB,
+by contamination from a background EB, and by cer-
+tain configurations of a hierarchical companion EB. For
+each scenario, we showed that it is highly unlikely for
+that scenario to cause our transits. In addition, we use
+triceratops (Giacalone & Dressing 2020) to indepen-
+dently check the false positive probabilities (FPPs) and
+nearby false positive probabilities (NFPPs) for each sig-
+nal. After 20 runs for each signal, we calculate mean
+and standard deviation FPPs of: 0.00211 ± 0.00018 and
+0.00320±0.00027 and NFPPs of: 0.0000138±0.0000013
+and 0.000387 ± 0.000026. Finally, Lissauer et al. (2012)
+and Guerrero et al. (2021) found that multi-candidate
+systems have lower false positive rates than single-
+candidate systems, so we receive a “multiplicity boost”,
+further decreasing the false positive probabilities and
+increasing the likelihood of having real planets. Alto-
+gether, we conclude our signals are statistically valid
+exoplanet transits.
+4. RESULTS AND DISCUSSION
+In this work, we statistically validated a pair of sub-
+Neptunes around M0V dwarf TOI 4342. In this section,
+we discuss this system in the context of other planetary
+systems.
+4.1. Mass and Atmosphere Follow-Up Characterization
+The best fit parameters from Section 3.2.1 show the
+planets have radii of 2.266+0.038
+−0.038 and 2.415+0.043
+−0.040 R⊕, and
+periods of 5.538 and 10.689 days. Given their radii, these
+planets are most likely sub-Neptunes with a significant
+fraction of H/He in their atmospheres (Rogers 2015).
+Using sub-Neptune mass-radius relationship from Wolf-
+gang et al. (2016), we expect the planets to have masses
+of 7.83+0.93
+−0.88 and 8.53+0.90
+−0.94 M⊕.
+This corresponds to expected radial velocity semi-
+amplitudes of 3.9 and 3.4 m s−1, meaning it is feasible
+to measure the precise masses using a high precision ra-
+dial velocity instrument mounted on a large telescope
+for TOI 4342 (V = 12.67 mag). With these masses, we
+will be able to compare bulk densities of the planets.
+Given the radii similarity (within 10% of each other),
+this will show the influence of different levels of irradia-
+tion (incident fluxes of ∼ 27 and ∼ 11 S⊕) on the planet
+atmospheres.
+Using the expected masses, we can also calculate
+transmission spectroscopy metrics (TSMs, Kempton
+et al. 2018), measures of how promising planets are for
+atmospheric characterization studies. For each planet,
+we find values of 36 and 32, both of which are above the
+updated follow-up threshold of ∼ 25 recommended by
+Guerrero et al. (2021) for the 100 best atmospherically
+characterizable sub-Neptunes (updated from Kempton
+et al. (2018)).
+More notably, TOI 4342 has multiple (more than one)
+high-TSM planets, making it one of the best M-dwarf
+systems for atmospheric comparison studies. Figure 7
+shows the TSM values for multi-planet M-dwarf sys-
+tems sorted by second highest TSM value of planets in
+each system. Characterizing and comparing the atmo-
+spheres of both planets will allow us to perform com-
+parative exoplanetology, and TOI 4342 is one of the
+few systems where multiple planets have characteriz-
+able atmospheres. Additionally, given the two planets
+are near a mean-motion resonance (MMR) of 2:1, it’s
+likely they migrated together to their current orbits and
+have similar primordial compositions. This would mean
+any differences detected in their atmospheric properties
+can probably be attributed to the differing levels of stel-
+lar irradiation between the planets. This will help us
+gain insights on planetary atmosphere evolution and re-
+sponses to different intensities of stellar irradiation.
+4.2. Small Planet Radius Gap
+Fulton et al. (2017) identified a radius gap for small
+planets roughly between 1.5 and 2.0 R⊕ separating
+rocky super-Earths and gaseous sub-Neptunes for sun-
+like stars. Cloutier & Menou (2020) showed this gap per-
+sisted around low-mass stars. One predominant cause of
+
+12
+Tey et al.
+Figure 7. TSM, as defined in Kempton et al. (2018), for TOIs in multi-planet systems around bright M-dwarfs (Teff < 4000 K
+and Tmag < 11.5) with Rp < 4R⊕ as of July 7, 2022. TICs are sorted decreasingly by 2nd-largest TSM value in the system. Color
+corresponds to host Tmag and size corresponds to planet radius. The gray dashed line is the atmospheric follow-up threshold
+recommended by Guerrero et al. (2021) for sub-Neptunes. TOI 4342 is among the top 10 systems with multiple planets that
+are well-suited for atmospheric characterization.
+this gap may be photoevaporation: the stripping away
+of a planet’s atmosphere as it undergoes heavy irradia-
+tion from its star (Owen & Wu 2013). Highly irradiated
+planets would be left as bare rocky cores, while less irra-
+diated planets would keep their atmospheres with larger
+mass and radii, leading to a bimodal radius distribu-
+tion. With radii of 2.27 and 2.41 R⊕, both TOI 4342
+b and TOI 4342 c appear to fall on the upper mode of
+the valley. Figure 8 shows each TOI 4342 b and TOI
+4342 c in cyan as a function of irradiation level and
+planetary radius over relative occurrence contours from
+Fulton & Petigura (2018).
+We see both planets have
+relatively low irradiation levels of ∼ 27 and ∼ 11 S⊕ re-
+spectively, meaning they are likely good fits for the low
+mass atmosphere-retaining sub-Neptune planet descrip-
+tion on the upper side of the gap.
+4.3. Transit Timing Variation
+With periods of 5.538 days and 10.689 days, TOI 4342
+b and TOI 4342 c also fall within 5% of the first order
+MMR of 2:1. Given the close distance to the 2:1 reso-
+nance, we expect the system would show transit timing
+variation (TTV) signals. Based on formulas in Lithwick
+et al. (2012), the super period of the TTV is about 157
+days, the amplitude of the TTV signal is expected to
+be on the order of a few minutes using the estimated
+mass from empirical relations. Calculations using TTV-
+Fast (Deck et al. 2014) assuming the expected planet
+masses and eccentricities smaller than 0.1 for both plan-
+ets show similar results to Lithwick et al. (2012). For
+TOI 4342, the photometric observations from ground-
+based 1-meter telescopes were able to achieve transit
+time measurements at a similar or slightly better preci-
+sion than TESS. With these observations, we can search
+for evidence of TTVs over a baseline of > 800 days via
+exoplanet’s TTVOrbit. First, we subtract the best fit
+background trends found in Section 3.2.1 from each light
+curve. We use the global best fit ephemerides to set nor-
+mal priors on transit times with standard deviations of
+7 minutes.
+The rest of the parameters (limb darken-
+ing, stellar properties, radii ratios, impact parameter)
+are initialized following the best fit model.
+Sampling
+parameters similarly followed the best global fit model.
+Based on the TTV fit, we do not observe significant de-
+viation from linear ephemerides by more than 5 mins for
+a majority of the observations, indicating that the ec-
+centricities of both planets are likely to be close to zero.
+TESS will reobserve TOI 4342 in Sectors 66 & 67 (June
+– July 2023), adding new transits that will help further
+constrain the TTV amplitudes. Additional photometry
+observations that can achieve transit center timing with
+precision better than 1 min can also provide stronger
+
+Rp(RE)
+1.2
+1.8
+100
+2.4
+3.0
+50
+TSM
+25
+10
+Tmag
+8
+9
+10
+80
+11
+1
+270
+256
+1730
+776
+233
+4342
+1266
+4599
+904
+175
+732
+1468
+969
+74
+2079
+406
+4643
+2095
+700
+2
+2
+TOI IDTESS Discovery of Twin Planets Around TOI 4342
+13
+10
+30
+100
+300
+1000
+3000
+Stellar light intensity relative to Earth
+1.0
+1.5
+2.4
+3.5
+Planet Size [Earth radii]
+typical
+uncert.
+0.000
+0.005
+0.010
+0.015
+0.020
+Relative Occurrence
+Figure 8. Insolation and orbital period vs planet size. TOI 4342 b and TOI 4342 c are shown in cyan over occurence contours
+for host stars with M∗ < 0.97M⊙ from Fulton & Petigura (2018). M-dwarf TOIs as of July 7, 2022 are plotted as well, with
+multi-planet system planets highlighted in magenta. TOI 4342 b and TOI 4342 c both lie in the upper mode of the radius valley.
+
+14
+Tey et al.
+constraints on the TTV amplitudes. If measured, these
+amplitudes could help derive constrain planet masses.
+4.4. QLP and Extended Mission FFIs
+This discovery showcases our recent improvements
+to QLP on Extended Mission FFIs.
+By adding a
+multi-planet search, short-timescale systematic correc-
+tion, and improved difference images to QLP, we were
+able to detect this new M-dwarf system using the 1st Ex-
+tended Mission FFIs. We saw signal-to-noise improve-
+ments when comparing transit searches between the
+original detrending method and the improved method.
+We also saw improvements when comparing searches be-
+tween the Primary Mission and Extended Mission FFIs.
+On top of this, the multi-planet search, together with
+the longer light curve baseline, let us discover both plan-
+ets in the system, where the original QLP would only
+have found one. Finally, we were able to use the im-
+proved difference images in localizing the source of the
+transit events to our particular target.
+TOI 4342 is just one example of systems we will be
+able to find with all these improvements.
+In the fu-
+ture, we expect these upgrades to yield even more multi-
+planet M-dwarf systems as they continue to be used with
+every new sector of the standard QLP planet detection
+procedures at MIT. This will help build up our popula-
+tions of small planets that are suitable for follow-up.
+ACKNOWLEDGEMENTS
+This paper includes data collected by the TESS mis-
+sion, which are publicly available from the Mikulski
+Archive for Space Telescopes (MAST) (Team 2021;
+Fausnaugh 2021; Huang 2020)). Funding for the TESS
+mission is provided by NASA’s Science Mission direc-
+torate. We acknowledge the use of public TESS data
+from pipelines at the TESS Science Office and at the
+TESS Science Processing Operations Center. Resources
+supporting this work were provided by the NASA High-
+End Computing (HEC) Program through the NASA
+Advanced Supercomputing (NAS) Division at Ames Re-
+search Center for the production of the SPOC data
+products.
+This research has made use of the Exo-
+planet Follow-up Observation Program website (Exo-
+FOP 2019), which is operated by the California Institute
+of Technology, under contract with the National Aero-
+nautics and Space Administration under the Exoplanet
+Exploration Program. This research has made use of the
+NASA Exoplanet Archive, which is operated by the Cal-
+ifornia Institute of Technology, under contract with the
+National Aeronautics and Space Administration under
+the Exoplanet Exploration Program.
+Some of the observations in the paper made use of
+the High-Resolution Imaging instrument Zorro obtained
+under Gemini LLP Proposal Number: GN/S-2021A-LP-
+105. Zorro was funded by the NASA Exoplanet Explo-
+ration Program and built at the NASA Ames Research
+Center by Steve B. Howell, Nic Scott, Elliott P. Horch,
+and Emmett Quigley. Zorro was mounted on the Gem-
+ini South telescope of the international Gemini Observa-
+tory, a program of NSF’s OIR Lab, which is managed by
+the Association of Universities for Research in Astron-
+omy (AURA) under a cooperative agreement with the
+National Science Foundation. on behalf of the Gemini
+partnership: the National Science Foundation (United
+States), National Research Council (Canada), Agencia
+Nacional de Investigaci´on y Desarrollo (Chile), Minis-
+terio de Ciencia, Tecnolog´ıa e Innovaci´on (Argentina),
+Minist´erio da Ciˆencia, Tecnologia, Inova¸c˜oes e Comu-
+nica¸c˜oes (Brazil), and Korea Astronomy and Space Sci-
+ence Institute (Republic of Korea).
+This work makes use of observations from the LCOGT
+network.
+Part of the LCOGT telescope time was
+granted by NOIRLab through the Mid-Scale Innovations
+Program (MSIP). MSIP is funded by NSF.
+Facility:
+TESS,
+Gaia,
+CTIO:1.5m
+(CHIRON),
+LCO:1.0m (Sinistro), SOAR:4.1m (HRCam), Gemini
+South:8m (Zorro)
+Software:
+AstroImageJ (Collins et al. 2017b), As-
+tropy (Collaboration et al. 2013; Astropy Collaboration
+et al. 2018), exoplanet (Foreman-Mackey et al. 2021; Agol
+et al. 2020; Kumar et al. 2019; Kipping 2013a; Luger et al.
+2019; Salvatier et al. 2016; Team et al. 2016), H5py, Mat-
+plotlib (Hunter 2007), MIT Quick Look Pipeline (Huang
+et al. 2020a), Numpy (Harris et al. 2020), TESS SPOC
+Pipeline (Jenkins et al. 2016; Li et al. 2018; Twicken et al.
+2018), Pandas (Reback et al. 2020), Scipy (Virtanen et al.
+2020), Vartools (Hartman & Bakos 2016).
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+Astronomical Journal, 159, 19
+
+18
+Tey et al.
+
+TESS Discovery of Twin Planets Around TOI 4342
+19
+Table 3.
+Stellar and Planet Parameters for TOI 4342
+Parameter
+Value
+Source
+Catalog Information
+R.A. (h:m:s)
+21:37:33.48
+Gaia DR3
+Dec. (d:m:s)
+-77:58:44.9743
+Gaia DR3
+Epoch
+2016.0
+Gaia DR3
+Parallax (mas)
+16.249 ± 0.018
+Gaia DR3
+µra (mas yr−1)
+120.333 ± 0.019
+Gaia DR3
+µdec (mas yr−1)
+−91.503 ± 0.018
+Gaia DR3
+Gaia DR3 ID
+6355915466181029376
+TIC ID
+354944123
+TOI ID
+4342
+Photometric properties
+T ESS (mag). . . . . . . . . . .
+11.0318 ± 0.0074
+TIC v8.2
+Gaia (mag) . . . . . . . . . . . .
+11.97403 ± 0.00032
+Gaia DR3
+Gaia RP (mag) . . . . . . . .
+11.02247 ± 0.00081
+Gaia DR3
+Gaia BP (mag) . . . . . . . .
+12.8963 ± 0.0014
+Gaia DR3
+VJ (mag) . . . . . . . . . . . . . .
+12.669 ± 0.057
+UCAC4 a
+BJ (mag) . . . . . . . . . . . . . .
+14.055 ± 0.011
+APASS DR9 b
+J (mag). . . . . . . . . . . . . . . .
+9.832 ± 0.024
+2MASS
+H (mag) . . . . . . . . . . . . . . .
+9.179 ± 0.024
+2MASS
+Ks (mag) . . . . . . . . . . . . . .
+9.018 ± 0.021
+2MASS
+Derived properties
+M⋆ (M⊙) . . . . . . . . . . . . . .
+0.6296 ± 0.0086
+Parallax +Benedict et al. (2016)c
+R⋆ (R⊙) . . . . . . . . . . . . . . .
+0.599 ± 0.013
+Parallax +Mann et al. (2015) d
+log g⋆ (cgs) . . . . . . . . . . . .
+4.6878+0.0086
+−0.0096
+empirical relation + LC e
+L⋆ (L⊙) . . . . . . . . . . . . . . .
+0.0746 ± 0.0053
+Mann et al. (2015)
+Teff⋆ (K). . . . . . . . . . . . . . .
+3901 ± 69
+f
+MV (mag) . . . . . . . . . . . . .
+14.39±0.02
+Parallax
+MK (mag) . . . . . . . . . . . . .
+8.272±0.015
+Parallax
+Distance (pc) . . . . . . . . . .
+61.543 ± 0.070
+Parallax
+ρ⋆(g cm−3) . . . . . . . . . . . .
+2.985+0.089
+−0.097
+empirical relation + LC e
+Limb-darkening coefficients
+u1, T ESS . . . . . . . . . . . . . .
+0.27+0.13
+−0.11
+u2, T ESS . . . . . . . . . . . . . .
+0.37+0.17
+−0.18
+u1, i′ . . . . . . . . . . . . . . . . . . .
+0.44+0.13
+−0.14
+u2, i′ . . . . . . . . . . . . . . . . . . .
+0.08+0.18
+−0.17
+Light curve parameters
+TOI 4342 b
+TOI 4342 c
+P (days) . . . . . . . . . . . . . . .
+5.5382498+0.0000057
+−0.0000058
+10.688716+0.000015
+−0.000015
+Tc (BJD − 2457000) . . .
+1654.53559+0.00059
+−0.00061
+1659.34623+0.00096
+−0.00094
+T14 (hr) . . . . . . . . . . . . . . . .
+2.215+0.020
+−0.018
+2.820+0.024
+−0.025
+T12 = T34 (min) . . . . . . .
+4.88+0.15
+−0.13
+6.31+0.16
+−0.14
+a/R⋆ . . . . . . . . . . . . . . . . . .
+18.97+0.19
+−0.21
+29.40+0.29
+−0.32
+Rp/R⋆ . . . . . . . . . . . . . . . . .
+0.03491+0.00047
+−0.00049
+0.03719+0.00054
+−0.00057
+b ≡ a cos i/R⋆ . . . . . . . . . .
+0.290+0.042
+−0.050
+0.187+0.060
+−0.061
+i (deg) . . . . . . . . . . . . . . . . .
+89.13+0.15
+−0.13
+89.63+0.12
+−0.12
+Planetary parameters
+Rp (R⊕) . . . . . . . . . . . . . . .
+2.266+0.038
+−0.038
+2.415+0.043
+−0.040
+a (AU) . . . . . . . . . . . . . . . . .
+0.05251+0.00011
+−0.00011
+0.08140+0.00017
+−0.00017
+Teq (K) . . . . . . . . . . . . . . . .
+633.6+6.2
+−6.3
+508.9+5.0
+−5.0
+⟨F ⟩ (S⊕) . . . . . . . . . . . . . .
+26.9+1.1
+−1.0
+11.18+0.45
+−0.44
+a Zacharias et al. (2013)
+b Henden et al. (2016)
+c We adopt error based on the scatter in the empirical relations from Benedict et al. (2016)
+d We adopt error based on the scatter in the empirical relations from Mann et al. (2015)
+e We fitted the transit light curves with a prior constraint on the stellar density.
+f Teff was determined from the bolometric luminosity and the stellar radius.
+
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+page_content=' Atmospheric and Planetary Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 77 Massachusetts Ave,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  MA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 02139,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' USA  14Department of Aeronautics and Astronautics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 77 Massachusetts Avenue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' MA 02139,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  USA  15Department of Astrophysical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Princeton University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 4 Ivy Lane,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Princeton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' NJ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 08544,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' USA  16SETI Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Moffett Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Mountain View,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' CA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 94035,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' USA  ABSTRACT  With data from the Transiting Exoplanet Survey Satellite (TESS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' we showcase improvements to the  MIT Quick-Look Pipeline (QLP) through the discovery and validation of a multi-planet system around  M-dwarf TOI 4342 (Tmag = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='032, M⋆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='63 M⊙, R⋆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='60 R⊙, Teff = 3900 K, d=61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='54 pc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' With  updates to QLP, including a new multi-planet search, as well as faster cadence data from TESS ’ First  Extended Mission, we discovered two sub-Neptunes (Rb = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='266+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='038  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='038 R⊕ and Rc = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='415+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='043  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='040 R⊕;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Pb = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='538 days and Pc = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='689 days) and validated them with ground-based photometry, spectra,  and speckle imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Both planets notably have high transmission spectroscopy metrics (TSMs) of 36  and 32, making TOI 4342 one of the best systems for comparative atmospheric studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' This system  demonstrates how improvements to QLP, along with faster cadence Full-Frame Images (FFIs), can  lead to the discovery of new multi-planet systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Keywords: planetary systems, planets and satellites: detection, stars: individual (TOI 4342)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' INTRODUCTION  The Transiting Exoplanet Survey Satellite (TESS,  Ricker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2015) launched on April 18, 2018 with a  goal of discovering transiting exoplanets around bright  stars across the entire sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' During every sector (∼ 27  days), TESS observes a 24◦ × 96◦ swath of the sky dur-  ing two elongated orbits around the Earth before shift-  ing to the next sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' During its Primary Mission (2018  Jul 25 – 2020 Jul 04), TESS collected photometry at a  2 minute cadence for ∼ 20,000 targets each sector pre-  selected from the Candidate Target List (CTL, Stassun  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2018), while Full-Frame Images (FFIs) were col-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='01370v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='EP]  3 Jan 2023  ID2  Tey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  lected at a 30 minute cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' In these 26 sectors, TESS  covered 70% of the sky and found 2241 transiting planet  candidates (Guerrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  The MIT Quick Look Pipeline (QLP, Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  2020a) has been an important contributor to detecting  these planet candidates, expanding the search from the  ∼ 200, 000 CTL stars to millions of stars brighter than  Tmag = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='5 in the FFIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Every sector, QLP searches  for planet candidates around stars with Tmag < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='5,  making full use of all past sectors of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Notably,  ∼ 1000 candidates from the Primary Mission were found  around stars not on the CTL – a majority of which were  detected by QLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  With its 1st Extended Mission (2020 Jul 04 – 2022 Sep  01), TESS, and QLP in particular, was well-positioned  to yield even more candidates, especially as the FFI  recording cadence was reduced from 30 minutes to 10  minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' On 2022 Sep 01, TESS will have started its  Second Extended Mission, and the FFI recording ca-  dence will be reduced further to 200 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  These  reductions allow the flux-time series from the FFIs to  better resolve transit ingresses and egresses and there-  fore improve planet detection efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' QLP, however,  still has room for improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' We are introducing here  three major changes to the pipeline:  Systematic Multi-planet Search: In the Primary  Mission, QLP only sent the most promising tran-  sit candidate from each light curve through future  stages of vetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Additional candidates could be  reported if the TOI vetters noticed them during  visual inspection of the light curves, but no ex-  plicit search was otherwise done for these addi-  tional planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Based on our knowledge of close-in  exoplanet systems from the Kepler mission, a ma-  jority of the small planets reside in multiple plane-  tary systems (Winn & Fabrycky 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Adding an  automatic search for multiple transit signals will  potentially introduce new interesting systems for  follow-up studies and allow future statistical stud-  ies of system architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Improved Difference Images:  Difference images  (Bryson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2013) are an effective method of  determining the source location of a transit signal  in the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' For the Primary Mission, QLP used a  simplistic algorithm for creating difference images  by directly subtracting the median stacked frames  in the in-transit and out-of-transit time windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  By more intelligently selecting frames to avoid sys-  tematics, transit ingress / egress, and additional  planets in the system, we improve the robustness  of the difference images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' This reduces QLP’s false  positive rate and allows us to reduce human-review  times or alternatively expand our search to more  stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Quaternion Detrending:  Previously,  QLP de-  trended light curves by fitting and removing ba-  sis splines to correct for long-timescale systemat-  ics (Vanderburg & Johnson 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' However, there  are residual systematics on shorter timescales due  to space craft jitter motions that can have signif-  icant effects on light curves, especially for bright  stars (Vanderburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  We can measure and correct for this jitter with  TESS quaternion data – 3-vector time series data  describing spacecraft attitude every two seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1  For transit signals with relatively small depth and  short duration, correcting these systematics is im-  portant to improving QLP’s detection sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Altogether, these improvements to QLP – along with  extended light curves at faster cadence – will increase the  scientific output of TESS as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' One population  that will especially benefit from this are multi-planet  systems around M-dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  M-dwarfs are known to  frequently host multi-planetary systems (Ballard 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Dressing & Charbonneau 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' The duration of plane-  tary transits around M-dwarfs are often relatively short,  making these transits easily diluted in the 30 minute ca-  dence TESS FFI light curves from the Primary Mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  M-dwarfs are also the most abundant type of star in our  galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' So, even though many have been selected for 2  minute cadence target pixel stamp observations, plenty  are only monitored by the FFIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  The regular multi-  sector processing approach of the QLP also enables us  to take the most advantage of all the available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Fi-  nally, M-dwarfs are also ideal host stars for atmospheric  characterization of their transiting planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Determining  atmospheric composition, especially for multiple planets  in a single system, allows us to compare formation and  evolutionary histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  As of July 2022, TESS has detected 373 candidate  transiting planetary systems around nearby M-dwarfs  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 36 of these systems host multiple planets, including  TOI 270 (G¨unther et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Van Eylen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2021),  TOI 175 (Kostov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Cloutier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2019), and  TOI 700 (Gilbert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Rodriguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Building off the original Kepler mission (Kepler Mission  1 Quaternion data is available online at https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='edu/  missions/tess/engineering/  2 The TESS candidate list was downloaded from the NASA Exo-  planet Archive on July 7, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  TESS Discovery of Twin Planets Around TOI 4342  3  2019), this doubles the number of discovered transiting  M-dwarf multi-planet systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  The discovery of TOI 4342, an M-dwarf system with  two transiting sub-Neptunes, showcases the power of  the Extended Mission FFIs with the newly added im-  provements to QLP for detecting multi-planet M-dwarf  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' With the longer baseline, we have more op-  portunities to catch transits and, together with the  faster cadence, we have better statistics to detect the  shorter-duration transits that come with M-dwarf sys-  tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Adding short-timescale systematic corrections  adds to this sensitivity, and the multi-planet search  is what allows us to discover this multi-planet system  where the previous version of QLP could not make this  discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  TOI 4342 (TIC 354944123;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Tmag = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='032;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' d =  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='54 pc) is an M0V-type star with two transiting sub-  Neptunes TOI 4342 b (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='27 R⊕) and TOI 4342 c (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='41  R⊕).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' The planets are near a 2:1 mean-motion resonance  with periods of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='538 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='689 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Both planets are  good targets for atmospheric characterization & com-  parison studies with transmission spectroscopy metrics  ≥ 30 (TSM, Kempton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Both planets were detected with TESS and followed  up with ground-based photometry, reconnaissance spec-  troscopy, and high resolution imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' In Section 2 we  describe these observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' In Section 3 we perform fits  and validate both signals as planetary transits around  an M0V host star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' In Section 4 we re-emphasize the im-  provements to QLP and describe the TOI 4342 system  in the context of small planets, multi-planet systems,  and planets orbiting M-dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' OBSERVATIONS AND DATA ANALYSIS  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' TESS  TOI 4342 was first observed by TESS in Sector 13  (Primary Mission, 2019 Jun 19 – Jul 18) then observed  again in Sector 27 (1st Extended Mission, 2020 Jul 4 –  30, 2020) as a 2-min target due to its brightness and  small radius (Stassun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2019), as well as in FFIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  The 2-min data were reduced by the Science Pro-  cessing Operations Center (SPOC) pipeline at NASA  (Jenkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' The planets’ transit signals were  detected during a SPOC multisector search of Sec-  tors 13 and 27 on 26 May 2021 with an adaptive,  noise-compensating matched filter (Jenkins 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Jenk-  ins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2010, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  In the multisector search, the  5-day signal was detected with SNR 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='8 and multi-  ple event statistic (MES) of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='5, and the 10-day signal  was detected with SNR 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0 and MES 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Both tran-  sit signatures passed all the diagnostic tests reported in  the SPOC data validation reports (Twicken et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2019), including the difference image centroid  tests, which located the source of the transits to within  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='7 +/- 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='5 arcsec and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='4 +/- 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='6 arc sec of the target  star image for TOI 4342 b and c, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Both can-  didates were classified as high quality candidates by the  TESS-ExoClass classifier 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  TESS-ExoClass applies a  series of tests based on the Kepler Robovetter (Coughlin  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Thompson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2018) to pass the best can-  didates on to the manual TOI vetting process (Guerrero  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' These exoplanet signatures were alerted as  TOI 4342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='01 and .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='02 on 28 July 2021 (Guerrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  The FFIs meanwhile were reduced by QLP, but TOI  4342’s TESS band brightness is below the threshold to  be released as a QLP TOI (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='5) (Guerrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  While developing the QLP improvements listed in Sec-  tion 1, we conducted an independent search for multi-  planet systems in QLP light curves and found two planet  candidates in TOI 4342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' For both sectors, our analysis  started with raw QLP light curves, which uses calibrated  FFIs from the MIT TESS image calibration software  (TICA, Fausnaugh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2020) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Figure 1 shows TOI  4342 and the surrounding field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  This multi-planet search consisted of iterative applica-  tions of the Box Least Squares algorithm (BLS, Kov´acs  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2002) as implemented in VARTOOLS (Hartman &  Bakos 2016), masking out transits for previously found  signals when searching for new planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' We then also  performed an additional step of light curve detrending  to remove short-timescale systematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' This was done  by decorrelating spacecraft motion from the light curve  using leading statistical moments (mean, standard devi-  ation, skewness) of and covariances between the quater-  nion time series components Q1, Q2, and Q3 calculated  within each exposure (Vanderburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Together  with basis splines to remove long-timescale trends (Van-  derburg & Johnson 2014), we perform iterative fits to  the light curve, removing 3-σ outliers until the fit con-  verged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' The resulting combined trend was subtracted  out to produce a final corrected QLP light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  With these adjustments, we found two signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' The  first signal has a period of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='538 days with signal-to-  noise ratio (SNR) of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='126 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='14 per transit), and the  second has a period of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='688 days with SNR 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='506  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='74 per transit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' To highlight the impact of the first  Extended Mission and QLP improvements, we specif-  ically searched for planets in Sector 13 (Primary Mis-  3 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='com/christopherburke/TESS-ExoClass  4 TICA FFIs are available as High Level Science Products at the  Mikulski Archive for Space Telescopes (MAST): https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='edu/hlsp/tica  4  Tey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Images of the field surrounding TOI 4342 in 1976 (left), 1992 (top right), and 1995 (bottom right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' In all images,  the orange circle shows the 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0 location of TOI 4342 and the blue circle shows the location on J2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Both circles have radii  of 2′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Nearby stars in the TIC are shown in green with marker size corresponding to brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' In red is a 10′′ scale (∼ half a  TESS pixel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  sion) and Sector 27 (first Extended Mission) separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Corrected for the number of transits seen in each sec-  tor, we saw average SNR per transit of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='70 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='83 in  Sector 13 for each signal respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' In Sector 27, this  improved to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='52 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' We also compared the SNRs  with the original QLP detrending method (only correct-  ing for long-timescale systematics) and found worse per-  formance without the quaternion correction (SNRs per  transit of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='07 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  For the remainder of the system modeling in this work,  we use the Simple Aperture Photometry (SAP) light  curve from the SPOC pipeline (Twicken et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Morris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2020) with the following pre-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  First we removed contamination from nearby stars by  scaling the light curves by the SPOC-provided CROWDSAP  values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' We also ignored data with nonzero SPOC quality  flags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' This included an anomalous event during Sector  13 (TJD 1665.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2983 to 1665.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='3501) where the spacecraft  fell out of fine pointing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' The resulting light curve can  be seen in Figure 3 with its Lomb-Scargle periodogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  In the periodogram, we see a peak signifying stellar  variability at a period of about 13 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  To remove  this variability and other instrumental systematics, we  conducted our own correction similar to our new QLP  correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' First, we excluded data from QLP-predicted  transit times, then we split the light curve into indi-  vidual spacecraft orbits (with two orbits per sector) to  be corrected separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' We again iteratively removed  short-timescale systematics with quaternion time se-  ries statistics and long-timescale systematics with basis  spline fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' This final fit was then divided out to produce  our corrected light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Figure 2 shows the detrended  2-minute cadence light curve with transits highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Ground-based Photometry  We obtained seeing-limited ground-based follow-up  observations from the TESS Follow-up Observing Pro-  gram Sub Group 1 (TFOP SG1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Collins 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  The  ground-based observations have much higher spatial res-  olution than TESS and can help confirm the source  location of a TESS transit signal and can provide ad-  ditional transit observations to refine ephemerides for  predicting future transits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' We used the TESS Transit  Finder, which is a customized version of the Tapir soft-  ware package (Jensen 2013), to schedule our transit ob-  servations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  SERC-J Survey (1976-08-29)  AAO-SES Survey (1992-08-21)  Blue Photographic Plate  Red Photographic Plate  TESS Magnitude  40"  20  18  17  45"  77°58\'20"  Dec  50"  77°58\'55\'  Dec  40"  SERC-I Survey (1995-09-03)  IR Photographic, Plate  40"  45"  Dec  59\'00"  50"  2000  2021  77°58\'55\'  21h37m42s  36s  30s  21h37m36s  34s  32s  RA  RATESS Discovery of Twin Planets Around TOI 4342  5  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Top: Detrended TESS light curve in gray, binned values in purple, with transits highlighted for each TOI 4342 b  (blue) and TOI 4342 c (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Middle: TESS and LCOGT light curves folded on the best fit period and epoch found in  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Different background-subtracted observations are shown in different colors for each transit with the best fit model  in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Bottom: Residual flux between the fully combined light curve and the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Between UT 2021-05-30 and 2021-09-14, 4 transits of  TOI 4342 b and 5 transits of TOI 4342 c were observed in  the Sloan i′ band using the Las Cumbres Observatory  Global Telescope (LCOGT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2013) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0 m  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  The observations were taken at the Siding  Spring Observatory (SSO), South Africa Astronomical  Observatory (SAAO) and Cerro Tololo Inter-American  Observatory (CTIO) nodes of the LCOGT network and  are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' We use 7 of the 9 transits  as 2 were cut short for bad weather conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' The 1 m  telescopes are equipped with 4096 × 4096 SINISTRO  cameras having an image scale of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='′′389 per pixel, re-  sulting in a 26′ × 26′ field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' The images were cal-  ibrated by the standard LCOGT BANZAI pipeline (Mc-  Cully et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Differential photometric data were  extracted with AstroImageJ (Collins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2017a) using  circular photometric apertures with radii 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='′′0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Thus, the  TOI 4342 aperture excludes flux from the nearest known  Gaia DR3 and TICv8 star (TIC 2025922721) 16′′ east  of TOI 4342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' As shown in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2, the transit sig-  nals are detected on-target relative to known Gaia DR3  stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  We also checked the light curves of Gaia DR3  sources within 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='′5 of TOI 4342 and found no evidence  of nearby eclipsing binary systems that could be causing  the TESS detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' All SG1 data can be found online  at ExoFOP (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Reconnaissance Spectroscopy  We obtained 9 spectra of TOI 4342 over two seasons  in slicer mode with the fiber-fed high resolution echelle  spectrograph CHIRON (Tokovinin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  CHI-  3000  2000  : Flux (ppm)  1000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='Relative I  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='3000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1655  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1660  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1665  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1670  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1675  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1680  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2040  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2045  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2050  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2055  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2060  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='BJD-2457000 [Days]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='BJD-2457000 [Days]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='TOI 4342 b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='TOI 4342 c  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='3000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='3000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='TESS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='TESS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='SAAO 2021-06-19 UT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='CTIOa 2021-08-13 UT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='(ppm)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='CTIO 2021-07-12 UT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='CTIOb 2021-08-13 UT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='SSO 2021-08-14 UT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='CTIO 2021-09-14 UT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=': Flux  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='CTIO 2021-08-31 UT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='Relative  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='3000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='3000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='Hours frorm mid-transit  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='Hours from mid-transit  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='3000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='3000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='Combined  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='(wdd)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='Residual flux (  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  1000  1000  2000  2000  Combined  3000  3000  0  2  2  4  4  0  2  Hours frorm mid-transit  Hours from mid-transit6  Tey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  SG1 Follow-Up Observations  Target  Instrument  Date (UT)  Filter  Aperture  Observing Notes  TOI 4342 c  LCO-CTIO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0m  2021-09-14  i′  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1′′  TOI 4342 b  LCO-CTIO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0m  2021-08-31  i′  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2′′  TOI 4342 b  LCO-SSO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0m  2021-08-14  i′  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1′′  TOI 4342 c  LCO-CTIO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0m a  2021-08-13  i′  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='9′′  Simultaneous observation a  TOI 4342 c  LCO-CTIO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0m b  2021-08-13  i′  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='9′′  Simultaneous observation a  TOI 4342 b  LCO-CTIO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0m  2021-07-12  i′  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2′′  TOI 4342 c  LCO-SAAO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0m  2021-07-11  i′  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0′′  Noisy, cut short by weather  TOI 4342 b  LCO-SAAO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0m  2021-06-19  i′  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='6′′  TOI 4342 c  LCO-CTIO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0m  2021-05-30  i′  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='8”  Partial, cut short by weather  aTransit observed simultaneously by two distinct telescopes at CTIO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Top: SPOC SAP light curve with our correction trend (described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1) in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' This trend, formed from  quaternion data and basis splines to remove systematics and long timescale variability, is divided out of the raw light curve  to create our final flattened light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Bottom: Lomb-Scargle periodograms of the SAP SPOC light curve for each sector of  observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' We see peaks around 14 days marking stellar variability, but we cannot pin down a specific period due to the large  gap between observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  RON is mounted on the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='5 m SMARTS telescope, lo-  cated at the CTIO, Chile, and has a spectral resolving  power of 80,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' The spectra were taken using 1 hour  long exposures and were extracted by the standard CH-  IRON pipeline (Paredes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  We derived the radial velocities using a cross corre-  lation against a median combined template spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  The template spectrum is composed of a median combi-  nation of all CHIRON spectra, each shifted to rest after  an approximate velocity measurement via a cross corre-  lation against a synthetic template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' The measured ve-  locity of each spectrum is that of the mean velocity from  each spectral order, weighted by their cross correlation  function heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' The velocity uncertainties were esti-  mated from the scatter of the per-order velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' We  find a mean internal uncertainty of ∼ 15 m s−1, with an  RMS of ∼ 22 m s−1 between the 9 measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' The  full dataset can be found in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' High-Resolution Speckle Imaging  “Third-light” flux contamination from a close stellar  companion can lead to an underestimated planetary ra-  dius if not accounted for in the transit model (Ciardi  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2015) and even cause non-detections of small plan-  ets residing within the same exoplanetary system (Lester  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='98  1654  1658  1662  1666  1669  1673  1677  1681  2037  2041  2045  2049  2050  2054  2058  2062  BJD - 2457000 [Days]  BJD - 2457000 [Days]  BJD - 2457000 [Days]  BJD - 2457000 [Days]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='8  S13  S27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0 -  0  2  4  6  8  10  12  14  16  18  20  22  24  26  DaysTESS Discovery of Twin Planets Around TOI 4342  7  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Radial Velocities of TOI 4342  BJD  RV  σRV  Instrument  (km s−1)  (km s−1)  2459359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='91837  −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='726  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='013  CHIRON  2459360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='85475  −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='760  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='018  CHIRON  2459379.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='77461  −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='795  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='020  CHIRON  2459384.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='82649  −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='770  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='011  CHIRON  2459409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='75314  −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='788  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='012  CHIRON  2459700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='86421  −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='734  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='018  CHIRON  2459725.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='87319  −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='747  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='016  CHIRON  2459740.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='83584  −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='737  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='016  CHIRON  2459742.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='79459  −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='761  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='012  CHIRON  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' The discovery of close, bound companion  stars, which exist in nearly one-half of FGK type stars  (Matson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2018) and less so for M class stars, pro-  vides crucial information toward our understanding of  exoplanetary formation, dynamics and evolution (How-  ell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Thus, to search for close-in bound  companions unresolved by TESS, Gaia, or ground-based  seeing-limited follow-up observations, we obtained high-  resolution imaging speckle observations of TOI 4342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  TOI 4342 was observed at the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1-m Southern Astro-  physical Research (SOAR) telescope (Tokovinin 2018)  on 1 October 2021 UT, in Cousins I-band, a similar vis-  ible bandpass as TESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' This observation was sensitive to  a 5-magnitude-fainter star at an angular distance of 1′′  from the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' More details of the observations within  this survey are available in Ziegler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' The 5σ  detection sensitivity and speckle auto-correlation func-  tions from the observations are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' No  nearby stars were detected within 3′′ of TOI 4342 in the  SOAR observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  TOI 4342 was also observed on July 23 2021 UT using  the Zorro speckle instrument on the Gemini South 8-m  telescope5 (Scott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Zorro provides simultane-  ous speckle imaging in two bands (562 nm and 832 nm)  with output data products including a reconstructed im-  age with robust contrast limits on companion detections  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=', Howell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' During this observing run, the  blue channel was inoperable, thus only 832 nm observa-  tions were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Thirteen sets of 1000 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='06 sec  exposures were collected and subjected to Fourier anal-  ysis in the standard reduction pipeline (see Howell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Figure 5 shows our final contrast curve and the  5 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='gemini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='edu/sciops/instruments/alopeke-zorro/  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
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+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0  ∆arcsec  0  1  2  3  4  5  6  ∆magnitude (I-band)  2  0  2  ∆α [arcsec]  2  0  2  ∆δ [arcsec]  SOAR Speckle ACF  TIC354944123  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  SOAR speckle auto-correlation function (inset  plot) and its 5 sigma detection sensitivity curve (main plot)  in the Cousins-I band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2  angular separation (arcsec)  0  1  2  3  4  5  6  7  m  832 nm  TOI 4342  1"  832 nm  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Gemini South Zorro Speckle Imaging 5 sigma con-  trast curve (full plot) and reconstructed images (inset plot)  in the 832 nm band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' The diffraction limit of the instrument  is 20 mas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  832 nm reconstructed speckle image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' We find that TOI  4342 is a single star with no companion brighter than  5-6 magnitudes below that of the target star from very  close in (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1′′) out to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' At the distance of TOI 4342  (61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='54 pc) these angular distances correspond to spatial  distances of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2 to 74 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' ANALYSIS  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Stellar parameters  We re-derive stellar parameters according to the fol-  lowing empirical relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' First we take the observed  Ks magnitude from 2MASS and the Gaia Data Release  3 (DR3) (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2022) parallax to find  8  Tey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  the absolute Ks magnitude of TOI 4342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Then, using  the relation from Benedict et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' (2016), we get a stellar  mass of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='6296 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0086 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' With this mass, we use  the mass-radius relation from Boyajian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' (2012) to  find a stellar radius of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='599 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='013 R⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' We check this  with the absolute Ks magnitude-radius relation from  Mann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' (2015) where we find a consistent value of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='598 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='018 R⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  From Mann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' (2015), we also  calculate a bolometric luminosity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0746 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0053 L⊙  via the observed V and J magnitudes and the resulting  bolometric correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Using the Stefan-Boltzmann law,  we find an effective temperature of 3901 ± 69 K, giving  TOI 4342 a spectral type around M0V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' These values are  consistent with the stellar parameters listed in TIC 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='587±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='020 M⊙, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='598±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='018 R⊙, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='073±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='016 L⊙,  and 3880 ± 160 K, which were derived using the older  Gaia DR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Light curve modeling  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Best Fit Model  We simultaneously fit the detrended TESS light curve  and all 7 of the 9 SG1 light curves unaffected by weather  (see Table 1) using exoplanet (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  For TESS light curves, we approximate the per point  measured uncertainty as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='4826 times the median abso-  lute deviation (MAD) of the flux within each TESS or-  bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' For each SG1 light curve, we use the reported flux  uncertainties, and on top of that, fit for a jitter term  (added in quadrature) to capture additional errors in  the observation’s error budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' We also simultaneously  fit a second order polynomial to the SG1 light curves  to account for nightly trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' To model the stellar limb  darkening we use quadratic limb darkening models with  uninformative priors following Kipping (2013a) for each  observation band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  For both orbits, we assume an eccentricity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  6  Periods and epochs for each planet were given uniform  priors centered on the values found via BLS search to  ±10% of the period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' The ratios of planet to stellar radii  had uniform priors from 0 to 1, and impact parameters  had uniform priors from −(1+Rp/R∗) to +(1+Rp/R∗)  (though when reported, we take the absolute value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Fi-  nally, the stellar mass and radius were given normal pri-  ors using the parameters derived in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  exoplanet uses PyMC3 (Salvatier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2016) to per-  form No U-turn Sampling (Hoffman & Gelman 2011)  6 Based on priors from Kipping (2013b) and Eylen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' (2019), we  expect the eccentricities to be low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' We also performed a separate  fit with an uninformed prior on the eccentricity and argument of  periapsis, and did not see significant changes in the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  from the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' We sampled five inde-  pendent chains with 5000 tuning steps and 5000 draws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  All parameters converged with Gelman-Rubin statistic  ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='01 (Gelman & Rubin 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Table 3 shows the  median sampled and derived values with 1 sigma confi-  dence intervals, and Figure 2 shows the median posterior  model with residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Transit Shape Model  Separately, using exoplanet’s SimpleTransitOrbit  we also performed a TESS-only fit to best constrain each  candidate’s transit shape – the ratio between the dura-  tion of the flat part of the transit (tF ) to total transit  duration (tT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Again, we assume a Gaussian noise model on top of a  constant baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Priors on period and epoch were set  to uniform distributions centered around the BLS peri-  ods and epochs with bounds of ±10% of the period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Du-  rations were given uniform priors from 0 to 2 times the  BLS durations, Rp/R⋆ were given uniform priors from 0  to 1, and impact parameters were given uniform priors  from −(1 + Rp/R∗) to +(1 + Rp/R∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Since we want  to find the best characterization of the transit shape for  each planet independently from the host star, we use two  independent star models with loose stellar radius priors  from 0 to 2 R⊙ and quadratic limb darkening with Kip-  ping (2013a)’s priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Following the same fitting configuration as Section  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1, we can calculate the transit shape following Equa-  tion 15 from (Seager & Mall´en-Ornelas 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  �tF  tT  �2  = (1 − Rp/R∗)2 − b2  (1 + Rp/R∗)2 − b2  (1)  where Rp is planet radius, R∗ is stellar radius, and b  is impact parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' For TOI 4342 b and TOI 4342 c  respectively, we find median and one sigma confidence  intervals of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='912+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='013  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='026 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='907+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='016  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='039.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Radial Velocity Modeling  Using the CHIRON data (Table 2), we can place mass  upper limits on both planets by fitting for the ampli-  tudes of simple sinusoids assuming circular orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' This  puts an upper bound on the radial velocity variations,  meaning we can constrain any transiting companion  masses to be planetary rather than stellar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  We define our model with three unknowns: the base-  line radial velocity, Kb, and Kc, where the K’s are  semi-amplitudes of sinusoids set to the median periods  and epochs from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' After fitting, we find semi-  amplitudes of 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='4+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1  −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0 m s−1 and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='4+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='6  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='6 m s−1 for each  signal respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' These posteriors give us 3-σ mass  upper limits of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='321 MJup and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='289 MJup both much  smaller than stellar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  TESS Discovery of Twin Planets Around TOI 4342  9  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Photocenter motion  Photocenter motion analysis can help determine if the  location of a transit signal matches the location of the  target star on the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  An effective method of determining both of these loca-  tions is with the difference imaging technique (Bryson  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2013), designed initially for the Kepler Mission  and inherited by the SPOC pipeline, whereby the dif-  ference of averaged in- and out-of-transit pixel images is  found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Assuming stellar variability and/or instrumental  systematics are negligible on transit timescales, the dif-  ference image should appear star-like at the location of  the transit signal source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Meanwhile, the out-of-transit  image should represent a direct image of the field sur-  rounding the target star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' If the field is relatively un-  crowded and the target star is indeed the source of the  transit signal, the difference and direct images should  appear similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  To produce difference and direct images, we used  TESS-plots7, a publicly available Python package for  robust pixel-level analysis of TESS FFIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' For planets  in multi-planet systems such as TOI 4342, TESS-plots  masks out all cadences corresponding to other planet  transits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' This ensures that the primary source of vari-  ability in the difference image is due to the planet of  interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' TESS-plots also puts more care in choosing  which in- and out-of-transit frames are used compared  to the original difference images in QLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' It ignores the  first and last 5% of the transit duration to better avoid  ingress / egress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' It also places a buffer between in- and  out-of-transit frames to handle underestimated transit  durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Finally, “bad” transits (transits with lots of  missing or poor quality datapoints) are discarded to pre-  vent contamination of the overall difference image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Al-  together, with these improvements, TESS-plots marks  an important update over the Primary Mission QLP dif-  ference images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Figure 6 shows difference images for TOI 4342 b and  TOI 4342 c next to the direct image for TOI 4342, us-  ing Sector 27 FFIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' The difference images confidently  rule out the transit signals as coming from other nearby  sources listed in the TIC, consistent with the centroid  analysis derived by the SPOC pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' For TOI 4342  b, the SPOC-derived photo centroid is only 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='5 σ away  from the location of TIC 2025922721 (T=19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='805 mag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  However, this star is too faint to produce the transit  depths observed on TOI 4342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Possible False Positive Scenarios  7 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='com/mkunimoto/TESS-plots  In this section we rule out possible false positive sce-  narios where the signals are not coming from a multi-  planet system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' TOI 4342 is an eclipsing star system  One possible source of false positives could be signals  from eclipsing stellar companions rather than planetary  companions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' We can rule out this scenario by consider-  ing our radial velocity model in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' We found  3-σ upper limits on semi-amplitude magnitudes of 50  m s−1 and 36 m s−1 which translate to mass limits of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='321 MJup and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='289 MJup for TOI 4342 b and TOI  4342 c respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' In other words, if our transit sig-  nals are caused by gravitationally bound companions  that block out light from TOI 4342, those companions  must have sub-Jupiter masses and cannot be from stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Contamination from a Nearby Eclipsing Binary  Another major source of false positives is the signal of  an eclipsing binary (EB) in the field near our target of  interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Because photometers measure all light within a  specific aperture, eclipses from a nearby EB (NEB) can  contaminate the target aperture and cause transit-like  events in the light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' These false positives account  for as much as 40% of transit-like signals at the lowest  Galactic latitudes in the Kepler field (Morton & Johnson  2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Bryson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  We can start ruling out NEBs by restricting the sig-  nal source to be near TOI 4342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Our photocenter mo-  tion analysis in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='4 rules out signals from known  TICv8 stars, constraining the signal to be within ∼ 21′′  (one TESS pixel) of TOI 4342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2, SG1 ob-  servations rule out signals from the nearest Gaia DR3  stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' In DR3 (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Fabricius  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2021), Gaia has a resolution down to ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='7′′, so  the signal must be on-target or from an NEB within an  arcsecond of TOI 4342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Next, we can rule out potential NEBs by showing they  must be brighter than certain magnitudes to cause either  transit signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' First, we note the observed transit depth  δobs is  δobs = δEB ·  f  1 + f  (2)  where δEB the “true” depth (the NEB’s primary  eclipse depth if TOI 4342 were not present), and f is  the flux ratio between the NEB and TOI 4342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Then, we can place an upper bound on δEB by assum-  ing b = 0 as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 21 from Seager & Mall´en-Ornelas  (2003):  δEB ≤  (1 − tF  tT )2  (1 + tF  tT )2  (3)  10  Tey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Difference images for TOI 4342 b and TOI 4342 c, made using 20×20 pixel cutouts of the Sector 27 TESS FFIs (top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  A close-up of the central 5×5 pixels is also shown (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' The third column shows the average of out-of-transit images, which  should represent a direct image of the field near the target star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' The target TIC-354944123 is indicated with a pink star, while  nearby stars down to ∆T = 4 mag are plotted as white circles with sizes scaled by brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' The difference images for both  planet candidates indicate the transit sources are co-located with the target star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' The color bars are all in units of electrons per  second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Together with Equation 2, this places a lower bound  on f purely as a function of δobs and transit shape ( tF  tT ):  δobs/δEB  1 − δobs/δEB  ≤ f  (4)  which we can rearrange to an upper bound on NEB  magnitude (mEB − m∗ = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='5 log10 f where m∗ is the  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='032, the Tmag of TOI 4342).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Using the result of the transit shape model in Section  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2, we can derive 3-σ lower bounds on transit shape  for each signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' These correspond to NEB magnitude  upper bounds of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='38 and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='70 (∆T = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='35 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='67  mag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' In other words, for an NEB to cause the TOI  4342 b signal, it is likely within 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='35 mag of TOI 4342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  In Figure 4, we see no stars within 5-6 mag of TOI  4342 at separations of 1′′ – 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' This supports the rul-  ing out of Gaia sources down to around 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' In Figure 5,  no neighbors within 5-6 mag of TOI 4342 were detected  from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1′′ – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' So, altogether we can rule out NEBs  greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1′′away from TOI 4342 as sources of the  transit signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Lastly, taking advantage of the relatively high proper-  motion of TOI 4342, we can use archival images to rule  out NEBs within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1′′of TOI 4342’s current location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Figure 1 shows the field surrounding TOI 4342 in 1976  along with all known nearby TIC stars for magnitude  reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' We see that in 1976, the 2021 location of TOI  4342 is clear of any stars with Tmag ≲ 17 (∆T ≃ 6 mag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  This rules out NEBs within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1′′of TOI 4342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Altogether, using our photocenter motion analysis,  high-resolution speckle imaging, and archival field im-  ages, we are able to rule out contamination from an  NEB as the source of either transit signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' TOI 4342 is a hierarchical triple  The final scenario we consider is an EB gravitationally  bound to TOI 4342 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' a hierarchical triple system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  This EB would cause the same aperture contamination  described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2 (with the same magnitude lim-  its), but could have evaded detection in Figure 1 because  it would stay close to TOI 4342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  In this scenario, if we assume the EB’s orbit is within  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1 arcsec of TOI 4342, its semimajor axis would be at  most ∼ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2 AU (at a distance of 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='54 pc via Gaia DR3  parallax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' With Kepler’s third law and the stellar mass  from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1, this gives the NEB an orbital period  of ∼ 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2 years around TOI 4342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  From Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2, we saw that NEBs should be  brighter than ∆T ≃ 5 mag to cause the transit signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Direct Image  Difference Image  Difference Image  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0  1500  775  775  775  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='5  Pixel  1000  770  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0  770  770  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0  Row  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='5 765  765  F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='5  765  500  C  TOI-4342 b  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0  TO1-4342  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0  TO1-4342  760  760  760  595  600  595  600  605  610  595  600  605  605  610  610  772  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0  772  772  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0  1500  771  771  771  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='5  770  770  770  Pix  1000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0  F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0  ☆  Row  769  769  769  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='5  500  768  768  768  TOI-4342 b  TOI-4342 C  TOI-4342  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0  767  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0 767  767  602  604  606  602  604  606  602  604  606  Column Pixel  Column Pixel  Column PixelTESS Discovery of Twin Planets Around TOI 4342  11  This translates to a minimum luminosity of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='5×10−4L⊙  and, using L ∝ M 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='5, a minimum mass of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='13M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  For a gravitationally bound NEB with a mass of at  least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='13M⊙, we would expect an edge-on radial veloc-  ity semiamplitude of ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='8 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Gaia DR3 observed  TOI 4342 over a 34 month baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' In this timeframe,  if there were a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='13 M⊙ companion at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2 AU, we would  expect an RV shift of ∼ 1 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' The reported mean  RV error though is only ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='36 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Similarly, from  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='3, our RV data spans more than a year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Based  on this observation timeline, we expect an RV scatter  from a companion NEB at a distance of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2 AU with  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='13M⊙ would be larger than our observed 22 m s−1 at  least 95% of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Together, since the actual RV error is lower than ex-  pected from a companion EB for both our RV data  and Gaia data, we conclude that these scenarios of EBs  graviationally bound to TOI 4342 are unlikely causes of  the transit signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' TRICERATOPS & Summary  Using two sectors of TESS data along with additional  photometric and spectroscopic observations, we were  able to rule out most astrophysical false positives as  sources of our transit signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' We considered the pos-  sibilities that our signals are caused directly by an EB,  by contamination from a background EB, and by cer-  tain configurations of a hierarchical companion EB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' For  each scenario, we showed that it is highly unlikely for  that scenario to cause our transits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' In addition, we use  triceratops (Giacalone & Dressing 2020) to indepen-  dently check the false positive probabilities (FPPs) and  nearby false positive probabilities (NFPPs) for each sig-  nal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' After 20 runs for each signal, we calculate mean  and standard deviation FPPs of: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='00211 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='00018 and  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='00320±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='00027 and NFPPs of: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0000138±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0000013  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='000387 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='000026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Finally, Lissauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' (2012)  and Guerrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' (2021) found that multi-candidate  systems have lower false positive rates than single-  candidate systems, so we receive a “multiplicity boost”,  further decreasing the false positive probabilities and  increasing the likelihood of having real planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Alto-  gether, we conclude our signals are statistically valid  exoplanet transits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' RESULTS AND DISCUSSION  In this work, we statistically validated a pair of sub-  Neptunes around M0V dwarf TOI 4342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' In this section,  we discuss this system in the context of other planetary  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Mass and Atmosphere Follow-Up Characterization  The best fit parameters from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1 show the  planets have radii of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='266+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='038  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='038 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='415+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='043  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='040 R⊕, and  periods of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='538 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='689 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Given their radii, these  planets are most likely sub-Neptunes with a significant  fraction of H/He in their atmospheres (Rogers 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Using sub-Neptune mass-radius relationship from Wolf-  gang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' (2016), we expect the planets to have masses  of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='83+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='93  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='88 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='53+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='90  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='94 M⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  This corresponds to expected radial velocity semi-  amplitudes of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='9 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='4 m s−1, meaning it is feasible  to measure the precise masses using a high precision ra-  dial velocity instrument mounted on a large telescope  for TOI 4342 (V = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='67 mag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' With these masses, we  will be able to compare bulk densities of the planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Given the radii similarity (within 10% of each other),  this will show the influence of different levels of irradia-  tion (incident fluxes of ∼ 27 and ∼ 11 S⊕) on the planet  atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Using the expected masses, we can also calculate  transmission spectroscopy metrics (TSMs, Kempton  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2018), measures of how promising planets are for  atmospheric characterization studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' For each planet,  we find values of 36 and 32, both of which are above the  updated follow-up threshold of ∼ 25 recommended by  Guerrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' (2021) for the 100 best atmospherically  characterizable sub-Neptunes (updated from Kempton  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  More notably, TOI 4342 has multiple (more than one)  high-TSM planets, making it one of the best M-dwarf  systems for atmospheric comparison studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Figure 7  shows the TSM values for multi-planet M-dwarf sys-  tems sorted by second highest TSM value of planets in  each system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Characterizing and comparing the atmo-  spheres of both planets will allow us to perform com-  parative exoplanetology, and TOI 4342 is one of the  few systems where multiple planets have characteriz-  able atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Additionally, given the two planets  are near a mean-motion resonance (MMR) of 2:1, it’s  likely they migrated together to their current orbits and  have similar primordial compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' This would mean  any differences detected in their atmospheric properties  can probably be attributed to the differing levels of stel-  lar irradiation between the planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' This will help us  gain insights on planetary atmosphere evolution and re-  sponses to different intensities of stellar irradiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Small Planet Radius Gap  Fulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' (2017) identified a radius gap for small  planets roughly between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0 R⊕ separating  rocky super-Earths and gaseous sub-Neptunes for sun-  like stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Cloutier & Menou (2020) showed this gap per-  sisted around low-mass stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' One predominant cause of  12  Tey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' TSM, as defined in Kempton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' (2018), for TOIs in multi-planet systems around bright M-dwarfs (Teff < 4000 K  and Tmag < 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='5) with Rp < 4R⊕ as of July 7, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' TICs are sorted decreasingly by 2nd-largest TSM value in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Color  corresponds to host Tmag and size corresponds to planet radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' The gray dashed line is the atmospheric follow-up threshold  recommended by Guerrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' (2021) for sub-Neptunes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' TOI 4342 is among the top 10 systems with multiple planets that  are well-suited for atmospheric characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  this gap may be photoevaporation: the stripping away  of a planet’s atmosphere as it undergoes heavy irradia-  tion from its star (Owen & Wu 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Highly irradiated  planets would be left as bare rocky cores, while less irra-  diated planets would keep their atmospheres with larger  mass and radii, leading to a bimodal radius distribu-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' With radii of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='27 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='41 R⊕, both TOI 4342  b and TOI 4342 c appear to fall on the upper mode of  the valley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Figure 8 shows each TOI 4342 b and TOI  4342 c in cyan as a function of irradiation level and  planetary radius over relative occurrence contours from  Fulton & Petigura (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  We see both planets have  relatively low irradiation levels of ∼ 27 and ∼ 11 S⊕ re-  spectively, meaning they are likely good fits for the low  mass atmosphere-retaining sub-Neptune planet descrip-  tion on the upper side of the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Transit Timing Variation  With periods of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='538 days and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='689 days, TOI 4342  b and TOI 4342 c also fall within 5% of the first order  MMR of 2:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Given the close distance to the 2:1 reso-  nance, we expect the system would show transit timing  variation (TTV) signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Based on formulas in Lithwick  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' (2012), the super period of the TTV is about 157  days, the amplitude of the TTV signal is expected to  be on the order of a few minutes using the estimated  mass from empirical relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Calculations using TTV-  Fast (Deck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2014) assuming the expected planet  masses and eccentricities smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1 for both plan-  ets show similar results to Lithwick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' For  TOI 4342, the photometric observations from ground-  based 1-meter telescopes were able to achieve transit  time measurements at a similar or slightly better preci-  sion than TESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' With these observations, we can search  for evidence of TTVs over a baseline of > 800 days via  exoplanet’s TTVOrbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' First, we subtract the best fit  background trends found in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1 from each light  curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' We use the global best fit ephemerides to set nor-  mal priors on transit times with standard deviations of  7 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  The rest of the parameters (limb darken-  ing, stellar properties, radii ratios, impact parameter)  are initialized following the best fit model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Sampling  parameters similarly followed the best global fit model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Based on the TTV fit, we do not observe significant de-  viation from linear ephemerides by more than 5 mins for  a majority of the observations, indicating that the ec-  centricities of both planets are likely to be close to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  TESS will reobserve TOI 4342 in Sectors 66 & 67 (June  – July 2023), adding new transits that will help further  constrain the TTV amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Additional photometry  observations that can achieve transit center timing with  precision better than 1 min can also provide stronger  Rp(RE)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='8  100  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0  50  TSM  25  10  Tmag  8  9  10  80  11  1  270  256  1730  776  233  4342  1266  4599  904  175  732  1468  969  74  2079  406  4643  2095  700  2  2  TOI IDTESS Discovery of Twin Planets Around TOI 4342  13  10  30  100  300  1000  3000  Stellar light intensity relative to Earth  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='5  Planet Size [Earth radii]  typical  uncert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='020  Relative Occurrence  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Insolation and orbital period vs planet size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' TOI 4342 b and TOI 4342 c are shown in cyan over occurence contours  for host stars with M∗ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='97M⊙ from Fulton & Petigura (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' M-dwarf TOIs as of July 7, 2022 are plotted as well, with  multi-planet system planets highlighted in magenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' TOI 4342 b and TOI 4342 c both lie in the upper mode of the radius valley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  14  Tey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  constraints on the TTV amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' If measured, these  amplitudes could help derive constrain planet masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' QLP and Extended Mission FFIs  This discovery showcases our recent improvements  to QLP on Extended Mission FFIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  By adding a  multi-planet search, short-timescale systematic correc-  tion, and improved difference images to QLP, we were  able to detect this new M-dwarf system using the 1st Ex-  tended Mission FFIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' We saw signal-to-noise improve-  ments when comparing transit searches between the  original detrending method and the improved method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  We also saw improvements when comparing searches be-  tween the Primary Mission and Extended Mission FFIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  On top of this, the multi-planet search, together with  the longer light curve baseline, let us discover both plan-  ets in the system, where the original QLP would only  have found one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Finally, we were able to use the im-  proved difference images in localizing the source of the  transit events to our particular target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  TOI 4342 is just one example of systems we will be  able to find with all these improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  In the fu-  ture, we expect these upgrades to yield even more multi-  planet M-dwarf systems as they continue to be used with  every new sector of the standard QLP planet detection  procedures at MIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' This will help build up our popula-  tions of small planets that are suitable for follow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  This paper includes data collected by the TESS mis-  sion, which are publicly available from the Mikulski  Archive for Space Telescopes (MAST) (Team 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Fausnaugh 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Huang 2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Funding for the TESS  mission is provided by NASA’s Science Mission direc-  torate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' We acknowledge the use of public TESS data  from pipelines at the TESS Science Office and at the  TESS Science Processing Operations Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Resources  supporting this work were provided by the NASA High-  End Computing (HEC) Program through the NASA  Advanced Supercomputing (NAS) Division at Ames Re-  search Center for the production of the SPOC data  products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  This research has made use of the Exo-  planet Follow-up Observation Program website (Exo-  FOP 2019), which is operated by the California Institute  of Technology, under contract with the National Aero-  nautics and Space Administration under the Exoplanet  Exploration Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' This research has made use of the  NASA Exoplanet Archive, which is operated by the Cal-  ifornia Institute of Technology, under contract with the  National Aeronautics and Space Administration under  the Exoplanet Exploration Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Some of the observations in the paper made use of  the High-Resolution Imaging instrument Zorro obtained  under Gemini LLP Proposal Number: GN/S-2021A-LP-  105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Zorro was funded by the NASA Exoplanet Explo-  ration Program and built at the NASA Ames Research  Center by Steve B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Howell, Nic Scott, Elliott P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Horch,  and Emmett Quigley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Zorro was mounted on the Gem-  ini South telescope of the international Gemini Observa-  tory, a program of NSF’s OIR Lab, which is managed by  the Association of Universities for Research in Astron-  omy (AURA) under a cooperative agreement with the  National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' on behalf of the Gemini  partnership: the National Science Foundation (United  States), National Research Council (Canada), Agencia  Nacional de Investigaci´on y Desarrollo (Chile), Minis-  terio de Ciencia, Tecnolog´ıa e Innovaci´on (Argentina),  Minist´erio da Ciˆencia, Tecnologia, Inova¸c˜oes e Comu-  nica¸c˜oes (Brazil), and Korea Astronomy and Space Sci-  ence Institute (Republic of Korea).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  This work makes use of observations from the LCOGT  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Part of the LCOGT telescope time was  granted by NOIRLab through the Mid-Scale Innovations  Program (MSIP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' MSIP is funded by NSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Facility:  TESS,  Gaia,  CTIO:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='5m  (CHIRON),  LCO:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0m (Sinistro), SOAR:4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='1m (HRCam), Gemini  South:8m (Zorro)  Software:  AstroImageJ (Collins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2017b), As-  tropy (Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Astropy Collaboration  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2018), exoplanet (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Agol  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Kipping 2013a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Luger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Salvatier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Team et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2016), H5py, Mat-  plotlib (Hunter 2007), MIT Quick Look Pipeline (Huang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2020a), Numpy (Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2020), TESS SPOC  Pipeline (Jenkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
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+page_content=' 2020), Scipy (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  2020), Vartools (Hartman & Bakos 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
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+page_content=' Observed minus calculated transit times for all observed transits across TESS and SG1 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' Observed transit times  were modeled with exoplanet’s TTVOrbit (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='3) and the expected transit times were linearly propagated from the best  fit ephemerides (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
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+page_content=' Based on the current data, we see no evidence for significant deviations from expected transit  times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' With expected TTV amplitudes of only ∼ 5 minutes, future observations could help could pinpoint TTV amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
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+page_content=', Gommers, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=', Oliphant, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2020,  Nature Methods, 17, 261  Winn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=', & Fabrycky, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2015, Annual Review of  Astronomy and Astrophysics, 53, 409  Wolfgang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=', Rogers, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=', & Ford, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2016, The  Astrophysical Journal, 825, 19  Zacharias, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=', Finch, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=', Girard, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2013, The  Astronomical Journal, 145, 44  Ziegler, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=', Tokovinin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=', Brice˜no, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' 2020, The  Astronomical Journal, 159, 19  18  Tey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  TESS Discovery of Twin Planets Around TOI 4342  19  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  Stellar and Planet Parameters for TOI 4342  Parameter  Value  Source  Catalog Information  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' (h:m:s)  21:37:33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='48  Gaia DR3  Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' (d:m:s)  77:58:44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='9743  Gaia DR3  Epoch  2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0  Gaia DR3  Parallax (mas)  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='249 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='018  Gaia DR3  µra (mas yr−1)  120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='333 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='019  Gaia DR3  µdec (mas yr−1)  −91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='503 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='018  Gaia DR3  Gaia DR3 ID  6355915466181029376  TIC ID  354944123  TOI ID  4342  Photometric properties  T ESS (mag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0318 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0074  TIC v8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='2  Gaia (mag) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='97403 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='00032  Gaia DR3  Gaia RP (mag) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='02247 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='00081  Gaia DR3  Gaia BP (mag) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='8963 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0014  Gaia DR3  VJ (mag) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='669 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='057  UCAC4 a  BJ (mag) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
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+page_content='  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='055 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='011  APASS DR9 b  J (mag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
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+page_content='832 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='024  2MASS  H (mag) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
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+page_content='179 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='024  2MASS  Ks (mag) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
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+page_content='018 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='021  2MASS  Derived properties  M⋆ (M⊙) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
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+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='6296 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0086  Parallax +Benedict et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' (2016)c  R⋆ (R⊙) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
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+page_content='599 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='013  Parallax +Mann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' (2015) d  log g⋆ (cgs) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
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+page_content='0096  empirical relation + LC e  L⋆ (L⊙) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
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+page_content='0746 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content='0053  Mann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' (2015)  Teff⋆ (K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfafyY/content/2301.01370v1.pdf'}
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diff --git a/hNE2T4oBgHgl3EQfHgYi/content/tmp_files/2301.03668v1.pdf.txt b/hNE2T4oBgHgl3EQfHgYi/content/tmp_files/2301.03668v1.pdf.txt
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+Consistent Query Answering without Repairs in
+Tables with Nulls and Functional Dependencies
+Dominique Laurent
+dominique.laurent@u-cergy.fr
+ETIS Laboratory - ENSEA, CY Cergy Paris University, CNRS
+F-95000 Cergy-Pontoise - FRANCE
+Nicolas Spyratos
+nicolas.spyratos@lri.fr
+LISN Laboratory - University Paris-Saclay, CNRS
+F-91405 Orsay - FRANCE
+January 11, 2023
+Abstract
+In this paper, we study consistent query answering in tables with nulls
+and functional dependencies. Given such a table T, we consider the set
+T of all tuples that can be built up from constants appearing in T; and
+we use set theoretic semantics for tuples and functional dependencies to
+characterize the tuples of T in two orthogonal ways: first as true or false
+tuples; and then as consistent or inconsistent tuples. Queries are issued
+against T and evaluated in T .
+In this setting, we consider a query Q: select X from T where Condition
+over T and define its consistent answer to be the set of tuples x in T such
+that: (a) x is a true and consistent tuple with schema X and (b) there
+exists a true super-tuple t of x in T satisfying the condition. We show
+that, depending on the ‘status’ that the super-tuple t has in T , there are
+different types of consistent answer to Q.
+The main contributions of the paper are: (a) a novel approach to con-
+sistent query answering not using table repairs; (b) polynomial algo-
+rithms for computing the sets of true/false tuples and the sets of con-
+sistent/inconsistent tuples of T ; (c) polynomial algorithms in the size of
+T for computing different types of consistent answer for both conjunctive
+and disjunctive queries; and (d) a detailed discussion of the differences
+between our approach and the approaches using table repairs.
+Keywords: database semantics, inconsistent database, functional dependency,
+null value, consistent query answering
+1
+arXiv:2301.03668v1  [cs.DB]  9 Jan 2023
+
+1
+Introduction
+In a relational database, each table is seen as a set of tuples that must satisfy a
+set of functional dependencies. Moreover, each table is assumed to be consistent
+with the dependencies and its tuples are assumed to be true when users query
+the table (these are basic assumptions in relational databases). Consistency is
+verified during updates in the following sense: if the update to be performed
+results in an inconsistent table then the update is rejected, otherwise it is ac-
+cepted.
+However, the consistency of a table in general is not always easy to verify, for
+example if the table is the result of integrating data coming from independent
+sources (e.g., web sources). In such cases the table may contain inconsistent
+tuples, and the problem is: how to answer user queries so that the answers
+contain only tuples that are true and consistent. This kind of query answering
+process is usually referred to as consistent query answering [1, 3].
+T
+Emp
+Dept
+Addr
+1
+e
+d
+a
+2
+e
+d
+a′
+3
+e′
+d′
+a
+repairs
+====⇒
+R1
+Emp
+Dept
+Addr
+1
+e
+d
+a
+3
+e′
+d′
+a
+R2
+Emp
+Dept
+Addr
+2
+e
+d
+a′
+3
+e′
+d′
+a
+Figure 1: An inconsistent table and its two repairs
+To see an example, consider the table T of Figure 1, with dependencies
+Emp → Dept and Dept → Addr. The table T is defined over attributes Emp,
+Dept and Addr, standing for ‘employee identifier’, ‘department identifier’ and
+‘department address’, respectively.
+This table is inconsistent as the pair of tuples 1 and 2 violates the dependency
+Dept → Addr. However, if we ask the SQL query Q : select Emp, Dept from T,
+it makes sense to return the set of tuples {ed, e′d′} as the answer. Indeed, there
+is no reason to reject this answer as it is a consistent answer, in the sense that
+it satisfies the dependency Emp → Dept. In other words, an inconsistent table
+might contain some consistent parts (i.e., some useful information) which can
+be extracted through queries.
+The traditional approach to alleviate the impact of inconsistent data on
+query answers is to introduce the notion of repair: a repair is a maximal consis-
+tent sub-set of the table, and a tuple t is in the consistent answer if t is present
+in the answer from every repair [1, 15]. To illustrate this approach, consider
+again the table T in Figure 1 with its two repairs, namely R1 = {eda, e′d′a}
+and R2 = {eda′, e′d′a}. Both these repairs are consistent with the given de-
+pendencies, and maximal with respect to set-theoretic inclusion. The answer
+to Q from R1 is {ed, e′d′} and so is the answer from R2. Therefore the consis-
+tent answer to Q is {ed, e′d′}. We note here that the complexity of the query
+2
+
+evaluation algorithm in the repairs approach is APX-complete1 [11].
+Now, in several applications today we have to deal with tables containing
+nulls (i.e., missing values) and having to satisfy a set of functional dependencies.
+The presence of nulls in a table is due to various reasons such as: ‘value does not
+exist’ (e.g., the maiden name of a male employee); ‘value exists but is currently
+unknown’ (e.g., the department of a newly hired employee currently following
+a training course before being assigned to a specific department); and so on. A
+survey of the different kinds of missing values considered in the literature can
+be found in [10]. In our approach we assume that missing values are of the kind
+‘value exists but is currently unknown’.
+In the context of the previous example, let T = {ed, da, ea′} in which
+the Addr-value in ed, the Emp-value in da and the Dept-value in ea′ are
+nulls that is they exist but are currently unknown.
+Considering the query
+Q : select Emp, Addr from T, if T is seen as a regular table, the consis-
+tent answer to Q is {ea′}, as the tuples in T ‘seem’ to satisfy the functional
+dependencies Emp → Dept and Dept → Addr.
+However, these functional dependencies allow to infer the missing values:
+from tuples ed and da, it can be inferred that eda is true, and from tuples ed,
+ea′ it can be inferred that eda′ is true. This shows that Dept → Addr is not
+satisfied, and thus {ea′} should not be returned as a consistent answer. The
+process of inferring missing values in a table is known in the literature as the
+chase procedure [8, 13, 14], and it is well-known that this procedure fails (i.e.,
+stops and returns no table) when encountering an inconsistency. Therefore, the
+repair-based approaches do not work in the presence of nulls.
+In the light of the previous example, we propose a novel approach to consis-
+tent query answering that does not rely on repairs, but that works in the case
+where the given set of tuples contains missing values.
+In our approach one starts again with a set of tuples T that are not required
+to be all of them defined over the same set of attributes. Therefore T is rep-
+resented as a table with nulls. However, nulls simply act as place holders that
+may receive values implied by functional dependencies, as shown in the previous
+example. What is significant in our approach is the set T , called the universe of
+discourse of T, containing all tuples that can be built up from values appearing
+in T. In our previous example, T consists of the tuples eda and eda′ together
+with all their sub-tuples.
+Queries are addressed to T and consistent answers are obtained from T . To
+achieve this we associate every tuple t in T with a set of identifiers, referred to as
+the interpretation of t. In doing so, we define set-theoretic semantics for tuples
+and for functional dependencies which allows us to characterize the tuples of T
+along two orthogonal dimensions: a tuple of T can be true or false on one hand
+and consistent or inconsistent on the other hand.
+In this setting, we consider a query Q : select X from T where Condition
+over T and we define its consistent answer as the set of tuples x in T such that:
+1We recall that APX is the set of NP optimization problems that allow polynomial-time
+approximation algorithms (source Wikipedia).
+3
+
+(a) x is a true and consistent tuple with schema X and (b) there exists a true
+super-tuple t of x in T satisfying the condition. We show that, depending on the
+‘status’ that the super-tuple t has in T , there are different types of consistent
+answer to Q.
+The important point to emphasize again is that, in our approach, queries are
+addressed to T and evaluated using the universe of discourse T . As mentioned
+earlier, T is the set of all tuples that one can define using values appearing in T.
+Actually, the fundamental difference between our approach and all approaches
+based on repairs is that their universe of discourse is the table T itself. Therefore
+our approach is simply not comparable to approaches based on repairs. To see
+this, consider again our previous example but this time with Emp → Addr as
+the only functional dependency.
+Now suppose that T = {eda, eda′, e′da′} and that we ask for the set of all
+addresses. Then the answer in the repairs approach is {a′} because there are
+only two repairs, R1 = {eda, e′da′} and R2 = {eda′, e′da′}, and the answers
+from each repair are respectively {a, a′} and {a′}; whereas in our approach, the
+answer is empty because a and a′ are both inconsistent tuples of T . On the
+other hand, if T = {eda, ed′a′} and we ask for the set of all departments then
+the answer in the repairs approach is empty because there are only two repairs,
+R1 = {eda} and R2 = {ed′a′} and none of d, d′ is in the answer from both
+repairs; whereas, in our approach, the answer is {d, d′} because both d and d′
+are true and consistent tuples of T .
+Our work builds upon earlier work on partition semantics [6, 13] and on
+inconsistent tables with missing values [9]. The main contributions of this paper
+are as follows:
+1. We propose a novel approach to consistent query answering in tables with
+nulls and functional dependencies, without using table repairs.
+2. We provide polynomial algorithms for computing (a) the sets of true/false
+tuples and the sets of consistent/inconsistent tuples of T and (b) for com-
+puting the consistent answer to any conjunctive or disjunctive query over
+T.
+3. We offer a detailed discussion of the differences between our approach and
+other approaches including the approaches by table repairs.
+The remaining of the paper is organized as follows. In Section 2 we first in-
+troduce the basic notions and notation of our approach, and we present our
+set-theoretic semantics for tuples and functional dependencies; in Section 3 we
+address the issue of consistent query answering, introducing the notion of con-
+sistency with respect to a selection condition; in Section 4 we first define the
+m-Chase algorithm, a chase-like algorithm for computing the sets of true/false
+tuples and the sets of consistent/inconsistent tuples of T , and then we propose
+a polynomial algorithm for computing consistent answers; in Section 5 we com-
+pare our approach to the repair approaches and discuss other related work; and
+finally, in Section 6 we offer concluding remarks and outline current work and
+future perspectives.
+4
+
+2
+The Semantics of our Model
+In this section we recall basic definitions from the relational model regarding
+tuples and tables, and we present the set-theoretic semantics that we use for
+tuples and functional dependencies. Our approach builds upon the so-called
+“partition model” introduced in [13].
+2.1
+Terminology and Notation
+Following [13], we consider a universe U = {A1, . . . , An} in which every attribute
+Ai is associated with a set of atomic values called the domain of Ai and denoted
+by dom(Ai). An element of �
+A∈U dom(A) is called a constant.
+We call relation schema (or simply schema) any nonempty sub-set of U and
+we denote it by the concatenation of its elements; for example {A1, A2} is simply
+denoted by A1A2. Similarly, the union of schemes S1 and S2 is denoted as S1S2
+instead of S1 ∪ S2.
+We define a tuple t over U to be a partial function from U to �
+A∈U dom(A)
+such that, for every A in U, if t is defined over A then t(A) belongs to dom(A).
+The domain of definition of t is called the schema of t, denoted by sch(t). We
+note that tuples in our approach satisfy the First Normal Form [14] in the sense
+that each tuple component is an atomic value from the corresponding attribute
+domain.
+Regarding notation, we follow the usual convention that, whenever possible,
+lower-case characters denote domain constants and upper-case characters denote
+the corresponding attributes. Following this convention, the schema of a tuple
+t = ab is AB and more generally, we denote the schema of a tuple s as S.
+Assuming that the schema of a tuple t is understood, t is denoted by the
+concatenation of its values, that is: t = ai1 . . . aik means that for every j =
+1, . . . , k, t(Aij) = aij, where aij is in dom(Aij), and sch(t) = Ai1 . . . Aik.
+As in [13], we assume that for any two distinct attributes A and B, we have
+dom(A) ∩ dom(B) = ∅. However, it might be relevant for attribute domains
+to share values.
+For instance, in our introductory example, in the presence
+of attribute Mgr standing for ‘manager’, the domains of Emp and Mgr are
+employee identifiers.
+In such cases, in order to avoid confusion, we denote
+attribute values as pairs of the form ⟨attribute name, value⟩ and comparisons
+are assessed with respect to their value component only. In order to keep the
+notation simple we shall omit attribute names as prefixes whenever no ambiguity
+is possible.
+We define a table T over U to be any finite set of tuples over U (therefore
+duplicates are not allowed). As tuples over U are partial functions over U, it
+follows that T may contain nulls.
+Given a table T over U, we denote by T the set of all tuples that can be built
+up from constants appearing in T; and we call T the universe of discourse of T
+as it contains all tuples of interest in query processing. Actually, as we shall see
+shortly, queries are issued against T and consistent answers are obtained from
+5
+
+T . For every relation schema X, we denote by T (X) the set of all tuples in T
+whose schema is X. Formally, T (X) = {t ∈ T | sch(t) = X}.
+For every A in U, the set of all values from dom(A) occurring in T is referred
+to as the active domain of A, denoted by adom(A). We denote by AD the set
+of all constants appearing in T that is: AD = �
+A∈U adom(A). We emphasize
+that even when attribute domains are infinite, active domains are always finite
+and therefore the sets AD and T are finite as well.
+Given a tuple t, for every A in sch(t), t(A) is also denoted by t.A and more
+generally, for every nonempty sub-set S of sch(t) the restriction of t to S, also
+called sub-tuple of t, is denoted by t.S. In other words, if S ⊆ sch(t), t.S is the
+tuple such that sch(t.S) = S, and for every A in S we have (t.S).A = t.A.
+Moreover, ⊑ denotes the ‘sub-tuple’ relation, defined over T as follows: for
+any tuples t1 and t2, t1 ⊑ t2 holds if t1 is a sub-tuple of t2. It is thus important to
+keep in mind that whenever t1 ⊑ t2 holds, it is understood that sch(t1) ⊆ sch(t2)
+also holds. The relation ⊑ is clearly a partial order over T . Given a table T,
+the set of all sub-tuples of the tuples in T is called the lower closure of T and
+it is defined by: LoCl(T) = {q ∈ T | (∃t ∈ T)(q ⊑ t)}.
+2.2
+Partition Semantics for Tuples
+In this work, we consider tables T over a set U of attributes, possibly with nulls,
+and we assume that every tuple t in T is associated with a unique identifier,
+id(t).
+We denote by TID the set of all identifiers of tuples in T ; that is:
+TID = {id(t) | t ∈ T }. In our examples, for simplicity, we assume that TID
+is a set of positive integers. Denoting by P(TID) the power-set of TID, the
+following definition of interpretation is in the spirit of [13].
+Definition 1 Let T be a table over U. We call interpretation of T any function
+I from T to P(TID) such that:
+• For every tuple t of T, I(t) ̸= ∅
+• For every tuple t = a1a2 . . . an of T , I(t) = I(a1) ∩ I(a2) ∩ . . . ∩ I(an)
+We say that a tuple t in T is true in I if I(t) ̸= ∅; otherwise we say that t is
+false in I. We denote by True(I) the set of tuples t that are true in I and by
+False(I) the set of tuples t that are false in I.
+�
+Example 1 Let T = {ab, a′c} and let I be a function from T to P(TID) defined
+as follows: I(a) = {1, 2}, I(a′) = {2}, I(b) = {1, 2}, I(c) = {2}, I(ab) = {1, 2},
+I(ac) = {2}, I(a′b) = {2}, I(a′c) = {2}, I(abc) = {2}, I(a′bc) = {2}. Then I is
+an interpretation of T as we have:
+• I(ab) ̸= ∅ and I(a′c) ̸= ∅ that is the two tuples of T are true in I
+• For every tuple t in T , I(t) is indeed the intersection of the interpretations
+of its components. For example, I(ac) = I(a) ∩ I(c) = {1, 2} ∩ {2} = {2}
+and one can verify easily for the remaining tuples of T .
+�
+6
+
+It is important to note that the first item in Definition 1 expresses a fundamental
+assumption regarding a relational table T, namely that every tuple t of T is
+assumed to be true. Two immediate consequences are that (a) I(a) ̸= ∅, for all
+a in AD (i.e., every a in AD is true in I) and (b) if a tuple t is true in I then
+so is every sub-tuple of t; and if t is false in I then so is every super-tuple of
+t. As a consequence the set T is partitioned into true and false tuples that is
+True(I) ∪ False(I) = T and True(I) ∩ False(I) = ∅.
+The interpretations of T can be compared according to the following defini-
+tion.
+Definition 2 Let T be a table and I, I′ two interpretations of T. Then we say
+that I is less than or equal to I′, denoted I ⪯ I′ if for every t in T , I(t) ⊆ I′(t).
+�
+It is easy to see that the relation ⪯ is a partial ordering over the set of all
+interpretations of T. Note that if we view the constants of AD as unary tuples
+then we have that: if I ⪯ I′ then for every a in AD, I(a) ⊆ I′(a) holds; and in
+the opposite direction, if I(a) ⊆ I′(a) holds for every a in AD then for every t
+in T , I(t) ⊆ I′(t) also holds that is I ⪯ I′ holds. Therefore we have that:
+• I ⪯ I′ holds if and only if for every a in AD, I(a) ⊆ I′(a).
+Based on this result, in order to verify whether I ⪯ I′, it is sufficient to do so
+for every constant in AD rather than for every tuple in T .
+Now, to see the intuition behind the above definitions, think of the identifiers
+of TID as being objects and of every a in AD as being an atomic property that
+each object may have. Then I(a) is the set of objects having property a; and
+similarly, the intuitive idea behind the interpretation of a tuple is that a tuple,
+say ab, is the ‘conjunction’ of the atomic properties a and b. Therefore I(ab) is
+the set of objects each having both properties a and b, hence I(ab) = I(a)∩I(b).
+Clearly, if I(ab) = ∅ then no object has both properties a and b so the tuple ab
+is false in I.
+As for the ordering of interpretations what I ⪯ I′ means is that, for every
+property t of T , the set of objects having property t in I is included in the set
+of objects having property t in I′.
+Another fundamental assumption made in the relational model is that the
+components of every tuple be atomic. Actually such a tuple is said to satisfy
+the First Normal Form [14]. To express this assumption using our definition of
+interpretation, suppose that there are a, a′ in adom(A) such that a ̸= a′ and
+I(a) ∩ I(a′) ̸= ∅. Then every tuple whose identifier is in I(a) ∩ I(a′) violates
+the First Normal Form as its A-component is {a, a′}, therefore not atomic. In
+the light of this observation we introduce the following definition of inconsistent
+tuple.
+Definition 3 Let T be a table over U and I an interpretation of T. A tuple t
+in T is said to be inconsistent in I if there exist A in sch(t) and a′ in adom(A)
+such that t.A ̸= a′ and I(t) ∩ I(a′) ̸= ∅.
+7
+
+The set of all inconsistent tuples in I is denoted by Inc(I). A tuple that is
+not inconsistent in I is said to be consistent in I, and the set of all consistent
+tuples in I is denoted by Cons(I). An interpretation I such that Inc(I) = ∅ is
+said to be in First Normal Form.
+�
+For instance, in Example 1, the tuple t = abc is inconsistent in I as A is in
+sch(t), and a′ is in adom(A) such that t.A ̸= a′ and I(t) ∩ I(a′) = {2} ̸= ∅.
+Therefore I is not in First Normal Form.
+In view of the above definition, the set T is partitioned into inconsistent
+and consistent tuples that is Inc(I) ∪ Cons(I) = T and Inc(I) ∩ Cons(I) = ∅.
+Roughly speaking, inconsistent tuples are those that violate the First Normal
+Form. This relationship between inconsistency and First Normal Form is stated
+formally in the following lemma.
+Lemma 1 Let I be an interpretation of T. Then I is in First Normal Form if
+and only if I satisfies the following constraint:
+Partition constraint (PC): For all A in U and for all a, a′ in adom(A),
+if a ̸= a′ then I(a) ∩ I(a′) = ∅.
+Proof. Assume that I satisfies PC and that Inc(I) ̸= ∅. Given t in Inc(I),
+using the notation of Definition 3 and denoting by a the A-value occurring in t,
+we have I(a) ∩ I(a′) ̸= ∅, because I(t) ⊆ I(a) ∩ I(a′). Hence, I does not satisfy
+PC, which is a contradiction.
+Conversely, if I does not satisfy PC, then there exist A in U and a and a′
+such that I(a)∩I(a′) ̸= ∅. Therefore, by Definition 3, a and a′ are in Inc(I), thus
+implying that I is not in First Normal Form. The proof is therefore complete.
+�
+Note that, as mentioned in [9], satisfaction of the partition constraint implies
+that, for all A in U, the set {I(a) | a ∈ adom(A)} is a partition of the set
+adom(A) (whence the term “partition constraint”).
+An important question remains regarding the definition of an interpretation:
+is there a systematic way for defining an interpretation I of a given table T
+which is in First Normal Form? The answer is yes, there is such a “canonical”
+interpretation for every table T; it is called the basic interpretation of T, denoted
+by Ib and defined as follows:
+Basic Interpretation Ib: For every A in U and every a in adom(A),
+Ib(a) = {id(t) | t ∈ T and t.A = a}.
+In the context of Example 1 where T = {ab, a′c}, if ab and a′c are respectively
+associated with identifiers 1 and 2, the associated basic interpretation Ib is
+defined by Ib(a) = Ib(b) = {1} and Ib(a′) = Ib(c) = {2}. Considering T1 =
+{ab, bc, ac} with respective identifiers 1, 2 and 3, the basic interpretation Ib
+1 is
+defined by Ib
+1(a) = {1, 3}, Ib
+1(b) = {1, 2} and Ib
+1(c) = {2, 3}.
+It is interesting to note that the definition of basic interpretation parallels
+that of inverted file [17, 16]. Indeed, we first recall that an inverted file is an
+8
+
+index data structure that maps content to its location within a database file, in
+a document or in a set of documents. Hence, if we assume that each tuple of T
+is implemented as a record (possibly with missing values) then we can view T
+as a file that is as a function I assigning to each value x appearing in the file
+record the set I(x) of all identifiers of records in which x appears.
+The following lemma summarizes important properties of the basic interpre-
+tation.
+Lemma 2 Let T be a table and Ib its basic interpretation as defined above.
+Then:
+1. Ib is an interpretation satisfying the partition constraint.
+2. True(Ib) = LoCl(T) and True(Ib) is the set of all tuples t that belong to
+True(I) for every interpretation I of T.
+3. Ib is minimal with respect to ⪯ among all interpretations I of T such that
+for every t in T, id(t) belongs to I(t).
+Proof. 1. Assume that there exist A in U and a and a′ in adom(A) such that
+Ib(a) ∩ Ib(a′) ̸= ∅. If i is in Ib(a) ∩ Ib(a′) ̸= ∅, let ti be the tuple in T such that
+id(ti) = i. This implies that the value of ti(A) is a and a′, which is impossible
+because, as ti is a function, ti(A) consists of at most one value in adom(A).
+This part of the proof is thus complete.
+2. Since for every t in T, Ib(t) contains id(t) and as for every t′ such that t′ ⊑ t,
+we have Ib(t) ⊆ Ib(t′), it follows that LoCl(T) ⊆ True(Ib). Conversely, let t not
+in LoCl(T) and assume that Ib(t) ̸= ∅. Since t is not in T, it is not associated
+with an identifier. Hence, Ib(t) contains identifiers of other tuples, and if q is
+such a tuple, then by definition of Ib, this implies that id(q) belongs to every
+component of t, that is that t is a sub-tuple of q.
+A contradiction showing
+that True(Ib) ⊆ LoCl(T). Hence we have True(Ib) = LoCl(T). Moreover, since
+for every interpretation I of T, True(I) must at least contain all tuples in T
+along with their sub-tuples (as argued just above), we have LoCl(T) ⊆ True(I).
+Hence, True(Ib) ⊆ True(I), which completes this part of the proof.
+3. Let I be an interpretation such that I ⪯ Ib and I ̸= Ib. Thus, there exists a
+in AD such that I(a) ⊂ Ib(a). However, since I is an interpretation such that
+for every t in T, id(t) belongs to I(t), if a occurs in t then id(t) belongs to I(a).
+In other words, Ib(a) ⊆ I(a) holds. A contradiction showing that Ib is minimal
+with respect to ⪯ among all interpretations I of T such that for every t in T,
+id(t) belongs to I(t). The proof is therefore complete.
+�
+By Definition 3 and Lemma 1, it turns out that, since Ib satisfies the partition
+constraint PC, Ib is in First Normal Form, therefore Inc(Ib) = ∅.
+From now on, in all our examples we shall use the following convention: the
+tuples of T will receive successive integer identifiers in the order of their appear-
+ance in T, starting with number 1. For example, if T = {ab, a′c} as in Example 1,
+then it is understood that id(ab) = 1 and id(a′c) = 2. Then, the basic inter-
+pretation of T is defined as earlier stated, that is: Ib(ab) = {1}, Ib(a′c) = {2},
+9
+
+Ib(a) = {1}, Ib(a′) = {2}, Ib(b) = {1}, Ib(c) = {2}.
+Thus, Ib(ac) = ∅,
+Ib(abc) = ∅, Ib(a′bc) = ∅. Therefore we have: True(Ib) = {a, a′, b, c, ab, a′c}
+and False(Ib) = {ac, abc, a′bc}. Note that Ib is an interpretation that satisfies
+the partition constraint.
+2.3
+Partition Semantics for Functional Dependencies
+Let T be a table over a set U of attributes together with a set FD of functional
+dependencies. As usual in relational databases [14], a functional dependency is
+an expression of the form X → Y where X and Y are relation schemes. The
+notion of functional dependency satisfaction in the context of tables with nulls is
+stated in the literature [7, 8] in the following terms. A table T satisfies X → Y
+if for all tuples t and t′ in T the following holds: if XY is a sub-set of sch(t)
+and of sch(t′) then t.X = t′.X implies t.Y = t′.Y .
+Moreover, it is well-known that a relation satisfies X → Y if and only if it
+satisfies the functional dependencies X → A, for every A in Y . This justifies
+that functional dependencies are usually assumed to be of the form X → A,
+where X is a relation schema and A is an attribute not in X. In what follows,
+we make this assumption.
+Now, the question that we answer in this section is the following: how can
+an interpretation I of T express the satisfaction of a functional dependency by
+T? The answer is provided by the following lemma.
+Lemma 3 Let I be an interpretation of table T in First Normal Form, and
+let X → A be a functional dependency of FD. Then T satisfies X → A if I
+satisfies the following constraint:
+Inclusion constraint (IC): For all t in T such that XA ⊆ sch(t),
+I(t.X) ∩ I(t.A) ̸= ∅ implies I(t.X) ⊆ I(t.A).
+Proof.
+Suppose that I satisfies the inclusion constraint and consider the
+relation f : adom(X) → adom(A) defined by: for every true tuple t in T,
+f(t.X) = t.A. We show that f is actually a function. Indeed, suppose there
+is a true tuple t′ in T such that t′.X = t.X and t′.A ̸= t.A.
+As I is in
+First Normal Form, by Lemma 1, I satisfies the partition constraint, and thus,
+I(t.A)∩I(t′.A) = ∅. On the other hand, as I satisfies the inclusion constraint IC
+for X → A, we have I(t.X) ⊆ I(t.A) and I(t′.X) ⊆ I(t′.A). As t.X = t′.X, we
+have I(t.X) ⊆ I(t.A)∩I(t′.A), thus that I(t.X) ⊆ ∅. It follows that I(t.X) = ∅,
+which is a contradiction to the fact that t.X is true, being a sub-tuple of t. The
+proof is therefore complete.
+�
+Example 2 Let T = {ab, bc} and FD = {A → B}. Then the basic interpreta-
+tion Ib of T is as follows: Ib(a) = {1}, Ib(b) = {1, 2}, Ib(c) = {2} and we have
+that Ib(a) ∩ Ib(b) = {1} ̸= ∅ and Ib(a) ⊆ Ib(b). As the only tuple of T to which
+the lemma applies is ab we conclude that T satisfies A → B. In this case Ib
+satisfies the partition constraint PC and the inclusion constraint IC.
+As another example, let T = {ab, ac} and FD = {A → B}.
+Then the
+interpretation Ib of T is as follows: Ib(a) = {1, 2}, Ib(b) = {1}, Ib(c) = {2} and
+10
+
+we have Ib(a) ∩ Ib(b) = {1} ̸= ∅ but Ib(a) ̸⊆ Ib(b), showing that Ib does not
+satisfy the inclusion constraint. Therefore, Ib satisfies the partition constraint
+PC but not the inclusion constraint IC.
+�
+In view of Lemma 3, Ib satisfies the partition constraint, the set of true tuples
+of Ib is equal to the set of true tuples of every interpretation I, and Ib satisfies
+a minimality property with respect to ⪯ among interpretations of T. Therefore
+the question that arises here is whether we can “expand” Ib to an interpretation
+I′ that satisfies the inclusion constraint as well, so that I′ satisfies the same
+properties as Ib, along with the inclusion constraint IC.
+In Example 2, the answer is yes. Indeed, if we add Ib(a) to Ib(b) then Ib(b)
+becomes Ib(a) ∪ Ib(b) and the resulting interpretation is: I′(a) = {1, 2}, I′(b) =
+{1, 2}, I′(c) = {2}; and I′ satisfies the inclusion constraint for A → B. However,
+the following example shows that expanding Ib may lead to non satisfaction of
+the partition constraint PC.
+Example 3 Let T = {ab, bc, ac′} with FD = {A → C, B → C}. Here, Ib
+is defined by Ib(a) = {1, 3}, Ib(b) = {1, 2}, Ib(c) = {2} and Ib(c′) = {3}.
+Thus, Ib does not satisfy the constraint IC because Ib(a) ∩ Ib(c) = {2} ̸= ∅ but
+Ib(a) ̸⊆ Ib(c).
+Notice that this example shows that the converse of Lemma 3 does not hold.
+It is so because T satisfies A → C and B → C, although Ib is an interpretation
+of T in First Normal Form that does not satisfy the inclusion constraint IC.
+Expanding Ib as in Example 2 yields the interpretation I′ such that I′(c) =
+I′(c′) = {1, 2, 3}, I′(a) = Ib(a) and I′(b) = Ib(b). It should be clear that I′
+satisfies the constraint IC, whereas I′ does not satisfy the constraint PC since
+I′(c) ∩ I′(c′) ̸= ∅. As a consequence, by Lemma 1, Inc(I′) ̸= ∅. More precisely,
+as True(I′) is the set containing abc and abc′ along with their sub-tuples, we
+have Inc(I′) = {abc, abc′, bc, bc′, ac, ac′, c, c′}.
+�
+In view of our discussion above, the following definition states how we can
+expand an interpretation I so that it satisfies a given functional dependency
+X → A.
+Definition 4 The expansion of I with respect to a functional dependency X →
+A and a tuple xa in T (XA) is the interpretation Exp(I, xa) defined as follows:
+• If I(x) ∩ I(a) ̸= ∅ and I(x) ̸⊆ I(a) then: Exp(I, xa)(a) = I(a) ∪ I(x), and
+Exp(I, xa)(α) = I(α) for α in AD different than a.
+• Otherwise, Exp(I, xa) = I.
+�
+Of course, the expansion processing should be iterated when several tuples sat-
+isfy the above condition and when several functional dependencies are consid-
+ered. As will be seen next, starting with the basic interpretation Ib and applying
+expansion repeatedly until a fixed point is reached, we obtain an interpretation
+I∗ such that: (a) I∗ satisfies the inclusion constraint IC (but not necessarily the
+partition constraint PC) (b) a tuple t of T is true in I∗ if and only if t is true in
+every interpretation I of T satisfying IC and (c) a tuple t of T is inconsistent
+in I∗ if and only if t is inconsistent in every interpretation I of T satisfying IC.
+11
+
+2.4
+The True and the Inconsistent Tuples
+To define the limit interpretation I∗ of T mentioned above, for a given a table
+T over universe U with a set FD of functional dependencies, we consider the
+sequence of interpretations (Ij)j≥0 of T defined by:
+• For j = 0, we set I0 to be the basic interpretation Ib as defined earlier.
+• For j ≥ 0, let Ij+1 be defined as follows:
+– If there exist X → A, x and a such that Exp(Ij, xa) ̸= Ij, then
+Ij+1 = Exp(Ij, xa)
+– Else Ij+1 = Ij.
+The theorem below states that this sequence has a limit with important proper-
+ties. In this theorem, given a nonempty sub-set S of AD and an interpretation
+I, we denote by I(S) the set �
+a∈S I(a).
+Theorem 1 The sequence (Ij)j≥0 is increasing with respect to ⪯ and bounded
+above, therefore it has a limit that we denote by I∗. This limit has the following
+properties:
+1. I∗ is unique in the sense that it is independent of the order in which
+expansions are applied (i.e., the construction of I∗ has the Church-Rosser
+property).
+2. I∗ satisfies the inclusion constraint IC.
+3. For every nonempty sub-set S of AD, I∗(S) ̸= ∅ holds if and only if
+I(S) ̸= ∅ holds for every interpretation I of T satisfying IC.
+Proof. First, it is clear that for j ≥ 0, Ij ⪯ Ij+1 holds by definition of the
+sequence. Now, as for every a in AD we have that: Ij(a) ⊆ TID for every j ≥ 0,
+the sequence is bounded by the interpretation IT ID defined by: IT ID(a) = TID
+for every a in AD. Hence, the sequence has a limit, denoted by I∗. The proof
+of the items in the theorem are as follows:
+1. Uniqueness of the limit I∗ is shown in Appendix A.
+2.
+Suppose that I∗ does not satisfy the inclusion constraint IC.
+It follows
+that there exist X → A in FD and t in T such that XA ⊆ sch(t), I∗(t.X) ∩
+I∗(t.A) ̸= ∅ and I∗(t.X) ̸⊆ I∗(a).
+Thus, assuming j is such that I∗ = Ij,
+we have Ij+1 ̸= Ij, which implies that I∗ is not the limit of the sequence, a
+contradiction.
+3. Let S be a nonempty sub-set of AD such that I(S) ̸= ∅ for every I satisfying
+IC. Since I∗ is an interpretation that satisfies IC, we also have I∗(S) ̸= ∅.
+Conversely, if I∗(S) ̸= ∅, we prove by induction that I(S) ̸= ∅ holds for
+every interpretation I satisfying IC.
+• If I0(S) ̸= ∅, then it follows from Lemma 2 that S does not contain two
+elements from the same active domain (because Ib satisfies the partition con-
+straint PC). Thus, the elements of S form a tuple t that belongs to True(I0).
+12
+
+It follows that t is a sub-tuple of a tuple q in T. Hence, t belongs to True(I)
+for every interpretation I of T, thus for every interpretation of T satisfying the
+inclusion constraint IC.
+• Assume that for every nonempty sub-set Σ of AD, if Ij(Σ) ̸= ∅ then I(Σ) ̸= ∅
+for every I satisfying IC, and let S be such that Ij+1(S) ̸= ∅. If Ij(S) ̸= ∅, then
+Ij+1(S) ̸= ∅ holds because Ij ⪯ IJ+1. Considering the case where Ij(S) = ∅,
+as Ij+1(S) ̸= Ij(S), S contains a constant a such that Ij(a) ̸= Ij+1(a). De-
+noting S \ {a} by Sa, we have Ij(Sa) = Ij+1(Sa) (because expansion changes
+the interpretation of only one constant, namely a in this case), and there exist
+X → A in FD and x in T (X) such that Ij(x) ∩ Ij(a) ̸= ∅ and Ij(x) ̸⊆ Ij(a).
+In this case, we have Ij+1(a) = Ij(a) ∪ Ij(x), and thus
+Ij+1(S)
+=
+Ij+1(a) ∩ Ij+1(Sa)
+=
+(Ij(a) ∪ Ij(x)) ∩ Ij(Sa)
+=
+(Ij(a) ∩ Ij(Sa)) ∪ (Ij(x) ∩ Ij(Sa))
+Since Ij(S) = Ij(a) ∩ Ij(Sa) is empty, it follows that Ij(x) ∩ Ij(Sa) ̸= ∅ and so,
+for every I satisfying IC, I(x) ∩ I(a) ̸= ∅ and I(x) ∩ I(Sa) ̸= ∅. Hence, I(x) ⊆
+I(a), and so I(x) ∩ I(Sa) ⊆ I(a) ∩ I(Sa). As this implies that I(a) ∩ I(Sa) ̸= ∅,
+we obtain that I(S) ̸= ∅, and the proof is complete.
+�
+We note that similar results have been obtained with a slightly different formal-
+ism in [9]. Moreover, as a consequence of Theorem 1, for every tuple t in T , we
+have:
+• t belongs to True(I∗) if and only if t belong to True(I) for every inter-
+pretation I of T satisfying IC. We denote the set True(I∗) by True(T )
+and its tuples are said to be true in T . The set False(I∗) is denoted by
+False(T ) and its tuples are said to be false in T .
+• t belongs to Inc(I∗) if and only if t belongs to Inc(I) for every interpre-
+tation I of T satisfying IC. We denote the set Inc(I∗) by Inc(T ) and its
+tuples are said to be inconsistent in T . The set Cons(I∗) is denoted by
+Cons(T ) and its tuples are said to be consistent in T .
+The following proposition lists basic properties of true, false, consistent and
+inconsistent tuples in T .
+Proposition 1 For every table T over universe U, the following holds:
+1. If t is in True(T ) then every t′ such that t′ ⊑ t belongs to True(T ).
+2. Inc(T ) ⊆ True(T ).
+However, the inclusions Cons(T ) ⊆ True(T ) and
+Cons(T ) ⊆ False(T ) do not hold.
+3. Let t be in Inc(T ) and A in sch(t). Let a′ be in adom(A) such that a′ ̸= t.A
+and I∗(t) ∩ I∗(a′) ̸= ∅. Then every tuple t′ such that t′ ⊑ t and A belongs
+to sch(t′) is in Inc(T ).
+4. It does not hold that for t in Inc(T ), every true super-tuple of t is in
+Inc(T ).
+13
+
+Proof. 1. Since t is in True(I∗), I∗(t) ̸= ∅. Thus, as I∗(t) ⊆ I∗(t′) (because
+t′ ⊑ t), we have I∗(t) ̸= ∅, showing that t′ is in True(T ).
+2. By Definition 3, if t is in Inc(T ) then there is A in sch(t) and a′ ̸= t.A in
+adom(A) such that I∗(t) ∩ I∗(a′) ̸= ∅. Thus I∗(t) ̸= ∅, which implies that t
+belong to True(T ).
+Regarding the two other inclusions in the item, for T = {ab, a′b′} and FD =
+∅, we have True(T ) = LoCl(T) and Inc(T ) = ∅. Thus, False(T ) = {ab′, a′b} and
+Cons(T ) = T , showing that Cons(T ) ̸⊆ True(T ) and Cons(T ) ̸⊆ False(T ).
+3. In this case, we have I∗(t) ⊆ I∗(t′) (because t ⊑ t′) and as I∗(t)∩I∗(a′) ̸= ∅,
+we obtain that I∗(t′) ∩ I∗(a′) ̸= ∅. Since t′.A = t.A (because A ∈ sch(t′) and
+t ⊑ t′), we have a′ ̸= t′.A, which implies that t′ is in Inc(T ).
+4. Let T = {ab, ab′, bc} with A → B. In this case I∗ is defined by I∗(a) =
+I∗(b′) = {1, 2}, I∗(b) = {1, 2, 3} and I∗(c) = {3}.
+Thus b is inconsistent,
+because I∗(b) ∩ I∗(b′) ̸= ∅, and we argue that bc is not inconsistent although bc
+is a true super-tuple of b. This is so because I∗(bc) = {3} but I∗(b′)∩I∗(bc) = ∅.
+�
+We illustrate now the construction of I∗ through the following example.
+Example 4 Referring to Example 3, we recall that T = {ab, bc, ac′} with FD =
+{A → C, B → C}, and that Ib was found to be defined by Ib(a) = {1, 3},
+Ib(b) = {1, 2}, Ib(c) = {2} and Ib(c′) = {3}. The construction of I∗ starts with
+I0 = Ib, and the following steps are performed:
+(1) Considering A → C, we have I0(a) ∩ I0(c) ̸= ∅ but I0(a) ̸⊆ I0(c). Hence
+I1(c) = I0(c)∪I0(a) = {1, 2, 3}, I1(a) = {1, 3}, I1(b) = {1, 2} and I1(c′) = {3}.
+(2) Considering again A → C, we have I1(a) ∩ I1(c′) ̸= ∅ but I1(a) ̸⊆ I1(c′).
+Hence I2(c′) is set to I1(c′) ∪ I1(a) = {1, 3}, and I2(a) = {1, 3}, I2(b) = {1, 2}
+and I2(c) = {1, 2, 3}.
+(3) For B → C, we have I2(b) ∩ I2(c′) ̸= ∅ but I2(b) ̸⊆ I2(c′). Hence I3(c′) is
+set to I2(c′) ∪ I2(b) = {1, 2, 3}, and I3(a) = {1, 3}, I3(b) = {1, 2} and I3(c) =
+{1, 2, 3}.
+Since I3 satisfies the inclusion constraint, we obtain I∗ = I3. We notice that
+I∗ is equal to the expected interpretation I′ defined in Example 3. Moreover, it
+can be seen that in this example, True(T ) contains the tuples abc and abc′ along
+with all their sub-tuples, meaning that True(T ) = T , and thus that False(T ) =
+∅.
+Regarding inconsistent tuples, as we have I∗(c)∩I∗(c′) ̸= ∅, it turns out that
+Inc(T ) = {abc, abc′, ac, ac′, bc, bc′, c, c′}, and thus that Cons(T ) = {ab, a, b}.
+�
+To further illustrate the impact of functional dependencies over true tuples and
+over inconsistent tuples, we mention the following two properties: given a table
+T over U, X → A in FD and a true tuple xa in T (XA), then:
+1. For every tuple t in True(T ) such that X ⊆ sch(t) and A ̸∈ sch(t), if
+t.X = x then the tuple ta is in True(T ). In other words, writing t as qx
+this result expresses the well-known property of relational lossless join: if
+14
+
+qx and xa are true, the functional dependency X → A implies that qxa is
+true as well.
+This is so because I∗(x) ⊆ I∗(a), I∗(t) ⊆ I∗(x) (because x ⊑ t) and
+I∗(ta) = I∗(t)∩I∗(a). Hence, I∗(t) ⊆ I∗(a), implying that I∗(ta) = I∗(t).
+Therefore, I∗(ta) ̸= ∅, because I∗(t) ̸= ∅.
+2. For every true super-tuple t of xa (i.e., t ∈ True(T ), XA ⊆ sch(t) and
+t.XA = xa) and for every a′ in adom(A) different than a such that xa′ is
+true (i.e., xa′ ∈ True(T )), if we write t as qxa then t′ = qxa′ also belongs
+to True(T ), and t and t′ both belong to Inc(T ).
+Indeed, since I∗(xa) and I∗(xa′) are nonempty, and I∗ satisfies the inclu-
+sion constraint, we have I∗(x) ⊆ I∗(a) ∩ I∗(a′) and I∗(x) ̸= ∅. It follows
+that I∗(a) ∩ I∗(a′) ̸= ∅. Moreover, since t = qxa and I∗(t) ̸= ∅, we have
+I∗(qx) ̸= ∅ and as I∗(qx) ⊆ I∗(x), it follows that I∗(qx) ⊆ I∗(a) ∩ I∗(a′),
+that is, I∗(qxa′) = I∗(qx). Consequently, I∗(qxa′) ̸= ∅, entailing that
+t′ = qxa′ is in True(T ).
+Since I∗(t) ̸= ∅, I∗(a′) ∩ I∗(xa) ̸= ∅ and
+I∗(t) ⊆ I∗(xa), t is in Inc(T ) and a similar reasoning shows that t′ is
+also in Inc(T ).
+3
+Consistent Query Answering
+In this section, we first define the syntax of queries that we consider and then
+we define and discuss four kinds of consistent answer to a query, based on the
+semantics presented in the previous section.
+3.1
+Syntax of Queries
+We use SQL as the query language and, as we query a single table T, the queries
+Q that we consider have one of the following two forms:
+Q : select X from T
+or
+Q : select X from T where Γ
+In either of these forms, X is an attribute list seen as a relation schema, and in
+the second form the where clause specifies a selection condition Γ. As in SQL
+the where clause in a query is optional, the generic form of a query Q is denoted
+by
+Q : select X from T [where Γ]
+The set of all attributes occurring in Γ is called the schema of Γ, denoted by
+sch(Γ); and the attribute set X ∪ sch(Γ) is called the schema of Q, denoted by
+sch(Q).
+A selection condition Γ is a well-formed formula involving the usual connec-
+tors ¬, ∨ and ∧ and built up from atomic Boolean comparisons of one of the
+following forms: A θ a or A θ A′, where θ is a comparison predicate, A and A′
+are attributes in U whose domain elements are comparable through θ, and a is
+in dom(A).
+Given a selection condition Γ, we denote by Sat(Γ) the set of all tuples in
+T (sch(Γ)) satisfying Γ, as defined below:
+15
+
+• if Γ is of the the form A θ a, Sat(Γ) = {t ∈ T (sch(Γ)) | t.A θ a},
+• if Γ is of the the form A θ B, Sat(Γ) = {t ∈ T (sch(Γ)) | t.A θ t.B},
+• if Γ is of the form Γ1 ∨ Γ2, Sat(Γ) = Sat(Γ1) ∪ Sat(Γ2),
+• if Γ is of the form Γ1 ∧ Γ2, Sat(Γ) = Sat(Γ1) ∩ Sat(Γ2),
+• if Γ is of the form ¬Γ1, Sat(Γ) = T (sch(Γ)) \ Sat(Γ1).
+Moreover, the set of tuples that do not satisfy Γ is defined by:
+• Sat−(Γ) = T (sch(Γ)) \ Sat(Γ).
+For example, if Sal is an attribute whose active domain is {s, s′}, we have:
+• for Γ4 = (Sal = s′), Sat(Γ4) = {s′} and Sat−(Γ4) = {s},
+• for Γ5 = (Sal = s ∨ Sal = s′), Sat(Γ5) = {s, s′} and Sat−(Γ5) = ∅,
+• for Γ6 = (Sal > 10K), if s = 5K and s′ = 20K, Sat(Γ6) = {s′} and
+Sat−(Γ6) = {s}, and if s = 15K and s′ = 20K, Sat(Γ6) = {s, s′} and
+Sat−(Γ6) = ∅.
+Next, we explain how, in our context, the above syntactic definition of satis-
+faction of a selection condition can be ‘coupled’ with semantic considerations,
+when it comes to defining the notion of consistent answer to a query Q.
+Intuitively, given Q : select X from T [where Γ], the least requirements
+for a tuple x to belong to a consistent answer to Q are the following:
+R1 x is in T (X), i.e., the schema of x is X,
+R2 x is in True(T ), i.e., x is a true tuple,
+R3 x is in Cons(T ), i.e., x is consistent, and
+R4 if Q involves a selection condition Γ then there exists t in True(T ) such
+that sch(Q) ⊆ sch(t), x ⊑ t, and t.sch(Γ) is in Sat(Γ) (that is, t is a true
+super-tuple of x whose restriction to attributes in sch(Γ) satisfies Γ).
+It is important to note that when all the above requirements are satisfied, then
+the consistent answer coincides with the standard notion of answer to projection-
+selection queries against a consistent table. Indeed, if T is a relational consistent
+table whose schema contains all attributes in sch(Q), then the answer to a query
+Q : select X from T [where Γ] is the set of all tuples x such that:
+• the schema of x is X (see requirement R1), and
+• T contains a tuple t such that t is a super-tuple of x and the restriction of
+t to all attributes occurring in Γ satisfies Γ (see requirement R4, knowing
+that t is true since all tuples in T are implicitly assumed to be true).
+16
+
+Moreover, in this case, requirement R2 is also satisfied because all tuples in a
+consistent relational table are implicitly assumed to be true, and requirement
+R3 is trivial because in a consistent table, every tuple is consistent.
+On the other hand, in the presence of inconsistencies, it should be noticed
+that, based on our semantics as earlier defined, the status of any tuple x in the
+consistent answer to Q is clearly defined because x is then a true and consistent
+tuple of T whose schema is X. However, the situation is less clear regarding
+the status of the super-tuple t in requirement R4. More precisely, the question
+here is whether t should be consistent or not and whether t should satisfy other
+criteria as well (e.g., is t required to be consistent or is t required to be maximal
+with respect to ⊑?). The following example illustrates this discussion, showing
+in particular that the status of t has a significant impact on the consistent
+answer.
+Example 5 Consider the universe U = {Emp, Sal, Dept} with the functional
+dependency Emp → Sal and the table T = {es, es′d, e′s′} over U whose tu-
+ples state that employee e has s and s′ as salaries, employee e works in de-
+partment d, and that employee e′ has salary s′.
+The content of T clearly
+does not satisfy the functional dependency Emp → Sal as employee e is as-
+signed two distinct salaries s and s′. The limit interpretation I∗ of T is de-
+fined by I∗(e) = {1, 2}, I∗(e′) = {3}, I∗(s) = {1, 2}, I∗(s′) = {1, 2, 3} and
+I∗(d) = {2}. Thus True(T ) consists of the sub-tuples of esd, es′d and of e′s′,
+and Inc(T ) = {esd, es′d, es, es′, sd, s′d, s, s′}. Let us now consider the following
+queries Q1, Q2, Q3, Q4, Q5 and Q6:
+Q1 : select Emp, Dept from T
+Q2 : select Emp, Sal from T
+Q3 : select Sal from T
+Q4 : select Emp from T where Sal = s′
+Q5 : select Emp from T where Sal = s ∨ Sal = s′
+Q6 : select Emp from T where Sal > 10K
+Regarding Q1, since ed is the only true and consistent tuple in T (Emp, Dept),
+ed is the only tuple satisfying the requirements R1, R2 and R3 above. Require-
+ment R4 is irrelevant because Q1 involves no selection condition. Hence ed is a
+candidate tuple to belong to the consistent answer to Q1. Notice however that
+one could object that ed is not a good candidate since all its maximal (with
+respect to ⊑) true super-tuples in T (namely esd and es′d) are inconsistent.
+Regarding Q2, es, es′ and e′s′ are the only tuples satisfying requirements R1
+(they all belong to T (Emp, Sal)) and R2 (they all belong to True(T )). However,
+since es and es′ are in Inc(T ), they do not satisfy requirement R3, whereas e′s′
+does since this tuple is in Cons(T ). We moreover notice that, contrary to Q1
+above, e′s′ has no true strict super-tuple. Hence the consistent answer to Q2
+should be {e′s′}.
+The case of Q3 is clearer because s and s′ are the only tuples satisfying
+requirements R1 and R2, and since these two tuples are in Inc(T ), none of them
+can satisfy requirement R3. Therefore, the consistent answer to Q3 should be
+∅.
+17
+
+Let us now turn to queries involving a selection condition. Regarding Q4,
+the two tuples satisfying requirements R1 and R2 are e and e′, and since e and
+e′ are in Cons(T ), they also satisfy requirement R3. Moreover, we notice that
+es′ is a true super-tuple of e satisfying Γ4 = (Sal = s′), showing that e satisfies
+requirement R4. However, it should be noticed that es′ is inconsistent and that
+es is another true super-tuple of e not satisfying Γ4. Hence, it can thought that
+e should not belong to the consistent answer to Q4. On the other hand, e′ has
+only one true super-tuple, namely e′s′, and since this tuple is in Sat(Γ4), e′
+satisfies requirement R4 and no further information allows us to think that the
+salary of e′ could be different than s′. As a consequence, the consistent answer
+to Q4 is expected to contain e′ and possibly e.
+Regarding Q5, for the same reasons as for Q4, e and e′ are the two possible
+tuples satisfying requirements R1, R2 and R3. Now, and contrary to the case
+of Q4, the two true super-tuples of e, namely es and es′, belong to Sat(Γ5),
+thus ensuring that whatever e’s salary it satisfies the selection condition. This
+remark supports the fact that e is a candidate to belong to the consistent answer
+to Q5. Of course, as for query Q1, one could object that since es and es′ are
+inconsistent, e has not to belong to the consistent answer to Q5. It should be
+noticed here that the case of e′ is treated as for Q4 above, because e′s′ is a true
+super-tuple of e′ such that s′ is in Sat(Γ5). Hence, the consistent answer to Q5 is
+expected to be {e, e′} or {e′}, following the choice made regarding inconsistency
+of the tuple t in requirement R4.
+A reasoning similar to those for Q4 or Q5 applies to Q6, depending on the
+actual values of s and s′. Assuming first that s = 5K and s′ = 20K, as for
+Q4, the expected consistent answer is {e′} or possibly {e, e′}, knowing that e’s
+unique salary may not satisfy the condition. However, if s = 15K and s′ = 20K,
+the consistent answer to Q6 is expected to be as for Q5, that is, {e, e′} or {e′},
+following the choice made regarding inconsistency of the tuple t in requirement
+R4.
+�
+3.2
+Consistent Answers
+To account for some of the remarks regarding Q5 and Q6 in Example 5, we
+introduce the notion of consistency with respect to a selection condition. To this
+end, for every relation schema X, we denote by T (X↑) the set of all tuples t in
+T such that X ⊆ sch(t). In other words, T (X↑) is the set of all super-tuples of
+tuples over X, and it follows that T (X) is a sub-set of T (X↑).
+Definition 5 Given a table T over U, and Γ a selection condition:
+A tuple t in T (sch(Γ)↑) is said to be inconsistent with respect to Sat(Γ) if
+t.sch(Γ) is in Sat(Γ) and there exists s in Sat−(Γ) such that I∗(t) ∩ I∗(s) ̸= ∅.
+We denote by Inc(Γ, T ) the set of all tuples inconsistent with respect to Sat(Γ).
+A tuple t in T (sch(Γ)↑) is said to be consistent with respect to Sat(Γ) if
+t.sch(Γ) is in Sat(Γ) and t is not in Inc(Γ, T ), that is if for every s in Sat−(Γ),
+I∗(t) ∩ I∗(s) = ∅. We denote by Cons(Γ, T ) the set of all tuples consistent with
+respect to Sat(Γ).
+�
+18
+
+The following proposition states basic properties implied by Definition 5.
+Proposition 2 Given a table T over U and a selection condition Γ, the follow-
+ing holds:
+1. Inc(Γ, T ) is a sub-set of Inc(T ), and Cons(T ) ∩ T (sch(Γ)↑) is a sub-set of
+Cons(Γ, T ). But Cons(Γ, T ) ⊆ Cons(T ) does not always hold.
+2. For every t in Cons(Γ, T ), every super-tuple t′ of t is also in Cons(Γ, T ).
+3. If I∗ is in First Normal Form, then Inc(Γ, T ) = ∅ and Cons(Γ, T ) is the
+set of all super-tuples of tuples in Sat(Γ).
+4. For a given tuple t, if Sat(Γ) = {t}, then t is in Inc(T ) if and only if t is
+in Inc(Γ, T ), and t is in Cons(T ) if and only if t is in Cons(Γ, T ).
+5. If Sat(Γ) = ∅ then Cons(Γ, T ) = Inc(Γ, T ) = ∅. If Sat(Γ) = T (X), then
+Cons(Γ, T ) is the set of all super-tuples of tuples in Sat(Γ), and Inc(Γ, T )
+is empty.
+Proof. 1. If t is in Inc(Γ, T ), there exists s in Sat−(Γ) such that I∗(t)∩I∗(s) ̸=
+∅. Since t.sch(Γ) is in Sat(Γ), t.sch(Γ) and s are distinct tuples in T (sch(Γ)).
+Hence, there exists A in sch(Γ) such that t.A ̸= s.A.
+As I∗(s) ⊆ I∗(s.A),
+I∗(t) ∩ I∗(s) ̸= ∅ implies that I∗(t) ∩ I∗(s.A) ̸= ∅. Therefore, by Definition 3, t
+is in Inc(T ).
+As for the inclusion Cons(T )∩T (sch(Γ)) ⊆ Cons(Γ, T ), given t in T (sch(Γ)),
+if t is in Cons(T ) then t is not in Inc(T ). Thus, t is not in Inc(Γ, T ), which implies
+by Definition 5 that t is in Cons(Γ, T ).
+To see that Cons(Γ, T ) ⊆ Cons(T ) does not always hold, consider the earlier
+example where T = {abc, ab′c, a′b′′c′} over U = {A, B, C} and C → B. In
+this case, b and b′ are clearly in Inc(T ), thus, not in Cons(T ). For Γ = (B =
+b ∨ B = b′), we have Sat(Γ) = {b, b′} and Sat−(Γ) = {b′′}. Since I∗ is defined
+by I∗(a) = I∗(b) = I∗(b′) = I∗(c) = {1, 2}, I∗(a′) = I∗(b′′) = I∗(c′) = {3},
+Definition 5 shows that b and b′ are in Cons(Γ, T ) (because I∗(b) ∩ I∗(b′′) =
+I∗(b′) ∩ I∗(b′′) = ∅).
+2. If t′ is a super-tuple of t, t′ is in T (sch(Γ)↑) because so is t. Moreover, as t is
+in Cons(Γ, T ), I∗(t) ∩ I∗(s) = ∅ holds for every s in Sat−(Γ). As I∗(t′) ⊆ I∗(t),
+it follows that I∗(t′) ∩ I∗(s) = ∅ holds as well, showing that t′ is in Cons(Γ, T ).
+3. Assuming that I∗ is in First Normal Form means that Inc(T ) = ∅. By (1)
+above, this implies that Inc(Γ, T ) = ∅ and thus, by Definition 5, Cons(Γ, T ) is
+the set of all super-tuples of tuples in Sat(Γ).
+4. By (1) above, we have Inc(Γ, T ) ⊆ Inc(T ) always holds.
+Conversely, if t
+is in Inc(T ) then, by Definition 3, there exist A in sch(t) and a′ in adom(A)
+such that a′ ̸= t.A and I∗(t) ∩ I∗(a′) ̸= ∅. Denoting t.A by a and writing t
+as qa, the tuple qa′ is such that sch(qa′) = sch(t) and qa′ ∈ Sat−(Γ). Hence,
+I∗(t) ∩ I∗(qa′) ̸= ∅, showing that t is in Inc(Γ, T ).
+If t is in Cons(Γ, T ), then Sat(Γ) = T (X) \ {t}, and for every s in Sat−(Γ),
+I∗(t) ∩ I∗(s) = ∅, implying that for every A in sch(Γ) and every a′ in adom(A)
+19
+
+such that a′ ̸= t.A, I∗(t) ∩ I∗(a′) = ∅. By Definition 3, this implies that t is in
+Cons(T ). Conversely, if t is in Cons(T ), by (1) above, as t belongs to T (sch(Γ)),
+t is in Cons(Γ, T ).
+5. The results being immediate consequences of Definition 5, their proofs are
+omitted. The proof of the proposition is therefore complete.
+�
+In what follows, in order to account for the remarks in Example 5, we propose
+four ways of defining the consistent answer to a query Q. These definitions are
+then illustrated by examples and compared to each other.
+Definition 6 Let T be a table over universe U, FD the set of associated func-
+tional dependencies. Given a query Q : select X from T [where Γ], we define
+the following:
+1. The weakly consistent answer to Q, denoted WC ans(Q), is the set of all
+tuples x in T (X) such that
+(a) x is in True(T ) ∩ Cons(T ),
+(b) there exists t is in True(T ) such that t is in T (sch(Q)↑), t.X = x,
+t.sch(Γ) is in Sat(Γ).
+2. The consistent answer to Q, denoted C ans(Q), is the set of all tuples x
+in T (X) such that
+(a) x is in True(T ) ∩ Cons(T ),
+(b) there exists t is in True(T ) such that t is in T (sch(Q)↑), t.X = x, t
+is in Cons(Γ, T ).
+3. The strongly consistent answer to Q, denoted SC ans(Q), is the set of all
+tuples x in T (X) such that
+(a) x is in True(T ) ∩ Cons(T ),
+(b) there exists t is in True(T ) ∩ Cons(T ) such that t is in T (sch(Q)↑),
+t.X = x, and t.sch(Γ) is in Sat(Γ).
+4. The max-strongly consistent answer to Q, denoted MSC ans(Q), is the set
+of all tuples x in T (X) such that
+(a) x is in True(T ) ∩ Cons(T ),
+(b) there exists t is in True(T ) ∩ Cons(T ) such that t is maximal with
+respect to ⊑ in True(T ), t is in T (sch(Q)↑), t.X = x, and t.sch(Γ)
+is in Sat(Γ).
+�
+The four kinds of consistent answer in Definition 6 relate to requirements R1,
+R2, R3 and R4 in the following respects:
+• Statements (1.a), (2.a), (3.a) and (4.a) clearly show that any tuple x
+in any of the four kinds of consistent answer satisfies requirements R1
+(x ∈ T (X)), R2 (x ∈ True(T )) and R3 (x ∈ Cons(T )).
+20
+
+• Statements (1.b), (2.b), (3.b) and (4.b) show that any tuple x in any of the
+four kinds of consistent answer satisfies the requirement R4 by ensuring
+the existence of a true super-tuple t of x satisfying Γ.
+Moreover, as announced in our earlier discussion, the status of the tuple t is
+clarified in Definition 6 as follows:
+• in the case of the weakly consistent answer, t is only requested to be true
+(as stated in requirement R4),
+• in the case of the consistent answer, t is requested to be true and consistent
+with respect to Γ (which does not disallow t to be inconsistent),
+• in the case of the strongly consistent answer, t is requested to be true and
+consistent (which of course disallows t to be inconsistent), and
+• in the case of the max-strongly consistent answer, t is requested to be
+true, consistent and maximal.
+We also emphasize that when the query Q in Definition 6 involves no selection
+condition, then statements (1.b), (2.b) and (3.b) are reduced to the existence
+of t such that t is in True(T ), t is in T (X↑) and t.X = x. As statements (1.a),
+(2.a) and (3.a), ensure the existence of such a tuple t, statements (1.b), (2.b)
+and (3.b) are redundant. However, such is not the case for statement (4.b),
+because the true and consistent super-tuple of x needed to satisfy (4.a) might
+not be maximal, and thus might not satisfy (4.b). We refer to query Q1 in the
+forthcoming Example 6 for an illustration of this case.
+Moreover, Definition 6 and the results in Proposition 2 lead to the following
+observations when the query Q involves a selection condition Γ:
+1. For every t in Inc(Γ, T ), the tuple t.sch(Γ) is such that its syntax satisfies
+the condition Γ whereas its semantics meets that of a tuple whose syntax
+does not satisfy the condition Γ. Since Inc(Γ, T ) is a sub-set of Inc(T ),
+this shows that the notion of satisfaction of a selection condition might
+concern inconsistent tuples. It should be noticed that any super-tuple of
+such tuples is discarded from max-strongly consistent answers and from
+strongly consistent answers, whereas they may occur in consistent answers
+or in weakly consistent answers.
+2. Since every super-tuple of a tuple in Cons(Γ, T ) is also in Cons(Γ, T ),
+C ans(Q) can be computed by scanning maximal true tuples in T (sch(Q)↑).
+Obviously, MSC ans(Q) should be computed by scanning only maximal
+true tuples in T (sch(Q)↑), whereas the scan should concern true tuples in
+T (sch(Q)), when computing SC ans(Q) or WC ans(Q). It will however be
+seen in the next section, that all consistent answers are computed based
+on the same scan.
+3. If I∗ is in First Normal Form, i.e., if Inc(T ) = ∅, then Cons(Γ, T ) is the
+set of all super-tuples of tuples in Sat(Γ). As moreover, t.X is in Cons(T ),
+21
+
+t.X is in C ans(Q). This remark fits the intuition that if T is consistent
+then the consistent answer to Q is equal to the standard answer to Q, that
+is the set of projections over X of true tuples in T (sch(Q)↑) that satisfy
+Γ. This comes in line with our earlier remark stating that requirements
+R1, R2, R3 and R4 are satisfied by the tuples in the answer to Q when T
+is consistent.
+4. If Sat(Γ) is reduced to one tuple s, then for every t in T (sch(Q)↑) ∩
+True(T ) such that t.X is in Cons(T ) ∩ True(T ) and t.sch(Γ) = s, t.X
+is in WC ans(Q), and moreover, t.X is in SC ans(Q) and in C ans(Q) if
+and only if s is in Cons(T ). The case of MSC ans(Q) is more involved
+because maximal true tuples must be considered.
+For example, it can
+happen that, for Q : select A from T where B = b, a is in SC ans(Q)
+and in C ans(Q), but not in MSC ans(Q) because ab is true and consistent,
+whereas abc is the only true super-tuple of ab and abc is inconsistent.
+5. If Sat(Γ) = ∅ (i.e., if Γ is not satisfiable in T ), then it is obvious that
+MSC ans(Q), SC ans(Q) and WC ans(Q) are empty.
+As in this case,
+Cons(Γ, T ) = ∅, C ans(Q) is empty as well.
+On the other hand, if Sat(Γ) = T (sch(Γ)), it is easy to see that SC ans(Q)
+and WC ans(Q) are equal to the set of all tuples in T (X) that belong to
+Cons(T )∩True(T ). Moreover, since Cons(Γ, T ) = T (sch(Γ)↑), C ans(Q) is
+also the set of all tuples in T (X) that belong to Cons(T ) ∩ True(T ). How-
+ever, as in the previous item, the case of MSC ans(Q) is more involved
+because maximal true tuples must be considered.
+The following proposition states an important relationship among the four kinds
+of consistent answers, namely that they form an increasing chain of inclusions.
+Proposition 3 Let T be a table over universe U, FD the set of associated
+functional dependencies. Given a query Q : select X from T [where Γ], the
+following holds: MSC ans(Q) ⊆ SC ans(Q) ⊆ C ans(Q) ⊆ WC ans(Q).
+Proof. To show that MSC ans(Q) ⊆ SC ans(Q) holds, let x in MSC ans(Q).
+Then statements (1.a) and (1.b) in Definition 6 hold, and thus, so does statement
+(2.a). The result comes form the fact that it is easily seen that statement (1.b)
+implies statement (2.b).
+Regarding the inclusion SC ans(Q) ⊆ C ans(Q), we notice that for every x
+in SC ans(Q), statements (2.a) and (2.b) in Definition 6 hold, and thus, that
+statement (3.a) holds as well.
+Moreover, since statement (2.b) holds, there
+exists t in T (sch(Q)↑), such that t.X = x and t.sch(Γ) is in Sat(Γ). Thus t is in
+Cons(T )∩T (sch(Γ)↑), which by Proposition 2(1) implies that t is in Cons(Γ, T ).
+Hence statement (3.b) is satisfied, showing that x is in C ans(Q).
+Regarding the last inclusion, for every x in C ans(Q), statement (4.a) clearly
+holds and moreover, as the tuple t in Cons(Γ, T )∩True(T ) is obviously in True(T )
+and in Sat(Γ), the proof is complete.
+�
+22
+
+We illustrate Definition 6 and Proposition 3 in the context of Example 5. More-
+over, through this example, we show that the inclusions in Proposition 3 might
+be strict.
+Example 6 We recall that in Example 5, U = {Emp, Sal, Dept}, FD =
+{Emp → Sal} and T = {es, es′d, e′s′}, which implies that True(T ) consists of
+the sub-tuples of esd, es′d and of e′s′ and that Inc(T ) = {esd, es′d, es, es′, sd,
+s′d, s, s′}. The queries we are interested in are Q1, Q2, Q3, Q4, Q5 and Q6 as
+shown below:
+Q1 : select Emp, Dept from T
+Q2 : select Emp, Sal from T
+Q3 : select Sal from T
+Q4 : select Emp from T where Sal = s′
+Q5 : select Emp from T where Sal = s ∨ Sal = s′
+Q6 : select Emp from T where Sal > 10K
+In the case of Q1, the only possible true tuple in any answer is ed.
+As all
+maximal true super-tuples of ed are esd and es′d, none of these tuples is con-
+sistent. Thus, we have MSC ans(Q1) = ∅. However, SC ans(Q1) = C ans(Q1) =
+WC ans(Q1) = {ed}, because this tuple is true and consistent and because no
+selection condition is involved. Regarding the inclusions in Proposition 3, this
+case shows that the first inclusion might be strict.
+Considering now Q2, we have MSC ans(Q2) = {e′s′}, because this tuple is
+true and consistent, and has no strict true super-tuple. For a similar reason, we
+also have SC ans(Q2) = C ans(Q2) = WC ans(Q2) = {e′s′} because no selection
+condition is involved in Q2. In this case the inclusions in Proposition 3 hold
+because MSC ans(Q2) = SC ans(Q2) = C ans(Q2) = WC ans(Q2) hold.
+The case of Q3 is easy because as earlier mentioned, no Sal-value is con-
+sistent, implying that statement (a) of the four consistent answers in Defini-
+tion 6 can not be satisfied.
+As Q3 involves no selection condition, we have
+MSC ans(Q3) = SC ans(Q3) = C ans(Q3) = WC ans(Q3) = ∅, showing again
+that the inclusions in Proposition 3 hold.
+Regarding Q4 involving the selection condition Γ4 = (Sal = s′), since e and
+e′ are both true and consistent tuples over Emp, they satisfy statements (1.a),
+(2.a), (3.a) and (4.a) in Definition 6. Since Sat(Γ3) = {s′}, es′ and e′s′ are
+the only tuples of interest regarding statements (1.b), (2.b), (3.b) and (4.b) in
+Definition 6. Thus WC ans(Q4) = {e, e′}. As es′d is maximal and inconsistent,
+statement (1b) is not satisfied and since es′ is also inconsistent, statement (2.b)
+is not satisfied either. Now, regarding statement (3.b), we have s in Sat−(Γ4)
+and I∗(es′) ∩ I∗(s) = {1, 2} (see Example 5). Thus, by Definition 5, es′ is in
+Inc(Γ4, T ), and so, es′ does not satisfy statement (3.b). As a consequence, e is
+not in MSC ans(Q4), SC ans(Q4) nor in C ans(Q4).
+On the other hand, e′s′ does satisfy statements (1.b), (2.b) and (3.b) in
+Definition 6, because e′s′ is a maximal true and consistent tuple such that s′ is in
+Sat(Γ4), and regarding (3.b), e′s′ is in Cons(Γ4, T ) (because I∗(e′s′)∩I∗(s) = ∅).
+Therefore, MSC ans(Q4) = SC ans(Q4) = C ans(Q4) = {e′} and WC ans(Q4) =
+23
+
+{e, e′}, showing that the inclusions in Proposition 3 hold, and that the last one
+might be strict.
+As for Q5 involving the selection condition Γ5 defined by (Sal = s∨Sal = s′),
+for the same reasons as for Q4, the two possible tuples in the answer are e
+and e′ and WC ans(Q5) = {e, e′}. Moreover, since Sat(Γ5) = {s, s′}, e′ is in
+MSC ans(Q5), SC ans(Q5) and C ans(Q5). Regarding now e, the same argu-
+ments as for Q4 apply regarding the statements (1.b) and (2.b) in Definition 6,
+showing that e is neither in MSC ans(Q5) nor in SC ans(Q5). However, contrary
+to the case of Q4, es is in Cons(Γ5, T ), and thus, statement (3.b) of Definition 6
+is satisfied. We therefore obtain that MSC ans(Q5) = SC ans(Q5) = {e′} and
+C ans(Q5) = WC ans(Q5) = {e, e′}, which also shows that the second inclusion
+in Proposition 3 might be strict.
+Consider now Q6 involving the selection condition Γ6 = (Sal > 10K) and
+suppose that s = 5K and s′ = 20K.
+Then, we have Sat(Γ6) = {s′} and
+Sat−(Γ6) = {s}, and thus, as in the case of Q4, MSC ans(Q4) = SC ans(Q4) =
+C ans(Q4) = {e′}, and WC ans(Q6) = {e, e′}.
+On the other hand, if s =
+15K and s′ = 20K, then Sat(Γ6) = {s, s′} and Sat−(Γ3) = ∅, and thus, as
+for Q5 above, we have MSC ans(Q5) = SC ans(Q5) = {e′} and C ans(Q5) =
+WC ans(Q6) = {e, e′}.
+�
+To end this section, we would like to stress that proposing distinct consistent
+answers as done in Definition 6 raises the question of which one should be cho-
+sen. Although we think that this choice should be made by the final user, the
+comparisons stated in Proposition 3 should help in making this choice. More-
+over, assuming the choice of C ans(Q) is made, the knowledge of the answers
+MSC ans(Q), SC ans(Q) and WC ans(Q) are likely to be useful to the user, re-
+garding the quality of the provided consistent answer.
+For instance, in Example 6 the fact that e is in C ans(Q5) but not in
+SC ans(Q5) implies that the database contains an inconsistency regarding e’s
+salary. This information shows that the presence of e in C ans(Q5) might be
+considered less reliable than that of e′ for which such inconsistency does not
+exist as e′ is in SC ans(Q5) and in SC ans(Q5). On the other hand, a tuple
+in WC ans(Q) but not in C ans(Q) shows that although this tuple occurs with
+values satisfying the selection condition, it also systematically occurs in an in-
+consistency involving values not satisfying the selection condition. This happens
+for e when answering Q4: although e occurs associated with s′, it also occurs
+associated with s.
+This example also shows that comparing the consistent answers MSC ans(Q),
+SC ans(Q) and WC ans(Q) with C ans(Q) has an impact not only on the quality
+of the answers, but also allows for explaining the presence or the absence of
+some tuples in a consistent answer. These issues related to data quality and to
+answer explanation are among the subjects of our future work.
+24
+
+4
+Computational Issues
+In this section, we present algorithms first for computing efficiently the sets
+Cons(T ) and Inc(T ) (see Section 4.1) and then the four consistent answers to a
+given query (see Section 4.2).
+4.1
+The Tabular Representation of I∗
+In this section, we show how to compute True(T ) and Inc(T ), using a modified
+version of the well-known chase algorithm [8, 14] that we call m-Chase. Our
+algorithm is based on the notion of multi-valued tuple, or m-tuple for short,
+which generalizes that of usual tuple in the following sense: instead of associating
+every attribute A with at most one value in adom(A), a multi-valued tuple
+associates every attribute A with a sub-set of adom(A), possibly empty. Multi-
+valued tuples are defined as follows.
+Definition 7 Given a universe U = {A1, . . . , An} and active domains adom(Ai),
+i = 1, . . . , n, a multi-valued tuple σ over U, or m-tuple for short, is a total func-
+tion from U to the cross product XA∈UP(adom(A)), where P(adom(A)) denotes
+the power set of the active domain of attribute A. The ‘empty’ m-tuple, denoted
+by ω, is defined by ω(A) = ∅ for every A in U.
+The set of all attributes A such that σ(A) ̸= ∅, is called the schema of σ,
+denoted by sch(σ), and the schema of ω is ∅. Given σ ̸= ω and a sub-set X
+of sch(σ), the restriction of σ to X, denoted σ(X), is the m-tuple defined by
+(σ(X))(A) = σ(A) for every A in X and (σ(X))(A) = ∅ for any A not in X.
+Given an m-tuple σ, the set tuples(σ) denotes the set of all tuples t such that
+sch(t) = sch(σ) and for every A in sch(t), t(A) is a value of adom(A) that
+belongs to σ(A). As for the empty m-tuple ω, we have tuples(ω) = ∅.
+The set of all m-tuples over U is denoted by MT , and a finite set of m-tuples
+is called an m-table.
+�
+It is easy to see that for every t in T , it holds that sch(σt) = sch(t) and
+tuples(σt) = {t}. In what follows, we use t and σt interchangeably, whenever no
+confusion is possible. To simplify notation, given an m-tuple σ, the set σ(A) is
+denoted by the concatenation of its elements between parentheses, and σ itself
+is denoted by the concatenation of all σ(A) such that σ(A) ̸= ∅.
+Based on Definition 7, a tuple t in T yields an m-tuple σt in MT such that
+for every A in sch(t), σt(A) = {t.A} (or σt(A) = (t.A) according to the notation
+just introduced), and σt(B) = ∅ for every B not in sch(t).
+For example if U = {A, B, C, D}, the m-tuple σ such that σ(A) = {a, a′},
+σ(B) = {b}, σ(C) = ∅ and σ(D) = {d1, d2, d3} is denoted by (a, a′)(b)(d1d2d3)
+and we have sch(σ) = {A, B, D} (or simply sch(σ) = ABD). Now, to find
+the set tuples(σ) we make all combinations of values, one from each of σ(A),
+σ(B) and σ(D). We find tuples(σ) = {abd1, abd2, abd3, a′bd1, a′bd2, a′bd3}.
+Moreover, we have σ(AB) = (aa′)(b). On the other hand, for t = abd1, we have
+σt = (a)(b)(d1), and thus tuples(σt) = {abd1}.
+25
+
+We draw attention to the fact that, although m-tables can be seen as a par-
+ticular case of embedded relations [14], we do not intend to elaborate on such
+relations in our approach. We rather use m-tuples as a tool to explicitly ac-
+count for situations where the First Normal Form (or equivalently the partition
+constraint) is not satisfied. We define the following operations for m-tuples:
+Comparison: σ1 ⊑ σ2 holds if for every A in U, σ1(A) ⊆ σ2(A)2. Thus if
+σ1 ⊑ σ2 then sch(σ1) ⊆ sch(σ2). Moreover, if sch(σ1) = sch(σ2) then
+tuples(σ1) ⊆ tuples(σ2). It also holds that ω ⊑ σ, for every m-tuple σ.
+Union: σ1 ⊔ σ2 is the m-tuple σ such that for every A in U, σ(A) = σ1(A) ∪
+σ2(A). Thus, sch(σ1 ⊔ σ2) = sch(σ1) ∪ sch(σ2).
+Intersection: σ1 ⊓ σ2 is the m-tuple σ such that for every A in U, σ(A) =
+σ1(A) ∩ σ2(A). Thus, sch(σ1 ⊓ σ2) ⊆ sch(σi) for i = 1, 2.
+We now introduce a new chase algorithm, called m-Chase whose definition is
+shown as Algorithm 1.
+The m-Chase Algorithm works on sets of m-tuples
+to provide an m-table from which true and inconsistent tuples can easily be
+retrieved. Roughly speaking, Algorithm 1 follows the same idea as the standard
+chase, but when a functional dependency cannot be satisfied, our algorithm does
+not stop, but rather collects all values inferred from the functional dependencies.
+Consequently, it may happen that for a given row and a given column (i.e., for
+a given attribute value) we may have to store more than one constant, which
+is achieved by means of m-tuples.
+It is important to note that the output
+of Algorithm 1 (denoted by Σ∗) allows to explicitly ‘show’ all violations of the
+partition constraint PC. We illustrate how Algorithm 1 works using an example.
+Example 7 Let T = {ab, ab′, a′b′} over universe U = {A, B} with functional
+dependencies A → B and B → A. The computation of Σ∗ based on Algorithm 1
+works as follows:
+1. Considering tuples in T as m-tuples yields the following m-tuples: (a)(b),
+(a)(b′) and (a′)(b′). Thus, Σ∗ is set to {(a)(b), (a)(b′), (a′)(b′)} on line 1.
+2. Considering A → B, with σ = (a)(b) and σ′ = (a)(b′), the m-tuples
+σ1 = (a)(bb′) and σ′
+1 = (a)(b′b) are generated. The variable change is set
+to true and since σ1 = σ′
+1, Σ∗ is set to {(a)(bb′), (a′)(b′)}. Moreover, fail is
+set to true line 11 because we have (b) ̸= (b′). This means intuitively that
+when running the standard chase algorithm, a failure would be reached at
+this step.
+Considering B → A, since the B-value b′ occurs in (a)(bb′) and (a′)(b′),
+these m-tuples generate respectively (aa′)(bb′) and (aa′)(b′). Moreover,
+since (aa′)(b′) ⊑ (aa′)(bb′), Σ∗ is set to {(aa′)(bb′)}.
+2With a slight abuse of notation, we use here the same symbol ⊑ introduced earlier for the
+sub-tuple relation between usual tuples.
+26
+
+Algorithm 1 The m-Chase Algorithm
+Input: A table T over U and a set FD of functional dependencies over U.
+Output: An m-table denoted by Σ∗ and the truth value of fail.
+1: Σ∗ := {σt | t ∈ T}
+2: change := true
+3: fail := false
+4: while change = true do
+5:
+change := false
+6:
+for all σ in Σ∗ do
+7:
+for all X → A in FD such that XA ⊆ sch(σ) do
+8:
+for all σ′ in Σ∗ such that X ⊆ sch(σ′) do
+9:
+if for every B in X, σ(B) ∩ σ′(B) ̸= ∅ then
+10:
+if A ∈ sch(σ) and A ∈ sch(σ′) and σ(A) ̸= σ′(A) and fail = false
+then
+11:
+fail := true
+12:
+σ1 := σ ⊔ σ′(A) ; σ′
+1 := σ′ ⊔ σ(A)
+13:
+if σ ̸= σ1 or σ′ ̸= σ′
+1 then
+14:
+change := true
+15:
+if σ1 ⊑ σ′
+1 then
+16:
+Σ∗ := (Σ∗ \ {σ, σ′}) ∪ {σ′
+1}
+17:
+else if σ′
+1 ⊑ σ1 then
+18:
+Σ∗ := (Σ∗ \ {σ, σ′}) ∪ {σ1}
+19:
+else
+20:
+Σ∗ := (Σ∗ \ {σ, σ′}) ∪ {σ1, σ′
+1}
+21: return Σ∗ and the truth value of fail
+3. As the second execution of the while-loop line 4 does not change Σ∗, the
+variable change has value false. Hence the processing stops, returning
+Σ∗ = {(aa′)(bb′)} along with the value true assigned to fail.
+�
+We now state a basic lemma based on which it will be shown how to compute
+True(T ) and Inc(T ). In what follows, the notation LoCl is extended to Σ∗ in a
+straightforward way: LoCl(Σ∗) = {t ∈ T | (∃σ ∈ Σ∗)(t ⊑ σ)}. Moreover, for
+every interpretation I and every m-tuple σ, I(σ) is defined by I(σ) = �
+a⊑σ I(a).
+It should be noticed that, as for tuples in T , if σ ⊑ σ′, then I(σ′) ⊆ I(σ) holds.
+Lemma 4 Let T be a table over universe U. Then the following hold:
+1. Algorithm 1 applied to T always terminates.
+2. For every m-tuple s in MT , I∗(s) ̸= ∅ holds if and only if there exists σ
+in Σ∗ such that s ⊑ σ.
+3. Inc(T ) = ∅ if and only if fail = false.
+4. For every σ in Σ∗ and all t and t′ in tuples(σ), I∗(t) = I∗(t′).
+Proof. See Appendix B.
+�
+Note that Algorithm 1 is an extension of the standard chase algorithm. Indeed,
+the standard chase algorithm works on tuples (and not on m-tuples) which is
+the case for T. Moreover, as long as σ and σ′ are usual tuples, our processing
+is similar to that of standard chase. In particular, for every X → A in FD if
+27
+
+XA is a sub-set of sch(σ) and sch(σ′) and if σ(X) and σ′(X) are equal tuples,
+while σ(A) and σ′(A) are distinct values, then, in Algorithm 1, fail is set to
+true, which is precisely a case of failure for the standard chase algorithm. Thus,
+based on Lemma 4, the returned value for fail is true if and only if the standard
+chase algorithm fails.
+Moreover, the statements lines 16, 18 and 20 in Algorithm 1 imply that when
+σ and σ′ are in Σ∗, for every X → A in FD such that XA is a sub-set of sch(σ)
+and sch(σ′), if tuples(σ(X)) ∩ tuples(σ′(X)) ̸= ∅, then σ(A) = σ′(A).
+This
+remark can be seen as an extension of the well-known property of the output of
+the standard chase, by which all rows equal over X must have the same A-value.
+As shown in the following proposition, the m-table Σ∗ allows to compute the
+sets True(T ) and Inc(T ).
+Proposition 4 Let T be a table over U, FD the associated set of functional
+dependencies, and Σ∗ the output of Algorithm 1 run with T and FD as input.
+Then:
+1. True(T ) = LoCl(Σ∗).
+2. Inc(T ) = {t ∈ T | (∃σ ∈ Σ∗)(∃A ∈ sch(t))(∃a, a′ ∈ adom(A))(a ⊑ t ∧ t ⊑
+σ ∧ (aa′) ⊑ σ)}.
+3. Cons(T )∩True(T ) = {t ∈ True(T ) | (∀σ ∈ Σ∗)(t ⊑ σ ⇒ tuples(σ(sch(t))) =
+{t})}.
+Proof. 1. The result is an immediate consequence of Lemma 4(2).
+2. The result follows from Lemma 4(2) and Definition 3.
+3. The result follows from Definition 3 and the previous two items.
+�
+In order to assess the complexity of Algorithm 1, for every X → A in FD
+and every tuple x in T (X), we let N(x) be the number of different A-values a
+such that I∗(xa) ̸= ∅ and a is inconsistent, and we denote by δ the maximal
+value of all N(x) for all X → A in FD and all x in True(T ); in other words
+δ = max({N(x) | X → A ∈ FD ∧ x ∈ True(T )}).
+Using this notation, the size of the m-tuples in Σ∗ is bounded by |U|.δ,
+and as Σ∗ contains at most |T| m-tuples, a run of the core of loop line 4 is in
+O(|T|2.(|U|.δ)). Since the number of runs of the loop line 4 is bounded by the
+changes in the sets in Σ∗, this number of runs is in O(|T|.|FD|.δ) (that is the
+maximum number of constants that can be added in a set of Σ∗ times the size
+of Σ∗). Therefore, the computation of Σ∗ is in O((|T|.|FD|.δ).(|T|2.(|U|.δ)),
+that is in O(|T|3.|FD|.|U|.δ2) or even in O(|T|3.δ2), since |U| and |FD| are
+independent from |T|.
+We draw attention to the following important points regarding this result:
+1. When the database is consistent, δ is equal to 1, thus yielding a complex-
+ity in O(|T|3). This result can be shown independently from the above
+computations as follows: In the case of traditional chase the maximum
+of nulls in T being bounded by |U|.|T|, the number of iterations when
+28
+
+running the algorithm is also bounded by |U|.|T|. Since the run of one
+iteration is in |T|2, the overall complexity is in O(|U|.|T|3), or in O(|T|3),
+as |U| is independent from |T|.
+2. The above complexity study can be seen as a data complexity study as
+it relies on the size of T. Thus, this study should be further investigated
+in order to provide more accurate results regarding the estimation of the
+number of actual tests necessary to the computation of Σ∗. The results in
+[6] are likely to be useful for such a more thorough study of this complexity.
+4.2
+Computing Consistent Answers
+In this sub-section, we show how the consistent answers to a query can be
+computed based on the m-table Σ∗. We stress that to state one of the results,
+it is assumed that Σ∗ is not redundant, that is for all σ1 and σ2 in Σ∗, neither
+σ1 ⊑ σ2 nor σ2 ⊑ σ1 holds. Notice that, if Σ∗ is not redundant then for all σ1
+and σ2 in Σ∗, I∗(σ1) ∩ I∗(σ2) = ∅.
+Indeed, if I∗(σ1) ∩ I∗(σ2) ̸= ∅, Lemma 4(2) implies that Σ∗ contains σ such
+that σi ⊑ σ for i = 1, 2. As this entails that Σ∗ contains redundancies, we obtain
+a contradiction. We also notice in this respect that redundancies are eliminated
+at each step of the computation of Σ∗ in Algorithm 1 (see the statements lines 16
+and 18). Thus, it turns out that if the table T contains no redundancy, so does
+Σ∗.
+Moreover, to state our results, we introduce the following additional no-
+tation. Let T be a non-redundant table T over universe U, and Σ∗ its asso-
+ciated non-redundant m-table, as computed by Algorithm 1. Given a query
+Q : select X from T [where Γ], and a tuple x in T (X), we respectively de-
+note by Σ(x) and ΣQ(x), the sets defined by Σ(x) = {σ ∈ Σ∗ | (x ⊑ σ)} and
+ΣQ(x) = {σ ∈ Σ(x) | (∃s ∈ Sat(Γ))(s ⊑ σ)}.
+Considering that ΣQ(x) = Σ(x) if Q involves no selection condition, the fol-
+lowing proposition characterizes the tuples in MSC ans(Q), SC ans(Q), C ans(Q)
+and WC ans(Q) in terms of m-tuples as follows.
+Proposition 5 Let T be a non-redundant table over universe U, FD the set of
+associated functional dependencies and Σ∗ the non-redundant m-table as com-
+puted by Algorithm 1. Given a query Q : select X from T [where Γ], and a
+tuple x in T (X):
+1. x is in MSC ans(Q) if and only if
+(a) for every σ in Σ(x), tuples(σ(X)) = {x},
+(b) there exists σ in ΣQ(x) such that |tuples(σ)| = 1.
+2. x is in SC ans(Q) if and only if
+(a) for every σ in Σ(x), tuples(σ(X)) = {x},
+(b) there exists s in Sat(Γ) ∩ True(T ) such that for every σ in ΣQ(x), if
+s is in σ(sch(Γ)) then tuples(σ(sch(Γ)))) = {s}.
+29
+
+3. x is in C ans(Q) if and only if
+(a) for every σ in Σ(x), tuples(σ(X)) = {x},
+(b) there exists σ in ΣQ(x) such that tuples(σ(sch(Γ))) ⊆ Sat(Γ).
+4. x is in WC ans(Q) if and only if
+(a) for every σ in Σ(x), tuples(σ(X)) = {x},
+(b) ΣQ(x) is not empty.
+Proof. See Appendix C.
+�
+Example 8 We illustrate Proposition 5 in the context of Example 5 where
+U = {Emp, Sal, Dept}, FD = {Emp → Sal} and T = {es, es′d, e′s}. It is easy
+to see that Algorithm 1 returns truth value true for fail and Σ∗ = {σ1, σ2}
+where σ1 = (e)(ss′)(d) and σ2 = (e′)(s′). We apply Proposition 5 to the queries
+Q1, Q2, Q3, Q4, Q5 and Q6 shown below:
+Q1 : select Emp, Dept from T
+Q2 : select Emp, Sal from T
+Q3 : select Sal from T
+Q4 : select Emp from T where Sal = s′
+Q5 : select Emp from T where Sal = s ∨ Sal = s′
+Q6 : select Emp from T where Sal > 10K
+Since Q1 involves no selection condition, and since σ1 is the only m-tuple in
+Σ∗ whose schema contains Emp and Dept, the tests concern only this m-tuple.
+As tuples(σ1(Emp, Dept)) = {ed}, we have SC ans(Q1) = C ans(Q1) = {ed}.
+Moreover, as ΣQ1(ed) = {ed}, WC ans(Q1) = {ed} and MSC ans(Q1) = ∅,
+because |tuples(σ1)| = 2.
+For Q2, as tuples(σ1(Emp, Sal)) = {es, es′} and tuples(σ2(Emp, Sal)) =
+{e′s′}, it is immediate to see that SC ans(Q2) = C ans(Q2) = WC ans(Q2) =
+{e′s′}. Moreover, as e′s′ is a maximal consistent and true tuple, we also have
+MSC ans(Q2) = {e′s′}. In the case of Q3, it is easily seen that the four answers
+are empty because tuples(σ1(Sal)) = {s, s′}.
+Considering Q4, as tuples(σ1(Emp)) = {e} and tuples(σ2(Emp)) = {e′}, and
+thus, these two values satisfy statement (a) in the four items in Proposition 5.
+As Sat(Γ4) = {s′}, it is easy to see that ΣQ4(e) = {σ1} and ΣQ4(e′) = {σ2}.
+Thus, WC ans(Q4) = {e, e′}. Then, as |tuples(σ1(Emp, Sal))| = 2, e does not
+belong to MSC ans(Q4) nor to SC ans(Q4). Since tuples(σ1(Sal)) = {s, s′}, e is
+not in C ans(Q4) either, because statement (3.b) is not satisfied (as s is not in
+Sat(Γ4)). On the other hand is is easy to see that σ2 satisfies statements (b),
+entailing that we have MSC ans(Q4) = SC ans(Q4) = C ans(Q4) = {e′}.
+Concerning Q5, the two candidate Emp-values are again e and e′.
+As
+Sat(Γ5) = {s, s′}, ΣQ5(e) = {σ1} and ΣQ5(e′) = {σ2}, thus WC ans(Q5) =
+{e, e′}. As for Q4, e does not belong to MSC ans(Q5) and SC ans(Q5) because
+we have |tuples(σ1(Emp, Sal))| = 2. However, σ1 satisfies item (3.b) in Propo-
+sition 5 because tuples(σ1(Sal)) is indeed a sub-set of Sat(Γ5). Thus, e belongs
+30
+
+to C ans(Q5). The case of e′ and σ2 being similar to the previous query, we
+obtain MSC ans(Q5) = SC ans(Q5) = {e′} and C ans(Q5) = {e, e′}.
+Regarding Q6, if s = 15K and s′ = 20K, MSC ans(Q6) = SC ans(Q6) = {e′}
+and C ans(Q6) = WC ans(Q6) = {e, e′} due to arguments similar to the previous
+case, and if s = 5K and s′ = 20K, we have MSC ans(Q6) = SC ans(Q6) =
+C ans(Q6) = {e′} and WC ans(Q6) = {e, e′}, due to arguments similar to the
+case of Q4 above.
+�
+Algorithm 2 Consistent answers
+Input: A query Q : select X from T [where Γ] and Σ∗
+Output: The sets MSC ans(Q), SC ans(Q), C ans(Q) and WC ans(Q)
+1: MSC ans(Q) := ∅ ; SC ans(Q) := ∅ ; C ans(Q) := ∅ ; WC ans(Q) := ∅
+2: ToRemove X := ∅ ; Candidate SC := ∅ ; ToRemove SC := ∅
+3: for all σ in Σ∗ do
+4:
+if X ⊆ sch(σ) then
+5:
+if |tuples(σ(X))| > 1 then
+6:
+ToRemove X := ToRemove X ∪ tuples(σ(X))
+7:
+else
+8:
+Let x denote the unique tuple in σ(X)
+9:
+if sch(Γ) ⊆ sch(σ) then
+10:
+if tuples(σ(sch(Γ))) ∩ Sat(Γ) ̸= ∅ then
+11:
+WC ans(Q) := WC ans(Q) ∪ {x}
+12:
+if |tuples(σ)| = 1 then
+13:
+MSC ans(Q) := MSC ans(Q) ∪ {x}
+14:
+SC ans(Q) := SC ans(Q) ∪ {x}
+15:
+C ans(Q) := C ans(Q) ∪ {x}
+16:
+else
+17:
+if |tuples(σ(sch(Γ)))| = 1 then
+18:
+Candidate SC := Candidate SC ∪ tuples(σ(sch(Q)))
+19:
+C ans(Q) := C ans(Q) ∪ {x}
+20:
+else
+21:
+ToRemove SC := ToRemove SC ∪ tuples(σ(sch(Q)))
+22:
+if tuples(σ(sch(Γ))) ⊆ Sat(Γ) then
+23:
+C ans(Q) := C ans(Q) ∪ {x}
+24: MSC ans(Q) := MSC ans(Q) \ ToRemove X
+25: SC ans(Q) := SC ans(Q) ∪ {x | (∃q ∈ Candidate SC \ ToRemove SC)(q.X = x)}
+26: SC ans(Q) := SC ans(Q) \ ToRemove X
+27: C ans(Q) := C ans(Q) \ ToRemove X
+28: WC ans(Q) := WC ans(Q) \ ToRemove X
+29: return MSC ans(Q), SC ans(Q), C ans(Q), WC ans(Q)
+Based on Proposition 5, it can be seen that, given a query Q, Algorithm 2
+computes the consistent answer to Q. Indeed, the test line 4 discards all m-
+tuples of Σ∗ whose schema does not contain X, because such m-tuples have no
+chance to provide an X-value. Then, the items in Proposition 5 are checked as
+follows:
+• The test line 5 allows to store in the set ToRemove X all X-values of
+m-tuples σ such that tuples(σ(X)) contains more than one tuple.
+All
+these tuples being inconsistent, are then removed from the set of re-
+trieved X-values in MSC ans(Q) (line 24), in SC ans(Q) (line 26), in
+C ans(Q) (line 27) and in WC ans(Q) (line 28). Therefore, the statements
+31
+
+(1.a), (2.a), (3.a) and (4.a) in Proposition 5 are satisfied by the returned
+sets. In what follows, we consider that the test line 5 fails, that is that
+tuples(σ(X)) = {x}.
+• Statement (4.b) is concerned when the tests lines 9 and 10 succeed. Indeed,
+in this case, there exists a tuple s in tuples(σ(sch(Γ))) such that s is in
+Sat(Γ), meaning that σ is in ΣQ(x). Therefore, in this case, x is inserted
+in WC ans(Q) (line 11).
+• Regarding the statement (1.b) in Proposition 5, on line 9 it is checked that
+the schema of Γ is a sub-set of that of σ. If yes, when the tests lines 10
+and 12 succeed, σ is in fact a tuple t satisfying Γ. As moreover, Σ∗ is not
+redundant, t occurs nowhere else in Σ∗, meaning by Proposition 4(3) that
+t is in Cons(T ). Hence, x is inserted in MSC ans(Q) (line 13). Notice that
+in this case, the inclusions stated in Proposition 3 justify that x be also
+inserted in SC ans(Q) (line 14) and in C ans(Q) (line 15).
+• As for statement (2.b), assuming that the test line 9 succeeds, the case
+when the test line 12 succeeds is as above. Now, if this test fails, then,
+if the test line 17 succeeds, then σ(sch(Q)) might be a consistent true
+tuple t satisfying Γ, unless t occurs in some other tuples(σ′(sch(Q))) such
+that |tuples(σ′(sch(Q)))| > 1, meaning that t is inconsistent.
+This is
+why t is inserted in the set Candidate SC when the test line 17 suc-
+ceeds, and when this test fails, the tuples in the set tuples(σ(sch(Q))) are
+stored in ToRemove SC. Proposition 4(3) then ensures that the tuples in
+Candidate SC \ ToRemove SC are indeed consistent true tuples satisfy-
+ing Γ (other than maximal ones earlier detected). The statement line 25
+adds the corresponding X-values in the set SC ans(Q).
+• Considering now statement (3.b) when the test line 9 succeeds, we first
+notice that, thanks to Proposition 3, the insertions in C ans(Q) in state-
+ments lines 15 and 19 are correct. Moreover, when the test line 22 succeeds,
+the condition in the statement (3.b) is satisfied, and so, x is inserted in
+C ans(Q) as stated line 23.
+We illustrate Algorithm 2 using the queries Q3 and Q4 of Example 8, respectively
+defined by Q3 : select Sal from T and Q5 : select Emp from T where
+Sal = s ∨ Sal = s′.
+Regarding Q3, when considering σ1, ToRemove X is set to {s, s′} because
+the test line 5 succeeds. Then, with σ2, the test line 5 fails and the test line 10
+succeeds, entailing that s′ is inserted in WC ans(Q3). Then, as the test line 12
+succeeds, s′ is inserted in MSC ans(Q) (line 13), SC ans(Q) (line 14) and in
+C ans(Q) (line 15). When processing the statements lines 24, 26 and 27, s′ is
+removed, thus producing empty answers, as expected.
+As for Q5, when processing σ1, the test line 5 fails for σ1 and σ2 be-
+cause |tuples(σi(Emp))| = 1 (i = 1, 2).
+Thus ToRemove X remains empty
+in this case. When processing σ1, the tests line 12 and line 17 fail (because
+|tuples(σ1)| = 2 and |tuples(σ1(Sal))| = 2) and the test line 22 succeeds (because
+32
+
+tuples(σ1(Sal)) = Sat(Γ5) = {s, s′}). Thus, ToRemove SC is set to {es, es′}
+line 21 and C ans(Q) is set to {e} line 23. Then, when processing σ2, all tests suc-
+ceed (because |tuples(σ2)| = |tuples(σ2(Sal))| = 1 and tuples(σ2(Sal)) = {s′}).
+Thus, e′ is inserted in WC ans(Q5), MSC ans(Q5), SC ans(Q5) and C ans(Q5).
+The loop line 3 returns ToRemove X = ∅, Candidate SC = ∅, ToRemove SC =
+∅, along with WC ans(Q5) = MSC ans(Q5) = SC ans(Q5) = C ans(Q5) =
+{e, e′}, which is not changed by the statements lines 24-28. We thus obtain
+all four consistent answers equal to {e, e′}, as expected.
+In the next example, we further illustrate the role of the sets Candidate SC
+and ToRemove SC in Algorithm 2.
+Example 9 Let U = {A, B, C, D} over which the functional dependencies C →
+B and C → D are assumed. For T = {abcd, abcd′, abc′, ab′c′}, Algorithm 1
+returns Σ∗ = {(a)(b)(c)(dd′), (a)(bb′)(c′)}, and the processing of SC ans(Q) for
+Q : select A from T where B = b, is as follows:
+• For σ = (a)(b)(c)(dd′), the tests line 4 and line 12 fail (as |tuples(σ(A))| = 1
+and |tuples(σ)| = 2), whereas the test line 10 succeeds (because Sat(Γ) =
+{b}). Thus ToRemove X remains empty and as the test line 17 succeeds
+(because tuples(σ(B)) = {b}), ab is inserted in Candidate SC on line 18,
+and SC ans(Q) remains empty.
+• For σ = (a)(bb′)(c′), as the test line 4 fails, ToRemove X remains empty,
+and as the tests line 12 and line 17 also fail, ab and ab′ are inserted in
+ToRemove SC due to the statement line 21.
+Thus, the loop line 3 generates ToRemove X = ∅, Candidate SC = {ab} and
+ToRemove SC = {ab, ab′}. Therefore, when processing the statement line 25,
+SC ans(Q) remains empty (because Candidate SC \ ToRemove SC = ∅), as
+expected based on Definition 6(2), because ab is inconsistent.
+Considering now T1 = {abcd, abcd′}, Algorithm 1 returns Σ∗
+1 = {(a)(b)(c)(dd′)}
+and the computations are as in the first item above, implying that the loop line 3
+generates ToRemove X = ∅, Candidate SC = {ab}, ToRemove SC = ∅ and
+SC ans(Q) = ∅. When processing the statement line 25, SC ans(Q) is set to
+{a} (because Candidate SC \ ToRemove SC = {ab}), as expected based on
+Definition 6(2), because ab is a consistent true tuple satisfying Γ.
+�
+To assess the complexity of Algorithm 2, we first notice that this algorithm is
+clearly linear in the size of Σ∗. Moreover, using the same notation as when
+assessing the complexity of Algorithm 1, for σ in Σ∗, the size of tuples(σ) is in
+O(|FD|.δ), that is in O(δ), because |FD| is independent from T. Hence, the
+complexity of computing the consistent answer to a query in our approach is in
+O(|Σ∗|.δ). Since it has been shown that |Σ∗| ≤ |T|, we state that the sizes of
+Σ∗ and of T are of the same order. Therefore, the complexity of computing all
+four consistent answers to a query in our approach is in O(|T|.δ), that is linear
+in the size of the table T.
+It is however important to stress that Algorithm 2 requires significant extra
+storage due to the sets ToRemove X, Candidate SC and ToRemove SC. It
+33
+
+should be noticed that the storage of ToRemove X and ToRemove SC can
+be avoided by implementing a second scan of Σ∗ whose role would be to de-
+tect the X-values to be removed (when |tuples(σ(X))| > 1) and the tuples in
+Candidate SC to be removed (when |tuples(σ(sch(Q)))| > 1). As usual, this
+option saves storage at the cost of further processing, namely an additional scan
+of Σ∗.
+5
+Related Work
+The problem of consistent query answering has attracted considerable attention
+in the last two decades [1] and continues to be an active area of research today
+[4]. The approach by repairs is one that has attracted most attention in recent
+years and thus has been the subject of many papers, whose overview lies outside
+the scope of this paper. We refer to [5] for a detailed survey covering the topic.
+We rather focus on the basic differences between our approach and repair-based
+approaches with the goal of showing that although the tackled problem is the
+same, our approach is hardly comparable with those from the literature.
+5.1
+Comparison with Repair-Based Approaches
+We first point out that repair-based approaches assume a table T with no nulls.
+Therefore to compare our approach to repair-based approaches we need to re-
+strict our attention to tables with functional dependencies but without nulls.
+In this context, we identify and discuss three fundamental differences be-
+tween our approach and the approach by repairs, namely : (D1) the two ap-
+proaches use different universes of discourse, (D2) the two approaches use dif-
+ferent assumptions when evaluating queries, and (D3) the two approaches use
+functional dependencies for different purposes.
+(D1) different universes of discourse.
+We recall from Section 2 that the universe of discourse of a table T is the space in
+which queries are evaluated. The universe of discourse in our approach, denoted
+by T , is the set of all tuples one can define using constants appearing in T. On
+the other hand, the universe of discourse in the repairs approach, that we shall
+denote by Tr, is the set of all tuples of T together with all their sub-tuples.
+For example, if T = {ab, a′b′} then T = {ab, a′b′, a′b, ab′, a, b, a′, b′} whereas
+Tr = {ab, a′b′, a, b, a′, b′}.
+(D2) different assumptions when evaluating queries.
+The repairs approach uses the Closed World Assumption [12] that is:
+the only true tuples are those in Tr (implying that tuples not in Tr are
+false)
+whereas our approach uses the Open World Assumption that is:
+the only true tuples are those in T together with all tuples derived using
+the functional dependencies (implying that all other tuples in T are false).
+34
+
+It follows that if a tuple t of Tr is true then t is true in T , that is True(Tr) ⊆
+True(T ), while the opposite does not hold.
+For example, consider the table T = {abc, ab′c′} with functional dependency
+A → B. In this case, ac is in T and in True(T ), and ac is in True(Tr) as well.
+On the other hand, since Σ∗ = {(a)(bb′)(c), (a)(bb′)(c′)}, the tuple abc′ is in T
+but not in Tr. Thus abc′ is not in True(Tr), whereas abc′ is in True(T ).
+Regarding inconsistent tuples, the comparison not as easy since, to the best
+of our knowledge, the notion of inconsistent tuple is not explicitly defined in
+repair-based approaches. However, it is clear that a table T is inconsistent if
+and only if there exists a functional dependency X → A such that the two
+tuples xa and xa′ in T (XA) occur in T. The pair (xa, xa′) is usually called a
+conflicting pair, and we define inconsistent tuples in this context as follows:
+t is inconsistent if there exists a conflicting pair (xa, xa′) such that t is a
+super-tuple of xa.
+Denoting the set of these tuples by Conf Inc(Tr), we argue that Conf Inc(Tr) ⊆
+Inc(T ) whereas the inverse inclusion does not always hold. Indeed, based on Def-
+inition 3, it is easy to see that, for every conflicting pair (xa, xa′), the tuples
+xa, xa′ and all their true super-tuples are in Inc(T ), thus showing the inclu-
+sion Conf Inc(Tr) ⊆ Inc(T ). The previous example with T = {abc, ab′c′} and
+A → B, shows that the inverse inclusion does not always hold, because b is
+clearly not in Conf Inc(Tr), whereas Definition 3 implies that b is in Inc(T ).
+(D3) different purposes in the use of functional dependencies.
+The repair-based approaches use functional dependencies as a means for check-
+ing the consistency of T (i.e., as a means of inference of inconsistent tuples). In
+our approach, we also use the functional dependencies as a means of inference of
+inconsistent tuples. However, additionally, we use the functional dependencies
+as a means of inference of true tuples. For example, if T = {abc, ab′c′} and
+FD = {A → B}, our approach allows us to infer that abc is true and inconsis-
+tent, as repair-based approaches do. However, additionally, our approach allows
+to infer that the tuple ab′c is also true and inconsistent, which repair-based ap-
+proaches fail to do since ab′c is not even in Tr.
+Now that we have stated the differences in the ways the two approaches
+operate, consider a query Q : select X from T where Γ, and recall that in
+repair-based approaches, the consistent answer to Q is defined as the set of true
+tuples x in Tr such that, every repair contains a tuple satisfying the condition Γ
+and whose restriction over X is x. Denoting this set by Rep ans(Q), we recall
+that, by Definition 5, all consistent answers to Q in our approach are sets of true
+and consistent tuples in T (X). It is thus not surprising that the two approaches
+to consistent query answering are not comparable.
+Summarising our discussion so far we cannot claim that either of the two ap-
+proaches is better or preferable than the other. They are simply different as they
+have different universes of discourse, they operate under different assumptions
+and they use the functional dependencies in different ways.
+35
+
+What we can reasonably claim in favor of our approach is that: (a) it provides
+a formal framework based on which different kinds of consistent answers can be
+defined, (b) it offers a polynomial algorithm for evaluating the consistent answer
+to a query, whether conjunctive or disjunctive, and (c) by allowing nulls in the
+table, our approach broadens the angle of its application to other important
+areas of research as explained in the following section.
+5.2
+Comparison with the Approach in [9]
+The approach presented in this paper and the approach in [9], both rely on par-
+tition semantics [13]. However, the two approaches have important differences.
+First, although the limit interpretation I∗ as defined in Section 2 is the same as
+the interpretation µ∗ in [9], we have proved here the uniqueness of I∗ which is
+not done for µ∗.
+Second, the notions of true/false tuples and of inconsistent/consistent tuples
+differ between the two approaches. More precisely, since in [9], truth values are
+defined inspired by the Four-valued logic of [2], it is not possible to have tuples
+that are, for instance, true and consistent, as in the present work.
+Third, the universe of discourse in [9], is potentially infinite as it consists of
+all tuples that can be built up using constants from the entire attribute domains;
+and this is an issue when it comes to computing the set of all inconsistent tuples.
+In contrast, the universe of discourse in the present work is finite as it consists
+of all tuples that can be built up using constants only from the active attribute
+domains.
+Fourth, the notion of inconsistent tuple is not defined in the same way in
+the two approaches. Indeed, in [9], a tuple is inconsistent if its interpretation
+is a sub-set of the interpretations of two constants from the same domain. For
+example, consider the (true) tuples xa and xa′ in the presence of X → A.
+These tuples are inconsistent in the present approach (see Definition 3), as well
+as in the approach of [9]. However, the tuple x is also an inconsistent tuple
+according to [9] (because its interpretation is a sub-set of those of a and a′),
+but is not inconsistent in the present approach (because A is not in sch(x)). As
+a consequence, the consistent answer to a query in [9] differs from that in the
+present work.
+Finally, the issue of consistent query answering has not been addressed in
+full details in [9]: it is restricted to the case where the table T contains no nulls.
+We also note in this respect that, based on the concept of m-table as introduced
+in Section 4, the present approach allows to consider also disjunctive queries,
+an issue not addressed in [9], nor in repair-based approaches, such as [5, 15].
+6
+Concluding Remarks and Future Work
+In this paper, we have presented a novel approach to consistent query answering
+in tables with nulls and functional dependencies.
+Given such a table T, we
+defined T , the universe of discourse of T, and using set-theoretic semantics for
+36
+
+tuples and functional dependencies we partitioned T in two orthogonal ways:
+into true and false tuples, on the one hand and into consistent and inconsistent
+tuples on the other hand.
+Queries are addressed to T and evaluated in T
+in order to obtain consistent answers. The consistent answer to a query Q :
+select X from T where Condition over T was defined to be the set of tuples
+x in T such that: (a) x is a true and consistent tuple with schema X and (b)
+there exists a true super-tuple t of x in T satisfying the condition. We have
+seen that, depending on the ‘status’ of the super-tuple t in T , there are four
+different types of consistent answer to Q and we have presented polynomial time
+algorithms for computing these answers. We have also seen that our approach
+is not comparable to the approaches based on table repairs or the approach
+of [9], as they have different universes of discourse and operate under different
+assumptions.
+By allowing the presence of nulls in the table, our approach opens the way for
+a number of important applications. We envisage three such applications of our
+approach. The first is related to universal relation interfaces [8, 13, 14]. Given a
+relational database with n tables T1, T2, . . . , Tn, over schemes S1, S2, . . . , Sn and
+sets of functional dependencies FD1, FD2, . . . , FDn, respectively, the universal
+relation is defined to be a table T defined as follows:
+• The schema of T is S = S1 ∪ S2 ∪ . . . ∪ Sn and the associated set of
+functional dependencies is FD = FD1 ∪ FD2 ∪ . . . ∪ FDn.
+• For each tuple t in Ti, t is placed in a new line of T with a new tuple
+identifier, for all i = 1, 2, . . . , n.
+For example, if n = 2, T1 = {ab} and T2 = {bc} then the universal relation
+operates from a table T with schema ABC and current instance T = {ab, bc}
+with two nulls, one under C and one under A.
+A prerequisite for defining the table T is “name consolidation” that is (a)
+if an attribute name appears in two different database tables then it has the
+same meaning in the two tables and (b) if two different attribute names in the
+database have the same meaning then one is renamed to the other (using the
+renaming operation of the relational model).
+Once the table T is put in place querying the interface is straightforward:
+the system computes a consistent answer using our polynomial Algorithm 2 and
+returns the result. However, updating through such an interface poses several
+hard problems that we are currently investigating.
+The second application that we envisage is query result explanation.
+As
+already mentioned, the m-table Σ∗ offers a way to ‘display’ inconsistencies. We
+therefore plan to investigate how m-tuples in Σ∗ can be used to ‘explain’ to the
+user the presence or the absence of a given tuple in a consistent answer. This
+issue seems particularly relevant in the context of data integration and data
+exchange, where it is crucial to ‘explain’ consistent answers in reference to the
+sources the data come from [4].
+The third application that we envisage is related to data quality. Based on
+the two orthogonal ways of characterizing tuples, namely consistent/inconsistent
+37
+
+and true/false, we consider that the inconsistent tuples are as good a source of
+information as they are the consistent tuples; and similarly, we consider that
+the false tuples are as good a source of information as they are the true tu-
+ples. In doing so we can develop a meaningful theory for the quality of data
+in a data-set (whether the data-set is a table or a query result). To this end,
+we plan to consider various quality measures such as percentages of consis-
+tent/inconsistent tuples in the data-set, percentages of true or false tuples, or
+combinations thereof.
+Acknowledgment
+Work conducted while the second author was visiting at FORTH Institute of
+Computer Science, Crete, Greece (https://www.ics.forth.gr/)
+Declarations
+The two authors contributed to the study, conception and design. Both read
+and approved the manuscript.
+No funds, grants, or other support was received for conducting this study.
+References
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+A
+On the Church-Rosser Property of Expansion
+We show that expansion has the Church-Rosser property, that is when iterating
+the process, the order in which the steps are preformed is irrelevant. We first
+recall the definition of expansion.
+Definition 4 The expansion of an interpretation I with respect to a functional
+dependency X → A and the tuple xa in T (XA) is the interpretation Exp(I, xa)
+defined as follows:
+• If I(x) ∩ I(a) ̸= ∅ and I(x) ̸⊆ I(a) then:
+Exp(I, xa)(a) = I(a) ∪ I(x), and Exp(I, xa)(α) = I(α) for α in AD
+different than a.
+39
+
+• Otherwise, Exp(I, xa) = I.
+We consider two expansions one with respect to the dependency X → A and
+the tuple xa and the other one with respect to the dependency Y → B and
+the tuple yb.
+Starting with an interpretation I1, let I2 = Exp(I1, xa) and
+I3 = Exp(I1, yb). Referring to Figure 2(a), the problem is to find sequences of
+expansions called E2 and E3 in the figure, such that I24 = E2(I2) and I34 =
+E3(I3) are equal.
+It should be clear that if I2 = I1 or I3 = I1, then the order in which the
+expansions are processed is irrelevant. For example, if I2 = I1 and I3 ̸= I1, we
+obtain I24 = I34 by setting E3 = ϵ (where ϵ denotes the empty sequence) and
+E2 = Exp(I1, yb).
+From now on, we assume that I2 ̸= I1 and I3 ̸= I1. Based on Definition 4,
+we have:
+• I2(a) = I1(a) ∪ I1(x) and for every α ̸= a, I2(α) = I1(α)
+• I3(b) = I1(b) ∪ I1(y) and for every β ̸= b, I2(β) = I1(β).
+As a particular case, if it is assumed that a = b, we have:
+• I2(a) = I1(a) ∪ I1(x) and for every α ̸= a, I2(α) = I1(α)
+• I3(a) = I1(a) ∪ I1(y) and for every α ̸= a, I3(α) = I1(α).
+Thus, as shown in Figure 2(b), for E2 = Exp(I2, ya) and E3 = Exp(I3, xa),
+I24 and I34 are such that I24(a) = I34(a) = I1(a) ∪ I1(x) ∪ I1(y) and for every
+α ̸= a, I24(α) = I34(α) = I1(α). We therefore have, in this case, I24 = I34.
+ 
+ 
+I1
+I3
+I24 = I34
+Exp xa
+Exp yb
+Exp yb
+I1
+I2
+I3
+I24 = I34
+Exp xa
+Exp xa
+Exp yb
+E2 ?
+E3 ?
+(a) Generic diagram
+(b) (a = b) or (y ≠ y'a  and  x ≠ x'b)
+I1
+Exp xa
+Exp xa
+Exp yb
+Exp yb
+Exp yb
+I2
+I3
+I31
+I24 = I34
+(c) y = y'a  and  x ≠ x'b
+I2
+Exp xa
+Exp xa
+Exp yb
+Exp yb
+Exp yb
+Exp xa 
+(d) y = y'a  and  x = x'b
+I1
+I2
+I21
+I31
+I3
+I24 = I34
+Figure 2: Expansion Diagrams
+40
+
+Now assuming that I2 ̸= I1 and I3 ̸= I1 and a ̸= b, if a does not occur in y
+and b does not occur in x, the two expansions are independent from each other.
+In this case, as illustrated in Figure 2(b), the following lemma holds.
+Lemma 5 If a ̸= b, if a does not occur in y and b does not occur in x, for
+E2 = Exp(I2, yb) and E3 = Exp(I3, xa), denoting respectively by I24 and I34
+the interpretations Exp(I2, yb) and Exp(I3, xa), we have I24 = I34, that is:
+Exp(Exp(I1, xa), yb) = Exp(Exp(I1, yb), xa).
+Proof.
+Since a does not occur in y, we have I2(y) = I1(y) and similarly,
+since b does not occur in x, we have I3(x) = I1(x). Therefore, for a and b,
+I24 = Exp(Exp(I1, xa), yb) and I34 = Exp(Exp(I1, yb), xa) are defined by:
+− I24(a) = Exp(I2, yb)(a) = I2(a) = I1(a) ∪ I1(x) (because a ̸= b)
+− I24(b) = Exp(I2, yb)(b) = I2(b) ∪ I2(y) = I1(b) ∪ I1(y)
+− I34(a) = Exp(I3, xa)(a) = I3(a) ∪ I3(x) = I1(a) ∪ I1(x)
+− I34(b) = Exp(I3, xa)(b) = I3(b) = I1(b) ∪ I1(y) (because a ̸= b).
+Hence, we have I24(a) = I34(a) and I24(b) = I34(b). Since no other symbol
+is changed by expansions, we have shown the commutativity property in this
+case, namely I24 = I34.
+�
+We now switch to the most general case, namely when each expansion has an
+impact on the other. This happens when a occurs in y and when b occurs in x.
+Indeed, as shown in Figure 2(d), in this case, writing x and y respectively as bx′
+and ay′, we notice the following: I2(y) ̸= I1(y) because I2(y) = I2(a) ∩ I2(y′)
+and I2(a) ̸= I1(a), and similarly, I3(x) ̸= I1(x) because I3(x) = I3(b) ∩ I3(x′)
+and I3(b) ̸= I1(b).
+An important remark regarding x′ and y′ is that neither a nor b occurs
+is these tuples, thus implying that, during the computations of the various
+expansions considered here, their interpretation will not change, thus remaining
+equal to I1(x′) and I1(y′), respectively. For example, I2(y) = I2(a) ∩ I2(y′) and
+I3(x) = I3(b) ∩ I3(x′) can be respectively written as I2(y) = I2(a) ∩ I1(y′) and
+I3(x) = I3(b) ∩ I1(x′). The following lemma holds in this case.
+Lemma 6 If a ̸= b, and if y = ay′ and x = bx′, for E2 = Exp(Exp(I2, yb), xa)
+and E3 = Exp(Exp(I3, xa), yb), denoting respectively by I24 and I34 the corre-
+sponding interpretations, we have I24 = I34, that is:
+Exp(Exp(Exp(I1, xa), yb), xa) = Exp(Exp(Exp(I1, yb), xa), yb).
+Proof. Referring to Figure 2(d), we first give the computations regarding I2,
+I21 and I24. In each case the computations of the interpretations of a, x, b and
+y are shown.
+• Interpretation I2 = Exp(I1, xa).
+First, we have I2(a) = I1(a) ∪ I1(x), due to expansion. Moreover, I2(b) = I1(b)
+and as I2(y) = I2(a) ∩ I2(y′), we obtain I2(y) = (I1(a) ∪ I1(x)) ∩ I1(y′). Since
+I1(y) = I1(a) ∩ I1(y′), we have I2(y) = I1(y) ∪ (I1(x) ∩ I1(y′)).
+Moreover,
+I2(x) = I2(b) ∩ I2(x′), and thus, I2(x) = I1(b) ∩ I1(x′) = I1(x). Therefore I2 is
+defined as shown below:
+41
+
+I2(a) = I1(a) ∪ I1(x)
+I2(x) = I1(x)
+I2(b) = I1(b)
+I2(y) = I1(y) ∪ (I1(x) ∩ I1(y′))
+• Interpretation I21 = Exp(I2, yb).
+Now, we have I21(b) = I2(b) ∪ I2(y), due to expansion. It follows that I21(b) =
+I1(b) ∪ I1(y) ∪ (I1(x) ∩ I1(y′)).
+Moreover, I21(a) = I2(a), that is I21(a) =
+I1(a) ∪ I1(x). As I21(x) = I21(b) ∩ I21(x′) and I21(x′) = I1(x′), we obtain
+I21(x)
+=
+(I1(b) ∪ I1(y) ∪ (I1(x) ∩ I1(y′))) ∩ I1(x′)
+=
+(I1(b) ∩ I1(x′)) ∪ (I1(y) ∩ I1(x′)) ∪ (I1(x) ∩ I1(y′) ∩ I1(x′))
+Considering that I1(b)∩I1(x′) = I1(x) and that I1(x) ⊆ I1(x′) (because x′ ⊑ x),
+we obtain: I21(x) = I1(x) ∪ (I1(x′) ∩ I1(y)) ∪ (I1(x) ∩ I1(y′)). Since (I1(x) ∩
+I1(y′)) ⊆ I1(x) always holds, we have: I21(x) = I1(x) ∪ (I1(x′) ∩ I1(y)).
+On the other hand, I21(y) = I21(a)∩I21(y′), and thus, I21(y) = I2(a)∩I1(y′).
+Therefore, I21(y) = (I1(a) ∪ I1(x)) ∩ I1(y′), which can be written as I21(y) =
+I1(y) ∪ (I1(x) ∩ I1(y′)). Hence I21 is defined as shown below:
+I21(a) = I1(a) ∪ I1(x)
+I21(x) = I1(x) ∪ (I1(x′) ∩ I1(y))
+I21(b) = I1(b) ∪ I1(y) ∪ (I1(x) ∩ I1(y′))
+I21(y) = I1(y) ∪ (I1(x) ∩ I1(y′))
+• Interpretation I24 = Exp(I21, xa).
+Here, we have I24(a) = I21(a) ∪ I21(x), due to expansion. It follows that:
+I24(a)
+=
+(I1(a) ∪ I1(x)) ∪ (I1(x) ∪ (I1(x′) ∩ I1(y)))
+=
+I1(a) ∪ I1(x) ∪ (I1(x′) ∩ I1(y))
+Moreover, I24(b) = I21(b), that is I24(b) = I1(b) ∪ I1(y) ∪ (I1(x) ∩ I1(y′)). As
+I24(x) = I24(b) ∩ I24(x′) and I24(x′) = I1(x′), we have:
+I24(x)
+=
+(I1(b) ∪ I1(y) ∪ (I1(x) ∩ I1(y′))) ∩ I1(x′)
+=
+(I1(b) ∩ I1(x′)) ∪ (I1(y) ∩ I1(x′)) ∪ (I1(x) ∩ I1(y′) ∩ I1(x′))
+=
+I1(x) ∪ (I1(x′) ∩ I1(y)) ∪ (I1(x) ∩ I1(y′))
+(because I1(x) ⊆ I1(x′) as x′ ⊑ x)
+=
+I1(x) ∪ (I1(x′) ∩ I1(y)) (because (I1(x) ∩ I1(y′)) ⊆ I1(x))
+As I24(y) = I24(a) ∩ I24(y′), we obtain
+I24(y)
+=
+(I1(a) ∪ I1(x) ∪ (I1(x′) ∩ I1(y))) ∩ I1(y′)
+=
+(I1(a) ∩ I1(y′)) ∪ (I1(x) ∩ I1(y′)) ∪ (I1(x′) ∩ I1(y) ∩ I1(y′))
+=
+I1(y) ∪ (I1(x) ∩ I1(y′)) ∪ (I1(x′) ∩ I1(y))
+(because I1(y) ⊆ I1(y′) as y′ ⊑ y)
+=
+I1(y) ∪ (I1(x) ∩ I1(y′)) (because (I1(x′) ∩ I1(y)) ⊆ I1(y))
+Hence I24 is defined as shown below:
+I24(a) = I1(a) ∪ I1(x) ∪ (I1(x′) ∩ I1(y))
+I24(x) = I1(x) ∪ (I1(x′) ∩ I1(y))
+I24(b) = I1(b) ∪ I1(y) ∪ (I1(x) ∩ I1(y′))
+I24(y) = I1(y) ∪ (I1(x) ∩ I1(y′))
+Referring again to Figure 2(d), we now turn to the computations regarding I3,
+I31 and I34. As the computations are similar to those above, some details are
+skipped.
+42
+
+• Interpretation I3 = Exp(I1, yb).
+Here, we have I3(b) = I1(b) ∪ I1(y), due to expansion. Computations similar to
+those in the case of I2 show that I3 is defined as shown below:
+I3(a) = I1(a)
+I3(x) = I1(x) ∪ (I1(x′) ∩ I1(y))
+I3(b) = I1(b) ∪ I1(y)
+I3(y) = I1(y)
+• Interpretation I31 = Exp(I3, xa).
+Here, we have I31(a) = I3(a) ∪ I3(x), due to expansion. Computations similar
+to those for I24 show that I31 is defined as shown below:
+I31(a) = I1(a) ∪ I1(x) ∪ (I1(x′) ∩ I1(y))
+I31(x) = I1(x) ∪ (I1(x′) ∩ I1(y))
+I31(b) = I1(b) ∪ I1(y)
+I31(y) = I1(y) ∪ (I1(x) ∩ I1(y′))
+• Interpretation I34 = Exp(I31, yb).
+Here, we have I34(b) = I31(b) ∪ I31(y), due to expansion. It follows that com-
+putations similar to those for I24 show that I34 is defined as shown below:
+I34(a) = I1(a) ∪ I1(x) ∪ (I1(x′) ∩ I1(y))
+I34(x) = I1(x) ∪ (I1(x′) ∩ I1(y))
+I34(b) = I1(b) ∪ I1(y) ∪ (I1(x) ∩ I1(y′))
+I34(y) = I1(y) ∪ (I1(x) ∩ I1(y′))
+It thus turns out that we indeed have I24 = I34, and the proof is complete.
+�
+The last case to be addressed is when y = ay′ and x ̸= bx′ or when x = bx′
+and y ̸= ay′. We only consider the former case since the latter can be dealt
+with in a similar way. Moreover, as shown in Figure 2(c), this case is in fact a
+simplification of the case of Lemma 6 where the terms involving x′ are omitted.
+We thus omit the proof and just state that is this case again I24 = I34. It can
+be checked that in case of Figure 2(c) (that is y = ay′ and x ̸= bx′), we obtain
+the following interpretations:
+I24(a) = I34(a) = I1(a) ∪ I1(x)
+I24(x) = I34(x) = I1(x)
+I24(b) = I34(b) = I1(b) ∪ I1(y) ∪ (I1(x) ∩ I1(y′))
+I24(y) = I34(y) = I1(y) ∪ (I1(x) ∩ I1(y′))
+We therefore state the following basic result regarding expansion:
+The process of expanding an interpretation I with respect to a set FD of
+functional dependencies satisfies the Church-Rosser property.
+Hence, given T and FD, the process of expanding Ib until a fixed point is
+reached does not depend on the order the expansions are processed. Thus, the
+limit is indeed unique.
+B
+Proof of Lemma 4
+Lemma 4 Let T be a table over universe U. Then the following holds:
+43
+
+1. Algorithm 1 applied to T always terminates.
+2. For every m-tuple s in MT , I∗(s) ̸= ∅ holds if and only if there exists σ
+in Σ∗ such that s ⊑ σ.
+3. Inc(T ) = ∅ if and only if fail = false.
+4. For every σ in Σ∗ and all t and t′ in tuples(σ), I∗(t) = I∗(t′).
+Proof. 1. The computation of Σ∗ terminates because (i) the sets under
+construction for a given attribute A are all bounded by the finite set adom(A),
+and (ii) the construction is monotonous, in the sense that if Σk and Σk+1 are
+two consecutive states, for every σ in Σk+1, there exists σ′ in Σk such that
+σ′ ⊑ σ.
+2. Given s in MT , we first show that I∗(s) is nonempty if there exists σ in Σ∗
+such that s ⊑ σ. The proof is conducted by induction on the steps in the runs
+of the loop line 4. We show that if (Σk)k≥0 denotes the sequence of the states
+of Σ∗ during the computation, for every σ in Σk, I∗(σ) ̸= ∅. We first note in
+this respect that since Σ0 = T, for every σ in Σ0, we have I∗(σ) ̸= ∅.
+Assuming now that for k ≥ 0, for every σ ∈ Σk, I∗(σ) ̸= ∅, we prove the
+result for every σ in Σk+1. Let σ in Σk+1 but not in Σk. Then σ occurs in
+Σk+1 because there exist X → A in FD, σ1 and σ2 in Σk such that for every
+B in X, σ1.B ∩ σ2.B ̸= ∅. Let σ′
+1 = σ1 ⊔ σ2(A) and σ′
+2 = σ2 ⊔ σ1(A). Then,
+either σ = σ′
+1 and σ′
+1 ̸= σ1 or σ = σ′
+2 and σ′
+2 ̸= σ2.
+The two cases being
+similar, we just consider the first one, that is σ = σ′
+1 and σ′
+1 ̸= σ1, which
+implies that σ2(A) ̸= ∅ and σ2(A) ̸⊑ σ1. By our induction hypothesis, we have
+I∗(σ2) ̸= ∅ and thus denoting by x an X-tuple occurring in tuples(σ1 ⊓ σ2),
+we have I∗(x) ∩ I∗(σ2(A)) ̸= ∅. Since I∗ satisfies X → A, I∗(x) ⊆ I∗(σ2(A)).
+As x ⊑ σ1, I∗(σ1) ⊆ I∗(x) and thus, I∗(σ1) ⊆ I∗(σ2(A)). Therefore, I∗(σ) =
+I∗(σ1) ∩ I∗(σ2(A)) = I∗(σ1). As by our induction hypothesis, I∗(σ1) ̸= ∅, we
+obtain that I∗(σ) ̸= ∅. Therefore, it holds that for every k ≥ 0, I∗(σ) ̸= ∅ for
+every σ in Σk, and so, it holds that I∗(σ) ̸= ∅ for every σ in Σ∗. Thus, if s
+satisfies that there exists σ in Σ∗ such that s ⊑ σ, we have I∗(σ) ⊆ I∗(s). Hence
+I∗(s) ̸= ∅, and this part of the proof is complete.
+Conversely, we show that for every s, if I∗(s) ̸= ∅ then there exists σ in Σ∗
+such that s ⊑ σ. The proof is done by induction on the construction of I∗.
+By definition of I0 = Ib, if I0(s) ̸= ∅ then s is a sub-tuple of a tuple t in
+T, i.e., s ⊑ t holds. Since Σ0 = T, there exists σ in Σ0 such that t ⊑ σ. By
+monotonicity of the construction of Σ∗ (i.e., for every σk+1 in Σk+1 there exists
+σk in Σk such that σk ⊑ σk+1), there exists σ∗ in Σ∗ such that σ ⊑ σ∗, which
+shows that t ⊑ σ∗, thus that s ⊑ σ∗.
+Now, assuming that for every k ≥ 0 and every s, if Ik(s) ̸= ∅ then there
+exists σ in Σ∗ such that s ⊑ σ, we prove that this result holds for Ik+1.
+Let s be such that Ik(s) = ∅ and Ik+1(s) ̸= ∅.
+For every a in AD we
+have Ik+1(a) = Ik(a) ∪ Ik(x) if x and a are the tuples involved in Exp(Ik, xa)
+and Ik+1(a) = Ik(a) otherwise. Since Ik+1(s) ̸= Ik(s) and since a is the only
+symbol for which Ik has changed, it must be that a ⊑ s. Therefore, writing s
+44
+
+as s′ ⊔ a, we have Ik+1(s) = Ik+1(s′) ∩ Ik+1(a) and Ik+1(s′) = Ik(s′). It follows
+that Ik+1(s) = Ik(s′) ∩ (Ik(a) ∪ Ik(x)), thus that Ik+1(s) = (Ik(s′) ∩ (Ik(a)) ∪
+(Ik(s′) ∩ Ik(x)), that is Ik+1(s) = Ik(s) ∪ (Ik(s′) ∩ Ik(x)). As Ik+1(s) ̸= ∅ and
+Ik(s) = ∅, we have Ik(s′) ∩ Ik(x) ̸= ∅, in which case our induction hypothesis
+entails that there exists σ in Σ∗ such that (s′ ⊔ x) ⊑ σ. Since by definition of
+expansion, we also have Ik(x) ∩ Ik(a) ̸= ∅, Σ∗ contains an m-tuple σ′ such that
+xa ⊑ σ′. In this case, Algorithm 1 shows that there exists σ∗ in Σ∗ such that
+(s′ ⊔ a) ⊑ σ∗, that is s ⊑ σ∗. This part of the proof is therefore complete.
+3. If the returned value for fail is false, the previous item shows that there exist
+X → A in FD, x in T (X) and a and a′ in adom(A) such that I∗(a)∩I∗(a′) ̸= ∅.
+Therefore in this case, Inc(T ) is not empty. Conversely, if Inc(T ) ̸= ∅, for every t
+in Inc(T ), there exist A in U and a and a′ in adom(A) such that I∗(a)∩I∗(a′) ̸=
+∅, and the previous item shows that there exists σ in Σ∗ such that (aa′) ⊑ σ.
+Hence, during the execution of Algorithm 1, the value of fail must have been
+turned to true line 11.
+4. The proof is by induction on the construction of Σ∗. As all m-tuples in Σ0
+can be seen as tuples, for every σ in Σ0, tuples(σ) consists of one tuple. The
+result thus trivially holds in this case. Now assume that the result holds for Σk,
+and let σ be in Σk+1 obtained by steps lines 16-20 of Algorithm 1. Thus there
+exist σ′ in Σ∗, X → A in FD, x0 in T (X) and a0 in adom(A) such that in Σk,
+x0 ∈ tuples(σ(X)) ∩ tuples(σ′(X)), a0 is in tuples(σ′(A)) \ tuples(σ(A)), and in
+Σk+1, σ(A) becomes σ1(A) such that a0 ∈ tuples(σ1(A)).
+Let t1 be in tuples(σ1) such that t1.A = a0, and let x denote t1.X. Then, t1
+is not in tuples(σ) (because a0 ̸∈ tuples(σ(A))), but tuples(σ) contains a tuple t
+such that t.X = x and a tuple t′ such that t′.X = x0. Thus tuples(σ1) contains
+t1 such that t1.XA = xa0 and a tuple t′
+1 such that t′
+1.XA = x0a0.
+On the other hand, the fact that x0a0 occurs in tuples(σ′) implies that
+I∗(x0a0) ̸= ∅, thus that I∗(x0) ⊆ I∗(a0).
+Hence, I∗(x0a0) = I∗(x0).
+We
+now argue that I∗(t′
+1) = I∗(t′).
+Indeed, if σ(A) = ∅, then t′
+1 can be writ-
+ten as t′a0 in which case I∗(t′
+1) = I∗(t′) ∩ I∗(a0) holds. Since it also holds
+that I∗(t′) ⊆ I∗(x0) ⊆ I∗(a0), we have I∗(t′) ∩ I∗(a0) = I∗(t′). Otherwise, if
+σ(A) ̸= ∅ then t′ is written as q′a′ and thus t′
+1 is written as q′a0 (a′ has been
+replaced by a0 to obtain t′
+1 from t′). In this case, I∗(t′
+1) = I∗(q′) = I∗(t′).
+As t and t′ are both in tuples(σ), by our induction hypothesis, in Σi we have
+I∗(t) = I∗(t′), which implies that I∗(t) = I∗(t′
+1) = I(t′). We therefore obtain
+that for every t1 in tuples(σ1), there exists t in tuples(σ) such that I∗(t1) = I∗(t).
+As all tuples in t tuples(σ) have the same I∗(t), all tuples t1 in tuples(σ1) have
+the same I∗(t1). The proof is therefore complete.
+�
+C
+Proof of Proposition 5
+Proposition 5
+Let T be a non-redundant table over universe U, FD the
+set of associated functional dependencies and Σ∗ the non-redundant m-table as
+computed by Algorithm 1. Given a query Q : select X from T [where Γ],
+and a tuple x in T (X):
+45
+
+1. x is in MSC ans(Q) if and only if
+(a) for every σ in Σ(x), tuples(σ(X)) = {x},
+(b) there exists σ in ΣQ(x) such that |tuples(σ)| = 1.
+2. x is in SC ans(Q) if and only if
+(a) for every σ in Σ(x), tuples(σ(X)) = {x},
+(b) there exists s in Sat(Γ) ∩ True(T ) such that for every σ in ΣQ(x), if
+s is in σ(sch(Γ)) then tuples(σ(sch(Γ)))) = {s}.
+3. x is in C ans(Q) if and only if
+(a) for every σ in Σ(x), tuples(σ(X)) = {x},
+(b) there exists σ in ΣQ(x) such that tuples(σ(sch(Γ))) ⊆ Sat(Γ).
+4. x is in WC ans(Q) if and only if
+(a) for every σ in Σ(x), tuples(σ(X)) = {x},
+(b) ΣQ(x) is not empty.
+Proof.
+In this proof, Q is assumed to involve a selection condition Γ,
+because the case where no selection condition is involved is a simplification, and
+thus, is omitted.
+1. By Proposition 4(1) and (2), statement (a) in the proposition is equivalent
+to statement (1.a) in Definition 6.
+Moreover, as Σ∗ is assumed to be non-
+redundant, for every σ in Σ∗ and every t in tuples(σ), there exists no tuple
+t′ in True(T ) such that t � t′. Thus, by Proposition 4(2), statement (b) in
+the proposition yields a tuple having the same properties as the tuple t in
+statement (1.b) of Definition 6 (i.e., t is a maximal true and consistent tuple
+whose restriction over sch(Γ) satisfies Γ). Therefore this part of the proof is
+complete.
+2. As above, by Proposition 4(1) and (2), statement (a) in the proposition is
+equivalent to statement (2.a) in Definition 6. Moreover, statement (b) in the
+proposition yields a tuple t such that t.X = x, t.sch(Γ) = s, which is in Sat(Γ).
+Moreover, Proposition 4(3) implies that this tuple t is true and consistent, thus
+satisfying the statement (2.b) of Definition 6 (i.e., t is a true and consistent
+tuple whose restriction over sch(Γ) satisfies Γ). Therefore this part of the proof
+is complete.
+3. Let us first assume that x is in C ans(Q). As x is in Cons(T ) ∩ True(T ),
+by Proposition 4, Σ∗ must contain at least one σ such that x ⊑ σ and every
+such m-tuple must be such that σ(X) = {x}. Hence Σ(x) is nonempty and
+the first item in the proposition holds. Moreover, by Definition 6, one of these
+m-tuples σ is also such that sch(Γ) ⊆ sch(σ) and tuples(σ) contains a tuple t
+of Cons(Γ, T ). By Definition 5, this implies that the following statement (*)
+holds: t.sch(Γ) is in Sat(Γ) and for every s in Sat−(Γ), I∗(t) ∩ I∗(s) = ∅. This
+in particular implies that σ is in ΣQ(x), because t.(sch(Γ)) ⊑ t ⊑ σ holds.
+46
+
+As ΣQ(x) ⊆ Σ(x), every σ in ΣQ(x) is such that σ(X) = {x}. Assume now
+that every σ in ΣQ(x) is such that tuples(σ(sch(Γ))) contains q0 not in Sat(Γ).
+Then q0 is in Sat−(Γ) and tuples(σ) contains a tuple t0 such that t0.sch(Γ) = q0.
+Thus we have two tuples t and t0 in tuples(σ) such that t.X = t0.X = x,
+t.sch(Γ) ∈ Sat(Γ) and t0.sch(Γ) ∈ Sat−(Γ). Lemma 4(4) implies that I∗(t) =
+I∗(t0) (because t and t0 are in tuples(σ)), and as I∗(t0) ⊆ I∗(q0) (because
+t0.sch(Γ) = q0), we have I∗(t0) ∩ I∗(q0) = I∗(t0). Thus I∗(t0) ∩ I∗(q0) ̸= ∅,
+implying that I∗(t) ∩ I∗(q0) ̸= ∅, which is a contradiction with statement (*)
+above. We therefore have shown that if x is in C ans(Q) then the items in the
+proposition hold.
+Conversely, if the items in the proposition are satisfied, Proposition 4 implies
+that x is in Cons(T ) ∩ True(T ). Thus by Definition 5, we have to prove the
+existence of a tuple t in Cons(Γ, T )∩True(T ) such that x ⊑ t. Let σ be in ΣQ(x)
+as stated in item (3.b) of the proposition, that is such that tuples(σ(sch(Γ))) ⊆
+Sat(Γ), and let t0 be in tuples(σ). As t0 is in True(T ), we have to show that t0
+is in Cons(Γ, T ), that is that there exists q in Sat(Γ) such that (i) q ⊑ t0, and
+(ii) for every q′ in Sat−(Γ), I∗(t0)∩I∗(q′) = ∅. Since we know that t0.sch(Γ) is
+in Sat(Γ), (i) holds for q = t0.sch(Γ). Regarding (ii), let q′
+0 be in Sat−(Γ) such
+that I∗(t0)∩I∗(q′
+0) ̸= ∅. As I∗(σ) = �
+u∈tuples(σ) I∗(u), Lemma 4(4) implies that
+I∗(σ) = I∗(t0), and thus that I∗(σ) ∩ I∗(q′
+0) ̸= ∅. Applying now Lemma 4(2),
+we obtain that Σ∗ contains an m-tuple σ′ involving all attribute values in σ or
+in q′
+0. Since q′
+0 is not in tuples(σ), we have σ′ ̸= σ, and thus σ � σ′ holds. This
+is a contradiction with our hypothesis that Σ∗ contains no redundancy, showing
+that t0 is in Cons(Γ, T ). This part of the proof is therefore complete.
+4. As in cases (1) and (2) above, by Proposition 4(1) and (2), statement (4.a)
+in the proposition is equivalent to statement (2.a) in Definition 6. Moreover,
+ΣQ(x) ̸= ∅ is equivalent to the existence of σ in Σ∗ such that σ(X) = {x} and
+σ(sch(Γ)) contains a tuple s of Sat(Γ). Hence, ΣQ(x) ̸= ∅ is equivalent to the
+existence of a tuple t in True(T ) (due to Proposition 4(1)) such that t.X = x
+and t.sch(Γ) is in Sat(Γ). The proof is therefore complete.
+�
+47
+
diff --git a/hNE2T4oBgHgl3EQfHgYi/content/tmp_files/load_file.txt b/hNE2T4oBgHgl3EQfHgYi/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..22927297012a21f405199623a792f19b94810b51
--- /dev/null
+++ b/hNE2T4oBgHgl3EQfHgYi/content/tmp_files/load_file.txt
@@ -0,0 +1,1486 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf,len=1485
+page_content='Consistent Query Answering without Repairs in  Tables with Nulls and Functional Dependencies  Dominique Laurent  dominique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='laurent@u-cergy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='fr  ETIS Laboratory - ENSEA, CY Cergy Paris University, CNRS  F-95000 Cergy-Pontoise - FRANCE  Nicolas Spyratos  nicolas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='spyratos@lri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='fr  LISN Laboratory - University Paris-Saclay, CNRS  F-91405 Orsay - FRANCE  January 11, 2023  Abstract  In this paper, we study consistent query answering in tables with nulls  and functional dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Given such a table T, we consider the set  T of all tuples that can be built up from constants appearing in T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' and  we use set theoretic semantics for tuples and functional dependencies to  characterize the tuples of T in two orthogonal ways: first as true or false  tuples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' and then as consistent or inconsistent tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Queries are issued  against T and evaluated in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  In this setting, we consider a query Q: select X from T where Condition  over T and define its consistent answer to be the set of tuples x in T such  that: (a) x is a true and consistent tuple with schema X and (b) there  exists a true super-tuple t of x in T satisfying the condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We show  that, depending on the ‘status’ that the super-tuple t has in T , there are  different types of consistent answer to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  The main contributions of the paper are: (a) a novel approach to con-  sistent query answering not using table repairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' (b) polynomial algo-  rithms for computing the sets of true/false tuples and the sets of con-  sistent/inconsistent tuples of T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' (c) polynomial algorithms in the size of  T for computing different types of consistent answer for both conjunctive  and disjunctive queries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' and (d) a detailed discussion of the differences  between our approach and the approaches using table repairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Keywords: database semantics, inconsistent database, functional dependency,  null value, consistent query answering  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='03668v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='DB]  9 Jan 2023  1  Introduction  In a relational database, each table is seen as a set of tuples that must satisfy a  set of functional dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Moreover, each table is assumed to be consistent  with the dependencies and its tuples are assumed to be true when users query  the table (these are basic assumptions in relational databases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Consistency is  verified during updates in the following sense: if the update to be performed  results in an inconsistent table then the update is rejected, otherwise it is ac-  cepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  However, the consistency of a table in general is not always easy to verify, for  example if the table is the result of integrating data coming from independent  sources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=', web sources).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' In such cases the table may contain inconsistent  tuples, and the problem is: how to answer user queries so that the answers  contain only tuples that are true and consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' This kind of query answering  process is usually referred to as consistent query answering [1, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  T  Emp  Dept  Addr  1  e  d  a  2  e  d  a′  3  e′  d′  a  repairs  ====⇒  R1  Emp  Dept  Addr  1  e  d  a  3  e′  d′  a  R2  Emp  Dept  Addr  2  e  d  a′  3  e′  d′  a  Figure 1: An inconsistent table and its two repairs  To see an example, consider the table T of Figure 1, with dependencies  Emp → Dept and Dept → Addr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The table T is defined over attributes Emp,  Dept and Addr, standing for ‘employee identifier’, ‘department identifier’ and  ‘department address’, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  This table is inconsistent as the pair of tuples 1 and 2 violates the dependency  Dept → Addr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' However, if we ask the SQL query Q : select Emp, Dept from T,  it makes sense to return the set of tuples {ed, e′d′} as the answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Indeed, there  is no reason to reject this answer as it is a consistent answer, in the sense that  it satisfies the dependency Emp → Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' In other words, an inconsistent table  might contain some consistent parts (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=', some useful information) which can  be extracted through queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  The traditional approach to alleviate the impact of inconsistent data on  query answers is to introduce the notion of repair: a repair is a maximal consis-  tent sub-set of the table, and a tuple t is in the consistent answer if t is present  in the answer from every repair [1, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' To illustrate this approach, consider  again the table T in Figure 1 with its two repairs, namely R1 = {eda, e′d′a}  and R2 = {eda′, e′d′a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Both these repairs are consistent with the given de-  pendencies, and maximal with respect to set-theoretic inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The answer  to Q from R1 is {ed, e′d′} and so is the answer from R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Therefore the consis-  tent answer to Q is {ed, e′d′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We note here that the complexity of the query  2  evaluation algorithm in the repairs approach is APX-complete1 [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Now, in several applications today we have to deal with tables containing  nulls (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=', missing values) and having to satisfy a set of functional dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  The presence of nulls in a table is due to various reasons such as: ‘value does not  exist’ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=', the maiden name of a male employee);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' ‘value exists but is currently  unknown’ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=', the department of a newly hired employee currently following  a training course before being assigned to a specific department);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' A  survey of the different kinds of missing values considered in the literature can  be found in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' In our approach we assume that missing values are of the kind  ‘value exists but is currently unknown’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  In the context of the previous example, let T = {ed, da, ea′} in which  the Addr-value in ed, the Emp-value in da and the Dept-value in ea′ are  nulls that is they exist but are currently unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Considering the query  Q : select Emp, Addr from T, if T is seen as a regular table, the consis-  tent answer to Q is {ea′}, as the tuples in T ‘seem’ to satisfy the functional  dependencies Emp → Dept and Dept → Addr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  However, these functional dependencies allow to infer the missing values:  from tuples ed and da, it can be inferred that eda is true, and from tuples ed,  ea′ it can be inferred that eda′ is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' This shows that Dept → Addr is not  satisfied, and thus {ea′} should not be returned as a consistent answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The  process of inferring missing values in a table is known in the literature as the  chase procedure [8, 13, 14], and it is well-known that this procedure fails (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=',  stops and returns no table) when encountering an inconsistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Therefore, the  repair-based approaches do not work in the presence of nulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  In the light of the previous example, we propose a novel approach to consis-  tent query answering that does not rely on repairs, but that works in the case  where the given set of tuples contains missing values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  In our approach one starts again with a set of tuples T that are not required  to be all of them defined over the same set of attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Therefore T is rep-  resented as a table with nulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' However, nulls simply act as place holders that  may receive values implied by functional dependencies, as shown in the previous  example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' What is significant in our approach is the set T , called the universe of  discourse of T, containing all tuples that can be built up from values appearing  in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' In our previous example, T consists of the tuples eda and eda′ together  with all their sub-tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Queries are addressed to T and consistent answers are obtained from T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' To  achieve this we associate every tuple t in T with a set of identifiers, referred to as  the interpretation of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' In doing so, we define set-theoretic semantics for tuples  and for functional dependencies which allows us to characterize the tuples of T  along two orthogonal dimensions: a tuple of T can be true or false on one hand  and consistent or inconsistent on the other hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  In this setting, we consider a query Q : select X from T where Condition  over T and we define its consistent answer as the set of tuples x in T such that:  1We recall that APX is the set of NP optimization problems that allow polynomial-time  approximation algorithms (source Wikipedia).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  3  (a) x is a true and consistent tuple with schema X and (b) there exists a true  super-tuple t of x in T satisfying the condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We show that, depending on the  ‘status’ that the super-tuple t has in T , there are different types of consistent  answer to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  The important point to emphasize again is that, in our approach, queries are  addressed to T and evaluated using the universe of discourse T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As mentioned  earlier, T is the set of all tuples that one can define using values appearing in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Actually, the fundamental difference between our approach and all approaches  based on repairs is that their universe of discourse is the table T itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Therefore  our approach is simply not comparable to approaches based on repairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' To see  this, consider again our previous example but this time with Emp → Addr as  the only functional dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Now suppose that T = {eda, eda′, e′da′} and that we ask for the set of all  addresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Then the answer in the repairs approach is {a′} because there are  only two repairs, R1 = {eda, e′da′} and R2 = {eda′, e′da′}, and the answers  from each repair are respectively {a, a′} and {a′};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' whereas in our approach, the  answer is empty because a and a′ are both inconsistent tuples of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' On the  other hand, if T = {eda, ed′a′} and we ask for the set of all departments then  the answer in the repairs approach is empty because there are only two repairs,  R1 = {eda} and R2 = {ed′a′} and none of d, d′ is in the answer from both  repairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' whereas, in our approach, the answer is {d, d′} because both d and d′  are true and consistent tuples of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Our work builds upon earlier work on partition semantics [6, 13] and on  inconsistent tables with missing values [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The main contributions of this paper  are as follows:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We propose a novel approach to consistent query answering in tables with  nulls and functional dependencies, without using table repairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We provide polynomial algorithms for computing (a) the sets of true/false  tuples and the sets of consistent/inconsistent tuples of T and (b) for com-  puting the consistent answer to any conjunctive or disjunctive query over  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We offer a detailed discussion of the differences between our approach and  other approaches including the approaches by table repairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  The remaining of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' In Section 2 we first in-  troduce the basic notions and notation of our approach, and we present our  set-theoretic semantics for tuples and functional dependencies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' in Section 3 we  address the issue of consistent query answering, introducing the notion of con-  sistency with respect to a selection condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' in Section 4 we first define the  m-Chase algorithm, a chase-like algorithm for computing the sets of true/false  tuples and the sets of consistent/inconsistent tuples of T , and then we propose  a polynomial algorithm for computing consistent answers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' in Section 5 we com-  pare our approach to the repair approaches and discuss other related work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' and  finally, in Section 6 we offer concluding remarks and outline current work and  future perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  4  2  The Semantics of our Model  In this section we recall basic definitions from the relational model regarding  tuples and tables, and we present the set-theoretic semantics that we use for  tuples and functional dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Our approach builds upon the so-called  “partition model” introduced in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='1  Terminology and Notation  Following [13], we consider a universe U = {A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' , An} in which every attribute  Ai is associated with a set of atomic values called the domain of Ai and denoted  by dom(Ai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' An element of �  A∈U dom(A) is called a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  We call relation schema (or simply schema) any nonempty sub-set of U and  we denote it by the concatenation of its elements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' for example {A1, A2} is simply  denoted by A1A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Similarly, the union of schemes S1 and S2 is denoted as S1S2  instead of S1 ∪ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  We define a tuple t over U to be a partial function from U to �  A∈U dom(A)  such that, for every A in U, if t is defined over A then t(A) belongs to dom(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  The domain of definition of t is called the schema of t, denoted by sch(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We  note that tuples in our approach satisfy the First Normal Form [14] in the sense  that each tuple component is an atomic value from the corresponding attribute  domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Regarding notation, we follow the usual convention that, whenever possible,  lower-case characters denote domain constants and upper-case characters denote  the corresponding attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Following this convention, the schema of a tuple  t = ab is AB and more generally, we denote the schema of a tuple s as S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Assuming that the schema of a tuple t is understood, t is denoted by the  concatenation of its values, that is: t = ai1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' aik means that for every j =  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' , k, t(Aij) = aij, where aij is in dom(Aij), and sch(t) = Ai1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Aik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  As in [13], we assume that for any two distinct attributes A and B, we have  dom(A) ∩ dom(B) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' However, it might be relevant for attribute domains  to share values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  For instance, in our introductory example, in the presence  of attribute Mgr standing for ‘manager’, the domains of Emp and Mgr are  employee identifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  In such cases, in order to avoid confusion, we denote  attribute values as pairs of the form ⟨attribute name, value⟩ and comparisons  are assessed with respect to their value component only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' In order to keep the  notation simple we shall omit attribute names as prefixes whenever no ambiguity  is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  We define a table T over U to be any finite set of tuples over U (therefore  duplicates are not allowed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As tuples over U are partial functions over U, it  follows that T may contain nulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Given a table T over U, we denote by T the set of all tuples that can be built  up from constants appearing in T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' and we call T the universe of discourse of T  as it contains all tuples of interest in query processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Actually, as we shall see  shortly, queries are issued against T and consistent answers are obtained from  5  T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' For every relation schema X, we denote by T (X) the set of all tuples in T  whose schema is X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Formally, T (X) = {t ∈ T | sch(t) = X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  For every A in U, the set of all values from dom(A) occurring in T is referred  to as the active domain of A, denoted by adom(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We denote by AD the set  of all constants appearing in T that is: AD = �  A∈U adom(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We emphasize  that even when attribute domains are infinite, active domains are always finite  and therefore the sets AD and T are finite as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Given a tuple t, for every A in sch(t), t(A) is also denoted by t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A and more  generally, for every nonempty sub-set S of sch(t) the restriction of t to S, also  called sub-tuple of t, is denoted by t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' In other words, if S ⊆ sch(t), t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='S is the  tuple such that sch(t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='S) = S, and for every A in S we have (t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Moreover, ⊑ denotes the ‘sub-tuple’ relation, defined over T as follows: for  any tuples t1 and t2, t1 ⊑ t2 holds if t1 is a sub-tuple of t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' It is thus important to  keep in mind that whenever t1 ⊑ t2 holds, it is understood that sch(t1) ⊆ sch(t2)  also holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The relation ⊑ is clearly a partial order over T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Given a table T,  the set of all sub-tuples of the tuples in T is called the lower closure of T and  it is defined by: LoCl(T) = {q ∈ T | (∃t ∈ T)(q ⊑ t)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='2  Partition Semantics for Tuples  In this work, we consider tables T over a set U of attributes, possibly with nulls,  and we assume that every tuple t in T is associated with a unique identifier,  id(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  We denote by TID the set of all identifiers of tuples in T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' that is:  TID = {id(t) | t ∈ T }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' In our examples, for simplicity, we assume that TID  is a set of positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Denoting by P(TID) the power-set of TID, the  following definition of interpretation is in the spirit of [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Definition 1 Let T be a table over U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We call interpretation of T any function  I from T to P(TID) such that:  For every tuple t of T, I(t) ̸= ∅  For every tuple t = a1a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' an of T , I(t) = I(a1) ∩ I(a2) ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' ∩ I(an)  We say that a tuple t in T is true in I if I(t) ̸= ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' otherwise we say that t is  false in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We denote by True(I) the set of tuples t that are true in I and by  False(I) the set of tuples t that are false in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  �  Example 1 Let T = {ab, a′c} and let I be a function from T to P(TID) defined  as follows: I(a) = {1, 2}, I(a′) = {2}, I(b) = {1, 2}, I(c) = {2}, I(ab) = {1, 2},  I(ac) = {2}, I(a′b) = {2}, I(a′c) = {2}, I(abc) = {2}, I(a′bc) = {2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Then I is  an interpretation of T as we have:  I(ab) ̸= ∅ and I(a′c) ̸= ∅ that is the two tuples of T are true in I  For every tuple t in T , I(t) is indeed the intersection of the interpretations  of its components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' For example, I(ac) = I(a) ∩ I(c) = {1, 2} ∩ {2} = {2}  and one can verify easily for the remaining tuples of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  �  6  It is important to note that the first item in Definition 1 expresses a fundamental  assumption regarding a relational table T, namely that every tuple t of T is  assumed to be true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Two immediate consequences are that (a) I(a) ̸= ∅, for all  a in AD (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=', every a in AD is true in I) and (b) if a tuple t is true in I then  so is every sub-tuple of t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' and if t is false in I then so is every super-tuple of  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As a consequence the set T is partitioned into true and false tuples that is  True(I) ∪ False(I) = T and True(I) ∩ False(I) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  The interpretations of T can be compared according to the following defini-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Definition 2 Let T be a table and I, I′ two interpretations of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Then we say  that I is less than or equal to I′, denoted I ⪯ I′ if for every t in T , I(t) ⊆ I′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  �  It is easy to see that the relation ⪯ is a partial ordering over the set of all  interpretations of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Note that if we view the constants of AD as unary tuples  then we have that: if I ⪯ I′ then for every a in AD, I(a) ⊆ I′(a) holds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' and in  the opposite direction, if I(a) ⊆ I′(a) holds for every a in AD then for every t  in T , I(t) ⊆ I′(t) also holds that is I ⪯ I′ holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Therefore we have that:  I ⪯ I′ holds if and only if for every a in AD, I(a) ⊆ I′(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Based on this result, in order to verify whether I ⪯ I′, it is sufficient to do so  for every constant in AD rather than for every tuple in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Now, to see the intuition behind the above definitions, think of the identifiers  of TID as being objects and of every a in AD as being an atomic property that  each object may have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Then I(a) is the set of objects having property a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' and  similarly, the intuitive idea behind the interpretation of a tuple is that a tuple,  say ab, is the ‘conjunction’ of the atomic properties a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Therefore I(ab) is  the set of objects each having both properties a and b, hence I(ab) = I(a)∩I(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Clearly, if I(ab) = ∅ then no object has both properties a and b so the tuple ab  is false in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  As for the ordering of interpretations what I ⪯ I′ means is that, for every  property t of T , the set of objects having property t in I is included in the set  of objects having property t in I′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Another fundamental assumption made in the relational model is that the  components of every tuple be atomic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Actually such a tuple is said to satisfy  the First Normal Form [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' To express this assumption using our definition of  interpretation, suppose that there are a, a′ in adom(A) such that a ̸= a′ and  I(a) ∩ I(a′) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Then every tuple whose identifier is in I(a) ∩ I(a′) violates  the First Normal Form as its A-component is {a, a′}, therefore not atomic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' In  the light of this observation we introduce the following definition of inconsistent  tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Definition 3 Let T be a table over U and I an interpretation of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' A tuple t  in T is said to be inconsistent in I if there exist A in sch(t) and a′ in adom(A)  such that t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A ̸= a′ and I(t) ∩ I(a′) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  7  The set of all inconsistent tuples in I is denoted by Inc(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' A tuple that is  not inconsistent in I is said to be consistent in I, and the set of all consistent  tuples in I is denoted by Cons(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' An interpretation I such that Inc(I) = ∅ is  said to be in First Normal Form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  �  For instance, in Example 1, the tuple t = abc is inconsistent in I as A is in  sch(t), and a′ is in adom(A) such that t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A ̸= a′ and I(t) ∩ I(a′) = {2} ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Therefore I is not in First Normal Form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  In view of the above definition, the set T is partitioned into inconsistent  and consistent tuples that is Inc(I) ∪ Cons(I) = T and Inc(I) ∩ Cons(I) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Roughly speaking, inconsistent tuples are those that violate the First Normal  Form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' This relationship between inconsistency and First Normal Form is stated  formally in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Lemma 1 Let I be an interpretation of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Then I is in First Normal Form if  and only if I satisfies the following constraint:  Partition constraint (PC): For all A in U and for all a, a′ in adom(A),  if a ̸= a′ then I(a) ∩ I(a′) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Assume that I satisfies PC and that Inc(I) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Given t in Inc(I),  using the notation of Definition 3 and denoting by a the A-value occurring in t,  we have I(a) ∩ I(a′) ̸= ∅, because I(t) ⊆ I(a) ∩ I(a′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Hence, I does not satisfy  PC, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Conversely, if I does not satisfy PC, then there exist A in U and a and a′  such that I(a)∩I(a′) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Therefore, by Definition 3, a and a′ are in Inc(I), thus  implying that I is not in First Normal Form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The proof is therefore complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  �  Note that, as mentioned in [9], satisfaction of the partition constraint implies  that, for all A in U, the set {I(a) | a ∈ adom(A)} is a partition of the set  adom(A) (whence the term “partition constraint”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  An important question remains regarding the definition of an interpretation:  is there a systematic way for defining an interpretation I of a given table T  which is in First Normal Form?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The answer is yes, there is such a “canonical”  interpretation for every table T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' it is called the basic interpretation of T, denoted  by Ib and defined as follows:  Basic Interpretation Ib: For every A in U and every a in adom(A),  Ib(a) = {id(t) | t ∈ T and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A = a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  In the context of Example 1 where T = {ab, a′c}, if ab and a′c are respectively  associated with identifiers 1 and 2, the associated basic interpretation Ib is  defined by Ib(a) = Ib(b) = {1} and Ib(a′) = Ib(c) = {2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Considering T1 =  {ab, bc, ac} with respective identifiers 1, 2 and 3, the basic interpretation Ib  1 is  defined by Ib  1(a) = {1, 3}, Ib  1(b) = {1, 2} and Ib  1(c) = {2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  It is interesting to note that the definition of basic interpretation parallels  that of inverted file [17, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Indeed, we first recall that an inverted file is an  8  index data structure that maps content to its location within a database file, in  a document or in a set of documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Hence, if we assume that each tuple of T  is implemented as a record (possibly with missing values) then we can view T  as a file that is as a function I assigning to each value x appearing in the file  record the set I(x) of all identifiers of records in which x appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  The following lemma summarizes important properties of the basic interpre-  tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Lemma 2 Let T be a table and Ib its basic interpretation as defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Then:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Ib is an interpretation satisfying the partition constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' True(Ib) = LoCl(T) and True(Ib) is the set of all tuples t that belong to  True(I) for every interpretation I of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Ib is minimal with respect to ⪯ among all interpretations I of T such that  for every t in T, id(t) belongs to I(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Assume that there exist A in U and a and a′ in adom(A) such that  Ib(a) ∩ Ib(a′) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' If i is in Ib(a) ∩ Ib(a′) ̸= ∅, let ti be the tuple in T such that  id(ti) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' This implies that the value of ti(A) is a and a′, which is impossible  because, as ti is a function, ti(A) consists of at most one value in adom(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  This part of the proof is thus complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Since for every t in T, Ib(t) contains id(t) and as for every t′ such that t′ ⊑ t,  we have Ib(t) ⊆ Ib(t′), it follows that LoCl(T) ⊆ True(Ib).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Conversely, let t not  in LoCl(T) and assume that Ib(t) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Since t is not in T, it is not associated  with an identifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Hence, Ib(t) contains identifiers of other tuples, and if q is  such a tuple, then by definition of Ib, this implies that id(q) belongs to every  component of t, that is that t is a sub-tuple of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  A contradiction showing  that True(Ib) ⊆ LoCl(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Hence we have True(Ib) = LoCl(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Moreover, since  for every interpretation I of T, True(I) must at least contain all tuples in T  along with their sub-tuples (as argued just above), we have LoCl(T) ⊆ True(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Hence, True(Ib) ⊆ True(I), which completes this part of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Let I be an interpretation such that I ⪯ Ib and I ̸= Ib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Thus, there exists a  in AD such that I(a) ⊂ Ib(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' However, since I is an interpretation such that  for every t in T, id(t) belongs to I(t), if a occurs in t then id(t) belongs to I(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  In other words, Ib(a) ⊆ I(a) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' A contradiction showing that Ib is minimal  with respect to ⪯ among all interpretations I of T such that for every t in T,  id(t) belongs to I(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The proof is therefore complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  �  By Definition 3 and Lemma 1, it turns out that, since Ib satisfies the partition  constraint PC, Ib is in First Normal Form, therefore Inc(Ib) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  From now on, in all our examples we shall use the following convention: the  tuples of T will receive successive integer identifiers in the order of their appear-  ance in T, starting with number 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' For example, if T = {ab, a′c} as in Example 1,  then it is understood that id(ab) = 1 and id(a′c) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Then, the basic inter-  pretation of T is defined as earlier stated, that is: Ib(ab) = {1}, Ib(a′c) = {2},  9  Ib(a) = {1}, Ib(a′) = {2}, Ib(b) = {1}, Ib(c) = {2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Thus, Ib(ac) = ∅,  Ib(abc) = ∅, Ib(a′bc) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Therefore we have: True(Ib) = {a, a′, b, c, ab, a′c}  and False(Ib) = {ac, abc, a′bc}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Note that Ib is an interpretation that satisfies  the partition constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='3  Partition Semantics for Functional Dependencies  Let T be a table over a set U of attributes together with a set FD of functional  dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As usual in relational databases [14], a functional dependency is  an expression of the form X → Y where X and Y are relation schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The  notion of functional dependency satisfaction in the context of tables with nulls is  stated in the literature [7, 8] in the following terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' A table T satisfies X → Y  if for all tuples t and t′ in T the following holds: if XY is a sub-set of sch(t)  and of sch(t′) then t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X = t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X implies t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='Y = t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Moreover, it is well-known that a relation satisfies X → Y if and only if it  satisfies the functional dependencies X → A, for every A in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' This justifies  that functional dependencies are usually assumed to be of the form X → A,  where X is a relation schema and A is an attribute not in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' In what follows,  we make this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Now, the question that we answer in this section is the following: how can  an interpretation I of T express the satisfaction of a functional dependency by  T?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The answer is provided by the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Lemma 3 Let I be an interpretation of table T in First Normal Form, and  let X → A be a functional dependency of FD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Then T satisfies X → A if I  satisfies the following constraint:  Inclusion constraint (IC): For all t in T such that XA ⊆ sch(t),  I(t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X) ∩ I(t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A) ̸= ∅ implies I(t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X) ⊆ I(t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Suppose that I satisfies the inclusion constraint and consider the  relation f : adom(X) → adom(A) defined by: for every true tuple t in T,  f(t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X) = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We show that f is actually a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Indeed, suppose there  is a true tuple t′ in T such that t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X and t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A ̸= t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  As I is in  First Normal Form, by Lemma 1, I satisfies the partition constraint, and thus,  I(t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A)∩I(t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' On the other hand, as I satisfies the inclusion constraint IC  for X → A, we have I(t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X) ⊆ I(t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A) and I(t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X) ⊆ I(t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X = t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X, we  have I(t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X) ⊆ I(t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A)∩I(t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A), thus that I(t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X) ⊆ ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' It follows that I(t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X) = ∅,  which is a contradiction to the fact that t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X is true, being a sub-tuple of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The  proof is therefore complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  �  Example 2 Let T = {ab, bc} and FD = {A → B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Then the basic interpreta-  tion Ib of T is as follows: Ib(a) = {1}, Ib(b) = {1, 2}, Ib(c) = {2} and we have  that Ib(a) ∩ Ib(b) = {1} ̸= ∅ and Ib(a) ⊆ Ib(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As the only tuple of T to which  the lemma applies is ab we conclude that T satisfies A → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' In this case Ib  satisfies the partition constraint PC and the inclusion constraint IC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  As another example, let T = {ab, ac} and FD = {A → B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Then the  interpretation Ib of T is as follows: Ib(a) = {1, 2}, Ib(b) = {1}, Ib(c) = {2} and  10  we have Ib(a) ∩ Ib(b) = {1} ̸= ∅ but Ib(a) ̸⊆ Ib(b), showing that Ib does not  satisfy the inclusion constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Therefore, Ib satisfies the partition constraint  PC but not the inclusion constraint IC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  �  In view of Lemma 3, Ib satisfies the partition constraint, the set of true tuples  of Ib is equal to the set of true tuples of every interpretation I, and Ib satisfies  a minimality property with respect to ⪯ among interpretations of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Therefore  the question that arises here is whether we can “expand” Ib to an interpretation  I′ that satisfies the inclusion constraint as well, so that I′ satisfies the same  properties as Ib, along with the inclusion constraint IC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  In Example 2, the answer is yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Indeed, if we add Ib(a) to Ib(b) then Ib(b)  becomes Ib(a) ∪ Ib(b) and the resulting interpretation is: I′(a) = {1, 2}, I′(b) =  {1, 2}, I′(c) = {2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' and I′ satisfies the inclusion constraint for A → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' However,  the following example shows that expanding Ib may lead to non satisfaction of  the partition constraint PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Example 3 Let T = {ab, bc, ac′} with FD = {A → C, B → C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Here, Ib  is defined by Ib(a) = {1, 3}, Ib(b) = {1, 2}, Ib(c) = {2} and Ib(c′) = {3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Thus, Ib does not satisfy the constraint IC because Ib(a) ∩ Ib(c) = {2} ̸= ∅ but  Ib(a) ̸⊆ Ib(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Notice that this example shows that the converse of Lemma 3 does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  It is so because T satisfies A → C and B → C, although Ib is an interpretation  of T in First Normal Form that does not satisfy the inclusion constraint IC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Expanding Ib as in Example 2 yields the interpretation I′ such that I′(c) =  I′(c′) = {1, 2, 3}, I′(a) = Ib(a) and I′(b) = Ib(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' It should be clear that I′  satisfies the constraint IC, whereas I′ does not satisfy the constraint PC since  I′(c) ∩ I′(c′) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As a consequence, by Lemma 1, Inc(I′) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' More precisely,  as True(I′) is the set containing abc and abc′ along with their sub-tuples, we  have Inc(I′) = {abc, abc′, bc, bc′, ac, ac′, c, c′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  �  In view of our discussion above, the following definition states how we can  expand an interpretation I so that it satisfies a given functional dependency  X → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Definition 4 The expansion of I with respect to a functional dependency X →  A and a tuple xa in T (XA) is the interpretation Exp(I, xa) defined as follows:  If I(x) ∩ I(a) ̸= ∅ and I(x) ̸⊆ I(a) then: Exp(I, xa)(a) = I(a) ∪ I(x), and  Exp(I, xa)(α) = I(α) for α in AD different than a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Otherwise, Exp(I, xa) = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  �  Of course, the expansion processing should be iterated when several tuples sat-  isfy the above condition and when several functional dependencies are consid-  ered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As will be seen next, starting with the basic interpretation Ib and applying  expansion repeatedly until a fixed point is reached, we obtain an interpretation  I∗ such that: (a) I∗ satisfies the inclusion constraint IC (but not necessarily the  partition constraint PC) (b) a tuple t of T is true in I∗ if and only if t is true in  every interpretation I of T satisfying IC and (c) a tuple t of T is inconsistent  in I∗ if and only if t is inconsistent in every interpretation I of T satisfying IC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  11  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='4  The True and the Inconsistent Tuples  To define the limit interpretation I∗ of T mentioned above, for a given a table  T over universe U with a set FD of functional dependencies, we consider the  sequence of interpretations (Ij)j≥0 of T defined by:  For j = 0, we set I0 to be the basic interpretation Ib as defined earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  For j ≥ 0, let Ij+1 be defined as follows:  – If there exist X → A, x and a such that Exp(Ij, xa) ̸= Ij, then  Ij+1 = Exp(Ij, xa)  – Else Ij+1 = Ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  The theorem below states that this sequence has a limit with important proper-  ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' In this theorem, given a nonempty sub-set S of AD and an interpretation  I, we denote by I(S) the set �  a∈S I(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Theorem 1 The sequence (Ij)j≥0 is increasing with respect to ⪯ and bounded  above, therefore it has a limit that we denote by I∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' This limit has the following  properties:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' I∗ is unique in the sense that it is independent of the order in which  expansions are applied (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=', the construction of I∗ has the Church-Rosser  property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' I∗ satisfies the inclusion constraint IC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' For every nonempty sub-set S of AD, I∗(S) ̸= ∅ holds if and only if  I(S) ̸= ∅ holds for every interpretation I of T satisfying IC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' First, it is clear that for j ≥ 0, Ij ⪯ Ij+1 holds by definition of the  sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Now, as for every a in AD we have that: Ij(a) ⊆ TID for every j ≥ 0,  the sequence is bounded by the interpretation IT ID defined by: IT ID(a) = TID  for every a in AD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Hence, the sequence has a limit, denoted by I∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The proof  of the items in the theorem are as follows:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Uniqueness of the limit I∗ is shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Suppose that I∗ does not satisfy the inclusion constraint IC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  It follows  that there exist X → A in FD and t in T such that XA ⊆ sch(t), I∗(t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X) ∩  I∗(t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A) ̸= ∅ and I∗(t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X) ̸⊆ I∗(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Thus, assuming j is such that I∗ = Ij,  we have Ij+1 ̸= Ij, which implies that I∗ is not the limit of the sequence, a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Let S be a nonempty sub-set of AD such that I(S) ̸= ∅ for every I satisfying  IC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Since I∗ is an interpretation that satisfies IC, we also have I∗(S) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Conversely, if I∗(S) ̸= ∅, we prove by induction that I(S) ̸= ∅ holds for  every interpretation I satisfying IC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  If I0(S) ̸= ∅, then it follows from Lemma 2 that S does not contain two  elements from the same active domain (because Ib satisfies the partition con-  straint PC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Thus, the elements of S form a tuple t that belongs to True(I0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  12  It follows that t is a sub-tuple of a tuple q in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Hence, t belongs to True(I)  for every interpretation I of T, thus for every interpretation of T satisfying the  inclusion constraint IC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Assume that for every nonempty sub-set Σ of AD, if Ij(Σ) ̸= ∅ then I(Σ) ̸= ∅  for every I satisfying IC, and let S be such that Ij+1(S) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' If Ij(S) ̸= ∅, then  Ij+1(S) ̸= ∅ holds because Ij ⪯ IJ+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Considering the case where Ij(S) = ∅,  as Ij+1(S) ̸= Ij(S), S contains a constant a such that Ij(a) ̸= Ij+1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' De-  noting S \\ {a} by Sa, we have Ij(Sa) = Ij+1(Sa) (because expansion changes  the interpretation of only one constant, namely a in this case), and there exist  X → A in FD and x in T (X) such that Ij(x) ∩ Ij(a) ̸= ∅ and Ij(x) ̸⊆ Ij(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  In this case, we have Ij+1(a) = Ij(a) ∪ Ij(x), and thus  Ij+1(S)  =  Ij+1(a) ∩ Ij+1(Sa)  =  (Ij(a) ∪ Ij(x)) ∩ Ij(Sa)  =  (Ij(a) ∩ Ij(Sa)) ∪ (Ij(x) ∩ Ij(Sa))  Since Ij(S) = Ij(a) ∩ Ij(Sa) is empty, it follows that Ij(x) ∩ Ij(Sa) ̸= ∅ and so,  for every I satisfying IC, I(x) ∩ I(a) ̸= ∅ and I(x) ∩ I(Sa) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Hence, I(x) ⊆  I(a), and so I(x) ∩ I(Sa) ⊆ I(a) ∩ I(Sa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As this implies that I(a) ∩ I(Sa) ̸= ∅,  we obtain that I(S) ̸= ∅, and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  �  We note that similar results have been obtained with a slightly different formal-  ism in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Moreover, as a consequence of Theorem 1, for every tuple t in T , we  have:  t belongs to True(I∗) if and only if t belong to True(I) for every inter-  pretation I of T satisfying IC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We denote the set True(I∗) by True(T )  and its tuples are said to be true in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The set False(I∗) is denoted by  False(T ) and its tuples are said to be false in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  t belongs to Inc(I∗) if and only if t belongs to Inc(I) for every interpre-  tation I of T satisfying IC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We denote the set Inc(I∗) by Inc(T ) and its  tuples are said to be inconsistent in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The set Cons(I∗) is denoted by  Cons(T ) and its tuples are said to be consistent in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  The following proposition lists basic properties of true, false, consistent and  inconsistent tuples in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Proposition 1 For every table T over universe U, the following holds:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' If t is in True(T ) then every t′ such that t′ ⊑ t belongs to True(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Inc(T ) ⊆ True(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  However, the inclusions Cons(T ) ⊆ True(T ) and  Cons(T ) ⊆ False(T ) do not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Let t be in Inc(T ) and A in sch(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Let a′ be in adom(A) such that a′ ̸= t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A  and I∗(t) ∩ I∗(a′) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Then every tuple t′ such that t′ ⊑ t and A belongs  to sch(t′) is in Inc(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' It does not hold that for t in Inc(T ), every true super-tuple of t is in  Inc(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  13  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Since t is in True(I∗), I∗(t) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Thus, as I∗(t) ⊆ I∗(t′) (because  t′ ⊑ t), we have I∗(t) ̸= ∅, showing that t′ is in True(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' By Definition 3, if t is in Inc(T ) then there is A in sch(t) and a′ ̸= t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A in  adom(A) such that I∗(t) ∩ I∗(a′) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Thus I∗(t) ̸= ∅, which implies that t  belong to True(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Regarding the two other inclusions in the item, for T = {ab, a′b′} and FD =  ∅, we have True(T ) = LoCl(T) and Inc(T ) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Thus, False(T ) = {ab′, a′b} and  Cons(T ) = T , showing that Cons(T ) ̸⊆ True(T ) and Cons(T ) ̸⊆ False(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' In this case, we have I∗(t) ⊆ I∗(t′) (because t ⊑ t′) and as I∗(t)∩I∗(a′) ̸= ∅,  we obtain that I∗(t′) ∩ I∗(a′) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Since t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A (because A ∈ sch(t′) and  t ⊑ t′), we have a′ ̸= t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A, which implies that t′ is in Inc(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Let T = {ab, ab′, bc} with A → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' In this case I∗ is defined by I∗(a) =  I∗(b′) = {1, 2}, I∗(b) = {1, 2, 3} and I∗(c) = {3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Thus b is inconsistent,  because I∗(b) ∩ I∗(b′) ̸= ∅, and we argue that bc is not inconsistent although bc  is a true super-tuple of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' This is so because I∗(bc) = {3} but I∗(b′)∩I∗(bc) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  �  We illustrate now the construction of I∗ through the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Example 4 Referring to Example 3, we recall that T = {ab, bc, ac′} with FD =  {A → C, B → C}, and that Ib was found to be defined by Ib(a) = {1, 3},  Ib(b) = {1, 2}, Ib(c) = {2} and Ib(c′) = {3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The construction of I∗ starts with  I0 = Ib, and the following steps are performed:  (1) Considering A → C, we have I0(a) ∩ I0(c) ̸= ∅ but I0(a) ̸⊆ I0(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Hence  I1(c) = I0(c)∪I0(a) = {1, 2, 3}, I1(a) = {1, 3}, I1(b) = {1, 2} and I1(c′) = {3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  (2) Considering again A → C, we have I1(a) ∩ I1(c′) ̸= ∅ but I1(a) ̸⊆ I1(c′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Hence I2(c′) is set to I1(c′) ∪ I1(a) = {1, 3}, and I2(a) = {1, 3}, I2(b) = {1, 2}  and I2(c) = {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  (3) For B → C, we have I2(b) ∩ I2(c′) ̸= ∅ but I2(b) ̸⊆ I2(c′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Hence I3(c′) is  set to I2(c′) ∪ I2(b) = {1, 2, 3}, and I3(a) = {1, 3}, I3(b) = {1, 2} and I3(c) =  {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Since I3 satisfies the inclusion constraint, we obtain I∗ = I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We notice that  I∗ is equal to the expected interpretation I′ defined in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Moreover, it  can be seen that in this example, True(T ) contains the tuples abc and abc′ along  with all their sub-tuples, meaning that True(T ) = T , and thus that False(T ) =  ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Regarding inconsistent tuples, as we have I∗(c)∩I∗(c′) ̸= ∅, it turns out that  Inc(T ) = {abc, abc′, ac, ac′, bc, bc′, c, c′}, and thus that Cons(T ) = {ab, a, b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  �  To further illustrate the impact of functional dependencies over true tuples and  over inconsistent tuples, we mention the following two properties: given a table  T over U, X → A in FD and a true tuple xa in T (XA), then:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' For every tuple t in True(T ) such that X ⊆ sch(t) and A ̸∈ sch(t), if  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X = x then the tuple ta is in True(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' In other words, writing t as qx  this result expresses the well-known property of relational lossless join: if  14  qx and xa are true, the functional dependency X → A implies that qxa is  true as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  This is so because I∗(x) ⊆ I∗(a), I∗(t) ⊆ I∗(x) (because x ⊑ t) and  I∗(ta) = I∗(t)∩I∗(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Hence, I∗(t) ⊆ I∗(a), implying that I∗(ta) = I∗(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Therefore, I∗(ta) ̸= ∅, because I∗(t) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' For every true super-tuple t of xa (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=', t ∈ True(T ), XA ⊆ sch(t) and  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='XA = xa) and for every a′ in adom(A) different than a such that xa′ is  true (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=', xa′ ∈ True(T )), if we write t as qxa then t′ = qxa′ also belongs  to True(T ), and t and t′ both belong to Inc(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Indeed, since I∗(xa) and I∗(xa′) are nonempty, and I∗ satisfies the inclu-  sion constraint, we have I∗(x) ⊆ I∗(a) ∩ I∗(a′) and I∗(x) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' It follows  that I∗(a) ∩ I∗(a′) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Moreover, since t = qxa and I∗(t) ̸= ∅, we have  I∗(qx) ̸= ∅ and as I∗(qx) ⊆ I∗(x), it follows that I∗(qx) ⊆ I∗(a) ∩ I∗(a′),  that is, I∗(qxa′) = I∗(qx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Consequently, I∗(qxa′) ̸= ∅, entailing that  t′ = qxa′ is in True(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Since I∗(t) ̸= ∅, I∗(a′) ∩ I∗(xa) ̸= ∅ and  I∗(t) ⊆ I∗(xa), t is in Inc(T ) and a similar reasoning shows that t′ is  also in Inc(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  3  Consistent Query Answering  In this section, we first define the syntax of queries that we consider and then  we define and discuss four kinds of consistent answer to a query, based on the  semantics presented in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='1  Syntax of Queries  We use SQL as the query language and, as we query a single table T, the queries  Q that we consider have one of the following two forms:  Q : select X from T  or  Q : select X from T where Γ  In either of these forms, X is an attribute list seen as a relation schema, and in  the second form the where clause specifies a selection condition Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As in SQL  the where clause in a query is optional, the generic form of a query Q is denoted  by  Q : select X from T [where Γ]  The set of all attributes occurring in Γ is called the schema of Γ, denoted by  sch(Γ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' and the attribute set X ∪ sch(Γ) is called the schema of Q, denoted by  sch(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  A selection condition Γ is a well-formed formula involving the usual connec-  tors ¬, ∨ and ∧ and built up from atomic Boolean comparisons of one of the  following forms: A θ a or A θ A′, where θ is a comparison predicate, A and A′  are attributes in U whose domain elements are comparable through θ, and a is  in dom(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Given a selection condition Γ, we denote by Sat(Γ) the set of all tuples in  T (sch(Γ)) satisfying Γ, as defined below:  15  if Γ is of the the form A θ a, Sat(Γ) = {t ∈ T (sch(Γ)) | t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A θ a},  if Γ is of the the form A θ B, Sat(Γ) = {t ∈ T (sch(Γ)) | t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A θ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='B},  if Γ is of the form Γ1 ∨ Γ2, Sat(Γ) = Sat(Γ1) ∪ Sat(Γ2),  if Γ is of the form Γ1 ∧ Γ2, Sat(Γ) = Sat(Γ1) ∩ Sat(Γ2),  if Γ is of the form ¬Γ1, Sat(Γ) = T (sch(Γ)) \\ Sat(Γ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Moreover, the set of tuples that do not satisfy Γ is defined by:  Sat−(Γ) = T (sch(Γ)) \\ Sat(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  For example, if Sal is an attribute whose active domain is {s, s′}, we have:  for Γ4 = (Sal = s′), Sat(Γ4) = {s′} and Sat−(Γ4) = {s},  for Γ5 = (Sal = s ∨ Sal = s′), Sat(Γ5) = {s, s′} and Sat−(Γ5) = ∅,  for Γ6 = (Sal > 10K), if s = 5K and s′ = 20K, Sat(Γ6) = {s′} and  Sat−(Γ6) = {s}, and if s = 15K and s′ = 20K, Sat(Γ6) = {s, s′} and  Sat−(Γ6) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Next, we explain how, in our context, the above syntactic definition of satis-  faction of a selection condition can be ‘coupled’ with semantic considerations,  when it comes to defining the notion of consistent answer to a query Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Intuitively, given Q : select X from T [where Γ], the least requirements  for a tuple x to belong to a consistent answer to Q are the following:  R1 x is in T (X), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=', the schema of x is X,  R2 x is in True(T ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=', x is a true tuple,  R3 x is in Cons(T ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=', x is consistent, and  R4 if Q involves a selection condition Γ then there exists t in True(T ) such  that sch(Q) ⊆ sch(t), x ⊑ t, and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='sch(Γ) is in Sat(Γ) (that is, t is a true  super-tuple of x whose restriction to attributes in sch(Γ) satisfies Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  It is important to note that when all the above requirements are satisfied, then  the consistent answer coincides with the standard notion of answer to projection-  selection queries against a consistent table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Indeed, if T is a relational consistent  table whose schema contains all attributes in sch(Q), then the answer to a query  Q : select X from T [where Γ] is the set of all tuples x such that:  the schema of x is X (see requirement R1), and  T contains a tuple t such that t is a super-tuple of x and the restriction of  t to all attributes occurring in Γ satisfies Γ (see requirement R4, knowing  that t is true since all tuples in T are implicitly assumed to be true).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  16  Moreover, in this case, requirement R2 is also satisfied because all tuples in a  consistent relational table are implicitly assumed to be true, and requirement  R3 is trivial because in a consistent table, every tuple is consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  On the other hand, in the presence of inconsistencies, it should be noticed  that, based on our semantics as earlier defined, the status of any tuple x in the  consistent answer to Q is clearly defined because x is then a true and consistent  tuple of T whose schema is X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' However, the situation is less clear regarding  the status of the super-tuple t in requirement R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' More precisely, the question  here is whether t should be consistent or not and whether t should satisfy other  criteria as well (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=', is t required to be consistent or is t required to be maximal  with respect to ⊑?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The following example illustrates this discussion, showing  in particular that the status of t has a significant impact on the consistent  answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Example 5 Consider the universe U = {Emp, Sal, Dept} with the functional  dependency Emp → Sal and the table T = {es, es′d, e′s′} over U whose tu-  ples state that employee e has s and s′ as salaries, employee e works in de-  partment d, and that employee e′ has salary s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  The content of T clearly  does not satisfy the functional dependency Emp → Sal as employee e is as-  signed two distinct salaries s and s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The limit interpretation I∗ of T is de-  fined by I∗(e) = {1, 2}, I∗(e′) = {3}, I∗(s) = {1, 2}, I∗(s′) = {1, 2, 3} and  I∗(d) = {2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Thus True(T ) consists of the sub-tuples of esd, es′d and of e′s′,  and Inc(T ) = {esd, es′d, es, es′, sd, s′d, s, s′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Let us now consider the following  queries Q1, Q2, Q3, Q4, Q5 and Q6:  Q1 : select Emp, Dept from T  Q2 : select Emp, Sal from T  Q3 : select Sal from T  Q4 : select Emp from T where Sal = s′  Q5 : select Emp from T where Sal = s ∨ Sal = s′  Q6 : select Emp from T where Sal > 10K  Regarding Q1, since ed is the only true and consistent tuple in T (Emp, Dept),  ed is the only tuple satisfying the requirements R1, R2 and R3 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Require-  ment R4 is irrelevant because Q1 involves no selection condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Hence ed is a  candidate tuple to belong to the consistent answer to Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Notice however that  one could object that ed is not a good candidate since all its maximal (with  respect to ⊑) true super-tuples in T (namely esd and es′d) are inconsistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Regarding Q2, es, es′ and e′s′ are the only tuples satisfying requirements R1  (they all belong to T (Emp, Sal)) and R2 (they all belong to True(T )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' However,  since es and es′ are in Inc(T ), they do not satisfy requirement R3, whereas e′s′  does since this tuple is in Cons(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We moreover notice that, contrary to Q1  above, e′s′ has no true strict super-tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Hence the consistent answer to Q2  should be {e′s′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  The case of Q3 is clearer because s and s′ are the only tuples satisfying  requirements R1 and R2, and since these two tuples are in Inc(T ), none of them  can satisfy requirement R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Therefore, the consistent answer to Q3 should be  ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  17  Let us now turn to queries involving a selection condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Regarding Q4,  the two tuples satisfying requirements R1 and R2 are e and e′, and since e and  e′ are in Cons(T ), they also satisfy requirement R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Moreover, we notice that  es′ is a true super-tuple of e satisfying Γ4 = (Sal = s′), showing that e satisfies  requirement R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' However, it should be noticed that es′ is inconsistent and that  es is another true super-tuple of e not satisfying Γ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Hence, it can thought that  e should not belong to the consistent answer to Q4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' On the other hand, e′ has  only one true super-tuple, namely e′s′, and since this tuple is in Sat(Γ4), e′  satisfies requirement R4 and no further information allows us to think that the  salary of e′ could be different than s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As a consequence, the consistent answer  to Q4 is expected to contain e′ and possibly e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Regarding Q5, for the same reasons as for Q4, e and e′ are the two possible  tuples satisfying requirements R1, R2 and R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Now, and contrary to the case  of Q4, the two true super-tuples of e, namely es and es′, belong to Sat(Γ5),  thus ensuring that whatever e’s salary it satisfies the selection condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' This  remark supports the fact that e is a candidate to belong to the consistent answer  to Q5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Of course, as for query Q1, one could object that since es and es′ are  inconsistent, e has not to belong to the consistent answer to Q5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' It should be  noticed here that the case of e′ is treated as for Q4 above, because e′s′ is a true  super-tuple of e′ such that s′ is in Sat(Γ5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Hence, the consistent answer to Q5 is  expected to be {e, e′} or {e′}, following the choice made regarding inconsistency  of the tuple t in requirement R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  A reasoning similar to those for Q4 or Q5 applies to Q6, depending on the  actual values of s and s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Assuming first that s = 5K and s′ = 20K, as for  Q4, the expected consistent answer is {e′} or possibly {e, e′}, knowing that e’s  unique salary may not satisfy the condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' However, if s = 15K and s′ = 20K,  the consistent answer to Q6 is expected to be as for Q5, that is, {e, e′} or {e′},  following the choice made regarding inconsistency of the tuple t in requirement  R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  �  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='2  Consistent Answers  To account for some of the remarks regarding Q5 and Q6 in Example 5, we  introduce the notion of consistency with respect to a selection condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' To this  end, for every relation schema X, we denote by T (X↑) the set of all tuples t in  T such that X ⊆ sch(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' In other words, T (X↑) is the set of all super-tuples of  tuples over X, and it follows that T (X) is a sub-set of T (X↑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Definition 5 Given a table T over U, and Γ a selection condition:  A tuple t in T (sch(Γ)↑) is said to be inconsistent with respect to Sat(Γ) if  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='sch(Γ) is in Sat(Γ) and there exists s in Sat−(Γ) such that I∗(t) ∩ I∗(s) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  We denote by Inc(Γ, T ) the set of all tuples inconsistent with respect to Sat(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  A tuple t in T (sch(Γ)↑) is said to be consistent with respect to Sat(Γ) if  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='sch(Γ) is in Sat(Γ) and t is not in Inc(Γ, T ), that is if for every s in Sat−(Γ),  I∗(t) ∩ I∗(s) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We denote by Cons(Γ, T ) the set of all tuples consistent with  respect to Sat(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  �  18  The following proposition states basic properties implied by Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Proposition 2 Given a table T over U and a selection condition Γ, the follow-  ing holds:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Inc(Γ, T ) is a sub-set of Inc(T ), and Cons(T ) ∩ T (sch(Γ)↑) is a sub-set of  Cons(Γ, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' But Cons(Γ, T ) ⊆ Cons(T ) does not always hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' For every t in Cons(Γ, T ), every super-tuple t′ of t is also in Cons(Γ, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' If I∗ is in First Normal Form, then Inc(Γ, T ) = ∅ and Cons(Γ, T ) is the  set of all super-tuples of tuples in Sat(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' For a given tuple t, if Sat(Γ) = {t}, then t is in Inc(T ) if and only if t is  in Inc(Γ, T ), and t is in Cons(T ) if and only if t is in Cons(Γ, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' If Sat(Γ) = ∅ then Cons(Γ, T ) = Inc(Γ, T ) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' If Sat(Γ) = T (X), then  Cons(Γ, T ) is the set of all super-tuples of tuples in Sat(Γ), and Inc(Γ, T )  is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' If t is in Inc(Γ, T ), there exists s in Sat−(Γ) such that I∗(t)∩I∗(s) ̸=  ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Since t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='sch(Γ) is in Sat(Γ), t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='sch(Γ) and s are distinct tuples in T (sch(Γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Hence, there exists A in sch(Γ) such that t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A ̸= s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  As I∗(s) ⊆ I∗(s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A),  I∗(t) ∩ I∗(s) ̸= ∅ implies that I∗(t) ∩ I∗(s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Therefore, by Definition 3, t  is in Inc(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  As for the inclusion Cons(T )∩T (sch(Γ)) ⊆ Cons(Γ, T ), given t in T (sch(Γ)),  if t is in Cons(T ) then t is not in Inc(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Thus, t is not in Inc(Γ, T ), which implies  by Definition 5 that t is in Cons(Γ, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  To see that Cons(Γ, T ) ⊆ Cons(T ) does not always hold, consider the earlier  example where T = {abc, ab′c, a′b′′c′} over U = {A, B, C} and C → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' In  this case, b and b′ are clearly in Inc(T ), thus, not in Cons(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' For Γ = (B =  b ∨ B = b′), we have Sat(Γ) = {b, b′} and Sat−(Γ) = {b′′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Since I∗ is defined  by I∗(a) = I∗(b) = I∗(b′) = I∗(c) = {1, 2}, I∗(a′) = I∗(b′′) = I∗(c′) = {3},  Definition 5 shows that b and b′ are in Cons(Γ, T ) (because I∗(b) ∩ I∗(b′′) =  I∗(b′) ∩ I∗(b′′) = ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' If t′ is a super-tuple of t, t′ is in T (sch(Γ)↑) because so is t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Moreover, as t is  in Cons(Γ, T ), I∗(t) ∩ I∗(s) = ∅ holds for every s in Sat−(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As I∗(t′) ⊆ I∗(t),  it follows that I∗(t′) ∩ I∗(s) = ∅ holds as well, showing that t′ is in Cons(Γ, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Assuming that I∗ is in First Normal Form means that Inc(T ) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' By (1)  above, this implies that Inc(Γ, T ) = ∅ and thus, by Definition 5, Cons(Γ, T ) is  the set of all super-tuples of tuples in Sat(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' By (1) above, we have Inc(Γ, T ) ⊆ Inc(T ) always holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Conversely, if t  is in Inc(T ) then, by Definition 3, there exist A in sch(t) and a′ in adom(A)  such that a′ ̸= t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A and I∗(t) ∩ I∗(a′) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Denoting t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A by a and writing t  as qa, the tuple qa′ is such that sch(qa′) = sch(t) and qa′ ∈ Sat−(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Hence,  I∗(t) ∩ I∗(qa′) ̸= ∅, showing that t is in Inc(Γ, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  If t is in Cons(Γ, T ), then Sat(Γ) = T (X) \\ {t}, and for every s in Sat−(Γ),  I∗(t) ∩ I∗(s) = ∅, implying that for every A in sch(Γ) and every a′ in adom(A)  19  such that a′ ̸= t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A, I∗(t) ∩ I∗(a′) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' By Definition 3, this implies that t is in  Cons(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Conversely, if t is in Cons(T ), by (1) above, as t belongs to T (sch(Γ)),  t is in Cons(Γ, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The results being immediate consequences of Definition 5, their proofs are  omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The proof of the proposition is therefore complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  �  In what follows, in order to account for the remarks in Example 5, we propose  four ways of defining the consistent answer to a query Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' These definitions are  then illustrated by examples and compared to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Definition 6 Let T be a table over universe U, FD the set of associated func-  tional dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Given a query Q : select X from T [where Γ], we define  the following:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The weakly consistent answer to Q, denoted WC ans(Q), is the set of all  tuples x in T (X) such that  (a) x is in True(T ) ∩ Cons(T ),  (b) there exists t is in True(T ) such that t is in T (sch(Q)↑), t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X = x,  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='sch(Γ) is in Sat(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The consistent answer to Q, denoted C ans(Q), is the set of all tuples x  in T (X) such that  (a) x is in True(T ) ∩ Cons(T ),  (b) there exists t is in True(T ) such that t is in T (sch(Q)↑), t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X = x, t  is in Cons(Γ, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The strongly consistent answer to Q, denoted SC ans(Q), is the set of all  tuples x in T (X) such that  (a) x is in True(T ) ∩ Cons(T ),  (b) there exists t is in True(T ) ∩ Cons(T ) such that t is in T (sch(Q)↑),  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X = x, and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='sch(Γ) is in Sat(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The max-strongly consistent answer to Q, denoted MSC ans(Q), is the set  of all tuples x in T (X) such that  (a) x is in True(T ) ∩ Cons(T ),  (b) there exists t is in True(T ) ∩ Cons(T ) such that t is maximal with  respect to ⊑ in True(T ), t is in T (sch(Q)↑), t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X = x, and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='sch(Γ)  is in Sat(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  �  The four kinds of consistent answer in Definition 6 relate to requirements R1,  R2, R3 and R4 in the following respects:  Statements (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='a), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='a), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='a) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='a) clearly show that any tuple x  in any of the four kinds of consistent answer satisfies requirements R1  (x ∈ T (X)), R2 (x ∈ True(T )) and R3 (x ∈ Cons(T )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  20  Statements (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b) show that any tuple x in any of the  four kinds of consistent answer satisfies the requirement R4 by ensuring  the existence of a true super-tuple t of x satisfying Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' as announced in our earlier discussion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' the status of the tuple t is  clarified in Definition 6 as follows:  in the case of the weakly consistent answer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' t is only requested to be true  (as stated in requirement R4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  in the case of the consistent answer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' t is requested to be true and consistent  with respect to Γ (which does not disallow t to be inconsistent),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  in the case of the strongly consistent answer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' t is requested to be true and  consistent (which of course disallows t to be inconsistent),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' and  in the case of the max-strongly consistent answer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' t is requested to be  true,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' consistent and maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  We also emphasize that when the query Q in Definition 6 involves no selection  condition, then statements (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b) are reduced to the existence  of t such that t is in True(T ), t is in T (X↑) and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As statements (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='a),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='a) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='a), ensure the existence of such a tuple t, statements (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b)  and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b) are redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' However, such is not the case for statement (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b),  because the true and consistent super-tuple of x needed to satisfy (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='a) might  not be maximal, and thus might not satisfy (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We refer to query Q1 in the  forthcoming Example 6 for an illustration of this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Moreover, Definition 6 and the results in Proposition 2 lead to the following  observations when the query Q involves a selection condition Γ:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' For every t in Inc(Γ, T ), the tuple t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='sch(Γ) is such that its syntax satisfies  the condition Γ whereas its semantics meets that of a tuple whose syntax  does not satisfy the condition Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Since Inc(Γ, T ) is a sub-set of Inc(T ),  this shows that the notion of satisfaction of a selection condition might  concern inconsistent tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' It should be noticed that any super-tuple of  such tuples is discarded from max-strongly consistent answers and from  strongly consistent answers, whereas they may occur in consistent answers  or in weakly consistent answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Since every super-tuple of a tuple in Cons(Γ, T ) is also in Cons(Γ, T ),  C ans(Q) can be computed by scanning maximal true tuples in T (sch(Q)↑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Obviously, MSC ans(Q) should be computed by scanning only maximal  true tuples in T (sch(Q)↑), whereas the scan should concern true tuples in  T (sch(Q)), when computing SC ans(Q) or WC ans(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' It will however be  seen in the next section, that all consistent answers are computed based  on the same scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' If I∗ is in First Normal Form, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=', if Inc(T ) = ∅, then Cons(Γ, T ) is the  set of all super-tuples of tuples in Sat(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As moreover, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X is in Cons(T ),  21  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X is in C ans(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' This remark fits the intuition that if T is consistent  then the consistent answer to Q is equal to the standard answer to Q, that  is the set of projections over X of true tuples in T (sch(Q)↑) that satisfy  Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' This comes in line with our earlier remark stating that requirements  R1, R2, R3 and R4 are satisfied by the tuples in the answer to Q when T  is consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' If Sat(Γ) is reduced to one tuple s, then for every t in T (sch(Q)↑) ∩  True(T ) such that t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X is in Cons(T ) ∩ True(T ) and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='sch(Γ) = s, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X  is in WC ans(Q), and moreover, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X is in SC ans(Q) and in C ans(Q) if  and only if s is in Cons(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The case of MSC ans(Q) is more involved  because maximal true tuples must be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  For example, it can  happen that, for Q : select A from T where B = b, a is in SC ans(Q)  and in C ans(Q), but not in MSC ans(Q) because ab is true and consistent,  whereas abc is the only true super-tuple of ab and abc is inconsistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' If Sat(Γ) = ∅ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=', if Γ is not satisfiable in T ), then it is obvious that  MSC ans(Q), SC ans(Q) and WC ans(Q) are empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  As in this case,  Cons(Γ, T ) = ∅, C ans(Q) is empty as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  On the other hand, if Sat(Γ) = T (sch(Γ)), it is easy to see that SC ans(Q)  and WC ans(Q) are equal to the set of all tuples in T (X) that belong to  Cons(T )∩True(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Moreover, since Cons(Γ, T ) = T (sch(Γ)↑), C ans(Q) is  also the set of all tuples in T (X) that belong to Cons(T ) ∩ True(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' How-  ever, as in the previous item, the case of MSC ans(Q) is more involved  because maximal true tuples must be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  The following proposition states an important relationship among the four kinds  of consistent answers, namely that they form an increasing chain of inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Proposition 3 Let T be a table over universe U, FD the set of associated  functional dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Given a query Q : select X from T [where Γ], the  following holds: MSC ans(Q) ⊆ SC ans(Q) ⊆ C ans(Q) ⊆ WC ans(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' To show that MSC ans(Q) ⊆ SC ans(Q) holds, let x in MSC ans(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Then statements (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='a) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b) in Definition 6 hold, and thus, so does statement  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The result comes form the fact that it is easily seen that statement (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b)  implies statement (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Regarding the inclusion SC ans(Q) ⊆ C ans(Q), we notice that for every x  in SC ans(Q), statements (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='a) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b) in Definition 6 hold, and thus, that  statement (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='a) holds as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Moreover, since statement (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b) holds, there  exists t in T (sch(Q)↑), such that t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X = x and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='sch(Γ) is in Sat(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Thus t is in  Cons(T )∩T (sch(Γ)↑), which by Proposition 2(1) implies that t is in Cons(Γ, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Hence statement (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b) is satisfied, showing that x is in C ans(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Regarding the last inclusion, for every x in C ans(Q), statement (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='a) clearly  holds and moreover, as the tuple t in Cons(Γ, T )∩True(T ) is obviously in True(T )  and in Sat(Γ), the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  �  22  We illustrate Definition 6 and Proposition 3 in the context of Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' More-  over, through this example, we show that the inclusions in Proposition 3 might  be strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Example 6 We recall that in Example 5, U = {Emp, Sal, Dept}, FD =  {Emp → Sal} and T = {es, es′d, e′s′}, which implies that True(T ) consists of  the sub-tuples of esd, es′d and of e′s′ and that Inc(T ) = {esd, es′d, es, es′, sd,  s′d, s, s′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The queries we are interested in are Q1, Q2, Q3, Q4, Q5 and Q6 as  shown below:  Q1 : select Emp, Dept from T  Q2 : select Emp, Sal from T  Q3 : select Sal from T  Q4 : select Emp from T where Sal = s′  Q5 : select Emp from T where Sal = s ∨ Sal = s′  Q6 : select Emp from T where Sal > 10K  In the case of Q1, the only possible true tuple in any answer is ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  As all  maximal true super-tuples of ed are esd and es′d, none of these tuples is con-  sistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Thus, we have MSC ans(Q1) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' However, SC ans(Q1) = C ans(Q1) =  WC ans(Q1) = {ed}, because this tuple is true and consistent and because no  selection condition is involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Regarding the inclusions in Proposition 3, this  case shows that the first inclusion might be strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Considering now Q2, we have MSC ans(Q2) = {e′s′}, because this tuple is  true and consistent, and has no strict true super-tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' For a similar reason, we  also have SC ans(Q2) = C ans(Q2) = WC ans(Q2) = {e′s′} because no selection  condition is involved in Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' In this case the inclusions in Proposition 3 hold  because MSC ans(Q2) = SC ans(Q2) = C ans(Q2) = WC ans(Q2) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  The case of Q3 is easy because as earlier mentioned, no Sal-value is con-  sistent, implying that statement (a) of the four consistent answers in Defini-  tion 6 can not be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  As Q3 involves no selection condition, we have  MSC ans(Q3) = SC ans(Q3) = C ans(Q3) = WC ans(Q3) = ∅, showing again  that the inclusions in Proposition 3 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Regarding Q4 involving the selection condition Γ4 = (Sal = s′), since e and  e′ are both true and consistent tuples over Emp, they satisfy statements (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='a),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='a), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='a) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='a) in Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Since Sat(Γ3) = {s′}, es′ and e′s′ are  the only tuples of interest regarding statements (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b) in  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Thus WC ans(Q4) = {e, e′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As es′d is maximal and inconsistent,  statement (1b) is not satisfied and since es′ is also inconsistent, statement (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b)  is not satisfied either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Now, regarding statement (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b), we have s in Sat−(Γ4)  and I∗(es′) ∩ I∗(s) = {1, 2} (see Example 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Thus, by Definition 5, es′ is in  Inc(Γ4, T ), and so, es′ does not satisfy statement (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As a consequence, e is  not in MSC ans(Q4), SC ans(Q4) nor in C ans(Q4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  On the other hand, e′s′ does satisfy statements (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b) in  Definition 6, because e′s′ is a maximal true and consistent tuple such that s′ is in  Sat(Γ4), and regarding (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b), e′s′ is in Cons(Γ4, T ) (because I∗(e′s′)∩I∗(s) = ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Therefore, MSC ans(Q4) = SC ans(Q4) = C ans(Q4) = {e′} and WC ans(Q4) =  23  {e, e′}, showing that the inclusions in Proposition 3 hold, and that the last one  might be strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  As for Q5 involving the selection condition Γ5 defined by (Sal = s∨Sal = s′),  for the same reasons as for Q4, the two possible tuples in the answer are e  and e′ and WC ans(Q5) = {e, e′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Moreover, since Sat(Γ5) = {s, s′}, e′ is in  MSC ans(Q5), SC ans(Q5) and C ans(Q5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Regarding now e, the same argu-  ments as for Q4 apply regarding the statements (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b) in Definition 6,  showing that e is neither in MSC ans(Q5) nor in SC ans(Q5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' However, contrary  to the case of Q4, es is in Cons(Γ5, T ), and thus, statement (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b) of Definition 6  is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We therefore obtain that MSC ans(Q5) = SC ans(Q5) = {e′} and  C ans(Q5) = WC ans(Q5) = {e, e′}, which also shows that the second inclusion  in Proposition 3 might be strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Consider now Q6 involving the selection condition Γ6 = (Sal > 10K) and  suppose that s = 5K and s′ = 20K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Then, we have Sat(Γ6) = {s′} and  Sat−(Γ6) = {s}, and thus, as in the case of Q4, MSC ans(Q4) = SC ans(Q4) =  C ans(Q4) = {e′}, and WC ans(Q6) = {e, e′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  On the other hand, if s =  15K and s′ = 20K, then Sat(Γ6) = {s, s′} and Sat−(Γ3) = ∅, and thus, as  for Q5 above, we have MSC ans(Q5) = SC ans(Q5) = {e′} and C ans(Q5) =  WC ans(Q6) = {e, e′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  �  To end this section, we would like to stress that proposing distinct consistent  answers as done in Definition 6 raises the question of which one should be cho-  sen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Although we think that this choice should be made by the final user, the  comparisons stated in Proposition 3 should help in making this choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' More-  over, assuming the choice of C ans(Q) is made, the knowledge of the answers  MSC ans(Q), SC ans(Q) and WC ans(Q) are likely to be useful to the user, re-  garding the quality of the provided consistent answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  For instance, in Example 6 the fact that e is in C ans(Q5) but not in  SC ans(Q5) implies that the database contains an inconsistency regarding e’s  salary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' This information shows that the presence of e in C ans(Q5) might be  considered less reliable than that of e′ for which such inconsistency does not  exist as e′ is in SC ans(Q5) and in SC ans(Q5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' On the other hand, a tuple  in WC ans(Q) but not in C ans(Q) shows that although this tuple occurs with  values satisfying the selection condition, it also systematically occurs in an in-  consistency involving values not satisfying the selection condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' This happens  for e when answering Q4: although e occurs associated with s′, it also occurs  associated with s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  This example also shows that comparing the consistent answers MSC ans(Q),  SC ans(Q) and WC ans(Q) with C ans(Q) has an impact not only on the quality  of the answers, but also allows for explaining the presence or the absence of  some tuples in a consistent answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' These issues related to data quality and to  answer explanation are among the subjects of our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  24  4  Computational Issues  In this section, we present algorithms first for computing efficiently the sets  Cons(T ) and Inc(T ) (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='1) and then the four consistent answers to a  given query (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='1  The Tabular Representation of I∗  In this section, we show how to compute True(T ) and Inc(T ), using a modified  version of the well-known chase algorithm [8, 14] that we call m-Chase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Our  algorithm is based on the notion of multi-valued tuple, or m-tuple for short,  which generalizes that of usual tuple in the following sense: instead of associating  every attribute A with at most one value in adom(A), a multi-valued tuple  associates every attribute A with a sub-set of adom(A), possibly empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Multi-  valued tuples are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Definition 7 Given a universe U = {A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' , An} and active domains adom(Ai),  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' , n, a multi-valued tuple σ over U, or m-tuple for short, is a total func-  tion from U to the cross product XA∈UP(adom(A)), where P(adom(A)) denotes  the power set of the active domain of attribute A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The ‘empty’ m-tuple, denoted  by ω, is defined by ω(A) = ∅ for every A in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  The set of all attributes A such that σ(A) ̸= ∅, is called the schema of σ,  denoted by sch(σ), and the schema of ω is ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Given σ ̸= ω and a sub-set X  of sch(σ), the restriction of σ to X, denoted σ(X), is the m-tuple defined by  (σ(X))(A) = σ(A) for every A in X and (σ(X))(A) = ∅ for any A not in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Given an m-tuple σ, the set tuples(σ) denotes the set of all tuples t such that  sch(t) = sch(σ) and for every A in sch(t), t(A) is a value of adom(A) that  belongs to σ(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As for the empty m-tuple ω, we have tuples(ω) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  The set of all m-tuples over U is denoted by MT , and a finite set of m-tuples  is called an m-table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  �  It is easy to see that for every t in T , it holds that sch(σt) = sch(t) and  tuples(σt) = {t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' In what follows, we use t and σt interchangeably, whenever no  confusion is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' To simplify notation, given an m-tuple σ, the set σ(A) is  denoted by the concatenation of its elements between parentheses, and σ itself  is denoted by the concatenation of all σ(A) such that σ(A) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Based on Definition 7, a tuple t in T yields an m-tuple σt in MT such that  for every A in sch(t), σt(A) = {t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A} (or σt(A) = (t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A) according to the notation  just introduced), and σt(B) = ∅ for every B not in sch(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  For example if U = {A, B, C, D}, the m-tuple σ such that σ(A) = {a, a′},  σ(B) = {b}, σ(C) = ∅ and σ(D) = {d1, d2, d3} is denoted by (a, a′)(b)(d1d2d3)  and we have sch(σ) = {A, B, D} (or simply sch(σ) = ABD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Now, to find  the set tuples(σ) we make all combinations of values, one from each of σ(A),  σ(B) and σ(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We find tuples(σ) = {abd1, abd2, abd3, a′bd1, a′bd2, a′bd3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Moreover, we have σ(AB) = (aa′)(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' On the other hand, for t = abd1, we have  σt = (a)(b)(d1), and thus tuples(σt) = {abd1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  25  We draw attention to the fact that, although m-tables can be seen as a par-  ticular case of embedded relations [14], we do not intend to elaborate on such  relations in our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We rather use m-tuples as a tool to explicitly ac-  count for situations where the First Normal Form (or equivalently the partition  constraint) is not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We define the following operations for m-tuples:  Comparison: σ1 ⊑ σ2 holds if for every A in U, σ1(A) ⊆ σ2(A)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Thus if  σ1 ⊑ σ2 then sch(σ1) ⊆ sch(σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Moreover, if sch(σ1) = sch(σ2) then  tuples(σ1) ⊆ tuples(σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' It also holds that ω ⊑ σ, for every m-tuple σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Union: σ1 ⊔ σ2 is the m-tuple σ such that for every A in U, σ(A) = σ1(A) ∪  σ2(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Thus, sch(σ1 ⊔ σ2) = sch(σ1) ∪ sch(σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Intersection: σ1 ⊓ σ2 is the m-tuple σ such that for every A in U, σ(A) =  σ1(A) ∩ σ2(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Thus, sch(σ1 ⊓ σ2) ⊆ sch(σi) for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  We now introduce a new chase algorithm, called m-Chase whose definition is  shown as Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  The m-Chase Algorithm works on sets of m-tuples  to provide an m-table from which true and inconsistent tuples can easily be  retrieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Roughly speaking, Algorithm 1 follows the same idea as the standard  chase, but when a functional dependency cannot be satisfied, our algorithm does  not stop, but rather collects all values inferred from the functional dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Consequently, it may happen that for a given row and a given column (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=', for  a given attribute value) we may have to store more than one constant, which  is achieved by means of m-tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  It is important to note that the output  of Algorithm 1 (denoted by Σ∗) allows to explicitly ‘show’ all violations of the  partition constraint PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We illustrate how Algorithm 1 works using an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Example 7 Let T = {ab, ab′, a′b′} over universe U = {A, B} with functional  dependencies A → B and B → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The computation of Σ∗ based on Algorithm 1  works as follows:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Considering tuples in T as m-tuples yields the following m-tuples: (a)(b),  (a)(b′) and (a′)(b′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Thus, Σ∗ is set to {(a)(b), (a)(b′), (a′)(b′)} on line 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Considering A → B, with σ = (a)(b) and σ′ = (a)(b′), the m-tuples  σ1 = (a)(bb′) and σ′  1 = (a)(b′b) are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The variable change is set  to true and since σ1 = σ′  1, Σ∗ is set to {(a)(bb′), (a′)(b′)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Moreover, fail is  set to true line 11 because we have (b) ̸= (b′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' This means intuitively that  when running the standard chase algorithm, a failure would be reached at  this step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Considering B → A, since the B-value b′ occurs in (a)(bb′) and (a′)(b′),  these m-tuples generate respectively (aa′)(bb′) and (aa′)(b′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Moreover,  since (aa′)(b′) ⊑ (aa′)(bb′), Σ∗ is set to {(aa′)(bb′)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  2With a slight abuse of notation, we use here the same symbol ⊑ introduced earlier for the  sub-tuple relation between usual tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  26  Algorithm 1 The m-Chase Algorithm  Input: A table T over U and a set FD of functional dependencies over U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Output: An m-table denoted by Σ∗ and the truth value of fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  1: Σ∗ := {σt | t ∈ T}  2: change := true  3: fail := false  4: while change = true do  5:  change := false  6:  for all σ in Σ∗ do  7:  for all X → A in FD such that XA ⊆ sch(σ) do  8:  for all σ′ in Σ∗ such that X ⊆ sch(σ′) do  9:  if for every B in X, σ(B) ∩ σ′(B) ̸= ∅ then  10:  if A ∈ sch(σ) and A ∈ sch(σ′) and σ(A) ̸= σ′(A) and fail = false  then  11:  fail := true  12:  σ1 := σ ⊔ σ′(A) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' σ′  1 := σ′ ⊔ σ(A)  13:  if σ ̸= σ1 or σ′ ̸= σ′  1 then  14:  change := true  15:  if σ1 ⊑ σ′  1 then  16:  Σ∗ := (Σ∗ \\ {σ, σ′}) ∪ {σ′  1}  17:  else if σ′  1 ⊑ σ1 then  18:  Σ∗ := (Σ∗ \\ {σ, σ′}) ∪ {σ1}  19:  else  20:  Σ∗ := (Σ∗ \\ {σ, σ′}) ∪ {σ1, σ′  1}  21: return Σ∗ and the truth value of fail  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As the second execution of the while-loop line 4 does not change Σ∗, the  variable change has value false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Hence the processing stops, returning  Σ∗ = {(aa′)(bb′)} along with the value true assigned to fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  �  We now state a basic lemma based on which it will be shown how to compute  True(T ) and Inc(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' In what follows, the notation LoCl is extended to Σ∗ in a  straightforward way: LoCl(Σ∗) = {t ∈ T | (∃σ ∈ Σ∗)(t ⊑ σ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Moreover, for  every interpretation I and every m-tuple σ, I(σ) is defined by I(σ) = �  a⊑σ I(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  It should be noticed that, as for tuples in T , if σ ⊑ σ′, then I(σ′) ⊆ I(σ) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Lemma 4 Let T be a table over universe U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Then the following hold:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Algorithm 1 applied to T always terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' For every m-tuple s in MT , I∗(s) ̸= ∅ holds if and only if there exists σ  in Σ∗ such that s ⊑ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Inc(T ) = ∅ if and only if fail = false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' For every σ in Σ∗ and all t and t′ in tuples(σ), I∗(t) = I∗(t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' See Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  �  Note that Algorithm 1 is an extension of the standard chase algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Indeed,  the standard chase algorithm works on tuples (and not on m-tuples) which is  the case for T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Moreover, as long as σ and σ′ are usual tuples, our processing  is similar to that of standard chase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' In particular, for every X → A in FD if  27  XA is a sub-set of sch(σ) and sch(σ′) and if σ(X) and σ′(X) are equal tuples,  while σ(A) and σ′(A) are distinct values, then, in Algorithm 1, fail is set to  true, which is precisely a case of failure for the standard chase algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Thus,  based on Lemma 4, the returned value for fail is true if and only if the standard  chase algorithm fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Moreover, the statements lines 16, 18 and 20 in Algorithm 1 imply that when  σ and σ′ are in Σ∗, for every X → A in FD such that XA is a sub-set of sch(σ)  and sch(σ′), if tuples(σ(X)) ∩ tuples(σ′(X)) ̸= ∅, then σ(A) = σ′(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  This  remark can be seen as an extension of the well-known property of the output of  the standard chase, by which all rows equal over X must have the same A-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  As shown in the following proposition, the m-table Σ∗ allows to compute the  sets True(T ) and Inc(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Proposition 4 Let T be a table over U, FD the associated set of functional  dependencies, and Σ∗ the output of Algorithm 1 run with T and FD as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Then:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' True(T ) = LoCl(Σ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Inc(T ) = {t ∈ T | (∃σ ∈ Σ∗)(∃A ∈ sch(t))(∃a, a′ ∈ adom(A))(a ⊑ t ∧ t ⊑  σ ∧ (aa′) ⊑ σ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Cons(T )∩True(T ) = {t ∈ True(T ) | (∀σ ∈ Σ∗)(t ⊑ σ ⇒ tuples(σ(sch(t))) =  {t})}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The result is an immediate consequence of Lemma 4(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The result follows from Lemma 4(2) and Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The result follows from Definition 3 and the previous two items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  �  In order to assess the complexity of Algorithm 1, for every X → A in FD  and every tuple x in T (X), we let N(x) be the number of different A-values a  such that I∗(xa) ̸= ∅ and a is inconsistent, and we denote by δ the maximal  value of all N(x) for all X → A in FD and all x in True(T );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' in other words  δ = max({N(x) | X → A ∈ FD ∧ x ∈ True(T )}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Using this notation, the size of the m-tuples in Σ∗ is bounded by |U|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='δ,  and as Σ∗ contains at most |T| m-tuples, a run of the core of loop line 4 is in  O(|T|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='(|U|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='δ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Since the number of runs of the loop line 4 is bounded by the  changes in the sets in Σ∗, this number of runs is in O(|T|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='|FD|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='δ) (that is the  maximum number of constants that can be added in a set of Σ∗ times the size  of Σ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Therefore, the computation of Σ∗ is in O((|T|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='|FD|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='(|T|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' (|U|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='δ)),  that is in O(|T|3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='|FD|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='|U|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='δ2) or even in O(|T|3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='δ2), since |U| and |FD| are  independent from |T|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  We draw attention to the following important points regarding this result:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' When the database is consistent, δ is equal to 1, thus yielding a complex-  ity in O(|T|3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' This result can be shown independently from the above  computations as follows: In the case of traditional chase the maximum  of nulls in T being bounded by |U|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='|T|, the number of iterations when  28  running the algorithm is also bounded by |U|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='|T|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Since the run of one  iteration is in |T|2, the overall complexity is in O(|U|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='|T|3), or in O(|T|3),  as |U| is independent from |T|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The above complexity study can be seen as a data complexity study as  it relies on the size of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Thus, this study should be further investigated  in order to provide more accurate results regarding the estimation of the  number of actual tests necessary to the computation of Σ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The results in  [6] are likely to be useful for such a more thorough study of this complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='2  Computing Consistent Answers  In this sub-section, we show how the consistent answers to a query can be  computed based on the m-table Σ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We stress that to state one of the results,  it is assumed that Σ∗ is not redundant, that is for all σ1 and σ2 in Σ∗, neither  σ1 ⊑ σ2 nor σ2 ⊑ σ1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Notice that, if Σ∗ is not redundant then for all σ1  and σ2 in Σ∗, I∗(σ1) ∩ I∗(σ2) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Indeed, if I∗(σ1) ∩ I∗(σ2) ̸= ∅, Lemma 4(2) implies that Σ∗ contains σ such  that σi ⊑ σ for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As this entails that Σ∗ contains redundancies, we obtain  a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We also notice in this respect that redundancies are eliminated  at each step of the computation of Σ∗ in Algorithm 1 (see the statements lines 16  and 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Thus, it turns out that if the table T contains no redundancy, so does  Σ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Moreover, to state our results, we introduce the following additional no-  tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Let T be a non-redundant table T over universe U, and Σ∗ its asso-  ciated non-redundant m-table, as computed by Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Given a query  Q : select X from T [where Γ], and a tuple x in T (X), we respectively de-  note by Σ(x) and ΣQ(x), the sets defined by Σ(x) = {σ ∈ Σ∗ | (x ⊑ σ)} and  ΣQ(x) = {σ ∈ Σ(x) | (∃s ∈ Sat(Γ))(s ⊑ σ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Considering that ΣQ(x) = Σ(x) if Q involves no selection condition, the fol-  lowing proposition characterizes the tuples in MSC ans(Q), SC ans(Q), C ans(Q)  and WC ans(Q) in terms of m-tuples as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Proposition 5 Let T be a non-redundant table over universe U, FD the set of  associated functional dependencies and Σ∗ the non-redundant m-table as com-  puted by Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Given a query Q : select X from T [where Γ], and a  tuple x in T (X):  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' x is in MSC ans(Q) if and only if  (a) for every σ in Σ(x), tuples(σ(X)) = {x},  (b) there exists σ in ΣQ(x) such that |tuples(σ)| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' x is in SC ans(Q) if and only if  (a) for every σ in Σ(x), tuples(σ(X)) = {x},  (b) there exists s in Sat(Γ) ∩ True(T ) such that for every σ in ΣQ(x), if  s is in σ(sch(Γ)) then tuples(σ(sch(Γ)))) = {s}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  29  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' x is in C ans(Q) if and only if  (a) for every σ in Σ(x), tuples(σ(X)) = {x},  (b) there exists σ in ΣQ(x) such that tuples(σ(sch(Γ))) ⊆ Sat(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' x is in WC ans(Q) if and only if  (a) for every σ in Σ(x), tuples(σ(X)) = {x},  (b) ΣQ(x) is not empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' See Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  �  Example 8 We illustrate Proposition 5 in the context of Example 5 where  U = {Emp, Sal, Dept}, FD = {Emp → Sal} and T = {es, es′d, e′s}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' It is easy  to see that Algorithm 1 returns truth value true for fail and Σ∗ = {σ1, σ2}  where σ1 = (e)(ss′)(d) and σ2 = (e′)(s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We apply Proposition 5 to the queries  Q1, Q2, Q3, Q4, Q5 and Q6 shown below:  Q1 : select Emp, Dept from T  Q2 : select Emp, Sal from T  Q3 : select Sal from T  Q4 : select Emp from T where Sal = s′  Q5 : select Emp from T where Sal = s ∨ Sal = s′  Q6 : select Emp from T where Sal > 10K  Since Q1 involves no selection condition, and since σ1 is the only m-tuple in  Σ∗ whose schema contains Emp and Dept, the tests concern only this m-tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  As tuples(σ1(Emp, Dept)) = {ed}, we have SC ans(Q1) = C ans(Q1) = {ed}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Moreover, as ΣQ1(ed) = {ed}, WC ans(Q1) = {ed} and MSC ans(Q1) = ∅,  because |tuples(σ1)| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  For Q2, as tuples(σ1(Emp, Sal)) = {es, es′} and tuples(σ2(Emp, Sal)) =  {e′s′}, it is immediate to see that SC ans(Q2) = C ans(Q2) = WC ans(Q2) =  {e′s′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Moreover, as e′s′ is a maximal consistent and true tuple, we also have  MSC ans(Q2) = {e′s′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' In the case of Q3, it is easily seen that the four answers  are empty because tuples(σ1(Sal)) = {s, s′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Considering Q4, as tuples(σ1(Emp)) = {e} and tuples(σ2(Emp)) = {e′}, and  thus, these two values satisfy statement (a) in the four items in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  As Sat(Γ4) = {s′}, it is easy to see that ΣQ4(e) = {σ1} and ΣQ4(e′) = {σ2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Thus, WC ans(Q4) = {e, e′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Then, as |tuples(σ1(Emp, Sal))| = 2, e does not  belong to MSC ans(Q4) nor to SC ans(Q4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Since tuples(σ1(Sal)) = {s, s′}, e is  not in C ans(Q4) either, because statement (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b) is not satisfied (as s is not in  Sat(Γ4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' On the other hand is is easy to see that σ2 satisfies statements (b),  entailing that we have MSC ans(Q4) = SC ans(Q4) = C ans(Q4) = {e′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Concerning Q5, the two candidate Emp-values are again e and e′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  As  Sat(Γ5) = {s, s′}, ΣQ5(e) = {σ1} and ΣQ5(e′) = {σ2}, thus WC ans(Q5) =  {e, e′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As for Q4, e does not belong to MSC ans(Q5) and SC ans(Q5) because  we have |tuples(σ1(Emp, Sal))| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' However, σ1 satisfies item (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b) in Propo-  sition 5 because tuples(σ1(Sal)) is indeed a sub-set of Sat(Γ5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Thus, e belongs  30  to C ans(Q5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The case of e′ and σ2 being similar to the previous query, we  obtain MSC ans(Q5) = SC ans(Q5) = {e′} and C ans(Q5) = {e, e′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Regarding Q6, if s = 15K and s′ = 20K, MSC ans(Q6) = SC ans(Q6) = {e′}  and C ans(Q6) = WC ans(Q6) = {e, e′} due to arguments similar to the previous  case, and if s = 5K and s′ = 20K, we have MSC ans(Q6) = SC ans(Q6) =  C ans(Q6) = {e′} and WC ans(Q6) = {e, e′}, due to arguments similar to the  case of Q4 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  �  Algorithm 2 Consistent answers  Input: A query Q : select X from T [where Γ] and Σ∗  Output: The sets MSC ans(Q), SC ans(Q), C ans(Q) and WC ans(Q)  1: MSC ans(Q) := ∅ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' SC ans(Q) := ∅ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' C ans(Q) := ∅ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' WC ans(Q) := ∅  2: ToRemove X := ∅ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Candidate SC := ∅ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' ToRemove SC := ∅  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='3: for all σ in Σ∗ do  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='4:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='if X ⊆ sch(σ) then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='5:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='if |tuples(σ(X))| > 1 then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='6:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='ToRemove X := ToRemove X ∪ tuples(σ(X))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='7:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='else  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='8:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='Let x denote the unique tuple in σ(X)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='9:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='if sch(Γ) ⊆ sch(σ) then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='10:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='if tuples(σ(sch(Γ))) ∩ Sat(Γ) ̸= ∅ then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='11:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='WC ans(Q) := WC ans(Q) ∪ {x}  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='12:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='if |tuples(σ)| = 1 then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='13:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='MSC ans(Q) := MSC ans(Q) ∪ {x}  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='14:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='SC ans(Q) := SC ans(Q) ∪ {x}  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='15:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='C ans(Q) := C ans(Q) ∪ {x}  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='16:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='else  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='17:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='if |tuples(σ(sch(Γ)))| = 1 then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='18:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='Candidate SC := Candidate SC ∪ tuples(σ(sch(Q)))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='19:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='C ans(Q) := C ans(Q) ∪ {x}  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='20:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='else  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='21:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='ToRemove SC := ToRemove SC ∪ tuples(σ(sch(Q)))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='22:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='if tuples(σ(sch(Γ))) ⊆ Sat(Γ) then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='23:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='C ans(Q) := C ans(Q) ∪ {x}  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='24: MSC ans(Q) := MSC ans(Q) \\ ToRemove X  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='25: SC ans(Q) := SC ans(Q) ∪ {x | (∃q ∈ Candidate SC \\ ToRemove SC)(q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X = x)}  26: SC ans(Q) := SC ans(Q) \\ ToRemove X  27: C ans(Q) := C ans(Q) \\ ToRemove X  28: WC ans(Q) := WC ans(Q) \\ ToRemove X  29: return MSC ans(Q), SC ans(Q), C ans(Q), WC ans(Q)  Based on Proposition 5, it can be seen that, given a query Q, Algorithm 2  computes the consistent answer to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Indeed, the test line 4 discards all m-  tuples of Σ∗ whose schema does not contain X, because such m-tuples have no  chance to provide an X-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Then, the items in Proposition 5 are checked as  follows:  The test line 5 allows to store in the set ToRemove X all X-values of  m-tuples σ such that tuples(σ(X)) contains more than one tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  All  these tuples being inconsistent, are then removed from the set of re-  trieved X-values in MSC ans(Q) (line 24), in SC ans(Q) (line 26), in  C ans(Q) (line 27) and in WC ans(Q) (line 28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Therefore, the statements  31  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='a), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='a), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='a) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='a) in Proposition 5 are satisfied by the returned  sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' In what follows, we consider that the test line 5 fails, that is that  tuples(σ(X)) = {x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Statement (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b) is concerned when the tests lines 9 and 10 succeed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Indeed,  in this case, there exists a tuple s in tuples(σ(sch(Γ))) such that s is in  Sat(Γ), meaning that σ is in ΣQ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Therefore, in this case, x is inserted  in WC ans(Q) (line 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Regarding the statement (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b) in Proposition 5, on line 9 it is checked that  the schema of Γ is a sub-set of that of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' If yes, when the tests lines 10  and 12 succeed, σ is in fact a tuple t satisfying Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As moreover, Σ∗ is not  redundant, t occurs nowhere else in Σ∗, meaning by Proposition 4(3) that  t is in Cons(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Hence, x is inserted in MSC ans(Q) (line 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Notice that  in this case, the inclusions stated in Proposition 3 justify that x be also  inserted in SC ans(Q) (line 14) and in C ans(Q) (line 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  As for statement (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b), assuming that the test line 9 succeeds, the case  when the test line 12 succeeds is as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Now, if this test fails, then,  if the test line 17 succeeds, then σ(sch(Q)) might be a consistent true  tuple t satisfying Γ, unless t occurs in some other tuples(σ′(sch(Q))) such  that |tuples(σ′(sch(Q)))| > 1, meaning that t is inconsistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  This is  why t is inserted in the set Candidate SC when the test line 17 suc-  ceeds, and when this test fails, the tuples in the set tuples(σ(sch(Q))) are  stored in ToRemove SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Proposition 4(3) then ensures that the tuples in  Candidate SC \\ ToRemove SC are indeed consistent true tuples satisfy-  ing Γ (other than maximal ones earlier detected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The statement line 25  adds the corresponding X-values in the set SC ans(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Considering now statement (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b) when the test line 9 succeeds, we first  notice that, thanks to Proposition 3, the insertions in C ans(Q) in state-  ments lines 15 and 19 are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Moreover, when the test line 22 succeeds,  the condition in the statement (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b) is satisfied, and so, x is inserted in  C ans(Q) as stated line 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  We illustrate Algorithm 2 using the queries Q3 and Q4 of Example 8, respectively  defined by Q3 : select Sal from T and Q5 : select Emp from T where  Sal = s ∨ Sal = s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Regarding Q3, when considering σ1, ToRemove X is set to {s, s′} because  the test line 5 succeeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Then, with σ2, the test line 5 fails and the test line 10  succeeds, entailing that s′ is inserted in WC ans(Q3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Then, as the test line 12  succeeds, s′ is inserted in MSC ans(Q) (line 13), SC ans(Q) (line 14) and in  C ans(Q) (line 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' When processing the statements lines 24, 26 and 27, s′ is  removed, thus producing empty answers, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  As for Q5, when processing σ1, the test line 5 fails for σ1 and σ2 be-  cause |tuples(σi(Emp))| = 1 (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Thus ToRemove X remains empty  in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' When processing σ1, the tests line 12 and line 17 fail (because  |tuples(σ1)| = 2 and |tuples(σ1(Sal))| = 2) and the test line 22 succeeds (because  32  tuples(σ1(Sal)) = Sat(Γ5) = {s, s′}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Thus, ToRemove SC is set to {es, es′}  line 21 and C ans(Q) is set to {e} line 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Then, when processing σ2, all tests suc-  ceed (because |tuples(σ2)| = |tuples(σ2(Sal))| = 1 and tuples(σ2(Sal)) = {s′}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Thus, e′ is inserted in WC ans(Q5), MSC ans(Q5), SC ans(Q5) and C ans(Q5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  The loop line 3 returns ToRemove X = ∅, Candidate SC = ∅, ToRemove SC =  ∅, along with WC ans(Q5) = MSC ans(Q5) = SC ans(Q5) = C ans(Q5) =  {e, e′}, which is not changed by the statements lines 24-28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We thus obtain  all four consistent answers equal to {e, e′}, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  In the next example, we further illustrate the role of the sets Candidate SC  and ToRemove SC in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Example 9 Let U = {A, B, C, D} over which the functional dependencies C →  B and C → D are assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' For T = {abcd, abcd′, abc′, ab′c′}, Algorithm 1  returns Σ∗ = {(a)(b)(c)(dd′), (a)(bb′)(c′)}, and the processing of SC ans(Q) for  Q : select A from T where B = b, is as follows:  For σ = (a)(b)(c)(dd′), the tests line 4 and line 12 fail (as |tuples(σ(A))| = 1  and |tuples(σ)| = 2), whereas the test line 10 succeeds (because Sat(Γ) =  {b}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Thus ToRemove X remains empty and as the test line 17 succeeds  (because tuples(σ(B)) = {b}), ab is inserted in Candidate SC on line 18,  and SC ans(Q) remains empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  For σ = (a)(bb′)(c′), as the test line 4 fails, ToRemove X remains empty,  and as the tests line 12 and line 17 also fail, ab and ab′ are inserted in  ToRemove SC due to the statement line 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Thus, the loop line 3 generates ToRemove X = ∅, Candidate SC = {ab} and  ToRemove SC = {ab, ab′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Therefore, when processing the statement line 25,  SC ans(Q) remains empty (because Candidate SC \\ ToRemove SC = ∅), as  expected based on Definition 6(2), because ab is inconsistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Considering now T1 = {abcd, abcd′}, Algorithm 1 returns Σ∗  1 = {(a)(b)(c)(dd′)}  and the computations are as in the first item above, implying that the loop line 3  generates ToRemove X = ∅, Candidate SC = {ab}, ToRemove SC = ∅ and  SC ans(Q) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' When processing the statement line 25, SC ans(Q) is set to  {a} (because Candidate SC \\ ToRemove SC = {ab}), as expected based on  Definition 6(2), because ab is a consistent true tuple satisfying Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  �  To assess the complexity of Algorithm 2, we first notice that this algorithm is  clearly linear in the size of Σ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Moreover, using the same notation as when  assessing the complexity of Algorithm 1, for σ in Σ∗, the size of tuples(σ) is in  O(|FD|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='δ), that is in O(δ), because |FD| is independent from T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Hence, the  complexity of computing the consistent answer to a query in our approach is in  O(|Σ∗|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Since it has been shown that |Σ∗| ≤ |T|, we state that the sizes of  Σ∗ and of T are of the same order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Therefore, the complexity of computing all  four consistent answers to a query in our approach is in O(|T|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='δ), that is linear  in the size of the table T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  It is however important to stress that Algorithm 2 requires significant extra  storage due to the sets ToRemove X, Candidate SC and ToRemove SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' It  33  should be noticed that the storage of ToRemove X and ToRemove SC can  be avoided by implementing a second scan of Σ∗ whose role would be to de-  tect the X-values to be removed (when |tuples(σ(X))| > 1) and the tuples in  Candidate SC to be removed (when |tuples(σ(sch(Q)))| > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As usual, this  option saves storage at the cost of further processing, namely an additional scan  of Σ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  5  Related Work  The problem of consistent query answering has attracted considerable attention  in the last two decades [1] and continues to be an active area of research today  [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The approach by repairs is one that has attracted most attention in recent  years and thus has been the subject of many papers, whose overview lies outside  the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We refer to [5] for a detailed survey covering the topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  We rather focus on the basic differences between our approach and repair-based  approaches with the goal of showing that although the tackled problem is the  same, our approach is hardly comparable with those from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='1  Comparison with Repair-Based Approaches  We first point out that repair-based approaches assume a table T with no nulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Therefore to compare our approach to repair-based approaches we need to re-  strict our attention to tables with functional dependencies but without nulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  In this context, we identify and discuss three fundamental differences be-  tween our approach and the approach by repairs, namely : (D1) the two ap-  proaches use different universes of discourse, (D2) the two approaches use dif-  ferent assumptions when evaluating queries, and (D3) the two approaches use  functional dependencies for different purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  (D1) different universes of discourse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  We recall from Section 2 that the universe of discourse of a table T is the space in  which queries are evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The universe of discourse in our approach, denoted  by T , is the set of all tuples one can define using constants appearing in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' On  the other hand, the universe of discourse in the repairs approach, that we shall  denote by Tr, is the set of all tuples of T together with all their sub-tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  For example, if T = {ab, a′b′} then T = {ab, a′b′, a′b, ab′, a, b, a′, b′} whereas  Tr = {ab, a′b′, a, b, a′, b′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  (D2) different assumptions when evaluating queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  The repairs approach uses the Closed World Assumption [12] that is:  the only true tuples are those in Tr (implying that tuples not in Tr are  false)  whereas our approach uses the Open World Assumption that is:  the only true tuples are those in T together with all tuples derived using  the functional dependencies (implying that all other tuples in T are false).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  34  It follows that if a tuple t of Tr is true then t is true in T , that is True(Tr) ⊆  True(T ), while the opposite does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  For example, consider the table T = {abc, ab′c′} with functional dependency  A → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' In this case, ac is in T and in True(T ), and ac is in True(Tr) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  On the other hand, since Σ∗ = {(a)(bb′)(c), (a)(bb′)(c′)}, the tuple abc′ is in T  but not in Tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Thus abc′ is not in True(Tr), whereas abc′ is in True(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Regarding inconsistent tuples, the comparison not as easy since, to the best  of our knowledge, the notion of inconsistent tuple is not explicitly defined in  repair-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' However, it is clear that a table T is inconsistent if  and only if there exists a functional dependency X → A such that the two  tuples xa and xa′ in T (XA) occur in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The pair (xa, xa′) is usually called a  conflicting pair, and we define inconsistent tuples in this context as follows:  t is inconsistent if there exists a conflicting pair (xa, xa′) such that t is a  super-tuple of xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Denoting the set of these tuples by Conf Inc(Tr), we argue that Conf Inc(Tr) ⊆  Inc(T ) whereas the inverse inclusion does not always hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Indeed, based on Def-  inition 3, it is easy to see that, for every conflicting pair (xa, xa′), the tuples  xa, xa′ and all their true super-tuples are in Inc(T ), thus showing the inclu-  sion Conf Inc(Tr) ⊆ Inc(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The previous example with T = {abc, ab′c′} and  A → B, shows that the inverse inclusion does not always hold, because b is  clearly not in Conf Inc(Tr), whereas Definition 3 implies that b is in Inc(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  (D3) different purposes in the use of functional dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  The repair-based approaches use functional dependencies as a means for check-  ing the consistency of T (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=', as a means of inference of inconsistent tuples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' In  our approach, we also use the functional dependencies as a means of inference of  inconsistent tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' However, additionally, we use the functional dependencies  as a means of inference of true tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' For example, if T = {abc, ab′c′} and  FD = {A → B}, our approach allows us to infer that abc is true and inconsis-  tent, as repair-based approaches do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' However, additionally, our approach allows  to infer that the tuple ab′c is also true and inconsistent, which repair-based ap-  proaches fail to do since ab′c is not even in Tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Now that we have stated the differences in the ways the two approaches  operate, consider a query Q : select X from T where Γ, and recall that in  repair-based approaches, the consistent answer to Q is defined as the set of true  tuples x in Tr such that, every repair contains a tuple satisfying the condition Γ  and whose restriction over X is x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Denoting this set by Rep ans(Q), we recall  that, by Definition 5, all consistent answers to Q in our approach are sets of true  and consistent tuples in T (X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' It is thus not surprising that the two approaches  to consistent query answering are not comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Summarising our discussion so far we cannot claim that either of the two ap-  proaches is better or preferable than the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' They are simply different as they  have different universes of discourse, they operate under different assumptions  and they use the functional dependencies in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  35  What we can reasonably claim in favor of our approach is that: (a) it provides  a formal framework based on which different kinds of consistent answers can be  defined, (b) it offers a polynomial algorithm for evaluating the consistent answer  to a query, whether conjunctive or disjunctive, and (c) by allowing nulls in the  table, our approach broadens the angle of its application to other important  areas of research as explained in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='2  Comparison with the Approach in [9]  The approach presented in this paper and the approach in [9], both rely on par-  tition semantics [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' However, the two approaches have important differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  First, although the limit interpretation I∗ as defined in Section 2 is the same as  the interpretation µ∗ in [9], we have proved here the uniqueness of I∗ which is  not done for µ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Second, the notions of true/false tuples and of inconsistent/consistent tuples  differ between the two approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' More precisely, since in [9], truth values are  defined inspired by the Four-valued logic of [2], it is not possible to have tuples  that are, for instance, true and consistent, as in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Third, the universe of discourse in [9], is potentially infinite as it consists of  all tuples that can be built up using constants from the entire attribute domains;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  and this is an issue when it comes to computing the set of all inconsistent tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  In contrast, the universe of discourse in the present work is finite as it consists  of all tuples that can be built up using constants only from the active attribute  domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Fourth, the notion of inconsistent tuple is not defined in the same way in  the two approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Indeed, in [9], a tuple is inconsistent if its interpretation  is a sub-set of the interpretations of two constants from the same domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' For  example, consider the (true) tuples xa and xa′ in the presence of X → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  These tuples are inconsistent in the present approach (see Definition 3), as well  as in the approach of [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' However, the tuple x is also an inconsistent tuple  according to [9] (because its interpretation is a sub-set of those of a and a′),  but is not inconsistent in the present approach (because A is not in sch(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As  a consequence, the consistent answer to a query in [9] differs from that in the  present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Finally, the issue of consistent query answering has not been addressed in  full details in [9]: it is restricted to the case where the table T contains no nulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  We also note in this respect that, based on the concept of m-table as introduced  in Section 4, the present approach allows to consider also disjunctive queries,  an issue not addressed in [9], nor in repair-based approaches, such as [5, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  6  Concluding Remarks and Future Work  In this paper, we have presented a novel approach to consistent query answering  in tables with nulls and functional dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Given such a table T, we  defined T , the universe of discourse of T, and using set-theoretic semantics for  36  tuples and functional dependencies we partitioned T in two orthogonal ways:  into true and false tuples, on the one hand and into consistent and inconsistent  tuples on the other hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Queries are addressed to T and evaluated in T  in order to obtain consistent answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The consistent answer to a query Q :  select X from T where Condition over T was defined to be the set of tuples  x in T such that: (a) x is a true and consistent tuple with schema X and (b)  there exists a true super-tuple t of x in T satisfying the condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We have  seen that, depending on the ‘status’ of the super-tuple t in T , there are four  different types of consistent answer to Q and we have presented polynomial time  algorithms for computing these answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We have also seen that our approach  is not comparable to the approaches based on table repairs or the approach  of [9], as they have different universes of discourse and operate under different  assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  By allowing the presence of nulls in the table, our approach opens the way for  a number of important applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We envisage three such applications of our  approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The first is related to universal relation interfaces [8, 13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Given a  relational database with n tables T1, T2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' , Tn, over schemes S1, S2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' , Sn and  sets of functional dependencies FD1, FD2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' , FDn, respectively, the universal  relation is defined to be a table T defined as follows:  The schema of T is S = S1 ∪ S2 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' ∪ Sn and the associated set of  functional dependencies is FD = FD1 ∪ FD2 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' ∪ FDn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  For each tuple t in Ti, t is placed in a new line of T with a new tuple  identifier, for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  For example, if n = 2, T1 = {ab} and T2 = {bc} then the universal relation  operates from a table T with schema ABC and current instance T = {ab, bc}  with two nulls, one under C and one under A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  A prerequisite for defining the table T is “name consolidation” that is (a)  if an attribute name appears in two different database tables then it has the  same meaning in the two tables and (b) if two different attribute names in the  database have the same meaning then one is renamed to the other (using the  renaming operation of the relational model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Once the table T is put in place querying the interface is straightforward:  the system computes a consistent answer using our polynomial Algorithm 2 and  returns the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' However, updating through such an interface poses several  hard problems that we are currently investigating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  The second application that we envisage is query result explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  As  already mentioned, the m-table Σ∗ offers a way to ‘display’ inconsistencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We  therefore plan to investigate how m-tuples in Σ∗ can be used to ‘explain’ to the  user the presence or the absence of a given tuple in a consistent answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' This  issue seems particularly relevant in the context of data integration and data  exchange, where it is crucial to ‘explain’ consistent answers in reference to the  sources the data come from [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  The third application that we envisage is related to data quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Based on  the two orthogonal ways of characterizing tuples, namely consistent/inconsistent  37  and true/false, we consider that the inconsistent tuples are as good a source of  information as they are the consistent tuples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' and similarly, we consider that  the false tuples are as good a source of information as they are the true tu-  ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' In doing so we can develop a meaningful theory for the quality of data  in a data-set (whether the data-set is a table or a query result).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' To this end,  we plan to consider various quality measures such as percentages of consis-  tent/inconsistent tuples in the data-set, percentages of true or false tuples, or  combinations thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Acknowledgment  Work conducted while the second author was visiting at FORTH Institute of  Computer Science, Crete, Greece (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='gr/)  Declarations  The two authors contributed to the study, conception and design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Both read  and approved the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  No funds, grants, or other support was received for conducting this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
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+page_content='  [8] Ronald Fagin, Alberto O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
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+page_content=' A simplified  universal relation assumption and its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Database  Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=', 7(3):343–360, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  [9] Dominique Laurent and Nicolas Spyratos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Handling inconsistencies in ta-  bles with nulls and functional dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Intell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
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+page_content='  [10] Mark Levene and George Loizou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' A guided tour of relational databases and  beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Springer, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  [11] Ester Livshits, Benny Kimelfeld, and Sudeepa Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Computing optimal  repairs for functional dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Database Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=', 45(1):4:1–  4:46, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  [12] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Reiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' On closed World DataBases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Gallaire et Minker, Plenium Press,  New York, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  [13] Nicolas Spyratos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The partition model: A deductive database model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' ACM  Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Database Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=', 12(1):1–37, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  [14] Jeffrey D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Ullman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Principles of Databases and Knowledge-Base Systems,  volume 1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Computer Science Press, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  [15] Jef Wijsen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Database repairing using updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Database Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=',  30(3):722–768, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  [16] Justin Zobel and Alistair Moffat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Inverted files for text search engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=', 38(2):6, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  [17] Justin Zobel, Alistair Moffat, and Kotagiri Ramamohanarao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Inverted  files versus signature files for text indexing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Database Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=',  23(4):453–490, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  A  On the Church-Rosser Property of Expansion  We show that expansion has the Church-Rosser property, that is when iterating  the process, the order in which the steps are preformed is irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We first  recall the definition of expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Definition 4 The expansion of an interpretation I with respect to a functional  dependency X → A and the tuple xa in T (XA) is the interpretation Exp(I, xa)  defined as follows:  If I(x) ∩ I(a) ̸= ∅ and I(x) ̸⊆ I(a) then:  Exp(I, xa)(a) = I(a) ∪ I(x), and Exp(I, xa)(α) = I(α) for α in AD  different than a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  39  Otherwise, Exp(I, xa) = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  We consider two expansions one with respect to the dependency X → A and  the tuple xa and the other one with respect to the dependency Y → B and  the tuple yb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Starting with an interpretation I1, let I2 = Exp(I1, xa) and  I3 = Exp(I1, yb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Referring to Figure 2(a), the problem is to find sequences of  expansions called E2 and E3 in the figure, such that I24 = E2(I2) and I34 =  E3(I3) are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  It should be clear that if I2 = I1 or I3 = I1, then the order in which the  expansions are processed is irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' For example, if I2 = I1 and I3 ̸= I1, we  obtain I24 = I34 by setting E3 = ϵ (where ϵ denotes the empty sequence) and  E2 = Exp(I1, yb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  From now on, we assume that I2 ̸= I1 and I3 ̸= I1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Based on Definition 4,  we have:  I2(a) = I1(a) ∪ I1(x) and for every α ̸= a, I2(α) = I1(α)  I3(b) = I1(b) ∪ I1(y) and for every β ̸= b, I2(β) = I1(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  As a particular case, if it is assumed that a = b, we have:  I2(a) = I1(a) ∪ I1(x) and for every α ̸= a, I2(α) = I1(α)  I3(a) = I1(a) ∪ I1(y) and for every α ̸= a, I3(α) = I1(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Thus, as shown in Figure 2(b), for E2 = Exp(I2, ya) and E3 = Exp(I3, xa),  I24 and I34 are such that I24(a) = I34(a) = I1(a) ∪ I1(x) ∪ I1(y) and for every  α ̸= a, I24(α) = I34(α) = I1(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We therefore have, in this case, I24 = I34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  I1  I3  I24 = I34  Exp xa  Exp yb  Exp yb  I1  I2  I3  I24 = I34  Exp xa  Exp xa  Exp yb  E2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  E3 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content="  (a) Generic diagram  (b) (a = b) or (y ≠ y'a  and  x ≠ x'b)  I1  Exp xa  Exp xa  Exp yb  Exp yb  Exp yb  I2  I3  I31  I24 = I34  (c) y = y'a  and  x ≠ x'b  I2  Exp xa  Exp xa  Exp yb  Exp yb  Exp yb  Exp xa  (d) y = y'a  and  x = x'b  I1  I2  I21  I31  I3  I24 = I34  Figure 2: Expansion Diagrams  40  Now assuming that I2 ̸= I1 and I3 ̸= I1 and a ̸= b, if a does not occur in y  and b does not occur in x, the two expansions are independent from each other." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  In this case, as illustrated in Figure 2(b), the following lemma holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Lemma 5 If a ̸= b, if a does not occur in y and b does not occur in x, for  E2 = Exp(I2, yb) and E3 = Exp(I3, xa), denoting respectively by I24 and I34  the interpretations Exp(I2, yb) and Exp(I3, xa), we have I24 = I34, that is:  Exp(Exp(I1, xa), yb) = Exp(Exp(I1, yb), xa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Since a does not occur in y, we have I2(y) = I1(y) and similarly,  since b does not occur in x, we have I3(x) = I1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Therefore, for a and b,  I24 = Exp(Exp(I1, xa), yb) and I34 = Exp(Exp(I1, yb), xa) are defined by:  − I24(a) = Exp(I2, yb)(a) = I2(a) = I1(a) ∪ I1(x) (because a ̸= b)  − I24(b) = Exp(I2, yb)(b) = I2(b) ∪ I2(y) = I1(b) ∪ I1(y)  − I34(a) = Exp(I3, xa)(a) = I3(a) ∪ I3(x) = I1(a) ∪ I1(x)  − I34(b) = Exp(I3, xa)(b) = I3(b) = I1(b) ∪ I1(y) (because a ̸= b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Hence, we have I24(a) = I34(a) and I24(b) = I34(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Since no other symbol  is changed by expansions, we have shown the commutativity property in this  case, namely I24 = I34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  �  We now switch to the most general case, namely when each expansion has an  impact on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' This happens when a occurs in y and when b occurs in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Indeed, as shown in Figure 2(d), in this case, writing x and y respectively as bx′  and ay′, we notice the following: I2(y) ̸= I1(y) because I2(y) = I2(a) ∩ I2(y′)  and I2(a) ̸= I1(a), and similarly, I3(x) ̸= I1(x) because I3(x) = I3(b) ∩ I3(x′)  and I3(b) ̸= I1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  An important remark regarding x′ and y′ is that neither a nor b occurs  is these tuples, thus implying that, during the computations of the various  expansions considered here, their interpretation will not change, thus remaining  equal to I1(x′) and I1(y′), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' For example, I2(y) = I2(a) ∩ I2(y′) and  I3(x) = I3(b) ∩ I3(x′) can be respectively written as I2(y) = I2(a) ∩ I1(y′) and  I3(x) = I3(b) ∩ I1(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The following lemma holds in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Lemma 6 If a ̸= b, and if y = ay′ and x = bx′, for E2 = Exp(Exp(I2, yb), xa)  and E3 = Exp(Exp(I3, xa), yb), denoting respectively by I24 and I34 the corre-  sponding interpretations, we have I24 = I34, that is:  Exp(Exp(Exp(I1, xa), yb), xa) = Exp(Exp(Exp(I1, yb), xa), yb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Referring to Figure 2(d), we first give the computations regarding I2,  I21 and I24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' In each case the computations of the interpretations of a, x, b and  y are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Interpretation I2 = Exp(I1, xa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  First, we have I2(a) = I1(a) ∪ I1(x), due to expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Moreover, I2(b) = I1(b)  and as I2(y) = I2(a) ∩ I2(y′), we obtain I2(y) = (I1(a) ∪ I1(x)) ∩ I1(y′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Since  I1(y) = I1(a) ∩ I1(y′), we have I2(y) = I1(y) ∪ (I1(x) ∩ I1(y′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Moreover,  I2(x) = I2(b) ∩ I2(x′), and thus, I2(x) = I1(b) ∩ I1(x′) = I1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Therefore I2 is  defined as shown below:  41  I2(a) = I1(a) ∪ I1(x)  I2(x) = I1(x)  I2(b) = I1(b)  I2(y) = I1(y) ∪ (I1(x) ∩ I1(y′))  Interpretation I21 = Exp(I2, yb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Now, we have I21(b) = I2(b) ∪ I2(y), due to expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' It follows that I21(b) =  I1(b) ∪ I1(y) ∪ (I1(x) ∩ I1(y′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Moreover, I21(a) = I2(a), that is I21(a) =  I1(a) ∪ I1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As I21(x) = I21(b) ∩ I21(x′) and I21(x′) = I1(x′), we obtain  I21(x)  =  (I1(b) ∪ I1(y) ∪ (I1(x) ∩ I1(y′))) ∩ I1(x′)  =  (I1(b) ∩ I1(x′)) ∪ (I1(y) ∩ I1(x′)) ∪ (I1(x) ∩ I1(y′) ∩ I1(x′))  Considering that I1(b)∩I1(x′) = I1(x) and that I1(x) ⊆ I1(x′) (because x′ ⊑ x),  we obtain: I21(x) = I1(x) ∪ (I1(x′) ∩ I1(y)) ∪ (I1(x) ∩ I1(y′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Since (I1(x) ∩  I1(y′)) ⊆ I1(x) always holds, we have: I21(x) = I1(x) ∪ (I1(x′) ∩ I1(y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  On the other hand, I21(y) = I21(a)∩I21(y′), and thus, I21(y) = I2(a)∩I1(y′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Therefore, I21(y) = (I1(a) ∪ I1(x)) ∩ I1(y′), which can be written as I21(y) =  I1(y) ∪ (I1(x) ∩ I1(y′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Hence I21 is defined as shown below:  I21(a) = I1(a) ∪ I1(x)  I21(x) = I1(x) ∪ (I1(x′) ∩ I1(y))  I21(b) = I1(b) ∪ I1(y) ∪ (I1(x) ∩ I1(y′))  I21(y) = I1(y) ∪ (I1(x) ∩ I1(y′))  Interpretation I24 = Exp(I21, xa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Here, we have I24(a) = I21(a) ∪ I21(x), due to expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' It follows that:  I24(a)  =  (I1(a) ∪ I1(x)) ∪ (I1(x) ∪ (I1(x′) ∩ I1(y)))  =  I1(a) ∪ I1(x) ∪ (I1(x′) ∩ I1(y))  Moreover, I24(b) = I21(b), that is I24(b) = I1(b) ∪ I1(y) ∪ (I1(x) ∩ I1(y′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As  I24(x) = I24(b) ∩ I24(x′) and I24(x′) = I1(x′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' we have:  I24(x)  =  (I1(b) ∪ I1(y) ∪ (I1(x) ∩ I1(y′))) ∩ I1(x′)  =  (I1(b) ∩ I1(x′)) ∪ (I1(y) ∩ I1(x′)) ∪ (I1(x) ∩ I1(y′) ∩ I1(x′))  =  I1(x) ∪ (I1(x′) ∩ I1(y)) ∪ (I1(x) ∩ I1(y′))  (because I1(x) ⊆ I1(x′) as x′ ⊑ x)  =  I1(x) ∪ (I1(x′) ∩ I1(y)) (because (I1(x) ∩ I1(y′)) ⊆ I1(x))  As I24(y) = I24(a) ∩ I24(y′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' we obtain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='I24(y)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='(I1(a) ∪ I1(x) ∪ (I1(x′) ∩ I1(y))) ∩ I1(y′)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='(I1(a) ∩ I1(y′)) ∪ (I1(x) ∩ I1(y′)) ∪ (I1(x′) ∩ I1(y) ∩ I1(y′))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='I1(y) ∪ (I1(x) ∩ I1(y′)) ∪ (I1(x′) ∩ I1(y))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='(because I1(y) ⊆ I1(y′) as y′ ⊑ y)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='I1(y) ∪ (I1(x) ∩ I1(y′)) (because (I1(x′) ∩ I1(y)) ⊆ I1(y))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='Hence I24 is defined as shown below:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='I24(a) = I1(a) ∪ I1(x) ∪ (I1(x′) ∩ I1(y))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='I24(x) = I1(x) ∪ (I1(x′) ∩ I1(y))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='I24(b) = I1(b) ∪ I1(y) ∪ (I1(x) ∩ I1(y′))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='I24(y) = I1(y) ∪ (I1(x) ∩ I1(y′))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='Referring again to Figure 2(d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' we now turn to the computations regarding I3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  I31 and I34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As the computations are similar to those above, some details are  skipped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  42  Interpretation I3 = Exp(I1, yb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Here, we have I3(b) = I1(b) ∪ I1(y), due to expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Computations similar to  those in the case of I2 show that I3 is defined as shown below:  I3(a) = I1(a)  I3(x) = I1(x) ∪ (I1(x′) ∩ I1(y))  I3(b) = I1(b) ∪ I1(y)  I3(y) = I1(y)  Interpretation I31 = Exp(I3, xa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Here, we have I31(a) = I3(a) ∪ I3(x), due to expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Computations similar  to those for I24 show that I31 is defined as shown below:  I31(a) = I1(a) ∪ I1(x) ∪ (I1(x′) ∩ I1(y))  I31(x) = I1(x) ∪ (I1(x′) ∩ I1(y))  I31(b) = I1(b) ∪ I1(y)  I31(y) = I1(y) ∪ (I1(x) ∩ I1(y′))  Interpretation I34 = Exp(I31, yb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Here, we have I34(b) = I31(b) ∪ I31(y), due to expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' It follows that com-  putations similar to those for I24 show that I34 is defined as shown below:  I34(a) = I1(a) ∪ I1(x) ∪ (I1(x′) ∩ I1(y))  I34(x) = I1(x) ∪ (I1(x′) ∩ I1(y))  I34(b) = I1(b) ∪ I1(y) ∪ (I1(x) ∩ I1(y′))  I34(y) = I1(y) ∪ (I1(x) ∩ I1(y′))  It thus turns out that we indeed have I24 = I34, and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  �  The last case to be addressed is when y = ay′ and x ̸= bx′ or when x = bx′  and y ̸= ay′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We only consider the former case since the latter can be dealt  with in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Moreover, as shown in Figure 2(c), this case is in fact a  simplification of the case of Lemma 6 where the terms involving x′ are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  We thus omit the proof and just state that is this case again I24 = I34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' It can  be checked that in case of Figure 2(c) (that is y = ay′ and x ̸= bx′), we obtain  the following interpretations:  I24(a) = I34(a) = I1(a) ∪ I1(x)  I24(x) = I34(x) = I1(x)  I24(b) = I34(b) = I1(b) ∪ I1(y) ∪ (I1(x) ∩ I1(y′))  I24(y) = I34(y) = I1(y) ∪ (I1(x) ∩ I1(y′))  We therefore state the following basic result regarding expansion:  The process of expanding an interpretation I with respect to a set FD of  functional dependencies satisfies the Church-Rosser property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Hence, given T and FD, the process of expanding Ib until a fixed point is  reached does not depend on the order the expansions are processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Thus, the  limit is indeed unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  B  Proof of Lemma 4  Lemma 4 Let T be a table over universe U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Then the following holds:  43  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Algorithm 1 applied to T always terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' For every m-tuple s in MT , I∗(s) ̸= ∅ holds if and only if there exists σ  in Σ∗ such that s ⊑ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Inc(T ) = ∅ if and only if fail = false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' For every σ in Σ∗ and all t and t′ in tuples(σ), I∗(t) = I∗(t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The computation of Σ∗ terminates because (i) the sets under  construction for a given attribute A are all bounded by the finite set adom(A),  and (ii) the construction is monotonous, in the sense that if Σk and Σk+1 are  two consecutive states, for every σ in Σk+1, there exists σ′ in Σk such that  σ′ ⊑ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Given s in MT , we first show that I∗(s) is nonempty if there exists σ in Σ∗  such that s ⊑ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The proof is conducted by induction on the steps in the runs  of the loop line 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We show that if (Σk)k≥0 denotes the sequence of the states  of Σ∗ during the computation, for every σ in Σk, I∗(σ) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We first note in  this respect that since Σ0 = T, for every σ in Σ0, we have I∗(σ) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Assuming now that for k ≥ 0, for every σ ∈ Σk, I∗(σ) ̸= ∅, we prove the  result for every σ in Σk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Let σ in Σk+1 but not in Σk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Then σ occurs in  Σk+1 because there exist X → A in FD, σ1 and σ2 in Σk such that for every  B in X, σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='B ∩ σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='B ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Let σ′  1 = σ1 ⊔ σ2(A) and σ′  2 = σ2 ⊔ σ1(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Then,  either σ = σ′  1 and σ′  1 ̸= σ1 or σ = σ′  2 and σ′  2 ̸= σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  The two cases being  similar, we just consider the first one, that is σ = σ′  1 and σ′  1 ̸= σ1, which  implies that σ2(A) ̸= ∅ and σ2(A) ̸⊑ σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' By our induction hypothesis, we have  I∗(σ2) ̸= ∅ and thus denoting by x an X-tuple occurring in tuples(σ1 ⊓ σ2),  we have I∗(x) ∩ I∗(σ2(A)) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Since I∗ satisfies X → A, I∗(x) ⊆ I∗(σ2(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  As x ⊑ σ1, I∗(σ1) ⊆ I∗(x) and thus, I∗(σ1) ⊆ I∗(σ2(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Therefore, I∗(σ) =  I∗(σ1) ∩ I∗(σ2(A)) = I∗(σ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As by our induction hypothesis, I∗(σ1) ̸= ∅, we  obtain that I∗(σ) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Therefore, it holds that for every k ≥ 0, I∗(σ) ̸= ∅ for  every σ in Σk, and so, it holds that I∗(σ) ̸= ∅ for every σ in Σ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Thus, if s  satisfies that there exists σ in Σ∗ such that s ⊑ σ, we have I∗(σ) ⊆ I∗(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Hence  I∗(s) ̸= ∅, and this part of the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Conversely, we show that for every s, if I∗(s) ̸= ∅ then there exists σ in Σ∗  such that s ⊑ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The proof is done by induction on the construction of I∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  By definition of I0 = Ib, if I0(s) ̸= ∅ then s is a sub-tuple of a tuple t in  T, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=', s ⊑ t holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Since Σ0 = T, there exists σ in Σ0 such that t ⊑ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' By  monotonicity of the construction of Σ∗ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=', for every σk+1 in Σk+1 there exists  σk in Σk such that σk ⊑ σk+1), there exists σ∗ in Σ∗ such that σ ⊑ σ∗, which  shows that t ⊑ σ∗, thus that s ⊑ σ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Now, assuming that for every k ≥ 0 and every s, if Ik(s) ̸= ∅ then there  exists σ in Σ∗ such that s ⊑ σ, we prove that this result holds for Ik+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Let s be such that Ik(s) = ∅ and Ik+1(s) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  For every a in AD we  have Ik+1(a) = Ik(a) ∪ Ik(x) if x and a are the tuples involved in Exp(Ik, xa)  and Ik+1(a) = Ik(a) otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Since Ik+1(s) ̸= Ik(s) and since a is the only  symbol for which Ik has changed, it must be that a ⊑ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Therefore, writing s  44  as s′ ⊔ a, we have Ik+1(s) = Ik+1(s′) ∩ Ik+1(a) and Ik+1(s′) = Ik(s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' It follows  that Ik+1(s) = Ik(s′) ∩ (Ik(a) ∪ Ik(x)), thus that Ik+1(s) = (Ik(s′) ∩ (Ik(a)) ∪  (Ik(s′) ∩ Ik(x)), that is Ik+1(s) = Ik(s) ∪ (Ik(s′) ∩ Ik(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As Ik+1(s) ̸= ∅ and  Ik(s) = ∅, we have Ik(s′) ∩ Ik(x) ̸= ∅, in which case our induction hypothesis  entails that there exists σ in Σ∗ such that (s′ ⊔ x) ⊑ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Since by definition of  expansion, we also have Ik(x) ∩ Ik(a) ̸= ∅, Σ∗ contains an m-tuple σ′ such that  xa ⊑ σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' In this case, Algorithm 1 shows that there exists σ∗ in Σ∗ such that  (s′ ⊔ a) ⊑ σ∗, that is s ⊑ σ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' This part of the proof is therefore complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' If the returned value for fail is false, the previous item shows that there exist  X → A in FD, x in T (X) and a and a′ in adom(A) such that I∗(a)∩I∗(a′) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Therefore in this case, Inc(T ) is not empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Conversely, if Inc(T ) ̸= ∅, for every t  in Inc(T ), there exist A in U and a and a′ in adom(A) such that I∗(a)∩I∗(a′) ̸=  ∅, and the previous item shows that there exists σ in Σ∗ such that (aa′) ⊑ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Hence, during the execution of Algorithm 1, the value of fail must have been  turned to true line 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The proof is by induction on the construction of Σ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As all m-tuples in Σ0  can be seen as tuples, for every σ in Σ0, tuples(σ) consists of one tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The  result thus trivially holds in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Now assume that the result holds for Σk,  and let σ be in Σk+1 obtained by steps lines 16-20 of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Thus there  exist σ′ in Σ∗, X → A in FD, x0 in T (X) and a0 in adom(A) such that in Σk,  x0 ∈ tuples(σ(X)) ∩ tuples(σ′(X)), a0 is in tuples(σ′(A)) \\ tuples(σ(A)), and in  Σk+1, σ(A) becomes σ1(A) such that a0 ∈ tuples(σ1(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Let t1 be in tuples(σ1) such that t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='A = a0, and let x denote t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Then, t1  is not in tuples(σ) (because a0 ̸∈ tuples(σ(A))), but tuples(σ) contains a tuple t  such that t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X = x and a tuple t′ such that t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X = x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Thus tuples(σ1) contains  t1 such that t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='XA = xa0 and a tuple t′  1 such that t′  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='XA = x0a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  On the other hand, the fact that x0a0 occurs in tuples(σ′) implies that  I∗(x0a0) ̸= ∅, thus that I∗(x0) ⊆ I∗(a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Hence, I∗(x0a0) = I∗(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  We  now argue that I∗(t′  1) = I∗(t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Indeed, if σ(A) = ∅, then t′  1 can be writ-  ten as t′a0 in which case I∗(t′  1) = I∗(t′) ∩ I∗(a0) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Since it also holds  that I∗(t′) ⊆ I∗(x0) ⊆ I∗(a0), we have I∗(t′) ∩ I∗(a0) = I∗(t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Otherwise, if  σ(A) ̸= ∅ then t′ is written as q′a′ and thus t′  1 is written as q′a0 (a′ has been  replaced by a0 to obtain t′  1 from t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' In this case, I∗(t′  1) = I∗(q′) = I∗(t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  As t and t′ are both in tuples(σ), by our induction hypothesis, in Σi we have  I∗(t) = I∗(t′), which implies that I∗(t) = I∗(t′  1) = I(t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We therefore obtain  that for every t1 in tuples(σ1), there exists t in tuples(σ) such that I∗(t1) = I∗(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  As all tuples in t tuples(σ) have the same I∗(t), all tuples t1 in tuples(σ1) have  the same I∗(t1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The proof is therefore complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  �  C  Proof of Proposition 5  Proposition 5  Let T be a non-redundant table over universe U, FD the  set of associated functional dependencies and Σ∗ the non-redundant m-table as  computed by Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Given a query Q : select X from T [where Γ],  and a tuple x in T (X):  45  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' x is in MSC ans(Q) if and only if  (a) for every σ in Σ(x), tuples(σ(X)) = {x},  (b) there exists σ in ΣQ(x) such that |tuples(σ)| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' x is in SC ans(Q) if and only if  (a) for every σ in Σ(x), tuples(σ(X)) = {x},  (b) there exists s in Sat(Γ) ∩ True(T ) such that for every σ in ΣQ(x), if  s is in σ(sch(Γ)) then tuples(σ(sch(Γ)))) = {s}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' x is in C ans(Q) if and only if  (a) for every σ in Σ(x), tuples(σ(X)) = {x},  (b) there exists σ in ΣQ(x) such that tuples(σ(sch(Γ))) ⊆ Sat(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' x is in WC ans(Q) if and only if  (a) for every σ in Σ(x), tuples(σ(X)) = {x},  (b) ΣQ(x) is not empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  In this proof, Q is assumed to involve a selection condition Γ,  because the case where no selection condition is involved is a simplification, and  thus, is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' By Proposition 4(1) and (2), statement (a) in the proposition is equivalent  to statement (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='a) in Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Moreover, as Σ∗ is assumed to be non-  redundant, for every σ in Σ∗ and every t in tuples(σ), there exists no tuple  t′ in True(T ) such that t � t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Thus, by Proposition 4(2), statement (b) in  the proposition yields a tuple having the same properties as the tuple t in  statement (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b) of Definition 6 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=', t is a maximal true and consistent tuple  whose restriction over sch(Γ) satisfies Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Therefore this part of the proof is  complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As above, by Proposition 4(1) and (2), statement (a) in the proposition is  equivalent to statement (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='a) in Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Moreover, statement (b) in the  proposition yields a tuple t such that t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X = x, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='sch(Γ) = s, which is in Sat(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Moreover, Proposition 4(3) implies that this tuple t is true and consistent, thus  satisfying the statement (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b) of Definition 6 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=', t is a true and consistent  tuple whose restriction over sch(Γ) satisfies Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Therefore this part of the proof  is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Let us first assume that x is in C ans(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As x is in Cons(T ) ∩ True(T ),  by Proposition 4, Σ∗ must contain at least one σ such that x ⊑ σ and every  such m-tuple must be such that σ(X) = {x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Hence Σ(x) is nonempty and  the first item in the proposition holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Moreover, by Definition 6, one of these  m-tuples σ is also such that sch(Γ) ⊆ sch(σ) and tuples(σ) contains a tuple t  of Cons(Γ, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' By Definition 5, this implies that the following statement (*)  holds: t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='sch(Γ) is in Sat(Γ) and for every s in Sat−(Γ), I∗(t) ∩ I∗(s) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' This  in particular implies that σ is in ΣQ(x), because t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='(sch(Γ)) ⊑ t ⊑ σ holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  46  As ΣQ(x) ⊆ Σ(x), every σ in ΣQ(x) is such that σ(X) = {x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Assume now  that every σ in ΣQ(x) is such that tuples(σ(sch(Γ))) contains q0 not in Sat(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Then q0 is in Sat−(Γ) and tuples(σ) contains a tuple t0 such that t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='sch(Γ) = q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Thus we have two tuples t and t0 in tuples(σ) such that t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X = t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X = x,  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='sch(Γ) ∈ Sat(Γ) and t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='sch(Γ) ∈ Sat−(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Lemma 4(4) implies that I∗(t) =  I∗(t0) (because t and t0 are in tuples(σ)), and as I∗(t0) ⊆ I∗(q0) (because  t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='sch(Γ) = q0), we have I∗(t0) ∩ I∗(q0) = I∗(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Thus I∗(t0) ∩ I∗(q0) ̸= ∅,  implying that I∗(t) ∩ I∗(q0) ̸= ∅, which is a contradiction with statement (*)  above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' We therefore have shown that if x is in C ans(Q) then the items in the  proposition hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  Conversely, if the items in the proposition are satisfied, Proposition 4 implies  that x is in Cons(T ) ∩ True(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Thus by Definition 5, we have to prove the  existence of a tuple t in Cons(Γ, T )∩True(T ) such that x ⊑ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Let σ be in ΣQ(x)  as stated in item (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='b) of the proposition, that is such that tuples(σ(sch(Γ))) ⊆  Sat(Γ), and let t0 be in tuples(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As t0 is in True(T ), we have to show that t0  is in Cons(Γ, T ), that is that there exists q in Sat(Γ) such that (i) q ⊑ t0, and  (ii) for every q′ in Sat−(Γ), I∗(t0)∩I∗(q′) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Since we know that t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='sch(Γ) is  in Sat(Γ), (i) holds for q = t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='sch(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Regarding (ii), let q′  0 be in Sat−(Γ) such  that I∗(t0)∩I∗(q′  0) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As I∗(σ) = �  u∈tuples(σ) I∗(u), Lemma 4(4) implies that  I∗(σ) = I∗(t0), and thus that I∗(σ) ∩ I∗(q′  0) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Applying now Lemma 4(2),  we obtain that Σ∗ contains an m-tuple σ′ involving all attribute values in σ or  in q′  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Since q′  0 is not in tuples(σ), we have σ′ ̸= σ, and thus σ � σ′ holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' This  is a contradiction with our hypothesis that Σ∗ contains no redundancy, showing  that t0 is in Cons(Γ, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' This part of the proof is therefore complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' As in cases (1) and (2) above, by Proposition 4(1) and (2), statement (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='a)  in the proposition is equivalent to statement (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='a) in Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Moreover,  ΣQ(x) ̸= ∅ is equivalent to the existence of σ in Σ∗ such that σ(X) = {x} and  σ(sch(Γ)) contains a tuple s of Sat(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' Hence, ΣQ(x) ̸= ∅ is equivalent to the  existence of a tuple t in True(T ) (due to Proposition 4(1)) such that t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='X = x  and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='sch(Γ) is in Sat(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content=' The proof is therefore complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
+page_content='  �  47' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE2T4oBgHgl3EQfHgYi/content/2301.03668v1.pdf'}
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+An ensemble-based framework for mispronunciation detection of
+Arabic phonemes
+Sükrü Selim Calıka, Ayhan Kucukmanisab and Zeynep Hilal Kilimcic,∗
+aMaviay Consultancy Company, Kocaeli University Technopark, 41275, Kocaeli, Turkey
+bDepartment of Electronics and Communication Engineering, Kocaeli University, 41001, Kocaeli, Turkey
+cDepartment of Information Systems Engineering, Kocaeli University, 41001, Kocaeli, Turkey
+A R T I C L E I N F O
+Keywords:
+computer aided language learning
+Arabic pronunciation detection
+ensemble learning
+machine learning
+voting classifier
+A B S T R A C T
+Determination of mispronunciations and ensuring feedback to users are maintained by computer-
+assisted language learning (CALL) systems. In this work, we introduce an ensemble model that
+defines the mispronunciation of Arabic phonemes and assists learning of Arabic, effectively. To
+the best of our knowledge, this is the very first attempt to determine the mispronunciations of
+Arabic phonemes employing ensemble learning techniques and conventional machine learning
+models, comprehensively. In order to observe the effect of feature extraction techniques, mel-
+frequency cepstrum coefficients (MFCC), and Mel spectrogram are blended with each learning
+algorithm. To show the success of proposed model, 29 letters in the Arabic phonemes, 8 of
+which are hafiz, are voiced by a total of 11 different person. The amount of data set has been
+enhanced employing the methods of adding noise, time shifting, time stretching, pitch shifting.
+Extensive experiment results demonstrate that the utilization of voting classifier as an ensemble
+algorithm with Mel spectrogram feature extraction technique exhibits remarkable classification
+result with 95.9% of accuracy.
+1. Introduction
+Computer Aided Language Learning (CALL) systems are popular nowadays because they facilitate people to
+progress the ability of language and pronunciation. CALL mainly focus on fields such as pronunciation errors in
+non-native learners’ talk. CALL achieves tasks, such as detection of pronunciation error, speech recognition, and
+grading of pronunciation. Furthermore, there are many researches on speech processing, applied in various languages
+for the purpose of aiding language learning (Cucchiarini et al., 1998), (Neumeyer et al., 2000), Minematsu (2004).
+Advances in computer science, especially in artificial intelligence, have consented a great deal of research to be done
+with CALL (Cucchiarini et al., 1998), (Neumeyer et al., 2000), Minematsu (2004), (Adnan and Zamari, 2012), Hu
+et al. (2013), (Ehsani and Knodt, 1998), (Arias et al., 2010).
+For a certain language, speakers of distinct languages incline to compose pronunciation errors in their utterances
+because the muscles of mouth are not able to pronunciate the nuances of the specific language. For this reason,
+researchers generally focus on to investigate the mispronunciation for various languages such as English, Dutch, and
+French, however; literature studies on Arabic are limited. Nevertheless, studies on Arabic have been increasing in
+recent years (Abufanas, 2013), Khan et al. (2013), (Muaad et al., 2021), (Shaalan 1, 2005), (Nazir et al., 2019), (Ziafat
+et al., 2021). Arabic, the most widely spoken (Lane, 2019) with over 290 million native speakers and 132 million non-
+native speakers, and one of the six official languages of the United Nations (UN), occurs two main dialects, Classical
+Arabic (CA) and modern standard Arabic (MSA). Classical Arabic is the language of the Quran while MSA is the
+modified version of the Quran used in daily communication. The pronunciation rules in the language of the Quran
+are very well defined in order to preserve the correct meaning of the words. This work concentrates on detecting the
+mispronunciation of the phonemes in the Quran language employing both machine and ensemble learning models.
+Ensemble learning is a technique that employs multiple classifiers to acquire a better success compared to an
+individual classifier. In other words, a bunch of single classifiers is consolidated for the purpose of composing an
+ensemble-based classifier. The generation and consolidation steps are the main phases of ensemble learning strategy.
+∗Corresponding author: Zeynep Hilal Kilimci
+sselimcalik1@gmail.com (S.S. Calık); ayhan.kucukmanisa@kocaeli.edu.tr (A. Kucukmanisa);
+zeynep.kilimci@kocaeli.edu.tr (Z.H. Kilimci)
+ORCID(s):
+Selim et al.: Preprint submitted to Elsevier
+Page 1 of 13
+arXiv:2301.01378v1  [cs.SD]  3 Jan 2023
+
+aAn ensemble-based framework for mispronunciation detection of Arabic phonemes
+In generation phase of ensemble, a diversified data set of single learners is composed of the training set. The final
+decision is provided by consolidating each individual classifiers in the integration phase. The major approach in
+ensemble strategy is whence to produce many learners and combine decisions of each individual classifier such that
+the consolidation of base learners boosts the success of a single classifier (Rokach, 2010), (Polikar, 2006a), (Gopika
+and Azhagusundari, 2014), (Ren et al., 2016), Kilimci et al. (2016).
+In this study, we concentrate on the detection of mispronunciation of Arabic phonemes observing the effects of both
+conventional machine learning algorithms and ensemble learning techniques when blended with the feature extraction
+methods. For this purpose, support vector machine, k-nearest neighbors, decision tree, naive Bayes, random forest
+classifiers are evaluated diversifying with feature extraction techniques, namely mel-frequency cepstrum coefficients,
+and Mel spectrogram. After that, each decision of individual classifiers is consolidated majority voting, boosting,
+bagging, stacking with logistic regression, stacking with random forest, stacking with extra tree methods. Experiment
+results reveal that the usage of majority voting as an ensemble technique with Mel spectrogram feature extraction
+method gives considerable results with 95.3% of accuracy.
+The main contributions of the paper are as follows:
+∙ In order to accelerate future studies, our own dataset of Arabic letters is created with hafizes.
+∙ Providing better accuracy with low computational load compared to high-performance deep learning models.
+∙ Suitable for low capacity embedded platforms with its low complexity.
+The rest of paper is designed as follows: Section 2 introduces related literature studies on mispronunciation of
+Arabic phonemes. Section 3 presents individual machine learning classifiers and ensemble learning methods utilized
+in this work. Section 4 gives experimental results. The paper is concluded with discussion and conclusion in Section
+5.
+2. Literature Review
+In this section, literature studies on mispronunciation of Arabic phonemes are briefly introduced.
+Muhammad et al. (2010) present a non-real-time e-Hafiz system that proposes to avoid the Quran from being mis-
+pronunciation. For this purpose, data preparation, feature extraction, and modeling stages are performed. After silence
+removal and pre-emphasis methods are implemented at the data preparation step, framing, windowing, discrete Fourier
+transform, Mel Filter-bank, logarithm, and inverse discrete Fourier transformation techniques are applied in the fea-
+ture extraction phase, respectively. After that, similarity is calculated to find mispronunciation. The authors compare
+the introduced model with a counterpart web application on five people and inform that they exhibit advancement in
+their memorization. In other study, (Elsayed and Fathy, 2019) aim an automated model to assess the memorization of
+the Qur’an depending on Hafiz reading. Mel-frequency cepstral coefficient is assessed as feature extraction technique
+while vector quantization is employed for the purpose of dimension reduction by the proposed system. The authors
+present that the experiment results ensure remarkable accuracy scores to evaluate Quran memorization with the uti-
+lization of proposed system. In another study (Putra et al., 2012), a speech recognition model is developed to learn
+the Holy Quran. Gaussian mixture model is blended with MFCC feature extraction technique. Different speech signal
+processing techniques are applied namely, minimal sampling frequency, frame blocking, windowing, discrete Fourier
+transform, and dynamical 40 cepstral coefficient and spectrum energy in addition to MFCC. The authors present the
+experimental results provide 70% of accuracy performance for pronunciation, 90% of accuracy success for reading
+rules of tajweed, and 60% of accuracy score for the consolidation of both pronunciation and tajweed reading rules.
+(Arafa et al., 2018) introduce a speech recognition system that is able to discover mispronunciation. The dataset
+is composed of 89 students, which of 46 is female. 28 Arabic phonemes are voiced 10 times. After gathering 890
+utterances, MFCC coefficients are extracted for modeling with five different machine learning models. These are k-
+nearest neighbor, support vector machine, naive Bayes, multi layer perceptron, and random forest. The experimental
+results indicate that the random forest technique exhibits 85.02% of accuracy result, which is better than compared to
+other machine learning models. In another recent study, (Nazir et al., 2019) present a model depending on mispronun-
+ciation detection employing deep convolutional neural network properties for Arabic phonemes. As a first, features
+are extracted various layers of CNN ranges from 4 to 7. After that, k-nearest neighbor, support vector machine, and
+neural network classifier are employed for training phase. They compare the experiment result when hand-crafted
+features are employed instead of utilizing deep features. Finally, the authors emphasize that the proposed learning
+method considerably boosts the success of the mispronunciation determination process in terms of accuracy by 92.2%
+when deep features are employed. In a study, (Akhtar et al., 2020) introduce a model to develop the detection of
+Selim et al.: Preprint submitted to Elsevier
+Page 2 of 13
+
+An ensemble-based framework for mispronunciation detection of Arabic phonemes
+mispronunciation of Arabic words for non-native students utilizing features of deep convolutional neural network. To
+utilize deep features, features are extracted from layers 6, 7, and 8 of Alex Net. After that, n-nearest neighbor, support
+vector machine, and random forest are employed at the training phase. To show the effect of deep features, authors
+also evaluate the features extracted using mel frequency cepstral coefficients. The same three model are employed and
+the results of both approach is compared. They emphasize that the introduced model with 93.2% of accuracy achieves
+the determination of incorrect pronunciation of Arabic words. (Maqsood et al., 2016) concentrate on the detection
+of mispronunciation for Arabic phonemes employing acoustic phonetic features (APF) with support vector classifier.
+For this purpose, authors concentrate on the acoustic phonetic features instead of utilizing confidence measure-based
+scores. The data set is constructed with 500 utterances of 100 speakers. The features are composed of root mean
+square energy (RMSE), MFCCs,low energy, spectral features, zero-cross rate, pitch and statistical features. After that,
+SVM is employed to determine the mispronunciation of Arabic phonemes. The paper is concluded that support vector
+machine demonstrates 97.5% of accuracy result.
+In another study, (Farooq and Imran, 2021) focus on determination of mispronunciation in articulation points for
+Arabic letters. For this purpose, Relative Spectral Transform - Perceptual Linear Prediction feature extraction technique
+is blended with Hidden Markov Model (HMM). 1,486 samples are gathered for 29 Arabic letters. Various techniques
+are employed to perform with HMM, like for discovering the observation probability Gaussian mixture models or for-
+ward method is utilized. Authors conclude the study that proposed model is capable to recognize the mispronunciation
+performing 85% of recognition rate. In the study (Nazir et al., 2021), support vector machine-based system is inves-
+tigated for the aim of determining mispronunciation of Arabic phonemes for Asian speakers on 28 Arabic phonemes.
+The proposed system is accomplished 88% of accuracy. (Asif et al., 2021) concentrate on the correct pronunciation
+of Arabic vowels with the help of deep neural networks. The new data set is constructed and augmented with various
+techniques due to the limited number of samples. The data set is composed of 85 individual records which of 43 is
+female and only four of them are non-native speakers. After gathering 6,229 records, preprocessing stage is carried
+out. For this purpose, noise reduction, data segmentation, re-sampling, and silence truncation are implemented. To
+determine the mispronunciation of Arabic vowels, optimized convolutional neural network is modeled. The proposed
+model shows 95.77% of accuracy score on pronunciation classification of classical Arabic phonemes. (Maqsood et al.,
+2017) perform comparative analysis of different classifiers for the purpose of detecting mispronunciation for Arabic
+phonemes using acoustic phonetic features namely, MFCC. The following machine learning algorithms are employed:
+Random forest, naive Bayes, Ada-boost, and k-nn. Authors report that random forest model outperforms others with
+95.4% of accuracy.
+(Shareef and Al-Irhayim, 2022) concentrate on the detailed comparison of feature extraction models for the pur-
+pose of detecting impairment Arabic speech. The feature extraction technique is based on diverse versions of wavelet
+transformation. To detect the impairment Arabic speech, LSTM and CNN-LSTM models are constructed. The com-
+bination of MFCC and LSTM achives the best classification accuracy with 93% while it is followed by CNN-LSTM
+with 91% of accuracy. Mispronunciation detection system is designed by (Algabri et al., 2022) for non-native Arabic
+speakers for the purpose of providing them impairment feedback. The proposed system ensures feedbacks to the Ara-
+bic learners utilizing deep learning-based models in the level of word and sentence. Experiment results show that the
+best model performs nearly 4% of error rate for phoneme detection, and roughly 70% of F1-score for mispronunciation
+detection. (Yang et al., 2022) propose improved mispronunciation detection system based on online pseudo-labeling
+and wav2vec model for English. In details, pseudo-labeling procedure is performed using unlabeled L2 speech and
+pre-trained self-supervised learning model (wav2vec) is carried out for enhancing fine-tuning approach. Experiment
+results demonstrate remarkable score when compared to traditional offline pseudo-labeling techniques. The model
+exhibits reduction in phoneme error rate nearly 5% and approximately 2.5% enhancement in F1-score for detecting
+mispronunciation of L2 speech. (Peng et al., 2022) present a text aware model for detection of mispronunciation by
+enhancing weight of audio features compared to score of unrelated text features. For this purpose, contrastive learning
+procedure is carried out in TIMIT and L2-Arctic English data sets. They report that proposed model ensures roughly
+4% improvement comparing with the baselines.
+Our study differs from aforementioned literature works as well mainly because we concentrate on the ensemble-
+based strategy to enable better accuracy with low computational load compared to high-performance deep learning
+models. Furthermore, our own dataset of Arabic letters is composed for the purpose of accelerating future studies.
+Selim et al.: Preprint submitted to Elsevier
+Page 3 of 13
+
+An ensemble-based framework for mispronunciation detection of Arabic phonemes
+3. Proposed Framework
+In this work, an ensemble machine learning based mispronunciation detection method for Arabic phonemes is
+proposed. Flowchart of the proposed system is depicted in Figure 1. The proposed system has three main stages.
+First step is to utilize preprocessing to improve the sound quality and the performance of the next steps. Next, feature
+extraction process is applied to the raw audio signals obtained from the microphone. Finally, the performance of
+machine learning-based approaches on the problem is examined. Then, the performance is improved by using the
+ensemble learning approach on these methods. Arabic phonemes used in this work is Arabic letters which is given in
+Figure 2. In this figure isolated forms and pronunciation of letters are presented.
+Figure 1: The flowchart of the proposed model.
+3.1. Pre-processing
+The pre-processing step aims to separating the clean speech from a mixed sound of speech and noise and get
+trimming only the part with the audio signal. Noise reduction which relies on a method called “spectral gating” which
+is a form of Noise Gate is primarily used to improve speech recognition. Then, silence removal process is applied.
+The beginning of speech indicates that the sound level rises above a certain dB value. In this study, the segment where
+the audio signal first rises above 30 dB and then drops below 30 dB is trimmed and used as an input to the algorithms.
+Selim et al.: Preprint submitted to Elsevier
+Page 4 of 13
+
+AUDIO
+DATASET
+PRE-PROCESSING
+DATA
+NOISE
+SILENCE
+REMOVAL
+AUGMENTATION
+REDUCTION
+(ONLY IN TRAINING)
+EXTRACTION
+FEATURE
+MFCC
+MEL-SPECTOGRAM
+MACHINELEARNING
+ENSEMBLELEARNING
+METHODS
+METHODS
+SVM
+VOTING
+DETECTION
+K-NN
+STACKING
+DECISION
+BOOSTING
+TREE
+NAIVEBAYES
+BAGGING
+RANDOM
+FOREST
+FINAL
+DECISIONAn ensemble-based framework for mispronunciation detection of Arabic phonemes
+Figure 2: Isolated form and pronunciation of Arabic letters.
+Figure 3 shows a sample input signal before and after noise reduction and silence removal operations. As the last
+preprocessing operation, the number of data is increased by augmentation process. Noise adding, time shifting, time
+stretching and pitch shifting are the used augmentation techniques. This process is not included in the testing process,
+but only in the training process.
+3.2. Feature extraction
+Human speech has various distinctive features. However, processing raw audio signal is generally not preferred
+due to the high noise and data size. It is observed that extracting distinctive features from the audio signal and using
+it as input to the base machine learning model will produce much better performance than directly considering raw
+audio signal as input. MFCC (Dave, 2013) and Mel-Spectrogram (Stevens et al., 1937) are two widely used technique
+for extracting the features from the audio signal.
+The Mel-spectrogram consist of 128 Mel feature and Fast Fourier Transform (FFT) with 2048 window length. The
+dimensions of the obtained features differ according to the audio duration. Therefore, the feature vector size is scaled
+to a single dimension. After the Mel features are obtained, it is scaled to 1×128 by taking the averages (before scaling
+the dimensions are like 128×48, 128×40 for example). In the end, the problem evolves into the classification of the
+feature vectors obtained in size 1×128.
+MFCC features are the result of a series of applied processes. These processes are as follows: Windowing of the
+audio signal, applying Discrete Fourier Transform (DFT), obtaining the receipt of the logarithm of amplitude, distorting
+frequencies on a Mel scale, and applying inverse discrete cosine transform (DCT). As a result of these processes, 39
+features are obtained, of which the last 20 have more distinctiveness. In this study, these last 20 features are used as
+input features. Similar to the Mel-spectrogram, the audio file size causes a change in the dimensions of the features.
+For this reason, all features are averaged among themselves and the problem turns into the classification of the data in
+the size of 1×20.
+Figure 4 shows the scaling process of feature vectors and Figure 5 presents average Mel-Spectrogram and MFCC
+features for 2 letters (“Alif” and “Baa”). In Figure 5, different colours represent the speeches of different hafizes. As
+Selim et al.: Preprint submitted to Elsevier
+Page 5 of 13
+
+Isolated Form
+2
+Pronunciation
+haa
+jiim
+thaa
+taa
+baa
+alif
+Isolated Form
+3
+?
+Pronunciation
+siin
+zaay
+raa
+thaal
+daal
+kha
+Isolated Form
+&
+上
+CBCD
+Pronunciation
+ayn
+thaa
+taa
+daad
+saad
+shiin
+Isolated Form
+L
+m.
+Pronunciation
+miim
+laam
+kaaf
+qaaf
+faa
+ghayn
+R
+9
+Isolated Form
+Pronunciation
+lamelif
+yaa
+waaw
+ha
+unnuAn ensemble-based framework for mispronunciation detection of Arabic phonemes
+(a)
+(b)
+(c)
+(d)
+Figure 3: Noise reduction and silence removal pre-processing operations (a) Original audio signal (b) After noise reduction
+(c) After noise silence removal (d) After noise reduction and silence removal.
+seen from this figure, even if different hafizes say the same letter, it creates dissimilar characteristic on the formed
+feature vector. In addition, it is seen that the distinctiveness of the same people in different letters is not sufficient.
+Because of these difficulties, the performance of the classification method is become more important.
+3.3. Detection
+3.3.1. Machine learning methods
+Machine learning can be defined as artificial intelligence algorithms that can infer and predict from data to mimic
+the way humans learn. There are various machine learning algorithms that are capable of solving classification, re-
+gression and clustering tasks. In our work, popular methods suitable for classification problem are emphasized which
+are support vector machine (SVM), k-Nearest neigbour (k-NN), decision tree (DT), naïve Bayes, random forest.
+SVM (Cristianini and Shawe-Taylor, 2000) is a supervised learning approach. Kernel functions can also be used
+depending on the type of data during the operation of the algorithm. In this way, both linear and nonlinear classification
+operations can be performed. It is aimed to separate all data with a hyperplane. However, if the data cannot be fully
+separated, they cannot be classified with a single plane. Therefore, different kernel functions are used. A margin is
+determined around the hyperplane. Whether this margin is large or small directly affects the classification performance.
+Margin can be controlled with the “C” hyperparameter. The larger the C, the narrower the margin. Also, if the model
+is overfit, C needs to be reduced. In this work linear kernel and 0.02 used as C parameter.
+k-NN (Mucherino et al., 2009) is basically based on the determination of the class of the data whose class is
+unknown, according to the nearest “k” neighbor from the data in the training set. As a result of performing a distance
+measurement between the test data and the training data, the nearest “k” nearest neighbors are determined. Then, the
+class value of the tested data is determined according to these labels. In this work, 3,4 and 5 is evaluated as “k” value.
+Basic purpose of decision trees (Alp, 2019) is to divide the data set into smaller subgroups that are more visually
+Selim et al.: Preprint submitted to Elsevier
+Page 6 of 13
+
+0.75
+0.50
+0.25
+Amplitude
+0.00
+-0.25
+-0.50
+-0.75
+0
+5000
+10000
+15000
+20000
+25000
+Time0.6
+0.4
+0.2
+Amplitude
+0.0
+-0.2
+-0.4
+-0.6
+-0.8
+0
+5000
+10000
+15000
+20000
+25000
+Time0.75
+0.50
+0.25
+Amplitude
+0.00
+-0.25
+-0.50
+-0.75
+0
+2500
+5000
+7500
+10000
+12500
+15000
+17500
+Time0.6
+0.4
+0.2
+Amplitude
+0.0
+-0.2
+-0.4
+-0.6
+-0.8
+0
+2500
+5000
+7500
+10000
+12500
+15000
+17500
+TimeAn ensemble-based framework for mispronunciation detection of Arabic phonemes
+(a)
+(b)
+(c)
+(d)
+Figure 4: Scaling of feature vectors (a) Original Mel-spectrogram image (b) Averaged Mel-spectrogram image (c) Original
+MFCC image (d) Averaged MFCC image.
+understandable within the framework of certain rules (decision rules). Since the output of the algorithm is a flowchart
+that looks like a tree visually, it is called a decision tree. There are 4 basic structures on a decision tree: root node,
+nodes, branches and leaves (terminal node). The root node is where classification process starts from this point. If the
+observations are in a homogeneous structure, they will naturally be in the same class and the classification process will
+end without branching the root node. In heterogeneous observations, the root node divides into two or more branches
+according to the best quality that divides the observations into classes and creates new nodes. The last non-branching
+node of the tree is the terminal node and represents the classes to which the observations are assigned.
+Naïve Bayes (Webb et al., 2010) classification is based on Bayes theorem. It is used to estimate the probability
+that a particular set of features belongs to a particular class. It aims to select the decision with the highest probability
+using probability calculations. Each attribute is considered independent from other attributes in the class. different
+class based on various attributes. Naïve Bayes classifiers are extremely fast compared to more complex methods.
+Selim et al.: Preprint submitted to Elsevier
+Page 7 of 13
+
+100
+0
+-100
+Value
+-200
+-300
+-400
+mfcc1
+mfcc2
+-500
+mfcc3
+mfcc4
+-600
+0
+5
+10
+15
+CoefficientsNumber0
+-100
+Value
+-200
+-300
+-400
+averagemfcc
+0
+5
+10
+15
+Coefficients Number350
+mel1
+mel2
+300
+mel3
+mel4
+250
+200
+Value
+150
+100
+50
+0
+0
+25
+50
+75
+100
+125
+CoefficientsNumberaverage mel
+20
+15
+Value
+10
+5
+0
+0
+25
+50
+75
+100
+125
+CoefficientsNumberAn ensemble-based framework for mispronunciation detection of Arabic phonemes
+(a)
+(b)
+(c)
+(d)
+Figure 5: Average feature vectors of “Alif” and “Baa” letters (a) Average Mel-spectrogram feature of “Alif” (b) Average
+MFCC feature of “Alif” (c) Average Mel-spectrogram feature of “Baa” (d) Average MFCC feature of “Baa”.
+3.3.2. Ensemble learning techniques
+Ensemble learning is an approach to boost the overall accuracy of a classification framework by utilizing multi-
+ple learning algorithms. The strategy usually yields better results than a individual learning model. There are most
+commonly employed ensemble learning techniques in the literature namely, bagging, stacking, boosting, and voting.
+Bagging is one of the most widely-applied ensemble based algorithms (L., 1996). It is known as the abbreviation
+of bootstrap aggregating. In this approach, diversity is performed by re-sampling in which various training data subsets
+are randomly chosen with replacement from the whole training data set. Each subset is assessed to train a distinct base
+learner in the set of ensemble learners. Final decision is determined by combining decisions of individual learners by
+taking a majority vote.
+Boosting (Freund and Schapire, 1996) is another ensemble learning model that is asserted as an alternative to
+the bagging technique. The main approach behind of this approach is to produce a set of individual classifiers that
+utilizes a data subset in which each instance is consolidated with a weight. It is carried out iteratively running base
+learners on different distributions over the training data set. While all instances have same weight at the first step,
+the weights of miscategorized instances are updated at each iteration according as the training error of preceding base
+learners. Each learner employs a subset of instances acquired from an updated version of training data set. After that,
+instances that are incorrectly forecasted by preceding learners are picked up more frequently than the instances that
+are correctly forecasted. The final outcome is achieved by weighted majority voting of the categories forecasted by the
+base learner. AdaBoost, AdaBoost.M1, AdaBoost.M2, AdaBoost.R, Arcing and Real Adaboost (L., 2010), (Polikar,
+2006b), (Gopika D. Azhagusundari, 2014), (Ren Y., 2016) are the variations employed in the literature. AdaBoost.M1
+is used in this work.
+Stacking is the process of fitting multiple types of models to the same data and then employing metal level model
+to learn how to integrate the predictions in the best way possible (Džeroski, 2004). There are two major steps. The first
+one is consists of generating a set of individual classifiers utilizing training set. Thus, meta-data set is composed of the
+decisions of individual classifiers. After construction of the meta-data set, meta-level learner modeled with this data
+Selim et al.: Preprint submitted to Elsevier
+Page 8 of 13
+
+mel1
+80
+mel2
+mel3
+mel4
+60
+lue
+40
+20
+0
+100
+101
+102
+CoefficientsNumber75
+mfcc2
+mfcc3
+50
+mfcc4
+25
+lue
+0
+-25
+-50
+-75
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+Coefficients Number175
+mel1
+mel2
+150
+mel3
+mel4
+125
+100
+Value
+75
+50
+25
+0
+100
+101
+102
+Coefficients Number150
+mfcc1
+mfcc2
+mfcc3
+100
+mfcc4
+50
+Value
+0
+-50
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+Coefficients NumberAn ensemble-based framework for mispronunciation detection of Arabic phonemes
+set in order to get generalized estimations. In our work, the meta-data set comprises of both original training examples
+and decisions of individual classifiers. Moreover, stacking method is employed with different learning algorithms
+namely, logistic regression, random forest, and extra tree in our study.
+In majority voting, each model makes a prediction (vote) for each test sample, and the prediction of the final result
+is the model that receives the most votes.
+Random forest (Breiman, 2001) is a set of decision tree classifiers. It is a certain enforcement of bagging method
+in which each individual classifier is a decision tree. Bagging is utilized to choose training sub sets for each base
+decision tree. The division task employed in Random forests varies from well-known decision tree strategy where
+each node is divided by the best attribute among all other attributes. In Random forest approach, a random subset of
+attributes is chosen first and then the best division is determined on the random subset of attributes. This approach
+ensures an extra randomness to the model in addition to bagging. Random forests is able to cope with over-fitting
+problem due to randomness performed in both sample and attribute spaces.
+Figure 6 shows an overview of the ensemble architectures.
+(a)
+(b)
+(c)
+(d)
+Figure 6: Ensemble architectures (a) Voting (b) Bagging (c) Stacking (d) Boosting.
+4. Experimental results
+In this study, an original dataset is obtained by collecting audio samples from 11 speakers, 8 of whom are hafiz.
+Dataset consists of Arabic letters. All letters are recorded by asking the speakers to say each letter individually. Original
+dataset size is 290. Sample size in the dataset is increased to 1450 by using noise adding, time shifting, time stretching,
+Selim et al.: Preprint submitted to Elsevier
+Page 9 of 13
+
+DATASET
+MODEL1
+VODEL2
+MODEL3
+MODELN
+FINAL
+DECISIONDATASET
+SUBSET
+SUBSET
+SUBSET3
+SUBSETN
+MODEL1
+MODEL
+MODEL3
+MODELN
+FINAL
+DFCISIONDATASET
+MODEL1
+MODEL2
+MODEL3
+MODELN
+METAMODEL
+FINAL
+DECISIONSUBSET1
+SUBSET 2
+SUBSET3
+SUBSETN
+Update
+Update
+Update
+[Valuesor Weights)
+(Values or Weights)
+(Values or Weights)
+MODEL1
+MODEL2
+MODEL3
+MODELN
+FINAL
+DECISIONAn ensemble-based framework for mispronunciation detection of Arabic phonemes
+Table 1
+Evaluation results of machine learning methods with MFCC features
+Method
+Accuracy
+Precision
+Recall
+F1-Score
+SVM
+0.758
+0.829
+0.785
+0.806
+k-NN (k=3)
+0.466
+0.527
+0.466
+0.494
+k-NN (k=4)
+0.308
+0.310
+0.308
+0.309
+k-NN (k=5)
+0.307
+0.34
+0.307
+0.322
+Decision Tree
+0.714
+0.735
+0.710
+0.722
+Naïve Bayes
+0.392
+0.445
+0.392
+0.417
+Random Forest
+0.756
+0.78
+0.754
+0.766
+Table 2
+Evaluation results of machine learning methods with Mel-Spectrogram features
+Method
+Accuracy
+Precision
+Recall
+F1-Score
+SVM
+0.952
+0.970
+0.952
+0.960
+k-NN (k=3)
+0.953
+0.968
+0.953
+0.960
+k-NN (k=4)
+0.698
+0.734
+0.698
+0.716
+k-NN (k=5)
+0.670
+0.700
+0.67
+0.684
+Decision Tree
+0.827
+0.852
+0.823
+0.837
+Naïve Bayes
+0.538
+0.68
+0.538
+0.600
+Random Forest
+0.938
+0.947
+0.938
+0.942
+pitch shifting augmentation methods. The dataset is divided into 80% training and 20% test. 5-fold cross-validation
+procedure is used to ensure that the score of proposed models does not depend on the way we select train and test
+subsets. The performance of the proposed methods is measured using the formulas given in (1), (2), (3) and (4). In
+these equations, if we named every class as 퐶푥, TP (True Positive) represents audio belonging to the 퐶푥 is correctly
+classified as 퐶푥. FP (False Positive) is all non-퐶푥 samples classified as 퐶푥. TN (True Negative) denotes all non- 퐶푥
+samples not classified as 퐶푥. FN (False Negative) represents all 퐶푥 samples not classified as 퐶푥.
+퐴푐푐푢푟푎푐푦 =
+푇 푃 + 푇 푁
+푇 푃 + 푇 푁 + 퐹푃 + 퐹푁
+(1)
+푃 푟푒푐푖푠푖표푛 =
+푇 푃
+푇 푃 + 퐹푃
+(2)
+푅푒푐푎푙푙 =
+푇 푃
+푇 푃 + 퐹푁
+(3)
+퐹 − 푚푒푎푠푢푟푒 = 2 × 푃푟푒푐푖푠푖표푛 × 푅푒푐푎푙푙
+푃푟푒푐푖푠푖표푛 + 푅푒푐푎푙푙
+(4)
+Evaluation results of machine learning based methods are given in Table 1 and Table 2. Table 1 shows effect
+of MFCC features used as input of machine learning methods. Likewise, Table 2 shows effect of Mel-Spectrogram
+features as input. In these tables, values with bold font represent best results. According to Table 1 SVM has the best
+results among the other methods. In Table 2, it can be considered that k-NN where k is determined as 3 is successful.
+When Table 1 and Table 2 are evaluated together, the highest result is obtained with the k-NN method, in which
+Mel-Spectrogram features are used as input features.
+The ensemble learning approach is applied to the methods given in Table 1 and Table 2. Evaluation results of
+ensemble learning methods are given in Table 3 and Table 4. In these tables, values with bold font represent best
+results. When Table 3 is examined, it can be seen that stacking with random forest approach is the best compared to
+the other approaches. Table 4 clearly shows that voting approach has the best results.
+Selim et al.: Preprint submitted to Elsevier
+Page 10 of 13
+
+An ensemble-based framework for mispronunciation detection of Arabic phonemes
+Table 3
+Evaluation results of ensemble learning methods with MFCC features
+Method
+Accuracy
+Precision
+Recall
+F1-Score
+Voting
+0.772
+0.794
+0.764
+0.778
+Stacking (Logistic Regression)
+0.700
+0.720
+0.704
+0.711
+Stacking (Random Forest)
+0.761
+0.804
+0.768
+0.785
+Stacking (Extra Tree)
+0.731
+0.767
+0.747
+0.756
+Boosting
+0.744
+0.793
+0.740
+0.765
+Bagging
+0.750
+0.781
+0.762
+0.771
+Table 4
+Evaluation results of ensemble learning methods with Mel-Spectrogram features
+Method
+Accuracy
+Precision
+Recall
+F1-Score
+Voting
+0.959
+0.969
+0.957
+0.962
+Stacking (Logistic Regression)
+0.866
+0.893
+0.862
+0.877
+Stacking (Random Forest)
+0.951
+0.960
+0.947
+0.953
+Stacking (Extra Tree)
+0.938
+0.944
+0.939
+0.941
+Boosting
+0.871
+0.932
+0.855
+0.891
+Bagging
+0.943
+0.953
+0.935
+0.943
+Table 5
+Performance evaluation with deep learning based methods
+Method
+Accuracy
+Precision
+Recall
+F1-Score
+Time (s)
+(Abdoli et al., 2019)
+0.910
+0.920
+0.910
+0.915
+0.470
+(Choi et al., 2018)
+0.900
+0.900
+0.900
+0.900
+0.612
+Proposed Method
+0.959
+0.969
+0.957
+0.962
+0.091
+Figure 7: Letter-based performance evaluation with deep learning based methods.
+Table 5 shows the comparison of the recent and proposed methods for sound classification in the literature, in terms
+of processing speed and detection performance. The comparison is carried out on the dataset created within the scope
+of this study. The methods used in comparison in Table 5 are deep learning-based approaches. These methods are
+implemented using original network structure and parameters specified in the papers. In this table, proposed method
+refers voting ensemble method with Mel-Spectrogram features input. As seen in Table 5, proposed method has su-
+perior performance results and the lowest processing speed among the other methods. In addition, the letter-based
+performances of the methods given in Table 5 are analyzed. Figure 7 shows the performances of (Abdoli et al., 2019),
+Choi et al. (2018) and the proposed method for 10 letters. It is understood that the proposed method shows a steady
+performance despite the high-performance decrease in some letters in other methods. Table 5 and Figure 7 show that
+popular deep learning approaches can be surpassed in mispronunciation detection of Arabic phonemes with the pro-
+Selim et al.: Preprint submitted to Elsevier
+Page 11 of 13
+
+Proposed System
+[45]
+[46]
+1.0
+0.8
+0.4
+0.2
+0.0
+亡
+)
+LettersAn ensemble-based framework for mispronunciation detection of Arabic phonemes
+posed ensemble approach. In addition, it should be noted that generally deep learning-based approaches require more
+training time and processing (inference) time due to their complexity. Arabic mispronunciation detection is a chal-
+lenging problem. It is significant that the proposed framework ensures superior performance with less computational
+load to this problem.
+5. Discussion and Conclusion
+In this work, an ensemble learning approach is proposed to detect mispronunciation of Arabic phonemes. In the
+proposed method, firstly, feature extraction is performed with MFCC and mel-spectrogram methods. Then, traditional
+and ensemble learning-based approaches used these features as input and their performance is evaluated on the original
+data set created for this study. The method with the highest performance obtained from evaluation is also compared with
+the deep learning-based approaches. Experimental results show that the utilization of voting classifier as an ensemble
+algorithm with mel-spectrogram feature extraction technique reveals remarkable results with 95.9% of accuracy. Future
+studies will focus on increasing the data set and transfer learning techniques.
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+Nazir, F., Majeed, M.N., Ghazanfar, M.A., Maqsood, M., 2019. Mispronunciation detection using deep convolutional neural network features and
+transfer learning-based model for arabic phonemes. IEEE Access 7, 52589–52608.
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+Page 13 of 13
+
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+page_content='An ensemble-based framework for mispronunciation detection of  Arabic phonemes  Sükrü Selim Calıka,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Ayhan Kucukmanisab and Zeynep Hilal Kilimcic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='∗  aMaviay Consultancy Company,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Kocaeli University Technopark,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' 41275,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Kocaeli,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Turkey  bDepartment of Electronics and Communication Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Kocaeli University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' 41001,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Kocaeli,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Turkey  cDepartment of Information Systems Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Kocaeli University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' 41001,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Kocaeli,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Turkey  A R T I C L E I N F O  Keywords:  computer aided language learning  Arabic pronunciation detection  ensemble learning  machine learning  voting classifier  A B S T R A C T  Determination of mispronunciations and ensuring feedback to users are maintained by computer-  assisted language learning (CALL) systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' In this work, we introduce an ensemble model that  defines the mispronunciation of Arabic phonemes and assists learning of Arabic, effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' To  the best of our knowledge, this is the very first attempt to determine the mispronunciations of  Arabic phonemes employing ensemble learning techniques and conventional machine learning  models, comprehensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' In order to observe the effect of feature extraction techniques, mel-  frequency cepstrum coefficients (MFCC), and Mel spectrogram are blended with each learning  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' To show the success of proposed model, 29 letters in the Arabic phonemes, 8 of  which are hafiz, are voiced by a total of 11 different person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The amount of data set has been  enhanced employing the methods of adding noise, time shifting, time stretching, pitch shifting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  Extensive experiment results demonstrate that the utilization of voting classifier as an ensemble  algorithm with Mel spectrogram feature extraction technique exhibits remarkable classification  result with 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='9% of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Introduction  Computer Aided Language Learning (CALL) systems are popular nowadays because they facilitate people to  progress the ability of language and pronunciation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' CALL mainly focus on fields such as pronunciation errors in  non-native learners’ talk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' CALL achieves tasks, such as detection of pronunciation error, speech recognition, and  grading of pronunciation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Furthermore, there are many researches on speech processing, applied in various languages  for the purpose of aiding language learning (Cucchiarini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', 1998), (Neumeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', 2000), Minematsu (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  Advances in computer science, especially in artificial intelligence, have consented a great deal of research to be done  with CALL (Cucchiarini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', 1998), (Neumeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', 2000), Minematsu (2004), (Adnan and Zamari, 2012), Hu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' (2013), (Ehsani and Knodt, 1998), (Arias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  For a certain language, speakers of distinct languages incline to compose pronunciation errors in their utterances  because the muscles of mouth are not able to pronunciate the nuances of the specific language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' For this reason,  researchers generally focus on to investigate the mispronunciation for various languages such as English, Dutch, and  French, however;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' literature studies on Arabic are limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Nevertheless, studies on Arabic have been increasing in  recent years (Abufanas, 2013), Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' (2013), (Muaad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', 2021), (Shaalan 1, 2005), (Nazir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', 2019), (Ziafat  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Arabic, the most widely spoken (Lane, 2019) with over 290 million native speakers and 132 million non-  native speakers, and one of the six official languages of the United Nations (UN), occurs two main dialects, Classical  Arabic (CA) and modern standard Arabic (MSA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Classical Arabic is the language of the Quran while MSA is the  modified version of the Quran used in daily communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The pronunciation rules in the language of the Quran  are very well defined in order to preserve the correct meaning of the words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' This work concentrates on detecting the  mispronunciation of the phonemes in the Quran language employing both machine and ensemble learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  Ensemble learning is a technique that employs multiple classifiers to acquire a better success compared to an  individual classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' In other words, a bunch of single classifiers is consolidated for the purpose of composing an  ensemble-based classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The generation and consolidation steps are the main phases of ensemble learning strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  ∗Corresponding author: Zeynep Hilal Kilimci  sselimcalik1@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='com (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Calık);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' ayhan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='kucukmanisa@kocaeli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='tr (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Kucukmanisa);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  zeynep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='kilimci@kocaeli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='tr (Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Kilimci)  ORCID(s):  Selim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 1 of 13  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='01378v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='SD]  3 Jan 2023  aAn ensemble-based framework for mispronunciation detection of Arabic phonemes  In generation phase of ensemble, a diversified data set of single learners is composed of the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The final  decision is provided by consolidating each individual classifiers in the integration phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The major approach in  ensemble strategy is whence to produce many learners and combine decisions of each individual classifier such that  the consolidation of base learners boosts the success of a single classifier (Rokach, 2010), (Polikar, 2006a), (Gopika  and Azhagusundari, 2014), (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', 2016), Kilimci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  In this study, we concentrate on the detection of mispronunciation of Arabic phonemes observing the effects of both  conventional machine learning algorithms and ensemble learning techniques when blended with the feature extraction  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' For this purpose, support vector machine, k-nearest neighbors, decision tree, naive Bayes, random forest  classifiers are evaluated diversifying with feature extraction techniques, namely mel-frequency cepstrum coefficients,  and Mel spectrogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' After that, each decision of individual classifiers is consolidated majority voting, boosting,  bagging, stacking with logistic regression, stacking with random forest, stacking with extra tree methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Experiment  results reveal that the usage of majority voting as an ensemble technique with Mel spectrogram feature extraction  method gives considerable results with 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='3% of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  The main contributions of the paper are as follows:  In order to accelerate future studies, our own dataset of Arabic letters is created with hafizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  Providing better accuracy with low computational load compared to high-performance deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  Suitable for low capacity embedded platforms with its low complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  The rest of paper is designed as follows: Section 2 introduces related literature studies on mispronunciation of  Arabic phonemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Section 3 presents individual machine learning classifiers and ensemble learning methods utilized  in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Section 4 gives experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The paper is concluded with discussion and conclusion in Section  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Literature Review  In this section, literature studies on mispronunciation of Arabic phonemes are briefly introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  Muhammad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' (2010) present a non-real-time e-Hafiz system that proposes to avoid the Quran from being mis-  pronunciation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' For this purpose, data preparation, feature extraction, and modeling stages are performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' After silence  removal and pre-emphasis methods are implemented at the data preparation step, framing, windowing, discrete Fourier  transform, Mel Filter-bank, logarithm, and inverse discrete Fourier transformation techniques are applied in the fea-  ture extraction phase, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' After that, similarity is calculated to find mispronunciation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The authors compare  the introduced model with a counterpart web application on five people and inform that they exhibit advancement in  their memorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' In other study, (Elsayed and Fathy, 2019) aim an automated model to assess the memorization of  the Qur’an depending on Hafiz reading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Mel-frequency cepstral coefficient is assessed as feature extraction technique  while vector quantization is employed for the purpose of dimension reduction by the proposed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The authors  present that the experiment results ensure remarkable accuracy scores to evaluate Quran memorization with the uti-  lization of proposed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' In another study (Putra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', 2012), a speech recognition model is developed to learn  the Holy Quran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Gaussian mixture model is blended with MFCC feature extraction technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Different speech signal  processing techniques are applied namely, minimal sampling frequency, frame blocking, windowing, discrete Fourier  transform, and dynamical 40 cepstral coefficient and spectrum energy in addition to MFCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The authors present the  experimental results provide 70% of accuracy performance for pronunciation, 90% of accuracy success for reading  rules of tajweed, and 60% of accuracy score for the consolidation of both pronunciation and tajweed reading rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  (Arafa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', 2018) introduce a speech recognition system that is able to discover mispronunciation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The dataset  is composed of 89 students, which of 46 is female.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' 28 Arabic phonemes are voiced 10 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' After gathering 890  utterances, MFCC coefficients are extracted for modeling with five different machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' These are k-  nearest neighbor, support vector machine, naive Bayes, multi layer perceptron, and random forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The experimental  results indicate that the random forest technique exhibits 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='02% of accuracy result, which is better than compared to  other machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' In another recent study, (Nazir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', 2019) present a model depending on mispronun-  ciation detection employing deep convolutional neural network properties for Arabic phonemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' As a first, features  are extracted various layers of CNN ranges from 4 to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' After that, k-nearest neighbor, support vector machine, and  neural network classifier are employed for training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' They compare the experiment result when hand-crafted  features are employed instead of utilizing deep features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Finally, the authors emphasize that the proposed learning  method considerably boosts the success of the mispronunciation determination process in terms of accuracy by 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='2%  when deep features are employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' In a study, (Akhtar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', 2020) introduce a model to develop the detection of  Selim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 2 of 13  An ensemble-based framework for mispronunciation detection of Arabic phonemes  mispronunciation of Arabic words for non-native students utilizing features of deep convolutional neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' To  utilize deep features, features are extracted from layers 6, 7, and 8 of Alex Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' After that, n-nearest neighbor, support  vector machine, and random forest are employed at the training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' To show the effect of deep features, authors  also evaluate the features extracted using mel frequency cepstral coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The same three model are employed and  the results of both approach is compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' They emphasize that the introduced model with 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='2% of accuracy achieves  the determination of incorrect pronunciation of Arabic words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' (Maqsood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', 2016) concentrate on the detection  of mispronunciation for Arabic phonemes employing acoustic phonetic features (APF) with support vector classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  For this purpose, authors concentrate on the acoustic phonetic features instead of utilizing confidence measure-based  scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The data set is constructed with 500 utterances of 100 speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The features are composed of root mean  square energy (RMSE), MFCCs,low energy, spectral features, zero-cross rate, pitch and statistical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' After that,  SVM is employed to determine the mispronunciation of Arabic phonemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The paper is concluded that support vector  machine demonstrates 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='5% of accuracy result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  In another study, (Farooq and Imran, 2021) focus on determination of mispronunciation in articulation points for  Arabic letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' For this purpose, Relative Spectral Transform - Perceptual Linear Prediction feature extraction technique  is blended with Hidden Markov Model (HMM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' 1,486 samples are gathered for 29 Arabic letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Various techniques  are employed to perform with HMM, like for discovering the observation probability Gaussian mixture models or for-  ward method is utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Authors conclude the study that proposed model is capable to recognize the mispronunciation  performing 85% of recognition rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' In the study (Nazir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', 2021), support vector machine-based system is inves-  tigated for the aim of determining mispronunciation of Arabic phonemes for Asian speakers on 28 Arabic phonemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  The proposed system is accomplished 88% of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' (Asif et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', 2021) concentrate on the correct pronunciation  of Arabic vowels with the help of deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The new data set is constructed and augmented with various  techniques due to the limited number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The data set is composed of 85 individual records which of 43 is  female and only four of them are non-native speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' After gathering 6,229 records, preprocessing stage is carried  out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' For this purpose, noise reduction, data segmentation, re-sampling, and silence truncation are implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' To  determine the mispronunciation of Arabic vowels, optimized convolutional neural network is modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The proposed  model shows 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='77% of accuracy score on pronunciation classification of classical Arabic phonemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' (Maqsood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=',  2017) perform comparative analysis of different classifiers for the purpose of detecting mispronunciation for Arabic  phonemes using acoustic phonetic features namely, MFCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The following machine learning algorithms are employed:  Random forest, naive Bayes, Ada-boost, and k-nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Authors report that random forest model outperforms others with  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='4% of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  (Shareef and Al-Irhayim, 2022) concentrate on the detailed comparison of feature extraction models for the pur-  pose of detecting impairment Arabic speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The feature extraction technique is based on diverse versions of wavelet  transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' To detect the impairment Arabic speech, LSTM and CNN-LSTM models are constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The com-  bination of MFCC and LSTM achives the best classification accuracy with 93% while it is followed by CNN-LSTM  with 91% of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Mispronunciation detection system is designed by (Algabri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', 2022) for non-native Arabic  speakers for the purpose of providing them impairment feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The proposed system ensures feedbacks to the Ara-  bic learners utilizing deep learning-based models in the level of word and sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Experiment results show that the  best model performs nearly 4% of error rate for phoneme detection, and roughly 70% of F1-score for mispronunciation  detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', 2022) propose improved mispronunciation detection system based on online pseudo-labeling  and wav2vec model for English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' In details, pseudo-labeling procedure is performed using unlabeled L2 speech and  pre-trained self-supervised learning model (wav2vec) is carried out for enhancing fine-tuning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Experiment  results demonstrate remarkable score when compared to traditional offline pseudo-labeling techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The model  exhibits reduction in phoneme error rate nearly 5% and approximately 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='5% enhancement in F1-score for detecting  mispronunciation of L2 speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' (Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', 2022) present a text aware model for detection of mispronunciation by  enhancing weight of audio features compared to score of unrelated text features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' For this purpose, contrastive learning  procedure is carried out in TIMIT and L2-Arctic English data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' They report that proposed model ensures roughly  4% improvement comparing with the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  Our study differs from aforementioned literature works as well mainly because we concentrate on the ensemble-  based strategy to enable better accuracy with low computational load compared to high-performance deep learning  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Furthermore, our own dataset of Arabic letters is composed for the purpose of accelerating future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  Selim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 3 of 13  An ensemble-based framework for mispronunciation detection of Arabic phonemes  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Proposed Framework  In this work, an ensemble machine learning based mispronunciation detection method for Arabic phonemes is  proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Flowchart of the proposed system is depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The proposed system has three main stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  First step is to utilize preprocessing to improve the sound quality and the performance of the next steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Next, feature  extraction process is applied to the raw audio signals obtained from the microphone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Finally, the performance of  machine learning-based approaches on the problem is examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Then, the performance is improved by using the  ensemble learning approach on these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Arabic phonemes used in this work is Arabic letters which is given in  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' In this figure isolated forms and pronunciation of letters are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  Figure 1: The flowchart of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Pre-processing  The pre-processing step aims to separating the clean speech from a mixed sound of speech and noise and get  trimming only the part with the audio signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Noise reduction which relies on a method called “spectral gating” which  is a form of Noise Gate is primarily used to improve speech recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Then, silence removal process is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  The beginning of speech indicates that the sound level rises above a certain dB value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' In this study, the segment where  the audio signal first rises above 30 dB and then drops below 30 dB is trimmed and used as an input to the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  Selim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='Page 4 of 13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='AUDIO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='DATASET  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='PRE-PROCESSING  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='DATA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='NOISE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='SILENCE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='REMOVAL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='AUGMENTATION  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='REDUCTION  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='(ONLY IN TRAINING)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='EXTRACTION  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='FEATURE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='MFCC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='MEL-SPECTOGRAM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='MACHINELEARNING  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='ENSEMBLELEARNING  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='METHODS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='METHODS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='SVM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='VOTING  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='DETECTION  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='K-NN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='STACKING  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='DECISION  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='BOOSTING  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='TREE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='NAIVEBAYES  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='BAGGING  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='RANDOM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='FOREST  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='FINAL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='DECISIONAn ensemble-based framework for mispronunciation detection of Arabic phonemes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='Figure 2: Isolated form and pronunciation of Arabic letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  Figure 3 shows a sample input signal before and after noise reduction and silence removal operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' As the last  preprocessing operation, the number of data is increased by augmentation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Noise adding, time shifting, time  stretching and pitch shifting are the used augmentation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' This process is not included in the testing process,  but only in the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Feature extraction  Human speech has various distinctive features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' However, processing raw audio signal is generally not preferred  due to the high noise and data size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' It is observed that extracting distinctive features from the audio signal and using  it as input to the base machine learning model will produce much better performance than directly considering raw  audio signal as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' MFCC (Dave, 2013) and Mel-Spectrogram (Stevens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', 1937) are two widely used technique  for extracting the features from the audio signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  The Mel-spectrogram consist of 128 Mel feature and Fast Fourier Transform (FFT) with 2048 window length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The  dimensions of the obtained features differ according to the audio duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Therefore, the feature vector size is scaled  to a single dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' After the Mel features are obtained, it is scaled to 1×128 by taking the averages (before scaling  the dimensions are like 128×48, 128×40 for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' In the end, the problem evolves into the classification of the  feature vectors obtained in size 1×128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  MFCC features are the result of a series of applied processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' These processes are as follows: Windowing of the  audio signal, applying Discrete Fourier Transform (DFT), obtaining the receipt of the logarithm of amplitude, distorting  frequencies on a Mel scale, and applying inverse discrete cosine transform (DCT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' As a result of these processes, 39  features are obtained, of which the last 20 have more distinctiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' In this study, these last 20 features are used as  input features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Similar to the Mel-spectrogram, the audio file size causes a change in the dimensions of the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  For this reason, all features are averaged among themselves and the problem turns into the classification of the data in  the size of 1×20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  Figure 4 shows the scaling process of feature vectors and Figure 5 presents average Mel-Spectrogram and MFCC  features for 2 letters (“Alif” and “Baa”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' In Figure 5, different colours represent the speeches of different hafizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' As  Selim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 5 of 13  Isolated Form  2  Pronunciation  haa  jiim  thaa  taa  baa  alif  Isolated Form  3  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  Pronunciation  siin  zaay  raa  thaal  daal  kha  Isolated Form  &  上  CBCD  Pronunciation  ayn  thaa  taa  daad  saad  shiin  Isolated Form  L  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  Pronunciation  miim  laam  kaaf  qaaf  faa  ghayn  R  9  Isolated Form  Pronunciation  lamelif  yaa  waaw  ha  unnuAn ensemble-based framework for mispronunciation detection of Arabic phonemes  (a)  (b)  (c)  (d)  Figure 3: Noise reduction and silence removal pre-processing operations (a) Original audio signal (b) After noise reduction  (c) After noise silence removal (d) After noise reduction and silence removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  seen from this figure, even if different hafizes say the same letter, it creates dissimilar characteristic on the formed  feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' In addition, it is seen that the distinctiveness of the same people in different letters is not sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  Because of these difficulties, the performance of the classification method is become more important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Detection  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Machine learning methods  Machine learning can be defined as artificial intelligence algorithms that can infer and predict from data to mimic  the way humans learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' There are various machine learning algorithms that are capable of solving classification, re-  gression and clustering tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' In our work, popular methods suitable for classification problem are emphasized which  are support vector machine (SVM), k-Nearest neigbour (k-NN), decision tree (DT), naïve Bayes, random forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  SVM (Cristianini and Shawe-Taylor, 2000) is a supervised learning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Kernel functions can also be used  depending on the type of data during the operation of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' In this way, both linear and nonlinear classification  operations can be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' It is aimed to separate all data with a hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' However, if the data cannot be fully  separated, they cannot be classified with a single plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Therefore, different kernel functions are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' A margin is  determined around the hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Whether this margin is large or small directly affects the classification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  Margin can be controlled with the “C” hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The larger the C, the narrower the margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Also, if the model  is overfit, C needs to be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' In this work linear kernel and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='02 used as C parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  k-NN (Mucherino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', 2009) is basically based on the determination of the class of the data whose class is  unknown, according to the nearest “k” neighbor from the data in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' As a result of performing a distance  measurement between the test data and the training data, the nearest “k” nearest neighbors are determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Then, the  class value of the tested data is determined according to these labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' In this work, 3,4 and 5 is evaluated as “k” value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  Basic purpose of decision trees (Alp, 2019) is to divide the data set into smaller subgroups that are more visually  Selim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 6 of 13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='25  Amplitude  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
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+page_content='8  0  2500  5000  7500  10000  12500  15000  17500  TimeAn ensemble-based framework for mispronunciation detection of Arabic phonemes  (a)  (b)  (c)  (d)  Figure 4: Scaling of feature vectors (a) Original Mel-spectrogram image (b) Averaged Mel-spectrogram image (c) Original  MFCC image (d) Averaged MFCC image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
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+page_content=' Since the output of the algorithm is a flowchart  that looks like a tree visually, it is called a decision tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
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+page_content=' The root node is where classification process starts from this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
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+page_content='  Naïve Bayes (Webb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', 2010) classification is based on Bayes theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
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+page_content=' Each attribute is considered independent from other attributes in the class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' different  class based on various attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
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+page_content='  Selim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
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+page_content='Figure 5: Average feature vectors of “Alif” and “Baa” letters (a) Average Mel-spectrogram feature of “Alif” (b) Average  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
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+page_content=' Ensemble learning techniques  Ensemble learning is an approach to boost the overall accuracy of a classification framework by utilizing multi-  ple learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The strategy usually yields better results than a individual learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' There are most  commonly employed ensemble learning techniques in the literature namely, bagging, stacking, boosting, and voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  Bagging is one of the most widely-applied ensemble based algorithms (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' It is known as the abbreviation  of bootstrap aggregating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' In this approach, diversity is performed by re-sampling in which various training data subsets  are randomly chosen with replacement from the whole training data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Each subset is assessed to train a distinct base  learner in the set of ensemble learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Final decision is determined by combining decisions of individual learners by  taking a majority vote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  Boosting (Freund and Schapire, 1996) is another ensemble learning model that is asserted as an alternative to  the bagging technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The main approach behind of this approach is to produce a set of individual classifiers that  utilizes a data subset in which each instance is consolidated with a weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' It is carried out iteratively running base  learners on different distributions over the training data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' While all instances have same weight at the first step,  the weights of miscategorized instances are updated at each iteration according as the training error of preceding base  learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Each learner employs a subset of instances acquired from an updated version of training data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' After that,  instances that are incorrectly forecasted by preceding learners are picked up more frequently than the instances that  are correctly forecasted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The final outcome is achieved by weighted majority voting of the categories forecasted by the  base learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' AdaBoost, AdaBoost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='M1, AdaBoost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='M2, AdaBoost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='R, Arcing and Real Adaboost (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', 2010), (Polikar,  2006b), (Gopika D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Azhagusundari, 2014), (Ren Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', 2016) are the variations employed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' AdaBoost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='M1  is used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  Stacking is the process of fitting multiple types of models to the same data and then employing metal level model  to learn how to integrate the predictions in the best way possible (Džeroski, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' There are two major steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The first  one is consists of generating a set of individual classifiers utilizing training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Thus, meta-data set is composed of the  decisions of individual classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' After construction of the meta-data set, meta-level learner modeled with this data  Selim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 8 of 13  mel1  80  mel2  mel3  mel4  60  lue  40  20  0  100  101  102  CoefficientsNumber75  mfcc2  mfcc3  50  mfcc4  25  lue  0  25  50  75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='0  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='5  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='0  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='5  Coefficients Number175  mel1  mel2  150  mel3  mel4  125  100  Value  75  50  25  0  100  101  102  Coefficients Number150  mfcc1  mfcc2  mfcc3  100  mfcc4  50  Value  0  50  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='0  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='5  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='0  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='5  Coefficients NumberAn ensemble-based framework for mispronunciation detection of Arabic phonemes  set in order to get generalized estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' In our work, the meta-data set comprises of both original training examples  and decisions of individual classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Moreover, stacking method is employed with different learning algorithms  namely, logistic regression, random forest, and extra tree in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  In majority voting, each model makes a prediction (vote) for each test sample, and the prediction of the final result  is the model that receives the most votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  Random forest (Breiman, 2001) is a set of decision tree classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' It is a certain enforcement of bagging method  in which each individual classifier is a decision tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Bagging is utilized to choose training sub sets for each base  decision tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The division task employed in Random forests varies from well-known decision tree strategy where  each node is divided by the best attribute among all other attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' In Random forest approach, a random subset of  attributes is chosen first and then the best division is determined on the random subset of attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' This approach  ensures an extra randomness to the model in addition to bagging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Random forests is able to cope with over-fitting  problem due to randomness performed in both sample and attribute spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  Figure 6 shows an overview of the ensemble architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  (a)  (b)  (c)  (d)  Figure 6: Ensemble architectures (a) Voting (b) Bagging (c) Stacking (d) Boosting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Experimental results  In this study, an original dataset is obtained by collecting audio samples from 11 speakers, 8 of whom are hafiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  Dataset consists of Arabic letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' All letters are recorded by asking the speakers to say each letter individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Original  dataset size is 290.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Sample size in the dataset is increased to 1450 by using noise adding, time shifting, time stretching,  Selim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='Page 9 of 13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='DATASET  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='MODEL1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='VODEL2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='MODEL3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='MODELN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='FINAL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='DECISIONDATASET  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='SUBSET  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='SUBSET  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='SUBSET3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='SUBSETN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='MODEL1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='MODEL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='MODEL3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='MODELN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='FINAL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='DFCISIONDATASET  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='MODEL1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='MODEL2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='MODEL3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='MODELN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='METAMODEL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='FINAL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='DECISIONSUBSET1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='SUBSET 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='SUBSET3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='SUBSETN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='Update  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='Update  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='Update  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='[Valuesor Weights)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='(Values or Weights)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='(Values or Weights)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='MODEL1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='MODEL2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='MODEL3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='MODELN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='FINAL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='DECISIONAn ensemble-based framework for mispronunciation detection of Arabic phonemes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='Table 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='Evaluation results of machine learning methods with MFCC features  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='Method  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='Accuracy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='Precision  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='Recall  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='F1-Score  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='SVM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='758  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='829  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='785  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='806  k-NN (k=3)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='466  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='527  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
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+page_content='942  pitch shifting augmentation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The dataset is divided into 80% training and 20% test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' 5-fold cross-validation  procedure is used to ensure that the score of proposed models does not depend on the way we select train and test  subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The performance of the proposed methods is measured using the formulas given in (1), (2), (3) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' In  these equations, if we named every class as 퐶푥, TP (True Positive) represents audio belonging to the 퐶푥 is correctly  classified as 퐶푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' FP (False Positive) is all non-퐶푥 samples classified as 퐶푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' TN (True Negative) denotes all non- 퐶푥  samples not classified as 퐶푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' FN (False Negative) represents all 퐶푥 samples not classified as 퐶푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  퐴푐푐푢푟푎푐푦 =  푇 푃 + 푇 푁  푇 푃 + 푇 푁 + 퐹푃 + 퐹푁  (1)  푃 푟푒푐푖푠푖표푛 =  푇 푃  푇 푃 + 퐹푃  (2)  푅푒푐푎푙푙 =  푇 푃  푇 푃 + 퐹푁  (3)  퐹 − 푚푒푎푠푢푟푒 = 2 × 푃푟푒푐푖푠푖표푛 × 푅푒푐푎푙푙  푃푟푒푐푖푠푖표푛 + 푅푒푐푎푙푙  (4)  Evaluation results of machine learning based methods are given in Table 1 and Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Table 1 shows effect  of MFCC features used as input of machine learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Likewise, Table 2 shows effect of Mel-Spectrogram  features as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' In these tables, values with bold font represent best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' According to Table 1 SVM has the best  results among the other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' In Table 2, it can be considered that k-NN where k is determined as 3 is successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  When Table 1 and Table 2 are evaluated together, the highest result is obtained with the k-NN method, in which  Mel-Spectrogram features are used as input features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  The ensemble learning approach is applied to the methods given in Table 1 and Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Evaluation results of  ensemble learning methods are given in Table 3 and Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' In these tables, values with bold font represent best  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' When Table 3 is examined, it can be seen that stacking with random forest approach is the best compared to  the other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
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+page_content='  Selim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 10 of 13  An ensemble-based framework for mispronunciation detection of Arabic phonemes  Table 3  Evaluation results of ensemble learning methods with MFCC features  Method  Accuracy  Precision  Recall  F1-Score  Voting  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
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+page_content='091  Figure 7: Letter-based performance evaluation with deep learning based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  Table 5 shows the comparison of the recent and proposed methods for sound classification in the literature, in terms  of processing speed and detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The comparison is carried out on the dataset created within the scope  of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The methods used in comparison in Table 5 are deep learning-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' These methods are  implemented using original network structure and parameters specified in the papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' In this table, proposed method  refers voting ensemble method with Mel-Spectrogram features input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' As seen in Table 5, proposed method has su-  perior performance results and the lowest processing speed among the other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' In addition, the letter-based  performances of the methods given in Table 5 are analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Figure 7 shows the performances of (Abdoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', 2019),  Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' (2018) and the proposed method for 10 letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' It is understood that the proposed method shows a steady  performance despite the high-performance decrease in some letters in other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Table 5 and Figure 7 show that  popular deep learning approaches can be surpassed in mispronunciation detection of Arabic phonemes with the pro-  Selim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 11 of 13  Proposed System  [45]  [46]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
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+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='0  亡  )  LettersAn ensemble-based framework for mispronunciation detection of Arabic phonemes  posed ensemble approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' In addition, it should be noted that generally deep learning-based approaches require more  training time and processing (inference) time due to their complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Arabic mispronunciation detection is a chal-  lenging problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' It is significant that the proposed framework ensures superior performance with less computational  load to this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Discussion and Conclusion  In this work, an ensemble learning approach is proposed to detect mispronunciation of Arabic phonemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' In the  proposed method, firstly, feature extraction is performed with MFCC and mel-spectrogram methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Then, traditional  and ensemble learning-based approaches used these features as input and their performance is evaluated on the original  data set created for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' The method with the highest performance obtained from evaluation is also compared with  the deep learning-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Experimental results show that the utilization of voting classifier as an ensemble  algorithm with mel-spectrogram feature extraction technique reveals remarkable results with 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='9% of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Future  studies will focus on increasing the data set and transfer learning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
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+page_content=' Computer Assisted Language Learning 18,  81–109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
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+page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Comparison between features extraction techniques for impairments arabic speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
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+page_content=' A scale for the measurement of the psychological magnitude pitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Journal of the Acoustical Society  of America 8, 185–190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  Webb, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
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+page_content=', Keogh, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
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+page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Naive bayes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Encyclopedia of Machine Learning 15, 713–714.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
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+page_content=', Hirschi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', Looney, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', Kang, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', Hansen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Improving mispronunciation detection with wav2vec2-based momentum  pseudo-labeling for accentedness and intelligibility assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='15937 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  Ziafat, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', Ahmad, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', Fatima, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', Zia, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', Alhumam, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', Rajpoot, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Correct pronunciation detection of the arabic alphabet using deep  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' Applied Sciences 11, 2508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content='  Selim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 13 of 13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfa_yH/content/2301.01378v1.pdf'}
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+arXiv:2301.04996v1  [q-fin.MF]  12 Jan 2023
+EUROPEAN BASKETS IN DISCRETE-TIME CONTINUOUS-BINOMIAL
+MARKET MODELS
+JAREK KĘDRA, ASSAF LIBMAN, AND VICTORIA STEBLOVSKAYA
+Abstract. We consider a discrete-time incomplete multi-asset market model with contin-
+uous price jumps. For a wide class of contingent claims, including European basket call
+options, we compute the bounds of the interval containing the no-arbitrage prices. We prove
+that the lower bound coincides, in fact, with Jensen’s bound. The upper bound can be com-
+puted by restricting to a binomial model for which an explicit expression for the bound is
+known by an earlier work of the authors. We describe explicitly a maximal hedging strategy
+which is the best possible in the sense that its value is equal to the upper bound of the price
+interval of the claim. Our results show that for any c in the interval of the non-arbitrage
+contingent claim price at time 0, one can change the boundaries of the price jumps to obtain
+a model in which c is the upper bound at time 0 of this interval. The lower bound of this
+interval remains unaffected.
+1. The main results
+Discrete time continuous-binomial market model. We consider a discrete-time mar-
+ket model, see e.g [2, §4.5] or [8, §3], with m risky assets S1, . . . , Sm and a bond S0 whose
+prices at time k = 0, . . . , n denoted Si(k), are random processes described as follows. We fix
+the initial values Si(0) of the assets (i = 0, 1, . . . , m) and parameters R > 0 (the interest rate)
+and 0 < Di < R < Ui. For time k = 1, 2, . . . , n the random process is defined by
+• S0(k) = RkS0(0), and
+• Si(k) = Ψi(k)Si(k − 1), for i = 1, 2, . . . , m, where Ψi(k) are random variables with
+values in [Di, Ui].
+We call Ψi(k) the price jumps at time k. We emphasize that the price jumps are not assumed
+to be independent of each other nor identically distributed.
+A European basket call option is a contingent claim with pay-off given by
+(1)
+F =
+� m
+�
+i=0
+ci · Si(n) − K
+�+
+,
+where K > 0 and ci ≥ 0 for i = 1, 2, . . . , m and x+ := max{x, 0}.
+Notice that there is
+no assumption on c0. The rational values of F at time k are the possible market values of
+the option at time k so that no arbitrage occurs. They are known to form an open interval
+(Γmin(F, k), Γmax(F, k)), where Γmin(F, k) and Γmax(F, k) depend on the “state of the world”
+at time k, namely the history of the market up to time k, and in particular they depend on
+the (current) values of the assets Si at time k.
+The main result of this paper is the computation of Γmin(F, k) and Γmax(F, k). It extends
+the main results of the authors’ previous work [5] in which we consider discrete time binomial
+models, i.e ones in which the price jumps Ψi(k) take values in (the discrete) set {Di, Ui}
+rather than the entire interval [Di, Ui].
+1
+
+2
+JAREK KĘDRA, ASSAF LIBMAN, AND VICTORIA STEBLOVSKAYA
+1.A. Computation of Γmin(F, k) and Γmax(F, k). For every 1 ≤ i ≤ m set
+(2)
+bi = R − Di
+Ui − Di
+and reorder the assets S1, . . . , Sm, if necessary, so that
+b1 ≥ · · · ≥ bm.
+Define q1, . . . , qm by
+(3)
+qi =
+
+
+
+1 − b1
+if i = 1
+bi − bi+1
+if 1 < i < m
+bm
+if i = m
+For any 0 ≤ i ≤ m and any 0 ≤ j ≤ m define numbers χi(j) as follows.
+χi(j) =
+�
+Ui
+if i ≤ j
+Di
+if i > j
+for 1 ≤ i ≤ m, and
+(4)
+χ0(j) = R.
+To streamline the notation, set for every k ≥ 0
+(5)
+Pk(m) = {0, . . . , m}k.
+Thus, any J ∈ Pk(m) is a sequence J = (j1, . . . , jk) with 0 ≤ j1, . . . , jk ≤ m. For such J set
+qJ
+=
+�
+j∈J
+qj
+(6)
+χi(J)
+=
+�
+j∈J
+χi(j).
+Theorem 1.1. With the setup of the market model and notation above, the extremal values
+of the rational values of F at time 0 ≤ k ≤ n are given by
+Γmin(F, k)
+=
+Rk−n ·
+�
+Rn−k
+m
+�
+i=0
+ci · Si(k) − K
+�+
+Γmax(F, k)
+=
+Rk−n ·
+�
+J∈Pn−k(m)
+qJ ·
+� m
+�
+i=0
+ci · χi(J) · Si(k) − K
+�+
+.
+1.B. Hedging strategies. Consider a sequence of (time changing) portfolios
+Vα(k) =
+�
+i
+αi(k)Si(k),
+(0 ≤ k ≤ n − 1)
+for some choices of values for αi(k) at time 0 ≤ k ≤ n − 1. A maximum hedging strategy is a
+choice for αi(k) at time 0 ≤ k ≤ n − 1 (which depends on the state of the world at that time)
+which minimizes the value Vα(k) subject to the requirement that
+m
+�
+i=0
+αi(k) · Si(k + 1) ≥ Γmax(F, k + 1)
+for any subsequent state of the world at time k + 1. That is, a maximum hedging strategy
+is a time dependant portfolio of minimum possible cost whose value is guaranteed to exceed
+the future rational value of the contingent claim F.
+Our next result, Theorem 1.2, shows that the value of any hedging portfolio Vβ(k) must
+always exceed Γmax(F, k) and that there exists a hedging strategy αi(k) that attains this
+
+EUROPEAN BASKETS IN DISCRETE-TIME CONTINUOUS-BINOMIAL MARKET MODELS
+3
+bound. It is a minimum cost maximum hedging strategy. To make this result precise, given
+the state of the world at time 0 ≤ k ≤ n − 1, let Yi(k) where 0 ≤ i ≤ m be the value of
+Γmax(F, k + 1) at the state of the world at time k + 1 which is obtained from the present
+one (at time k) by having the assets S1, . . . , Si make their maximum price jumps U1, . . . , Ui
+and having Si+1, . . . , Sm make their minimum price jumps Di+1, . . . , Dm. Explicitly, for any
+0 ≤ t ≤ m:
+Yt(k) = Rk+1−n ·
+�
+J∈Pn−k−1(m)
+qJ
+� m
+�
+i=0
+ci · χi(J)χi(t) · Si(k) − K
+�+
+.
+Theorem 1.2. Consider the continuous binomial market model above and a European option
+F. Any hedging strategy βi(k) satisfies
+Vβ(k) ≥ Γmax(F, k).
+There exists a maximum hedging strategy αi(0), . . . , αi(n − 1) such that Vα(k) = Γmax(F, k)
+for all 0 ≤ k ≤ n − 1. In fact, the values of αi(k) at time k are computed as follows.
+
+
+α0(k)
+α1(k)
+...
+αm(k)
+
+ = W(k) · N · Q ·
+
+
+Y0(k)
+Y1(k)
+...
+Ym(k)
+
+
+Where W(k), N, T are the following (m + 1) × (m + 1) matrices. Set ∆i = Ui − Di.
+W(k) =
+
+
+1
+R·S0(k)
+1
+S1(k)
+...
+1
+Sm(k)
+
+
+Q =
+
+
+1
+0
+0
+· · · · · ·
+0
+0
+−1
+1
+0
+· · · · · ·
+0
+0
+0
+−1
+1
+· · · · · ·
+0
+0
+...
+...
+...
+· · · · · ·
+...
+0
+0
+0
+· · · · · ·
+1
+0
+0
+0
+0
+· · · · · ·
+−1
+1
+
+
+N =
+
+
+1
+− D1
+∆1
+− D2
+∆2
+· · · · · ·
+− Dm
+∆m
+0
+1
+∆1
+0
+· · · · · ·
+0
+0
+0
+1
+∆2
+· · · · · ·
+0
+...
+...
+...
+...
+0
+0
+0
+· · · · · ·
+1
+∆m
+
+
+(Notice that W(k) depends on the state of the world, but N and Q do not. Also, R · S0(k) =
+S0(0) · Rk+1).
+This result extends our previous result in [6] which computes a similar hedging strategy in
+discrete time binomial models, i.e models in which Ψi(k) ∈ {Di, Ui} ⊂ [Di, Ui].
+Changing the parameters of the model. Keeping R fixed, we may change the values of
+Ui and Di to obtain different models for the same market. This has the effect of changing
+the limits of the price jumps of the assets Si, and consequently the random processes Si are
+changed. Clearly the values of Γmin(F, k) and Γmax(F, k) depend on these parameters, and
+
+4
+JAREK KĘDRA, ASSAF LIBMAN, AND VICTORIA STEBLOVSKAYA
+we therefore write Γmin(F, k; Ui, Di) and Γmax(F, k; Ui, Di) to emphasise this dependence.
+We will be interested in the rational prices of F at time 0, namely Γmin(F, 0; Ui, Di) and
+Γmax(F, 0; Ui, Di).
+Theorem 1.3. Consider a market model with some 0 < Di < R < Ui and the European
+basket F in (1). Then
+(1) Γmin(F, 0; Ui, Di) = (Rn �m
+i=0 ciSi(0) − K)+ and in particular it is independent of the
+values of Ui, Di.
+(2) For every c in the open interval ( Γmin(F, 0; Ui, Di) , Γmax(F, 0; Ui, Di) ) there exist
+Di ≤ di < R < ui ≤ Ui such that Γmax(F, 0; ui, di) = c.
+More precisely, consider the functions ui, di : [0, 1) → R defined by
+di(s)
+=
+Di + (R − Di)s
+ui(s)
+=
+R − (1 − bi)di(s)
+bi
+Then ϕ(s) = Γmax(F, 0; ui(s), di(s)) is a continuous function of s ∈ [0, 1) such that
+ϕ(0) = Γmax(F, 0; Ui, Di) and limsր1 ϕ(s) = Γmin(F, 0; Ui, Di).
+2. Preliminaries: Random processes and conditional expectation
+2.A. Non-degenerate density functions. This section is concerned with some general
+results about probability measure spaces. Our standard reference for Measure theory and
+Lebesgue integration are Halmos [4] and Royden [9], and for Probability Theory it is Feller
+[1].
+Throughout this paper, once m ≥ 1 is fixed we will denote
+(7)
+Ω = [0, 1]m ⊆ Rm
+equipped with the usual Borel σ-algebra and probability measure µ.
+A probability density function (pdf) is a measurable p: Ω → [0, ∞) such that
+�
+Ω pdµ = 1.
+It gives rise in a standard way to a probability measure on Ω which by abuse of notation we
+also denote by p.
+A probability measure on Ω is called absolutely continuous with respect to µ, written ν ≪ µ,
+if for any Borel subset E we have µ(E) = 0 =⇒ ν(E) = 0. A probability measure ν on Ω is
+non-degenerate if ν ≪ µ and µ ≪ ν. We write ν ≈ µ.
+By the Radon-Nykodim theorem [9, §11.5] if ν ≪ µ then there exists a pdf p: Ω → [0, ∞)
+called the Radon-Nykodim derivative, such that ν(E) =
+�
+E p(x) dµ(x). It is easy to check
+that ν ≈ µ if and only if p > 0 almost everywhere1. We say that p is non-degenerate.
+2.B. Conditional probability. Consider Ω(i) = [0, 1]mi where i = 1, . . . , n. Set
+Ω =
+n
+�
+i=1
+Ω(i) = [0, 1]m,
+where m = �
+i mi, equipped with the standard Borel σ-algebra. Let X(i) denote the random
+vector
+X(i) : Ω
+proji
+−−−−→ Ω(i) ⊆ Rmi.
+1If p > 0 a.e then
+�
+E p du > 0 for any E with µ(E) > 0 is a standard fact. If p = 0 on E with µ(E) = 0
+then ν(E) =
+�
+E p(x) dµ(x) = 0 so µ is not absolutely continuous with respect to ν.
+
+EUROPEAN BASKETS IN DISCRETE-TIME CONTINUOUS-BINOMIAL MARKET MODELS
+5
+For a non-empty I ⊆ {1, . . . , n} set mI = �
+i∈I mi. Then set
+Ω(I) =
+�
+i∈I
+Ω(i)
+X(I) : Ω
+proj(I)
+−−−−−→ Ω(I) ⊆ RmI.
+Thus, Ω(I) = [0, 1]mI and X(I) is a random vector into RmI. We will denote elements of Ω(I)
+by ω(I). We will write n − I for the complement of I.
+For the remainder of this subsection we fix a non-degenerate (with respect to the Lebesgue
+measure on Ω) pdf p: Ω → [0, ∞) and equip Ω with the probability measure it induces which
+we abusively denote by p. Note that p is the joint density function of the random vectors
+X(1), . . . , X(n) (because X{1,...,n} is the inclusion Ω ⊆ Rm).
+Given a non-empty I ⊆ {1, . . . , n}, the (joint) density function of X(I) is the function
+pX(I) : Ω(I) → [0, ∞) given by (See e.g. [3, Chap. 2, Scet. 3])
+(8)
+pX(I)(ω(I)) =
+�
+τ∈Ω(n−I)
+p(ω(I), τ)dτ.
+Fubini’s theorem readily implies that pX(I) is non-degenerate (with respect to the Lebesgue
+measure on Ω(I)).
+Consider some disjoint I, J ⊆ {1, . . . , n}. The density function of X(I) given X(J) denoted
+pX(I)|X(J) : Ω(I∪J) → [0, ∞) is
+(9)
+pX(I)|X(J) (ω(I), ω(J)) =
+pX(I∪J)(ω(I), ω(J))
+pX(J)(ω(J))
+whenever this is defined. Since pX(J) is non-degenerate, pX(I)|X(J) is defined a.e. For any ω(J)
+we obtain a function, the (conditional) density of X(I) given the event {X(J) = ω(J)},
+pX(I)|X(J)=ω(J) : Ω(I) → [0, ∞)
+defined by pX(I)|X(J)=ω(J)(−) = pX(I)|X(J)(−, ω(J)). By Fubini’s theorem it is a (measurable)
+probability density function on Ω(I).
+Consider a random vector
+f : Ω → Rk.
+To avoid issues of convergence we assume that f ≥ 0, namely all the components of f are
+non-negative. The expectation of f given X(I) is the function
+Ep(f|X(I)): Ω(I) → R
+where Ep(f|X(I))(ω(I)) is the conditional expectation Ep(f|X(I) = ω(I)), namely
+Ep(f|X(I))(ω(I)) =
+�
+τ∈Ω(n−I)
+f(τ, ω(I))pX(n−I)|X(I)(τ, ω(I))dτ.
+By Fubini’s theorem Ep(f|X(I)) is a measurable function.
+Lemma 2.1. Keeping the notation above, let I, J ⊆ {1, . . . , n} be disjoint and consider some
+ω(I) ∈ Ω(I). Set p′ = pX(J)|X(I)=ω(I), a pdf on Ω(J). Let f ≥ 0 be a random vector on Ω. Then
+Ep(f|X(I) = ω(I)) = Ep′(τ �→ Ep(f|X(I∪J))(ω(I), τ)).
+In particular,
+Ep(X(J)|X(I) = ω(I)) = Ep′(X(J))
+where X(J) is viewed as a random vector from Ω(J).
+
+6
+JAREK KĘDRA, ASSAF LIBMAN, AND VICTORIA STEBLOVSKAYA
+Proof. We compute the right hand side using Fubini’s theorem as follows
+Ep′(τ �→Ep(f|X(I∪J))(ω(I), τ)) =
+=
+�
+τ∈Ω(J)
+pX(J)|X(I)=ω(I)(τ) · Ep(f|X(I∪J))(ω(I), τ) dτ
+=
+�
+τ∈Ω(J)
+�
+pX(I∪J)(ω(I),τ)
+pX(I)(ω(I))
+·
+�
+θ∈Ω(n−I∪J)
+f(ω(I), τ, θ) ·
+p(ω(I),τ,θ)
+pX(I∪J)(ω(I),τ) dθ
+�
+dτ
+=
+�
+τ∈Ω(J)
+�
+θ∈Ω(n−I∪J)
+f(ω(I), τ, θ) ·
+p(ω(I),τ,θ)
+pX(I)(ω(I))dθ dτ
+=
+�
+ω∈Ω(n−I)
+f(ω(I), ω) ·
+p(ω(I),ω)
+pX(I)(ω(I)) dω
+=
+�
+ω∈Ω(n−I)
+f(ω(I), ω) · pX(n−I)|X(I)=ω(I)(ω) dω
+= Ep(f|X(I) = ω(I))
+This establishes the first claim. We apply it to the random vector f = X(J) to obtain the
+second claim as follows
+Ep(X(J)|X(I) = ω(I)) = Ep′(τ �→ Ep(X(J)|X(I∪J))(ω(I), τ)) = Ep′(τ �→ X(J)(τ)) = Ep′(X(J)).
+Here we observe that Ep(X(J)|X(I∪J))(ω(I), τ) = X(J)(τ) because with the abuse of notation
+for the domain of X(J) we have X(J)(ω(I), τ, θ) = X(J)(τ) for any τ ∈ Ω(J) and any θ ∈
+Ω(n−I∪J).
+□
+Lemma 2.2. Let f : Ω → Rd be a function and I, J ⊆ {1, . . . , n} be disjoint, and assume
+that f ≥ 0. Suppose that f factors through the projection Ω
+πI∪J
+−−−→ Ω(I∪J), namely there exists
+g: Ω(I∪J) → Rm such that f = g ◦ πI∪J. Set p′ = pX(J)|X(I)=ω(I), pdf on Ω(J). Then
+Ep(f|X(I) = ω(I)) = Ep′(ω(J) �→ g(ω(I) ω(J)))
+Proof. Lemma 2.1 gives
+Ep(f|X(I) = ω(I)) = Ep′(ω(J) �→ Ep(f|X(I∪J))(ω(I), ω(J))) = Ep′(ωJ �→ g(ωI, ωJ))
+because f(ω(I), ω(J), τ) = g(ω(I), ω(J)) for any τ ∈ Ω(n−I∪J), so Ep(f|X(I∪J))(ω(I), ω(J)) =
+g(ω(I), ω(J)).
+□
+2.C. Tensoring. Given p: Ω → R and q: Ω′ → R we obtain a function p⊗q: Ω×Ω′ → R by
+(10)
+(p ⊗ q)(ω, ω′) = p(ω) · q(ω′).
+It is clear that if p, q are pdf’s then so is p ⊗ q and that it is non-degenerate if p and q are
+non-degenerate.
+Keeping the notation above for Ω(i) and the random vectors X(i), let p(i) : Ω(i) → [0, ∞) be
+non-degenerate pdf’s. Then p = p(1) ⊗ · · · ⊗ p(n) is a non-degenerate pdf on Ω = �
+i Ω(i). For
+any I ⊆ {1, . . . , n} we denote
+p(I) = ⊗
+i∈Ip(i).
+This is a non-degenerate pdf on Ω(I).
+Lemma 2.3. Let p(i) : Ω(i) → [0, ∞) be non-degenerate pdf’s, i = 1, . . . , n. Set p = p(1) ⊗
+· · · ⊗ p(n), non-degenerate pdf on Ω. Then
+
+EUROPEAN BASKETS IN DISCRETE-TIME CONTINUOUS-BINOMIAL MARKET MODELS
+7
+(1) pX(I) = p(I) for any I ⊆ {1, . . . , n}.
+(2) pX(J)|X(I)=ω(I) = p(J) for any disjoint I, J ⊆ {1, . . . , n}, and furthermore
+(3) Ep(X(J)|X(I) = ω(I)) = Ep(J)(X(J)) where X(J) is viewed as a random vector on Ω(J).
+Proof. (1) Given ω(I) we compute
+pX(I)(ω(I)) =
+�
+τ∈Ω(n−I)
+p(ω(I), τ) dτ =
+�
+τ∈Ω(n−I)
+p(I)(ω(I)) · p(n−I)(τ) dτ = p(I)(ω(I))
+because p(J) is a pdf on Ω(J) for any J.
+(2) Given ω(J) and ω(I) we use this to compute
+pX(J)|X(I)=ωI(ω(J)) =
+pX(I∪J)(ω(I), ω(J))
+pX(I)(ω(I))
+= p(I∪J)(ω(I), ω(J))
+p(I)(ω(I))
+= p(I)(ω(I)) · p(J)(ω(J))
+p(I)(ω(I))
+= p(J)(ω(J)).
+(3) By Lemma 2.1 and item (2)
+Ep(X(J)|X(I) = ω(I)) = EpX(J)|X(I)=ω(I)(ω(J) �→ Ep(X(J)|X(I∪J)(ω(I), ω(J)))
+= Ep(J)(ω(J) �→ X(J)(ω(J))).
+□
+2.D. Processes.
+Definition 2.4. Let Ω be a set and n ≥ 1. An Rd-valued process (over Ω) is a sequence of
+functions
+Y (0), . . . , Y (n) : Ωn → Rd
+such that for each 0 ≤ k ≤ n the function X(k) factors through the projection Ωn → Ωk to
+the first k factors.
+If Ωn is equipped with a probability measure, Y (0), . . . , Y (k) is called an Rd-valued random
+process.
+We frequently regard Y (k) as a function with domain Ωk. We will sometimes “trim” the
+process to Y (1), . . . , Y (n) or to X(0), . . . , X(n−1).
+If Ω = [0, 1]m ⊆ Rm the projections L(k) : Ωn → Ω ⊆ Rm to the k-th factor give a
+universal process in the sense that if Y (0), . . . , Y (n) is a process then each Y (k) is a function
+of L(1), . . . , L(k).
+2.E. Supports. Let ν be a probability measure on a set Ω.
+Definition 2.5. We say that ν is supported on a measurable set A if µ(A) = 1.
+Any probability measure ν on A ⊆ Ω extends to a probability measure ˜ν on Ω supported
+on A by ˜ν(E) = ν(A ∩ E). Conversely, if ν on Ω is supported by A then ν = �
+ν|A. If A is
+finite then for any f : Ω → R we have Eν(f) = �
+a∈A ν({a})f(a).
+
+8
+JAREK KĘDRA, ASSAF LIBMAN, AND VICTORIA STEBLOVSKAYA
+3. Maximum and minimum expectation of random variables on Ω = [0, 1]m
+Fix some m ≥ 1 and Ω = [0, 1]m equipped with the usual Lebesgue measure µ. Let
+L: Ω → Rm
+denote the inclusion. We think of it as a random vector with components L = (ℓ1, . . . , ℓm).
+Thus, ℓi : Ω → R is the projection to the ith factor
+ℓi(x1, . . . , xm) = xi.
+For convenience we also set
+ℓ0 = 1,
+the constant function (random variable).
+Definition 3.1. Let M(Ω) denote the set of all probability measures on Ω on the Borel
+σ-algebra. Let M+(Ω) denote the set of all the non-degenerate probability measures (with
+respect to the Lebesgue measure).
+We will often refer to the elements of M+(Ω) as pdf’s which are non-vanishing a.e.
+Consider a non decreasing b ∈ int(Ω) = (0, 1)m, namely
+1 > b1 ≥ · · · ≥ bm > 0.
+We will also denote for convenience
+b0 = 1
+and
+bm+1 = 0.
+Definition 3.2. The set of mean-b probability measures on Ω is
+M(Ω, b) = {P ∈ M(Ω) : EP (L) = b}.
+That is, EP(ℓi) = bi for all 1 ≤ i ≤ m. The set of non-degenerate mean-b probability measures
+is
+M+(Ω, b) = {P ∈ M+(Ω) : EP (L) = b}.
+Definition 3.3. Let L denote the set of vertices of the cube Ω = [0, 1]m. That is,
+L = {0, 1}m.
+There is a standard identification of L with ℘({1, . . . , m}) where λ ∈ L corresponds to
+supp(λ). This turns L into a lattice where the partial order ⪯ is induced by inclusion of sets
+and joins and meets are ∪ and ∩. The next concept is originally due to Lovász [7].
+Definition 3.4. A function f : L → R is called supermodular if for any a, b ∈ L
+f(a ∨ b) + f(a ∧ b) ≥ f(a) + f(b).
+It is called modular if equality holds.
+Definition 3.5. A function f : Ω → R is called convex-supermodular if f is convex and its
+restriction to L is supermodular.
+Example 3.6. Let f : Ω → R and suppose that f = h ◦ g for some g: [0, 1]m → R and
+h: R → R such that either
+(1) h is convex and g is affine with non-negative coefficients except the constant term,
+namely g = �m
+i=0 aiℓi where a1, . . . , am ≥ 0.
+(2) g is convex, g|L is modular, and h is convex and increasing.
+
+EUROPEAN BASKETS IN DISCRETE-TIME CONTINUOUS-BINOMIAL MARKET MODELS
+9
+Then f is convex-supermodular.
+Proof: Indeed, f is convex as composition of convex functions and f|L is supermodular by
+[Simchi-Levi, Theorem 2.2.6] for item (1) and [Simchi-Levi Proposition 2.2.5(c)] for item (2).
+♦
+For every 0 ≤ k ≤ m let ρk ∈ L denote the element corresponding to {1, . . . , k}, namely
+(11)
+ρk = (1, . . . , 1
+� �� �
+k times
+, 0, . . . , 0) ∈ {0, 1}m.
+Definition 3.7 (Compare [5]). The upper supermodular vertex is the probability density
+function q∗ : L → R supported on {ρ0, . . . , ρm} with
+q∗(ρk) = bk − bk+1,
+(0 ≤ k ≤ m).
+One easily checks that �m
+i=0 q∗(ρk) = 1 and that Eq∗(ℓi) = bi, thus
+Proposition 3.8. q∗ ∈ M(Ω, b) and it is supported on L ⊆ Ω.
+The main result of this section is the following theorem.
+Theorem 3.9. Let f : Ω → R be a continuous convex-supermodular function and assume that
+f ≥ 0. Let q∗ be the upper supermodular vertex of L (Definition (3.7)). Then
+sup
+p∈M+(Ω,b)
+Ep(f) = Eq∗(f|L) =
+max
+p∈M(Ω,b) Ep(f).
+Note that Eq∗(f|L) = �m
+i=0 q∗(ρi) · f(ρi).
+In the remainder of this section we prove this theorem.
+It relies on the following key
+observation. Equip Rm with the norm ∥ ∥∞ and restrict this norm to Ω = [0, 1]m.
+Lemma 3.10 (Approximation lemma). Consider some q ∈ M(Ω, b) supported on a finite
+subset {x1, . . . , xk} of Ω and set qi = q({xi}). Suppose that b ∈ int(Ω) = (0, 1)m. Then for
+any ǫ > 0 and δ > 0 there exists p ∈ M+(Ω, b) and β < ǫ such that for any continuous
+f : Ω → R
+Ep(f) = β
+�
+Ω
+f dµ +
+k
+�
+i=0
+(qi − β
+k ) · f(ξi)
+where ξ1, . . . , ξk ∈ Ω are such that ∥ξi − xi∥∞ < δ.
+Proof. Closed balls of radius r > 0 in Rm have the form B(y, r) = y + [−r, r]m so their
+volume, hence their Lebesgue measure, is (2r)m. If y = (y1, . . . , ym) then by inspection, for
+any 1 ≤ j ≤ m
+�
+B(y,r)
+xj dµ(x1, . . . , xm) = (2r)myj.
+Claim: There exist distinct y1, . . . , yk in int(Ω) = (0, 1)m such that ∥yi − xi∥∞ < δ
+3 and such
+that �k
+i=1 qiyi = b.
+Proof: We show how to perturb the vectors x1, . . . , xk in order to obtain y1, . . . , yk. First,
+�k
+i=1 qixi = Eq(L) = b since q ∈ M(Ω, b). Suppose that for some 1 ≤ j ≤ m not all of
+x1
+j, . . . , xk
+j are in the open interval (0, 1). If xi′
+j = 0 for some i′ then there must exist i′′ such
+that xi′′
+j
+> 0 because �k
+i=1 qixi
+j = bj > 0. Since qi > 0 for all i we can increase xi′
+j and
+decrease xi′′
+j
+by a small positive number < δ
+3 so that the sum remains bj and that the new
+
+10
+JAREK KĘDRA, ASSAF LIBMAN, AND VICTORIA STEBLOVSKAYA
+values of xi′
+j and xi′′
+j are in (0, 1). Similarly, if xi′
+j = 1 for some i′ then there must exist some
+i′′ such that xi′′
+j < 1 because �k
+i=1 qixi
+j = bj < 1. We can then decrease xi′
+j and increase xi′′
+j
+by at most δ
+3 so that the sum remains bj and the new values of xi′
+j and xi′′
+j are in (0, 1). By
+repeating this process we can perturb x1, . . . , xk into y1, . . . , yk in int(Ω) such that �
+i qiyi = b
+and ∥xi − yi∥∞ < δ
+3. Since the xi’s admit pairwise disjoint neighbourhoods, we can perturb
+the xi’s inside these neighbourhoods to make sure that the yi’s are distinct.
+q.e.d
+Since y1, . . . , yk are in the interior of Ω there exists r < δ
+3 such that B(yi, 2r) ⊆ (0, 1)m for
+all 1 ≤ i ≤ k. Thus, if γ ∈ Rm is such that ∥γ∥∞ < r then B(yi + γ, r) ⊆ (0, 1)m. Set
+Q = min{q1, . . . , qm}.
+For any β > 0 set
+γ1
+j :=
+β · ( 1
+k
+�k
+i=1 yi
+j − 1
+2)
+q1 − β
+k
+.
+Choose 0 < β < min{ǫ, kQ} sufficiently small such that for every 1 ≤ j ≤ m
+|γ1
+j | < r.
+Let γ1 ∈ Rm be the vector with the components γ1
+j defined above and let γ2, . . . , γk ∈ Rm be
+the zero vectors. By construction ∥γi∥∞ < r for all 1 ≤ i ≤ k, hence B(yi + γi, r) ⊆ (0, 1)m.
+Define p: Ω → R by
+p = β +
+k
+�
+i=1
+qi− β
+k
+(2r)m · 1B(yi+γi,r)
+where 1B(yi+γi,r) is the characteristic function.
+Observe that p > 0 because β > 0 and
+qi − β
+k > qi − Q ≥ 0. Next, p is a pdf since
+�
+Ω
+p dµ = β +
+k
+�
+i=1
+qi − β
+k
+(2r)m
+�
+Ω
+1B(yi+γi,r)dµ =
+k
+�
+i=1
+qi = 1.
+We check that p ∈ M+(Ω, b). Indeed, since γi = 0 for all i ≥ 2
+Ep(ℓj) =
+�
+x∈Ω
+ℓj(x) · p(x) dµ(x)
+= β
+�
+Ω
+xj dµ +
+1
+(2r)m
+k
+�
+i=1
+(qi − β
+k)
+�
+B(yi+γi,r)
+xj dµ
+= 1
+2β +
+k
+�
+i=1
+(qi − β
+k )(yi
+j + γi
+j)
+= 1
+2β +
+k
+�
+i=1
+qiyi
+j + (q1 − β
+k )γ1
+j − β
+k
+k
+�
+i=1
+yi
+j
+= 1
+2β + bj + β( 1
+k
+k
+�
+i=1
+yi
+j − 1
+2) − β
+k
+k
+�
+i=1
+yi
+j
+= bj.
+
+EUROPEAN BASKETS IN DISCRETE-TIME CONTINUOUS-BINOMIAL MARKET MODELS
+11
+Suppose that f : Ω → R is continuous. By the mean value theorem there exist ξi ∈ B(yi+γi, r)
+such that
+Ep(f) =
+�
+x∈Ω
+f(x) · p(x) dµ(x) = β
+�
+Ω
+f dµ +
+k
+�
+i=1
+1
+(2r)m (qi − β
+k )
+�
+B(yi+γi,r)
+f dµ =
+β
+�
+Ω
+f dµ +
+k
+�
+i=1
+(qi − β
+k ) · f(ξi).
+By our choice β < ǫ and ∥ξi −xi∥∞ ≤ ∥ξi −(yi +γi)∥∞ +∥yi −xi∥∞ +∥γi∥∞ < r + δ
+3 +r < δ.
+This completes the proof.
+□
+Proof of Theorem 3.9. For i = 0, . . . , m set
+α0 = f(ρ0)
+αi = f(ρi) − f(ρi−1).
+Observe that since by definition b0 = 1 and bm+1 = 0,
+Eq∗(f|L) =
+m
+�
+i=0
+q∗(ρi) · f(ρi)
+=
+m
+�
+i=0
+(bi − bi+1)f(ρi)
+= f(ρ0) +
+m
+�
+i=1
+bi(f(ρi) − f(ρi−1))
+= α0 +
+m
+�
+i=1
+biαi.
+Let ˇf : Rm → R be the affine function ˇf = �m
+i=0 αiℓi, namely
+ˇf(x1, . . . , xm) = α0 +
+m
+�
+i=1
+αixi.
+Claim 1: ˇf dominates f on L, namely ˇf(λ) ≥ f(λ) for all λ ∈ L.
+Proof: By construction of ˇf and by the definition of ρj in (11), for any 0 ≤ j ≤ m
+ˇf(ρj) =
+j
+�
+i=0
+αi = f(ρj).
+So f and ˇf coincide on {ρ0, . . . , ρm} ⊆ L. Assume the statement of the claim is false, namely
+there exists λ ∈ L such that ˇf(λ) < f(λ). Among all these λ’s choose one which contains the
+longest leading run of 1, . . . , 1, namely λ with the largest possible k with ρk ⪯ λ. Notice that
+k < m because ˇf(ρm) = f(ρm) and ρm is maximal in L. In the lattice L set λ′ = λ ∨ ρk+1.
+Notice that since k is the largest such that ρk ⪯ λ it follows that λ ∧ ρk+1 = ρk. Since f|L is
+supermodular
+f(λ ∨ ρk+1) + f(ρk) ≥ f(λ) + f(ρk+1).
+Since ˇf is affine, it is modular, hence
+ˇf(λ ∨ ρk+1) + ˇf(ρk) = ˇf(λ) + ˇf(ρk+1).
+
+12
+JAREK KĘDRA, ASSAF LIBMAN, AND VICTORIA STEBLOVSKAYA
+Subtract the first equation from the second, taking into account that ˇf(ρi) = f(ρi), to get
+ˇf(λ ∨ ρk+1) − f(λ ∨ ρk+1) ≤ ˇf(λ) − f(λ) < 0.
+Thus ˇf(λ′) < f(λ′) and ρk+1 ⪯ λ′, contradiction to the maximality of k.
+q.e.d
+Claim 2: ˇf dominates f on Ω = [0, 1]m, namely ˇf(x) ≥ f(x) for all x ∈ Ω.
+Proof: Since Ω = [0, 1]m is the convex hull of L = {0, 1}m, every x ∈ Ω is a convex combination
+x =
+�
+λ∈L
+tλ · λ.
+Since ˇf is affine and f is convex, Claim 1 implies that
+ˇf(x) = ˇf(
+�
+λ∈L
+tλ · λ) =
+�
+λ∈L
+tλ ˇf(λ) ≥
+�
+λ∈L
+tλf(λ) ≥ f(
+�
+λ∈L
+tλλ) = f(x).
+q.e.d
+Claim 2 implies that for any p ∈ M(Ω, b)
+Ep(f) ≤ Ep( ˇf) = Ep(
+m
+�
+i=0
+αiℓi) =
+m
+�
+i=0
+αiEp(ℓi) = α0 +
+m
+�
+i=1
+αibi = Eq∗(f|L).
+However, q∗ extends to �q∗ ∈ M(Ω, b) and clearly E �
+q∗(f) = Eq∗(f|L) so we get
+max
+p∈M(Ω,b) Ep(f) = Eq∗(f|L).
+Notice that Eq∗(f|L) = �m
+i=0 q∗(ρi) · f(ρi). Lemma 3.10 applied to q∗ ∈ M(Ω, b) and the
+continuity of f easily imply that there exist λ ∈ M+(Ω, b) such that Eλ(f) is arbitrarily close
+to Eq∗(f). If follows that supM+(Ω,b) Ep(f) = Eq∗(f|L).
+□
+4. (L, b)-stable probability measures
+Main results. We consider n iterations of the random vector L = (ℓ1, . . . , ℓm) over Ω =
+[0, 1]m from Section 3. We obtain a sequence L1, . . . , Ln of random vectors in Rm with sample
+space Ωn. In fact
+Lk : Ωn → Ω ⊆ Rm
+is the projection to the kth factor. This is the universal process on Ωn, see Section 2.D. Denote
+by M+(Ω, n) the set of all non-degenerate probability measures on Ωn. We will identify these
+with the set of pdf’s p: Ωn → R which are non-vanishing a.e.
+With the notation and terminology of Section 2.B we make the following definition.
+Definition 4.1. Consider some b ∈ int(Ω) = (0, 1)m. A pdf p ∈ M+(Ω, n) is called (L, b)-
+stable if for any 0 ≤ k ≤ n − 1
+Ep(Lk+1|L1, . . . , Lk) = b.
+That is, the function Ep(Lk+1|L1, . . . , Lk): Ωk → Rm is constant with value b a.e. The set of
+all (L, b)-stable p ∈ M+(Ω, n) is denoted
+M+(Ω, n, b).
+Definition 4.2. A function f : Ωn → R is called fibrewise convex-supermodular if it is convex-
+supermodular at each fibre.
+Namely, for any τ ∈ Ωk−1 and any θ ∈ Ωn−k the function
+g: Ω → R defined by g: ω �→ f(τ, ω, θ) is convex-supermodular (Definition 3.5).
+
+EUROPEAN BASKETS IN DISCRETE-TIME CONTINUOUS-BINOMIAL MARKET MODELS
+13
+Example 4.3. For any J = (j1, . . . , jn) ∈ Pn(m), see (5), let ℓJ : Ωn → R denote the function
+ℓJ = ℓj1 ⊗ · · · ⊗ ℓjn, namely
+ℓJ : (ω1, . . . , ωn) �→ ℓj1(ω1) · · · ℓjn(ωn).
+Consider f : Ωn → R of the form f = h ◦ g where h: R → R is convex and g: Ωn → R is of
+the form
+g =
+�
+J∈Pn(m)
+aJ · ℓJ
+where aJ ≥ 0 for all J ̸= (0, . . . , 0). Then f is fibrewise convex-supermodular.
+Proof: It is clear that if τ ∈ Ωk−1 and θ ∈ Ωn−k then the restriction of ℓJ to the fibre
+{τ} × Ω × {θ} ⊆ Ωn is equal to �n
+i=0 biℓi where bi ≥ 0 for all i ≥ 1; In fact bi is the sum of
+all aJ in which the kth entry is equal to i. The result follows from Example 3.6(1).
+The main results of this section are the following two theorems.
+Theorem 4.4. Let f : Ωn → R be a continuous function of the form f = h ◦ g in Example
+4.3. Assume that f ≥ 0. Suppose that b ∈ int(Ω) = (0, 1)m. Then
+inf
+p∈M+(Ω,n,b) Ep(f|L1, . . . , Lk)(ω1, . . . , ωk) = f(ω1, . . . , ωk, b, · · · , b).
+Recall the upper supermodular vertex q∗ : L → R from Definition 3.7. It extends to an
+atomic probability measure on Ω supported on L which we abusively denote q∗. Let
+q∗⊗k
+be the obvious product probability measure on Lk as well as its extension to Ωk.
+Set qj = q∗(ρj) for all 0 ≤ j ≤ m. For any J ∈ Pk(m) denote
+qJ
+=
+�
+j∈J
+qj
+ρJ
+=
+(ρj1, · · · , ρjk) ∈ Ωk.
+If f : Ωn → R is measurable and (ω1, . . . , ωk) ∈ Ωk, we obtain a measurable function
+g: Ωn−k → R by g(−) = f(ω1, . . . , ωk, −). If Q is a probability measure on Ωn−k we will
+write EQ(f(ω1, . . . , ωk, −)) for EQ(g).
+Theorem 4.5. Let f : Ωn → R be a continuous fibrewise convex-supermodular, f ≥ 0. Then
+sup
+p∈M+(Ω,n,b)
+Ep(f|L1, . . . , Lk)(ω1, . . . , ωk) = Eq∗⊗n−k(f(ω1, . . . , ωk, −))
+=
+�
+J∈Pn−k(m)
+qJ · f(ω1, . . . , ωk, ρJ).
+In the remainder of this section we prove Theorems 4.4 and 4.5. Throughout we fix b ∈ (0, 1)m
+and assume that it is non-increasing, namely b1 ≥ · · · ≥ bm.
+Lemma 4.6. Consider some p ∈ M+(Ω, n, b). Suppose that 0 ≤ k ≤ n − 1 and consider
+ω1, . . . , ωk ∈ Ω. Set p′ = pLk+1|(L1,...,Lk)=(ω1,...,ωk); See Section 2.B. Then p′ ∈ M+(Ω, b).
+Proof. First, p′ is a pdf and p′ > 0 a.e., see (9) in Section 2.B. Set I = {1, . . . , k} and
+J = {k + 1}, subsets of {1, . . . , n}. Write LI for the random vector (L1, . . . , Lk) and ωI =
+
+14
+JAREK KĘDRA, ASSAF LIBMAN, AND VICTORIA STEBLOVSKAYA
+(ω1, . . . , ωk) ∈ Ωk. We use Lemma 2.1 and the fact that p ∈ M+(Ω, n, b) to compute
+Ep′(L) = Ep′(ω �→ Lk+1(ω)) = Ep′(ω �→ Ep(Lk+1|LI∪J)(ωI, ω))
+= Ep(Lk+1|LI = ωI) = Ep(Lk+1|L1, . . . , Lk)(ω1, . . . , ωk) = b.
+By definition, then, p′ ∈ M+(Ω, b).
+□
+Lemma 4.7. Let p1, . . . , pn ∈ M+(Ω, b). Then p1 ⊗ · · · ⊗ pn ∈ M+(Ω, n, b).
+Proof. It is clear that p1 ⊗ · · · ⊗ pn is a non degenerate pdf on Ωn. By Lemma 2.3(3) and
+since by definition Epk+1(L) = b,
+Ep1⊗···⊗pn(Lk+1|L1, . . . , Lk)(ω1, . . . , ωk) = Epk+1(Lk+1) = Epk+1(L) = b.
+Since this holds for all 0 ≤ k ≤ n − 1, by definition p1 ⊗ · · · ⊗ pn ∈ M+(Ω, n, b).
+□
+Lemma 4.8 (n-fold Approximation Lemma). Let b ∈ (0, 1)n and consider q ∈ M(Ω, b) with
+finite support {x1, . . . , xr} and set qi = q({xi}). Let q⊗k denote the induced product measure
+on Ωk. Let f : Ωn → R be continuous with f ≥ 0. Then for any ǫ > 0 and any 0 ≤ k ≤ n
+there exists P ∈ M+(Ω, n, b) such that
+���EP (f|L1, . . . , Lk)(ω1, . . . , ωk) − Eq⊗(n−k)(f(ω1, . . . , ωk, −))
+��� < ǫ
+for all ω1, . . . , ωk ∈ Ω.
+Proof. Since Ωn is compact, f is bounded, i.e ∥f∥∞ < ∞. Since f is uniformly continuous,
+we choose δ > 0 suitable for
+ǫ
+3n.
+Apply Lemma 3.10 with δ and with
+ǫ
+3n∥f∥∞ to obtain
+p ∈ M+(Ω, b) and β <
+ǫ
+3n∥f∥∞ such that for any continuous g: Ω → R where g ≥ 0,
+(12)
+Ep(g) = β
+�
+Ω
+g dµ +
+r
+�
+i=1
+(qi − β
+r )g(ξi)
+for some ξ1, . . . , ξr ∈ Ω such that ∥xi − ξi∥∞ < δ.
+Set [r] = {1, . . . , r}. For any I = (i1, . . . , in−k) ∈ [r]n−k set
+qI = qi1 · · · qin−k
+and
+xI = (xi1, . . . , xin−k) ∈ Ωn−k
+and let fI : Ωk → R be the function
+fI : (ω1, . . . , ωk) �→ f(ω1, . . . , ωk, xI).
+Observe that
+(13)
+Eq⊗(n−k)(f(ω1, . . . , ωk, −)) =
+�
+I∈[r]n−k
+qI · fI(ω1, . . . , ωk).
+It is clear that ∥fI∥∞ ≤ ∥f∥∞ and that fI is uniformly continuous with the same δ suitable
+for
+ǫ
+3n as that for f. Recall p ∈ M+(Ω, b) that we chose at the start of the proof.
+Claim: Consider some 0 ≤ k < n and some I ∈ [r]n−k−1. Then for any ω1, . . . , ωk ∈ Ω
+�����Ep
+�
+ω �→ fI(ω1, . . . , ωk, ω)
+�
+−
+r
+�
+i=1
+qi · fI(ω1, . . . , ωk, xi)
+����� < ǫ
+n.
+
+EUROPEAN BASKETS IN DISCRETE-TIME CONTINUOUS-BINOMIAL MARKET MODELS
+15
+Proof: Set g(w) = fI(ω1, . . . , ωk, ω). Clearly g: Ω → R is continuous and ∥g∥∞ ≤ ∥f∥∞.
+Moreover, it is uniformly continuous and clearly the same δ we chose for f suitable for
+ǫ
+3n
+works for g. Since Eq(g) = �r
+i=1 qig(xi) and since (12) holds
+|Ep(g) − Eq(g)| ≤ β
+�
+Ω
+g dµ + β
+r
+r
+�
+i=1
+|g(ξi)| +
+r
+�
+i=1
+qi|g(ξi) − g(xi)|.
+Since ∥g∥∞ ≤ ∥f∥∞ and β <
+ǫ
+3n∥f∥∞ the first and second terms in this sum are less than
+ǫ
+3n.
+Since ∥ξi − xi∥∞ < δ, the uniform continuity of g implies that the same is true for the last
+term since �
+i qi = 1. This completes the proof of the claim.
+q.e.d
+Set P = p⊗n. Then P ∈ M+(Ω, n, b) by Lemma 4.7. In light of (13), we complete the
+proof of the lemma by showing by downward induction on 0 ≤ k ≤ n that
+(14)
+������
+Ep⊗n(f|L1, . . . , Lk)(ω1, . . . , ωk) −
+�
+I∈[r]n−k
+qI · fI(ω1, . . . , ωk)
+������
+≤ (n − k) ǫ
+n.
+The base of induction k = n is a triviality since
+Ep⊗n(f|L1, . . . , Ln)(ω1, . . . , ωn) = f(ω1, . . . , ωn)
+and since fI = f and qI = 1 for the the only I ∈ [r]0.
+Assume inductively that (14) holds for some 1 ≤ k ≤ n. Fix some ω1, . . . , ωk−1 ∈ Ω.
+Lemmas 2.1 and 2.3(2) imply that for any
+Ep(τ �→ Ep⊗n(f|L1, . . . , Lk)(ω1, . . . , ωk−1, τ)) = Ep⊗n(f|L1, . . . , Lk−1)(ω1, . . . , ωk−1).
+Viewing the left hand side of (14) with ω1, . . . , ωk−1 fixed as a function of τ ∈ Ω, the linearity
+of expectation Ep(−) implies
+������
+Ep⊗n(f|L1, . . . , Lk−1)(ω1, . . . ωk−1) −
+�
+I∈[r]n−k
+qIEp(fI(ω1, . . . , ωk−1, −))
+������
+< (n − k) ǫ
+n.
+Thanks to (13), in order to complete the induction step (to k − 1) it remains to show by that
+������
+�
+I∈[r]n−k
+qI · Ep(fI(ω1, . . . , ωk−1, −)) −
+�
+J∈[r]n−k+1
+qJ · fJ(ω1, . . . , ωk−1)
+������
+< ǫ
+n.
+Given J = (i1, . . . , in−k+1) ∈ [r]n−k+1 set I = (i2, . . . , in−k+1) ∈ [r]n−k and observe that
+qJ = qIqi1 and that fJ(ω1, . . . , ωk−1) = fI(ω1, . . . , ωk−1, xi1). By the Claim above
+���
+�
+I∈[r]n−k
+qIEp(fI(ω1, . . . , ωk−1, −)) −
+�
+J∈[r]n−k+1
+qJfJ(ω1, . . . , ωk−1)
+��� =
+=
+���
+�
+I∈[r]n−k
+qIEp(fI(ω1, . . . , ωk−1, −)) −
+�
+I∈[r]n−k
+r
+�
+i=1
+qIqifI(ω1, . . . , ωk−1, xi)
+���
+≤
+�
+I∈[r]n−k
+qI ·
+���Ep(fI(ω1, . . . , ωk−1, −)) −
+r
+�
+i=1
+qifI(ω1, . . . , ωk−1, xi)
+���
+<
+�
+I∈[r]n−k−1
+qI · ǫ
+n = ǫ
+n.
+This completes the induction step.
+□
+
+16
+JAREK KĘDRA, ASSAF LIBMAN, AND VICTORIA STEBLOVSKAYA
+Proof of Theorem 4.4. For any J = (j1, . . . , jn) ∈ Pn(m) and any ω1, . . . , �
+ωk, . . . , ωn ∈ Ω
+(meaning ωk is omitted) we have
+ℓJ(ω1, . . . , ωk−1, −, ωk+1, . . . , ωn) =
+�
+i̸=k
+ℓji(ωi) · ℓjk(−).
+Therefore, if λ ∈ M+(Ω, b) we get
+Eλ(ℓJ(ω1, . . . , ωk−1, −, ωk+1, . . . , ωn)) = Eλ(ℓjk) ·
+�
+i̸=k
+ℓji(ωi) = bjk ·
+�
+i̸=k
+ℓji(ωi) =
+ℓjk(b) ·
+�
+i̸=k
+ℓji(ωi) = ℓJ(ω1, . . . , ωk−1, b, ωk+1, . . . , ωn).
+We use downward induction on 0 ≤ k ≤ n to show that for any p ∈ M+(Ω, n, b)
+Ep(f|L1, . . . , Lk)(ω1, . . . , ωk) ≥ f(ω1, . . . , ωk, b, . . . , b)
+almost everywhere. The base of induction k = n is a triviality (and in fact, equality holds a.e).
+Assume the inequality holds for k+1 ≤ n. Set p′ = pLk+1|L1=ω1,...,Lk=ωk. Then p′ ∈ M+(Ω, b)
+by Lemma 4.6. Lemma 2.1 and the induction hypothesis imply
+Ep(f|L1, . . . , Lk)(ω1, . . . , ωk) = Ep′(ω �→ Ep(f|L1, . . . , Lk+1)(ω1, . . . , ωk, ω))
+≥ Ep′(ω �→ f(ω1, . . . , ωk, ω, b, . . . , b).
+Since f = h ◦ g with h convex and g as in Example 4.3, Jensen’s inequality allows us to
+continue the inequality
+≥ h(
+�
+J
+aJEp′(ℓJ(ω1, . . . , ωk, −, b, . . . , b))
+= h(
+�
+J
+aJℓJ(ω1, . . . , ωk, b, . . . , b))
+= h(g(ω1, . . . , ωk, b, . . . , b))
+= f(ω1, . . . , ωk, b, . . . , b).
+This completes the induction step.
+We deduce that in the statement of the theorem the right hand side is a lower bound
+for the left hand side and it remains to show equality.
+Let ν be the probability mea-
+sure on Ω supported on {b}, i.e ν({b}) = 1. It is clear that for any measurable function
+g: Ωk → R we have Eν⊗k(g) = g(b, . . . , b). By Lemma 4.8, for any ǫ > 0 there exists P ∈
+M+(Ω, n, b) such that EP (f|L1, . . . , Lk)(ω1, . . . , ωk) is ǫ-close to Eν⊗(n−k)(f(ω1, . . . , ωk, −)) =
+f(ω1, . . . , ωk, b, . . . , b). This completes the proof.
+□
+Proof of Theorem 4.5. First, observe that
+(15)
+Eq∗⊗(n−k)(f(ω1, . . . , ωk, −)) =
+�
+I∈Pn−k(m)
+qI · f(ω1, . . . , ωk, ρI).
+Next, we prove that for any p ∈ M+(Ω, n, b) and any 0 ≤ k ≤ n
+(16)
+Ep(f|L1, . . . , Lk)(ω1, . . . , ωk) ≤ E(q∗)⊗n−k(f(ω1, . . . , ωk, −)).
+Fix some p and use downward induction on k. The base of induction k = n is a triviality
+since Ep(f|L1, . . . , Ln) = f a.e.
+Assume inductively that (16) holds for k + 1 ≤ n.
+Set
+
+EUROPEAN BASKETS IN DISCRETE-TIME CONTINUOUS-BINOMIAL MARKET MODELS
+17
+p′ = pLk+1|(L1,...,Lk)=(ω1,...,ωk). Lemma 2.1 and the induction hypothesis together with (15)
+imply that
+Ep(f|L1, . . . , Lk)(ω1, . . . , ωk) = Ep′(ω �→ Ep(f|L1, . . . , Lk+1)(ω1, . . . , ωk, ω))
+≤ Ep′(ω �→
+�
+I∈Pn−k−1
+qIf(ω1, . . . , ωk, ω, ρI))
+=
+�
+I∈Pn−k−1(m)
+qI · Ep′(ω �→ f(ω1, . . . , ωk, ω, ρI)).
+By the assumption on f, each function f(ω1, . . . , ωk, −, ρI) is convex-supermodular and con-
+tinuous and. Lemma 4.6 and Theorem 3.9 allow us to continue the estimate
+≤
+�
+I∈Pn−k−1(m)
+qI ·
+m
+�
+j=0
+qj · f(ω1, . . . , ωk, ρj, ρI)) =
+�
+I∈Pn−k(m)
+qI · f(ω1, . . . , ωk, ρI).
+Together with (15), this completes the induction step.
+We deduce that for any 0 ≤ k ≤ n the right hand side in the statement of the theorem
+is an upper bound for the left hand side. By Lemma 4.8 there exist P ∈ M+(Ω, n, b) such
+that EP(f|L1, . . . , Lk)(ω1, . . . , ωk) are arbitrarily close to Eq∗⊗(n−k)(f(ω1, . . . , ωk, −). This
+completes the proof.
+□
+5. Proofs of the main results
+In this section we prove the results in Section 1. We start by setting up a formal framework
+for the discrete-time continuous-binomial market model presented there.
+We begin with the “one-step” process, namely description of the price jumps Ψi where
+0 ≤ i ≤ m. By definition Ψ0 = R and Ψi are chosen at random from the interval [Di, Ui]. By
+choosing a linear homeomorphisms [Di, Ui] ∼= [0, 1], a natural sample space for the probability
+space underlying a single step is Ω = [0, 1]m and
+Ψi(x1, . . . , xm) = Di + (Ui − Di)xi,
+Ψ0(x1, . . . , xm) = R
+With the notation of Section 3, for any 1 ≤ i ≤ m
+Ψi = Diℓ0 + (Ui − Di)ℓi.
+We will write Ψ: Ω → Rm+1 for the random vector
+Ψ = (Ψ0, . . . , Ψm).
+The natural sample space for the n-step model is Ωn. We obtain a process Ψ1, . . . , Ψn of
+the price changes at time k:
+Ψk : Ωn Lk
+−→ Ω Ψ
+−→ Rm+1
+where Lk is the projection to the k-th factor and L1, . . . , Lk form the universal process on
+Ωn, see Section 2.D. Thus,
+Ψk
+i = Di + (Ui − Di)Lk
+i
+where Lk
+i is the ith component of Lk : Ωn → Ω ⊆ Rm+1 and we observe that (since ℓ0 =
+1: Ω → R)
+Lk
+i = ℓ⊗(k−1)
+0
+⊗ ℓi ⊗ ℓ⊗(n−k−1)
+0
+.
+Recall that we assume that 0 < Di < R < Ui so in particular Ψk
+i > 0 for all i and all k.
+
+18
+JAREK KĘDRA, ASSAF LIBMAN, AND VICTORIA STEBLOVSKAYA
+The prices of the assets form an Rm+1-valued process
+S0, . . . , Sn : Ωn → Rm+1
+where Sk = (Sk
+0, . . . , Sk
+m) is the vector of prices of the assets at time k. It is assumed by the
+model that
+Sk
+i > 0
+for all 0 ≤ i ≤ m and 0 ≤ k ≤ n.
+By construction of the model, the processes S0, . . . , Sn and Ψ1, . . . , Ψn satisfy the relation
+Sk
+i = Sk−1
+i
+· Ψk
+i
+(0 ≤ i ≤ m and 1 ≤ k ≤ n).
+It is therefore clear that for any 0 ≤ k ≤ n
+Sk
+i = S0
+i · Ψ1
+i · · · Ψk
+i .
+Recall the definition of Pk(m) from (5) in Section 1.
+For J = (j1, . . . , jk) ∈ Pk(m) set
+ℓJ = ℓj1 ⊗ · · · ⊗ ℓjk. It follows that
+Sk
+i =
+�
+J∈Pk(m)
+aJ · L1
+j1 · · · Lk
+jk =
+�
+J∈Pk(m)
+aJ · ℓJ ⊗ 1 ⊗ · · · ⊗ 1
+�
+��
+�
+n − k times
+for some aJ ≥ 0.
+Comment: In Section 1 the processes Ψk and Sk were denoted Ψ(k) and S(k).
+The European basket in Section 1 is the function (random variable) F : Ωn → R
+F =
+� m
+�
+i=0
+ci · Sn
+i − K
+�
+��
+�
+G
+�+
+where ci ≥ 0 for 1 ≤ i ≤ m and K > 0 is some number. Notice that
+G =
+�
+J∈Pn(m)
+aJℓJ
+where aJ ≥ 0 for all J ̸= (0, . . . , 0). It follows from Example 4.3 and since h(x) = x+ is
+continuous, convex and non-negative that
+Proposition 5.1. F is continuous and fibrewise convex-supermodular and F ≥ 0.
+Recall that a non-degenerate probability measure p on Ωn is called risk neutral if for any
+k ≥ 0 and any j ≥ 1 such that k + j ≤ n
+Ep(Sk+j|L1, . . . , Lk)(ω1, . . . , ωk) = Rj · Sk(ω1, . . . , ωk).
+We denote the set of these probability measures by RN.
+Proposition 5.2. RN = M+(Ω, n, b) where b = (b1, . . . , bm) is defined in (2) in Section 1.
+Proof. Since Sk+j
+i
+= Sk
+i · Ψk+1
+i
+· · · Ψk+j
+i
+and since Sk
+i > 0, it is clear that the condition for p
+being a risk neutral measure is equivalent to the condition
+Ep(Ψk+1
+i
+· · · Ψk+j
+i
+|L1, . . . , Lk) = Rj
+(almost everywhere constant function Ωn−k → R). It is easily verified using induction and
+Lemmas 2.2 and 2.1 that this condition (for any k, j such that k + j ≤ n) is equivalent to the
+single step condition, namely
+Ep(Ψk+1
+i
+|L1, . . . , Lk) = R
+
+EUROPEAN BASKETS IN DISCRETE-TIME CONTINUOUS-BINOMIAL MARKET MODELS
+19
+for all 1 ≤ k ≤ n − 1. But Ψk+1
+i
+= Di + (Ui − Di) · Lk+1
+i
+. So the condition above is equivalent
+to
+Di + (Ui − Di) · Ep(Lk+1
+i
+|L1, . . . , Lk) = R.
+Using the definition of b1, . . . , bm in (2), this is equivalent to Ep(Lk+1
+i
+|L1, . . . , Lk) = bi, and
+collecting these for all 1 ≤ i ≤ m we get
+Ep(Lk+1|L1, . . . , Lk) = b
+which by definition is the condition for p ∈ M+(Ω, n, b).
+□
+Proof of Theorem 1.1. By Proposition 5.1 F is continuous convex-supermodular and F ≥ 0.
+The interval (Γmin(F, k) , Γmax(F, k)) of the rational values of F at some state of the world
+(ω1, . . . , ωk) ∈ Ωk is known to be the collection of numbers
+{ Rk−n · Ep(F|L1, . . . , Lk)(ω1, . . . , ωk) }p∈RN.
+Proposition 5.2 and Theorem 4.4 imply that
+Rn−k · Γmin(F, k)(ω1, . . . , ωk) = inf
+p∈RN Ep(F|L1, . . . , Lk)(ω1, . . . , ωk) =
+inf
+p∈M+(Ω,n,b) Ep(F|L1, . . . , Lk)(ω1, . . . , ωk) = F(ω1, . . . , ωk, b, . . . , b).
+By definition of b, see (2), and since Ψi = Diℓ0 + (Ui − Di)ℓi,
+Ψi(b) = R
+for all 1 ≤ i ≤ m.
+By definition, for any 1 ≤ i ≤ m
+Sn
+i (ω1, . . . , ωk, b, . . . , b) = S0
+i · Ψi(ω1) · · · ψi(ωk) · Ψi(b) · · · Ψi(b)
+�
+��
+�
+n − k times
+= Rn−kSk
+i (ω1, . . . , ωk).
+Also, for i = 0 we clearly get Sn
+0 = S0
+0 · Rn = Rn−kSk
+0. It follows that
+F(ω1, . . . , ωk, b, . . . , b) =
+�
+Rn−k
+m
+�
+i=0
+ciSk
+i − K
+�+
+(ω1, . . . , ωk).
+This establishes the formula for Γmin(F, k).
+We note that the numbers qi defined in (3) and used in the statement of the theorem are
+equal to q∗(ρi) of the upper supermodular vertex (Definition 3.7). Since by Proposition 5.1
+the conditions of Theorem 4.5 hold, it follows that
+Rn−k · Γmax(F, k)(ω1, . . . , ωk) = sup
+p∈RN
+Ep(F|L1, . . . , Lk)(ω1, . . . , ωk)
+=
+sup
+p∈M+(Ω,n,b)
+Ep(F|L1, . . . , Lk)(ω1, . . . , ωk)
+=
+�
+J∈Pn−k(m)
+qJ · F(ω1, . . . , ωk, ρj1, . . . , ρjn−k).
+Since ℓi(ρj) = 1 if i ≤ j and ℓi(ρj) = 0 if i > j it follows that Ψi(ρj) = Di +(Ui −Di)ℓi(ρj) =
+χi(j). Therefore, for any J ∈ Pn−k(m),
+Sn
+i (ω1, . . . , ωk, ρj1, . . . , ρjn−k) = S0
+i · Ψi(ω1) · · · Ψi(ωk) · Ψi(ρj1) · · · Ψi(ρjn−k) =
+χi(J) · Sk
+i (ω1, . . . , ωk).
+
+20
+JAREK KĘDRA, ASSAF LIBMAN, AND VICTORIA STEBLOVSKAYA
+For i = 0 we get, of course, Sn
+0 = S0
+0 · Rn = Sk
+0 · χ0(J). Substitution into the definition of F
+we get
+F(ω1, . . . , ωk, ρj1, . . . , ρjn−k) =
+� m
+�
+i=0
+ci · χi(J) · Sk
+i − K
+�+
+(ω1, . . . , ωk).
+This establishes the formula for Γmax(F, k).
+□
+Proof of Theorem 1.3. Recall that Ψi(b) = Diℓ0 − (Ui − Di)ℓi(b) = Di + (Ui − Di)bi = R for
+1 ≤ i ≤ m and that Ψ0 = R by definition. By Theorem 1.1
+Γmin(F, 0) = R−n · F(b, . . . , b) = R−n(
+m
+�
+i=0
+ciS0
+i · Ψi(b)n − K)+ = R−n ·
+�
+Rn
+m
+�
+i=0
+ciS0
+i − K
+�+
+.
+This is independent of Ui, Di.
+Set bi as in (2) and denote by bi(s) the values of bi in our market model with parameters
+ui(s) and di(s). Notice that di(s) < R and that ui(s) > R for all 0 ≤ s < 1. Moreover,
+di(0) = Di and ui(0) = Ui and limsր1 di(s) = R and limsր1 ui(s) = R. One checks that
+bi(s) =
+R − di(s)
+ui(s) − di(s) = bi.
+Therefore the values of qi = bi − bi+1 are independent of s. The values of χi(j) do depend on
+s where χi(j)(s) = ui(s) if i ≤ j and χi(j)(s) = di(s) if i > j. Thus, χi(j)(s) is a polynomial
+(of degree 1) in s. Moreover, limsր χi(j)(s) = R. By Theorem 1.1
+ϕ(s) = Γmax(F, 0; ui(s), di(s)) = R−n
+�
+J∈Pn(m)
+qJ ·
+
+
+m
+�
+i=0
+ci · S0
+i ·
+�
+j∈J
+χi(j)(s)
+
+
++
+.
+So ϕ is a continuous function of s ∈ [0, 1). Now, ϕ(0) = Γmax(F, 0; Ui, Di) since di(0) = Di
+and ui(0) = Ui. Since h: x �→ x+ is continuous and �m
+j=0 qj = 1 we get
+lim
+sր1 ϕ(s) = R−n
+�
+J∈Pn(m)
+qJ
+
+
+m
+�
+i=0
+ciS0
+i · lim
+sր1
+�
+j∈J
+χi(j)(s) − K
+
+
++
+= R−n
+�
+J∈Pn(m)
+qJ
+� m
+�
+i=0
+ciS0
+i · Rn − K
+�+
+= R−n
+� m
+�
+i=0
+ciS0
+i · Rn − K
+�+
+= Γmin(F, 0; Ui, Di).
+The “intermediate value” result in the theorem follows from the continuity of ϕ(s).
+□
+In the next lemma we will consider processes on Ω, see Section 2.D. Recall that the universal
+process is L1, . . . , Ln where Lk is the projection Ωn → Ω to the k-th factor followed by the
+inclusion to Rm.
+Lemma 5.3. Fix some n. Let q be a probability measure on Ω supported on {ρ0, . . . , ρm}, see
+(11). Consider R-valued processes
+r0, . . . , rn−1 > 0
+and
+X0, . . . , Xn
+
+EUROPEAN BASKETS IN DISCRETE-TIME CONTINUOUS-BINOMIAL MARKET MODELS
+21
+and Rm-valued processes
+d0, . . . , dn−1
+and
+∆0, . . . , ∆n−1 > 0.
+Further, consider an Rm+1-valued processes
+S0, . . . , Sn
+and
+Ψ1, . . . , Ψn.
+with components Sk = (Sk
+0, . . . , Sk
+m) and Ψk = (Ψk
+0, . . . , Ψk
+m). Assume that
+(1) Sk
+i = Sk−1
+i
+· Ψk
+i for all 1 ≤ k ≤ n and all 0 ≤ i ≤ m.
+(2) Ψk+1
+0
+= rk and Ψk+1
+i
+= dk
+i + ∆k
+i · Lk+1
+i
+for any 0 ≤ k ≤ n − 1, and for any ωK =
+(ω1, . . . , ωk) ∈ Ωk,
+Eq(ω �→ Ψk+1
+i
+(ωK, ω)) = rk(ωK).
+(3) Xk are fibrewise convex-supermodular, and for any 0 ≤ k ≤ n − 1 and any ωK ∈ Ωk
+Eq(ω �→ Xk+1(ωK, ω)) = rk(ωK) · Xk(ωK).
+Then among all Rm+1-valued processes β0, . . . , βn−1 there exists a process α0, . . . , αn−1 which
+minimises the R-valued random process
+(17)
+V k
+β =
+m
+�
+i=0
+βk
+i · Sk
+i
+subject to the condition
+(18)
+m
+�
+i=0
+βk
+i · Sk+1
+i
+≥ Xk+1.
+Moreover, V k
+α = Xk and the value of αk(ωK) at ωK ∈ Ωk can be computed by
+
+
+αk
+0(ωK)
+αk
+1(ωK)
+...
+αk
+m(ωK)
+
+ = T(ωK)−1 · M′(ωK)−1 · Q ·
+
+
+Xk+1(ωKρ0)
+Xk+1(ωKρ1)
+...
+Xk+1(ωKρm)
+
+
+where Q is the matrix in the statement of Theorem 1.2 and T, M′ : Ωk → Mat(m+1)×(m+1)(R)
+are the (m + 1) × (m + 1) matrices
+T =
+
+
+rk · Sk
+0
+Sk
+1
+...
+Sk
+m
+
+
+M′ =
+
+
+1
+dk
+1
+dk
+2
+· · ·
+dk
+m
+0
+∆k
+1
+...
+...
+0
+∆k
+m
+
+
+Proof. We prove the lemma in a sequence of claims.
+Claim 1: If a process β0, . . . , βn−1 satisfies (18) then V k
+β ≥ Xk for all 0 ≤ k ≤ n − 1.
+
+22
+JAREK KĘDRA, ASSAF LIBMAN, AND VICTORIA STEBLOVSKAYA
+Proof: Choose some k and some ωK ∈ Ωk.
+Then (18) becomes the following system of
+(infinitely many) inequalities in the unknowns βk
+i (ωK)
+(19)
+m
+�
+i=0
+βk
+i (ωK) · Sk+1
+i
+(ωK, ω) ≥ Xk+1(ωK, ω),
+(ω ∈ Ω).
+Let Φ(ω) denote the left hand side of (19) and Y (ω) denote its right hand side. These are
+random variables with domain Ω so Eq(Φ) ≥ Eq(Y ). By the hypotheses on Xk
+Eq(Y ) = Eq(ω �→ Xk+1(ωKω) = rk(ωK) · Xk(ωK).
+By the hypotheses on Sk and Ψk
+Eq(Φ) =
+m
+�
+i=0
+βk
+i (ωK) · Sk
+i (ωK) · Eq(ω �→ Ψk+1
+i
+(ωK, ω)) =
+rk(ωK) ·
+m
+�
+i=0
+βk
+i (ωK) · Sk
+i (ωK) = rk(ωK) · V k
+β (ωK).
+Since rk > 0 it follows that V k
+β (ωK) ≥ Xk(ωK).
+q.e.d
+Consider some 0 ≤ k ≤ n − 1 and some ωK = (ω, . . . , ωk) ∈ Ωk. We will show in Claim 5
+below that the system of m + 1 linear equations with m + 1 unknowns αk
+0(ωK), . . . , αk
+m(ωK)
+(20)
+m
+�
+i=0
+Sk
+i (ωK)Ψk+1
+i
+(ωK, ρj) · αk
+i (ωK) = Xk+1(ωK, ρj),
+0 ≤ j ≤ m
+has a unique solution given by the matrices Q, M′, T as in the statement of the lemma. Since
+Sk > 0 and dk and ∆k > 0 are measurable, we obtain measurable functions αk : Ωk → Rm+1
+which form a process over Ω
+α0, . . . , αn−1.
+We remark that (20) are the inequalities in (19) corresponding to ω = ρ0, . . . , ρm with in-
+equalities turned into equalities.
+Claim 2: Consider some ωK ∈ Ωk where 0 ≤ k ≤ n − 1. Then αk(ωK) solves the inequalities
+(19) for all λ ∈ L ⊆ Ω.
+Proof: As in Claim 1, write Φ(ω) for the left hand side of (19) and Y (ω) for the right. The
+claim is that Φ(λ) ≥ Y (λ) for all λ ∈ L. Assume this is false, namely Φ(λ) < Y (λ) for some
+λ ∈ L. Among all these λ’s choose one for which j is maximal with ρj ⪯ λ; see Definition
+3.3 and the discussion below it. Clearly j < m because by definition of αk(ωK) we have
+Φ(ρi) = Y (ρi) for all 0 ≤ i ≤ m and because ρm ∈ L is maximal. Set λ′ = λ ∨ ρj+1. By
+the choice of λ we get λ ∧ ρj+1 = ρj. Since Sk+1
+i
+(ωK, ω) = Sk
+i (ωK) · Ψk+1
+i
+(ω) and since the
+assignment ω �→ Ψk+1
+i
+(ωK, ω) is an affine function on Ω, it follows that Φ: Ω → R is affine.
+Therefore
+Φ(λ′) + Φ(ρj) = Φ(λ) + Φ(ρj+1).
+The assumption on Xk+1 implies that Y |L is supermodular, hence
+Y (λ′) + Y (ρj) ≥ Y (λ) + Y (ρj+1).
+Subtracting these inequalities, keeping in mind that by construction Φ(ρi) = Y (ρi), we get
+Φ(λ′) − Y (λ′) ≤ Φ(λ) − Y (λ) < 0.
+Therefore Φ(λ′) < Y (λ′) and ρj+1 ⪯ λ′. This contradicts the maximality of j.
+q.e.d
+Claim 3: αk(ωK) solves the inequalities (19) for all ω ∈ Ω.
+
+EUROPEAN BASKETS IN DISCRETE-TIME CONTINUOUS-BINOMIAL MARKET MODELS
+23
+Proof: Since Ω is the convex hull of L, any ω ∈ Ω is a convex combination ω = �
+λ∈L tλ · λ.
+The assumption on Xk+1 implies that Y : Ω → R is convex. Together with Claim 2 and since
+Φ is affine
+Φ(ω) = Φ(
+�
+λ∈L
+tλλ) =
+�
+λ∈L
+tλΦ(λ) ≥
+�
+λ∈L
+tλY (λ) ≥ Y (
+�
+λ∈L
+tλλ) = Y (ω).
+q.e.d
+Claim 4: V k
+α = Xk.
+Proof: Denote qj = q(ρj). Consider some ωK ∈ Ωk. Equation (20) defining αk(ωK) yields
+rk(ωK) · Xk(ωK) = Eq(ω �→ Xk+1(ωK, ω))
+=
+m
+�
+j=0
+qjXk+1(ωK, ρj)
+=
+m
+�
+j=0
+m
+�
+i=0
+αk
+i (ωK)Sk
+i (ωK) · qjΨk+1
+i
+(ωK, ρj)
+=
+m
+�
+i=0
+αk
+i (ωK)Sk
+i (ωK) · Eq(ω �→ Ψk+1
+i
+(ωK, ω))
+= rk(ωK) · V k
+α (ωK).
+Since rk > 0 it follows that V k
+α (ωK) = Xk(ωK).
+q.e.d
+Claim 3 implies that α0, . . . , αn−1 solve all the inequalities (19) and hence it solves the
+constraints (18). Claims 1 and 4 imply that V k
+α ≤ V k
+β for all k and all β0, . . . , βn−1 that
+satisfy (18). To complete the proof of the lemma it only remains to prove:
+Claim 5: The system of equations (20) has a unique solution given by the matrices Q, T, M′
+as in the statement of the lemma.
+Proof: Consider 0 ≤ k ≤ n−1 and ωK ∈ Ωk. For 0 ≤ i, j ≤ m set χk+1
+i
+(j) = Ψk+1
+i
+(ωK, ρj).
+Notice that by the hypotheses
+χk+1
+0
+(j) = rk(ωK).
+For 1 ≤ i ≤ m observe that Lk+1
+i
+(ρj) = 1 if i ≤ j and Lk+1
+i
+(ρj) = 0 if i > j.
+Since
+Ψk+1 = dk
+i + ∆k
+i (ωK)Lk+1
+i
+we get
+χk
+i (j) =
+�
+dk
+i (ωK) + ∆k
+i (ωK)
+i ≤ j
+dk
+i (ωK)
+i > j
+Since Sk+1
+i
+(ωK, ρj) = Sk
+i (ωK) · Ψk+1
+i
+(ωK, ρj), the matrix representing the system (20) is
+M =
+
+
+Sk
+0χk+1
+0
+(0)
+Sk
+1χk+1
+1
+(0)
+· · ·
+Sk
+mχk+1
+m (0)
+Sk
+0χk+1
+0
+(1)
+Sk
+1χk+1
+1
+(1)
+· · ·
+Sk
+mχk+1
+m (1)
+...
+...
+Sk
+0χk+1
+0
+(m)
+Sk
+1χk+1
+1
+(m)
+· · ·
+Sk
+mχk+1
+m (m)
+
+ =
+
+
+1
+χk+1
+1
+(0)
+· · ·
+χk+1
+m (0)
+1
+χk+1
+1
+(1)
+· · ·
+χk+1
+m (1)
+...
+...
+1
+χk+1
+1
+(m)
+· · ·
+χk+1
+m (m)
+
+
+�
+��
+�
+M′′
+·
+
+
+rkSk
+0
+Sk
+1
+...
+Sk
+m
+
+
+�
+��
+�
+T
+
+24
+JAREK KĘDRA, ASSAF LIBMAN, AND VICTORIA STEBLOVSKAYA
+with all entries evaluated at ωK. Thus, M, M′′, T are functions Ωk → Mat(m+1)×(m+1)(R).
+Oserve that for all 1 ≤ i ≤ m and 1 ≤ j ≤ m
+χk+1
+i
+(j) − χk+1
+i
+(j − 1) =
+�
+∆k+1
+i
+(ωK)
+if i = j
+0
+if i ̸= j
+It follows that
+Q · M′′ =
+
+
+1
+dk
+1
+dk
+2
+· · ·
+dk
+m
+0
+∆k
+1
+0
+∆k
+2
+...
+...
+0
+∆k
+m
+
+
+with these matrices evaluated at ωK. We denote the latter matrix by M′ and notice that
+it is invertible since ∆k
+i > 0. In particular M(ωK) is invertible for any ωK ∈ Ωk so (20)
+has a unique solution. Note that M−1 = T −1 · M′−1 · Q is a measurable function Ωn →
+Mat(m+1)×(m+1)(R) and the solution of (20) is therefore the one given in the statement of the
+lemma.
+□
+Proof of Theorem 1.2. We apply Lemma 5.3 with the following data. The probability measure
+q is the upper supervertex q∗ : L → R (Definition 3.7) extended to a probability measure on
+Ω. The processes S0, . . . , Sn and Ψ1, . . . , Ψn are the prices Sk = (Sk
+0, . . . , Sk
+m) of the assets
+and their price jumps Ψk = (Ψk
+0, . . . , Ψk
+m). The process r0, . . . , rn−1 consists of the constant
+functions with value R. The processes d0, . . . , dn−1 and ∆0, . . . , ∆n−1 have components dk
+i
+and ∆k
+i (1 ≤ i ≤ m) where dk
+i is constant with value Di and ∆k
+i is constant with value
+Ui − Di. The process X0, . . . , Xn are the upper bound of the option’s F price at time k,
+namely Xk = Γmax(F, k).
+We need to show that the conditions of the lemma are fulfilled.
+First, Sk
+i > 0 for all
+k.
+By construction of the model, Sk+1
+i
+= Sk
+i · Ψk+1
+i
+and Ψk+1
+0
+= R = rk and Ψk+1
+i
+=
+Di + (Ui − Di)Lk
+i = dk
+i + ∆k
+i Lk+1
+i
+for all 0 ≤ k ≤ n − 1. Also,
+Eq(ω �→ Ψk+1
+i
+(ωK, ω)) = Eq(Di + (Ui − Di)ℓi(ω)) = Di + (Ui − Di)bi = R = rk(ωK)
+by construction of q∗ (Definition 3.7). By Theorem 1.1
+Xk(ωK) = Rk−n
+�
+J∈Pn−k(m)
+qJ · F(ωK, ρJ).
+Since F is fibrewise convex-supermodular by Proposition 5.1, Xk is a linear combination with
+non-negative coefficients of fibrewise convex-supermodular functions, hence it is one as well.
+Finally, we check that
+Eq∗(ω �→ Xk+1(ωK, ω)) = Eq∗
+
+ω �→ Rk+1−n
+�
+J∈Pn−k−1(m)
+qJ · F(ωK, ω, ρJ)
+
+
+= Rk+1−n
+m
+�
+j=0
+qj
+�
+J∈Pn−k−1(m)
+qJF(ωK, ρj, ρJ)
+= Rk+1−n
+�
+J∈Pn−k(m)
+qJF(ωK, ρJ)
+= R · Xk(ωK)
+= rk(ωK) · Xk(ωK).
+
+EUROPEAN BASKETS IN DISCRETE-TIME CONTINUOUS-BINOMIAL MARKET MODELS
+25
+All the conditions of Lemma 5.3 are fulfilled and we obtain a process α0, . . . , αn−1 which
+minimises
+V k
+α =
+m
+�
+i=0
+αk
+i Sk
+i
+subject to the requirement that for all 0 ≤ k ≤ n − 1
+m
+�
+i=0
+αk
+i Sk+1
+i
+≥ Xk+1 = Γmax(F, k + 1).
+Thus, α(k) = αk is a minimum-cost maximal hedging strategy as required with the formulas
+for its value given in the statement of the theorem. It only remains to note that Yt(k) at the
+state of the world ωK ∈ Ωk used in the statement of the theorem is precisely Xk+1(ωK, ρt)
+because
+Yt(k)(ωK) = Rk+1−n
+�
+J∈Pn−k−1(m)
+qJ · (
+m
+�
+i=0
+ciχi(J)χi(t)Sk
+i (ωK) − K)+
+= Rk+1−n
+�
+J∈Pn−k−1(m)
+qJ · (
+m
+�
+i=0
+ciχi(J)Ψi(ρt)Sk
+i (ωK) − K)+
+= Rk+1−n
+�
+J∈Pn−k−1(m)
+qJ · (
+m
+�
+i=0
+ciχi(J)Sk
+i (ωK, ρt) − K)+
+= Γmax(F, k + 1)(ωK, ρt)
+= Xk+1(ωK, ρt).
+□
+References
+[1] William Feller. An Introduction to Probability Theory and Its Applications. Vol. I. John Wiley & Sons,
+Inc., New York, N.Y., 1950.
+[2] Jürgen Franke, Wolfgang Karl Härdle, and Christian Matthias Hafner. Statistics of financial markets.
+Universitext. Springer, Cham, 2019. An introduction, Fifth edition of [ MR2102757].
+[3] Allan Gut. Probability: a graduate course. Springer Texts in Statistics. Springer, New York, second edition,
+2013.
+[4] Paul R. Halmos. Measure Theory. D. Van Nostrand Co., Inc., New York, N. Y., 1950.
+[5] Jarek Kędra, Assaf Libman, and Victoria Steblovskaya. Pricing and hedging contingent claims in a multi-
+asset binomial market. arXiv:2106.13283, 2021.
+[6] Jarek Kędra, Assaf Libman, and Victoria Steblovskaya. Hedging of european type contingent claims in
+discrete time binomial market models. Preprint, 2022.
+[7] L. Lovász. Submodular functions and convexity. In Mathematical programming: the state of the art (Bonn,
+1982), pages 235–257. Springer, Berlin, 1983.
+[8] Stanley Pliska. Introduction to Mathematical Finance - Discrete Time Models. Blackwell Publishing, 1997.
+[9] H. L. Royden. Real analysis. Macmillan Publishing Company, New York, third edition, 1988.
+University of Aberdeen and University of Szczecin
+Email address: kedra@abdn.ac.uk
+University of Aberdeen
+Email address: a.libman@abdn.ac.uk
+Bentley University
+Email address: vsteblovskay@bentley.edu
+
diff --git a/hdE4T4oBgHgl3EQfSAzu/content/tmp_files/load_file.txt b/hdE4T4oBgHgl3EQfSAzu/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf,len=1579
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='04996v1  [q-fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='MF]  12 Jan 2023  EUROPEAN BASKETS IN DISCRETE-TIME CONTINUOUS-BINOMIAL  MARKET MODELS  JAREK KĘDRA, ASSAF LIBMAN, AND VICTORIA STEBLOVSKAYA  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' We consider a discrete-time incomplete multi-asset market model with contin-  uous price jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' For a wide class of contingent claims, including European basket call  options, we compute the bounds of the interval containing the no-arbitrage prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' We prove  that the lower bound coincides, in fact, with Jensen’s bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' The upper bound can be com-  puted by restricting to a binomial model for which an explicit expression for the bound is  known by an earlier work of the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' We describe explicitly a maximal hedging strategy  which is the best possible in the sense that its value is equal to the upper bound of the price  interval of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Our results show that for any c in the interval of the non-arbitrage  contingent claim price at time 0, one can change the boundaries of the price jumps to obtain  a model in which c is the upper bound at time 0 of this interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' The lower bound of this  interval remains unaffected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' The main results  Discrete time continuous-binomial market model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' We consider a discrete-time mar-  ket model, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='g [2, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='5] or [8, §3], with m risky assets S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Sm and a bond S0 whose  prices at time k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , n denoted Si(k), are random processes described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' We fix  the initial values Si(0) of the assets (i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , m) and parameters R > 0 (the interest rate)  and 0 < Di < R < Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' For time k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , n the random process is defined by  S0(k) = RkS0(0), and  Si(k) = Ψi(k)Si(k − 1), for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , m, where Ψi(k) are random variables with  values in [Di, Ui].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  We call Ψi(k) the price jumps at time k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' We emphasize that the price jumps are not assumed  to be independent of each other nor identically distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  A European basket call option is a contingent claim with pay-off given by  (1)  F =  � m  �  i=0  ci · Si(n) − K  �+  ,  where K > 0 and ci ≥ 0 for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , m and x+ := max{x, 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Notice that there is  no assumption on c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' The rational values of F at time k are the possible market values of  the option at time k so that no arbitrage occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' They are known to form an open interval  (Γmin(F, k), Γmax(F, k)), where Γmin(F, k) and Γmax(F, k) depend on the “state of the world”  at time k, namely the history of the market up to time k, and in particular they depend on  the (current) values of the assets Si at time k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  The main result of this paper is the computation of Γmin(F, k) and Γmax(F, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' It extends  the main results of the authors’ previous work [5] in which we consider discrete time binomial  models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='e ones in which the price jumps Ψi(k) take values in (the discrete) set {Di, Ui}  rather than the entire interval [Di, Ui].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  1  2  JAREK KĘDRA, ASSAF LIBMAN, AND VICTORIA STEBLOVSKAYA  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Computation of Γmin(F, k) and Γmax(F, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' For every 1 ≤ i ≤ m set  (2)  bi = R − Di  Ui − Di  and reorder the assets S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Sm, if necessary, so that  b1 ≥ · · · ≥ bm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Define q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , qm by  (3)  qi =  \uf8f1  \uf8f2  \uf8f3  1 − b1  if i = 1  bi − bi+1  if 1 < i < m  bm  if i = m  For any 0 ≤ i ≤ m and any 0 ≤ j ≤ m define numbers χi(j) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  χi(j) =  �  Ui  if i ≤ j  Di  if i > j  for 1 ≤ i ≤ m, and  (4)  χ0(j) = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  To streamline the notation, set for every k ≥ 0  (5)  Pk(m) = {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , m}k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Thus, any J ∈ Pk(m) is a sequence J = (j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , jk) with 0 ≤ j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , jk ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' For such J set  qJ  =  �  j∈J  qj  (6)  χi(J)  =  �  j∈J  χi(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' With the setup of the market model and notation above, the extremal values  of the rational values of F at time 0 ≤ k ≤ n are given by  Γmin(F, k)  =  Rk−n ·  �  Rn−k  m  �  i=0  ci · Si(k) − K  �+  Γmax(F, k)  =  Rk−n ·  �  J∈Pn−k(m)  qJ ·  � m  �  i=0  ci · χi(J) · Si(k) − K  �+  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Hedging strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Consider a sequence of (time changing) portfolios  Vα(k) =  �  i  αi(k)Si(k),  (0 ≤ k ≤ n − 1)  for some choices of values for αi(k) at time 0 ≤ k ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' A maximum hedging strategy is a  choice for αi(k) at time 0 ≤ k ≤ n − 1 (which depends on the state of the world at that time)  which minimizes the value Vα(k) subject to the requirement that  m  �  i=0  αi(k) · Si(k + 1) ≥ Γmax(F, k + 1)  for any subsequent state of the world at time k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' That is, a maximum hedging strategy  is a time dependant portfolio of minimum possible cost whose value is guaranteed to exceed  the future rational value of the contingent claim F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Our next result, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='2, shows that the value of any hedging portfolio Vβ(k) must  always exceed Γmax(F, k) and that there exists a hedging strategy αi(k) that attains this  EUROPEAN BASKETS IN DISCRETE-TIME CONTINUOUS-BINOMIAL MARKET MODELS  3  bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' It is a minimum cost maximum hedging strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' To make this result precise, given  the state of the world at time 0 ≤ k ≤ n − 1, let Yi(k) where 0 ≤ i ≤ m be the value of  Γmax(F, k + 1) at the state of the world at time k + 1 which is obtained from the present  one (at time k) by having the assets S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Si make their maximum price jumps U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Ui  and having Si+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Sm make their minimum price jumps Di+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Explicitly, for any  0 ≤ t ≤ m:  Yt(k) = Rk+1−n ·  �  J∈Pn−k−1(m)  qJ  � m  �  i=0  ci · χi(J)χi(t) · Si(k) − K  �+  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Consider the continuous binomial market model above and a European option  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Any hedging strategy βi(k) satisfies  Vβ(k) ≥ Γmax(F, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  There exists a maximum hedging strategy αi(0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , αi(n − 1) such that Vα(k) = Γmax(F, k)  for all 0 ≤ k ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' In fact, the values of αi(k) at time k are computed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  α0(k)  α1(k)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  αm(k)  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb = W(k) · N · Q ·  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  Y0(k)  Y1(k)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Ym(k)  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb  Where W(k), N, T are the following (m + 1) × (m + 1) matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Set ∆i = Ui − Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  W(k) =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  1  R·S0(k)  1  S1(k)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  1  Sm(k)  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  Q =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  1  0  0  · · · · ·  0  0  −1  1  0  · · · · ·  0  0  0  −1  1  · · · · ·  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  · · · · ·  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  0  0  0  · · · · ·  1  0  0  0  0  · · · · ·  −1  1  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  N =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  1  − D1  ∆1  − D2  ∆2  · · · · ·  − Dm  ∆m  0  1  ∆1  0  · · · · ·  0  0  0  1  ∆2  · · · · ·  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  0  0  0  · · · · ·  1  ∆m  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  (Notice that W(k) depends on the state of the world, but N and Q do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Also, R · S0(k) =  S0(0) · Rk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  This result extends our previous result in [6] which computes a similar hedging strategy in  discrete time binomial models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='e models in which Ψi(k) ∈ {Di, Ui} ⊂ [Di, Ui].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Changing the parameters of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Keeping R fixed, we may change the values of  Ui and Di to obtain different models for the same market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' This has the effect of changing  the limits of the price jumps of the assets Si, and consequently the random processes Si are  changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Clearly the values of Γmin(F, k) and Γmax(F, k) depend on these parameters, and  4  JAREK KĘDRA, ASSAF LIBMAN, AND VICTORIA STEBLOVSKAYA  we therefore write Γmin(F, k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Ui, Di) and Γmax(F, k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Ui, Di) to emphasise this dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  We will be interested in the rational prices of F at time 0, namely Γmin(F, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Ui, Di) and  Γmax(F, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Ui, Di).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Consider a market model with some 0 < Di < R < Ui and the European  basket F in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Then  (1) Γmin(F, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Ui, Di) = (Rn �m  i=0 ciSi(0) − K)+ and in particular it is independent of the  values of Ui, Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  (2) For every c in the open interval ( Γmin(F, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Ui, Di) , Γmax(F, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Ui, Di) ) there exist  Di ≤ di < R < ui ≤ Ui such that Γmax(F, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' ui, di) = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  More precisely, consider the functions ui, di : [0, 1) → R defined by  di(s)  =  Di + (R − Di)s  ui(s)  =  R − (1 − bi)di(s)  bi  Then ϕ(s) = Γmax(F, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' ui(s), di(s)) is a continuous function of s ∈ [0, 1) such that  ϕ(0) = Γmax(F, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Ui, Di) and limsր1 ϕ(s) = Γmin(F, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Ui, Di).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Preliminaries: Random processes and conditional expectation  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Non-degenerate density functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' This section is concerned with some general  results about probability measure spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Our standard reference for Measure theory and  Lebesgue integration are Halmos [4] and Royden [9], and for Probability Theory it is Feller  [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Throughout this paper, once m ≥ 1 is fixed we will denote  (7)  Ω = [0, 1]m ⊆ Rm  equipped with the usual Borel σ-algebra and probability measure µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  A probability density function (pdf) is a measurable p: Ω → [0, ∞) such that  �  Ω pdµ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  It gives rise in a standard way to a probability measure on Ω which by abuse of notation we  also denote by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  A probability measure on Ω is called absolutely continuous with respect to µ, written ν ≪ µ,  if for any Borel subset E we have µ(E) = 0 =⇒ ν(E) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' A probability measure ν on Ω is  non-degenerate if ν ≪ µ and µ ≪ ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' We write ν ≈ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  By the Radon-Nykodim theorem [9, §11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='5] if ν ≪ µ then there exists a pdf p: Ω → [0, ∞)  called the Radon-Nykodim derivative, such that ν(E) =  �  E p(x) dµ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' It is easy to check  that ν ≈ µ if and only if p > 0 almost everywhere1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' We say that p is non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Conditional probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Consider Ω(i) = [0, 1]mi where i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Set  Ω =  n  �  i=1  Ω(i) = [0, 1]m,  where m = �  i mi, equipped with the standard Borel σ-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Let X(i) denote the random  vector  X(i) : Ω  proji  −−−−→ Ω(i) ⊆ Rmi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  1If p > 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='e then  �  E p du > 0 for any E with µ(E) > 0 is a standard fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' If p = 0 on E with µ(E) = 0  then ν(E) =  �  E p(x) dµ(x) = 0 so µ is not absolutely continuous with respect to ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  EUROPEAN BASKETS IN DISCRETE-TIME CONTINUOUS-BINOMIAL MARKET MODELS  5  For a non-empty I ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , n} set mI = �  i∈I mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Then set  Ω(I) =  �  i∈I  Ω(i)  X(I) : Ω  proj(I)  −−−−−→ Ω(I) ⊆ RmI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Thus, Ω(I) = [0, 1]mI and X(I) is a random vector into RmI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' We will denote elements of Ω(I)  by ω(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' We will write n − I for the complement of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  For the remainder of this subsection we fix a non-degenerate (with respect to the Lebesgue  measure on Ω) pdf p: Ω → [0, ∞) and equip Ω with the probability measure it induces which  we abusively denote by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Note that p is the joint density function of the random vectors  X(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , X(n) (because X{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=',n} is the inclusion Ω ⊆ Rm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Given a non-empty I ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , n}, the (joint) density function of X(I) is the function  pX(I) : Ω(I) → [0, ∞) given by (See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' [3, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' 2, Scet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' 3])  (8)  pX(I)(ω(I)) =  �  τ∈Ω(n−I)  p(ω(I), τ)dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Fubini’s theorem readily implies that pX(I) is non-degenerate (with respect to the Lebesgue  measure on Ω(I)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Consider some disjoint I, J ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' The density function of X(I) given X(J) denoted  pX(I)|X(J) : Ω(I∪J) → [0, ∞) is  (9)  pX(I)|X(J) (ω(I), ω(J)) =  pX(I∪J)(ω(I), ω(J))  pX(J)(ω(J))  whenever this is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Since pX(J) is non-degenerate, pX(I)|X(J) is defined a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' For any ω(J)  we obtain a function, the (conditional) density of X(I) given the event {X(J) = ω(J)},  pX(I)|X(J)=ω(J) : Ω(I) → [0, ∞)  defined by pX(I)|X(J)=ω(J)(−) = pX(I)|X(J)(−, ω(J)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' By Fubini’s theorem it is a (measurable)  probability density function on Ω(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Consider a random vector  f : Ω → Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  To avoid issues of convergence we assume that f ≥ 0, namely all the components of f are  non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' The expectation of f given X(I) is the function  Ep(f|X(I)): Ω(I) → R  where Ep(f|X(I))(ω(I)) is the conditional expectation Ep(f|X(I) = ω(I)), namely  Ep(f|X(I))(ω(I)) =  �  τ∈Ω(n−I)  f(τ, ω(I))pX(n−I)|X(I)(τ, ω(I))dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  By Fubini’s theorem Ep(f|X(I)) is a measurable function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Keeping the notation above, let I, J ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , n} be disjoint and consider some  ω(I) ∈ Ω(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Set p′ = pX(J)|X(I)=ω(I), a pdf on Ω(J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Let f ≥ 0 be a random vector on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Then  Ep(f|X(I) = ω(I)) = Ep′(τ �→ Ep(f|X(I∪J))(ω(I), τ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  In particular,  Ep(X(J)|X(I) = ω(I)) = Ep′(X(J))  where X(J) is viewed as a random vector from Ω(J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  6  JAREK KĘDRA, ASSAF LIBMAN, AND VICTORIA STEBLOVSKAYA  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' We compute the right hand side using Fubini’s theorem as follows  Ep′(τ �→Ep(f|X(I∪J))(ω(I),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' τ)) =  =  �  τ∈Ω(J)  pX(J)|X(I)=ω(I)(τ) · Ep(f|X(I∪J))(ω(I),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' τ) dτ  =  �  τ∈Ω(J)  �  pX(I∪J)(ω(I),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='τ)  pX(I)(ω(I)) �  θ∈Ω(n−I∪J)  f(ω(I),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' θ) ·  p(ω(I),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='θ)  pX(I∪J)(ω(I),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='τ) dθ  �  dτ  =  �  τ∈Ω(J)  �  θ∈Ω(n−I∪J)  f(ω(I),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' θ) ·  p(ω(I),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='θ)  pX(I)(ω(I))dθ dτ  =  �  ω∈Ω(n−I)  f(ω(I),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' ω) ·  p(ω(I),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='ω)  pX(I)(ω(I)) dω  =  �  ω∈Ω(n−I)  f(ω(I),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' ω) · pX(n−I)|X(I)=ω(I)(ω) dω  = Ep(f|X(I) = ω(I))  This establishes the first claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' We apply it to the random vector f = X(J) to obtain the  second claim as follows  Ep(X(J)|X(I) = ω(I)) = Ep′(τ �→ Ep(X(J)|X(I∪J))(ω(I), τ)) = Ep′(τ �→ X(J)(τ)) = Ep′(X(J)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Here we observe that Ep(X(J)|X(I∪J))(ω(I), τ) = X(J)(τ) because with the abuse of notation  for the domain of X(J) we have X(J)(ω(I), τ, θ) = X(J)(τ) for any τ ∈ Ω(J) and any θ ∈  Ω(n−I∪J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  □  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Let f : Ω → Rd be a function and I, J ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , n} be disjoint, and assume  that f ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Suppose that f factors through the projection Ω  πI∪J  −−−→ Ω(I∪J), namely there exists  g: Ω(I∪J) → Rm such that f = g ◦ πI∪J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Set p′ = pX(J)|X(I)=ω(I), pdf on Ω(J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Then  Ep(f|X(I) = ω(I)) = Ep′(ω(J) �→ g(ω(I) ω(J)))  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='1 gives  Ep(f|X(I) = ω(I)) = Ep′(ω(J) �→ Ep(f|X(I∪J))(ω(I), ω(J))) = Ep′(ωJ �→ g(ωI, ωJ))  because f(ω(I), ω(J), τ) = g(ω(I), ω(J)) for any τ ∈ Ω(n−I∪J), so Ep(f|X(I∪J))(ω(I), ω(J)) =  g(ω(I), ω(J)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  □  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Tensoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Given p: Ω → R and q: Ω′ → R we obtain a function p⊗q: Ω×Ω′ → R by  (10)  (p ⊗ q)(ω, ω′) = p(ω) · q(ω′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  It is clear that if p, q are pdf’s then so is p ⊗ q and that it is non-degenerate if p and q are  non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Keeping the notation above for Ω(i) and the random vectors X(i), let p(i) : Ω(i) → [0, ∞) be  non-degenerate pdf’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Then p = p(1) ⊗ · · · ⊗ p(n) is a non-degenerate pdf on Ω = �  i Ω(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' For  any I ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , n} we denote  p(I) = ⊗  i∈Ip(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  This is a non-degenerate pdf on Ω(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Let p(i) : Ω(i) → [0, ∞) be non-degenerate pdf’s, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Set p = p(1) ⊗  · · ⊗ p(n), non-degenerate pdf on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Then  EUROPEAN BASKETS IN DISCRETE-TIME CONTINUOUS-BINOMIAL MARKET MODELS  7  (1) pX(I) = p(I) for any I ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  (2) pX(J)|X(I)=ω(I) = p(J) for any disjoint I, J ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , n}, and furthermore  (3) Ep(X(J)|X(I) = ω(I)) = Ep(J)(X(J)) where X(J) is viewed as a random vector on Ω(J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' (1) Given ω(I) we compute  pX(I)(ω(I)) =  �  τ∈Ω(n−I)  p(ω(I), τ) dτ =  �  τ∈Ω(n−I)  p(I)(ω(I)) · p(n−I)(τ) dτ = p(I)(ω(I))  because p(J) is a pdf on Ω(J) for any J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  (2) Given ω(J) and ω(I) we use this to compute  pX(J)|X(I)=ωI(ω(J)) =  pX(I∪J)(ω(I), ω(J))  pX(I)(ω(I))  = p(I∪J)(ω(I), ω(J))  p(I)(ω(I))  = p(I)(ω(I)) · p(J)(ω(J))  p(I)(ω(I))  = p(J)(ω(J)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  (3) By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='1 and item (2)  Ep(X(J)|X(I) = ω(I)) = EpX(J)|X(I)=ω(I)(ω(J) �→ Ep(X(J)|X(I∪J)(ω(I), ω(J)))  = Ep(J)(ω(J) �→ X(J)(ω(J))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  □  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Let Ω be a set and n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' An Rd-valued process (over Ω) is a sequence of  functions  Y (0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Y (n) : Ωn → Rd  such that for each 0 ≤ k ≤ n the function X(k) factors through the projection Ωn → Ωk to  the first k factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  If Ωn is equipped with a probability measure, Y (0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Y (k) is called an Rd-valued random  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  We frequently regard Y (k) as a function with domain Ωk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' We will sometimes “trim” the  process to Y (1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Y (n) or to X(0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , X(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  If Ω = [0, 1]m ⊆ Rm the projections L(k) : Ωn → Ω ⊆ Rm to the k-th factor give a  universal process in the sense that if Y (0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Y (n) is a process then each Y (k) is a function  of L(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , L(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Let ν be a probability measure on a set Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' We say that ν is supported on a measurable set A if µ(A) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Any probability measure ν on A ⊆ Ω extends to a probability measure ˜ν on Ω supported  on A by ˜ν(E) = ν(A ∩ E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Conversely, if ν on Ω is supported by A then ν = �  ν|A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' If A is  finite then for any f : Ω → R we have Eν(f) = �  a∈A ν({a})f(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  8  JAREK KĘDRA, ASSAF LIBMAN, AND VICTORIA STEBLOVSKAYA  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Maximum and minimum expectation of random variables on Ω = [0, 1]m  Fix some m ≥ 1 and Ω = [0, 1]m equipped with the usual Lebesgue measure µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Let  L: Ω → Rm  denote the inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' We think of it as a random vector with components L = (ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ℓm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Thus, ℓi : Ω → R is the projection to the ith factor  ℓi(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , xm) = xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  For convenience we also set  ℓ0 = 1,  the constant function (random variable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Let M(Ω) denote the set of all probability measures on Ω on the Borel  σ-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Let M+(Ω) denote the set of all the non-degenerate probability measures (with  respect to the Lebesgue measure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  We will often refer to the elements of M+(Ω) as pdf’s which are non-vanishing a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Consider a non decreasing b ∈ int(Ω) = (0, 1)m, namely  1 > b1 ≥ · · · ≥ bm > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  We will also denote for convenience  b0 = 1  and  bm+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' The set of mean-b probability measures on Ω is  M(Ω, b) = {P ∈ M(Ω) : EP (L) = b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  That is, EP(ℓi) = bi for all 1 ≤ i ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' The set of non-degenerate mean-b probability measures  is  M+(Ω, b) = {P ∈ M+(Ω) : EP (L) = b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Let L denote the set of vertices of the cube Ω = [0, 1]m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' That is,  L = {0, 1}m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  There is a standard identification of L with ℘({1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , m}) where λ ∈ L corresponds to  supp(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' This turns L into a lattice where the partial order ⪯ is induced by inclusion of sets  and joins and meets are ∪ and ∩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' The next concept is originally due to Lovász [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' A function f : L → R is called supermodular if for any a, b ∈ L  f(a ∨ b) + f(a ∧ b) ≥ f(a) + f(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  It is called modular if equality holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' A function f : Ω → R is called convex-supermodular if f is convex and its  restriction to L is supermodular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Let f : Ω → R and suppose that f = h ◦ g for some g: [0, 1]m → R and  h: R → R such that either  (1) h is convex and g is affine with non-negative coefficients except the constant term,  namely g = �m  i=0 aiℓi where a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , am ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  (2) g is convex, g|L is modular, and h is convex and increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  EUROPEAN BASKETS IN DISCRETE-TIME CONTINUOUS-BINOMIAL MARKET MODELS  9  Then f is convex-supermodular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Proof: Indeed, f is convex as composition of convex functions and f|L is supermodular by  [Simchi-Levi, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='6] for item (1) and [Simchi-Levi Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='5(c)] for item (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  ♦  For every 0 ≤ k ≤ m let ρk ∈ L denote the element corresponding to {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , k}, namely  (11)  ρk = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , 1  � �� �  k times  , 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , 0) ∈ {0, 1}m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='7 (Compare [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' The upper supermodular vertex is the probability density  function q∗ : L → R supported on {ρ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ρm} with  q∗(ρk) = bk − bk+1,  (0 ≤ k ≤ m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  One easily checks that �m  i=0 q∗(ρk) = 1 and that Eq∗(ℓi) = bi, thus  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' q∗ ∈ M(Ω, b) and it is supported on L ⊆ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  The main result of this section is the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Let f : Ω → R be a continuous convex-supermodular function and assume that  f ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Let q∗ be the upper supermodular vertex of L (Definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Then  sup  p∈M+(Ω,b)  Ep(f) = Eq∗(f|L) =  max  p∈M(Ω,b) Ep(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Note that Eq∗(f|L) = �m  i=0 q∗(ρi) · f(ρi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  In the remainder of this section we prove this theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  It relies on the following key  observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Equip Rm with the norm ∥ ∥∞ and restrict this norm to Ω = [0, 1]m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='10 (Approximation lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Consider some q ∈ M(Ω, b) supported on a finite  subset {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , xk} of Ω and set qi = q({xi}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Suppose that b ∈ int(Ω) = (0, 1)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Then for  any ǫ > 0 and δ > 0 there exists p ∈ M+(Ω, b) and β < ǫ such that for any continuous  f : Ω → R  Ep(f) = β  �  Ω  f dµ +  k  �  i=0  (qi − β  k ) · f(ξi)  where ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ξk ∈ Ω are such that ∥ξi − xi∥∞ < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Closed balls of radius r > 0 in Rm have the form B(y, r) = y + [−r, r]m so their  volume, hence their Lebesgue measure, is (2r)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' If y = (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ym) then by inspection, for  any 1 ≤ j ≤ m  �  B(y,r)  xj dµ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , xm) = (2r)myj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Claim: There exist distinct y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , yk in int(Ω) = (0, 1)m such that ∥yi − xi∥∞ < δ  3 and such  that �k  i=1 qiyi = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Proof: We show how to perturb the vectors x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , xk in order to obtain y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , yk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' First,  �k  i=1 qixi = Eq(L) = b since q ∈ M(Ω, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Suppose that for some 1 ≤ j ≤ m not all of  x1  j, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , xk  j are in the open interval (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' If xi′  j = 0 for some i′ then there must exist i′′ such  that xi′′  j  > 0 because �k  i=1 qixi  j = bj > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Since qi > 0 for all i we can increase xi′  j and  decrease xi′′  j  by a small positive number < δ  3 so that the sum remains bj and that the new  10  JAREK KĘDRA, ASSAF LIBMAN, AND VICTORIA STEBLOVSKAYA  values of xi′  j and xi′′  j are in (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Similarly, if xi′  j = 1 for some i′ then there must exist some  i′′ such that xi′′  j < 1 because �k  i=1 qixi  j = bj < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' We can then decrease xi′  j and increase xi′′  j  by at most δ  3 so that the sum remains bj and the new values of xi′  j and xi′′  j are in (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' By  repeating this process we can perturb x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , xk into y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , yk in int(Ω) such that �  i qiyi = b  and ∥xi − yi∥∞ < δ  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Since the xi’s admit pairwise disjoint neighbourhoods, we can perturb  the xi’s inside these neighbourhoods to make sure that the yi’s are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='d  Since y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , yk are in the interior of Ω there exists r < δ  3 such that B(yi, 2r) ⊆ (0, 1)m for  all 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Thus, if γ ∈ Rm is such that ∥γ∥∞ < r then B(yi + γ, r) ⊆ (0, 1)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Set  Q = min{q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , qm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  For any β > 0 set  γ1  j :=  β · ( 1  k  �k  i=1 yi  j − 1  2)  q1 − β  k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Choose 0 < β < min{ǫ, kQ} sufficiently small such that for every 1 ≤ j ≤ m  |γ1  j | < r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Let γ1 ∈ Rm be the vector with the components γ1  j defined above and let γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , γk ∈ Rm be  the zero vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' By construction ∥γi∥∞ < r for all 1 ≤ i ≤ k, hence B(yi + γi, r) ⊆ (0, 1)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Define p: Ω → R by  p = β +  k  �  i=1  qi− β  k  (2r)m · 1B(yi+γi,r)  where 1B(yi+γi,r) is the characteristic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Observe that p > 0 because β > 0 and  qi − β  k > qi − Q ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Next, p is a pdf since  �  Ω  p dµ = β +  k  �  i=1  qi − β  k  (2r)m  �  Ω  1B(yi+γi,r)dµ =  k  �  i=1  qi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  We check that p ∈ M+(Ω, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Indeed, since γi = 0 for all i ≥ 2  Ep(ℓj) =  �  x∈Ω  ℓj(x) · p(x) dµ(x)  = β  �  Ω  xj dµ +  1  (2r)m  k  �  i=1  (qi − β  k)  �  B(yi+γi,r)  xj dµ  = 1  2β +  k  �  i=1  (qi − β  k )(yi  j + γi  j)  = 1  2β +  k  �  i=1  qiyi  j + (q1 − β  k )γ1  j − β  k  k  �  i=1  yi  j  = 1  2β + bj + β( 1  k  k  �  i=1  yi  j − 1  2) − β  k  k  �  i=1  yi  j  = bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  EUROPEAN BASKETS IN DISCRETE-TIME CONTINUOUS-BINOMIAL MARKET MODELS  11  Suppose that f : Ω → R is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' By the mean value theorem there exist ξi ∈ B(yi+γi, r)  such that  Ep(f) =  �  x∈Ω  f(x) · p(x) dµ(x) = β  �  Ω  f dµ +  k  �  i=1  1  (2r)m (qi − β  k )  �  B(yi+γi,r)  f dµ =  β  �  Ω  f dµ +  k  �  i=1  (qi − β  k ) · f(ξi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  By our choice β < ǫ and ∥ξi −xi∥∞ ≤ ∥ξi −(yi +γi)∥∞ +∥yi −xi∥∞ +∥γi∥∞ < r + δ  3 +r < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  □  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' For i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , m set  α0 = f(ρ0)  αi = f(ρi) − f(ρi−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Observe that since by definition b0 = 1 and bm+1 = 0,  Eq∗(f|L) =  m  �  i=0  q∗(ρi) · f(ρi)  =  m  �  i=0  (bi − bi+1)f(ρi)  = f(ρ0) +  m  �  i=1  bi(f(ρi) − f(ρi−1))  = α0 +  m  �  i=1  biαi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Let ˇf : Rm → R be the affine function ˇf = �m  i=0 αiℓi, namely  ˇf(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , xm) = α0 +  m  �  i=1  αixi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Claim 1: ˇf dominates f on L, namely ˇf(λ) ≥ f(λ) for all λ ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Proof: By construction of ˇf and by the definition of ρj in (11), for any 0 ≤ j ≤ m  ˇf(ρj) =  j  �  i=0  αi = f(ρj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  So f and ˇf coincide on {ρ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ρm} ⊆ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Assume the statement of the claim is false, namely  there exists λ ∈ L such that ˇf(λ) < f(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Among all these λ’s choose one which contains the  longest leading run of 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , 1, namely λ with the largest possible k with ρk ⪯ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Notice that  k < m because ˇf(ρm) = f(ρm) and ρm is maximal in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' In the lattice L set λ′ = λ ∨ ρk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Notice that since k is the largest such that ρk ⪯ λ it follows that λ ∧ ρk+1 = ρk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Since f|L is  supermodular  f(λ ∨ ρk+1) + f(ρk) ≥ f(λ) + f(ρk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Since ˇf is affine, it is modular, hence  ˇf(λ ∨ ρk+1) + ˇf(ρk) = ˇf(λ) + ˇf(ρk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  12  JAREK KĘDRA, ASSAF LIBMAN, AND VICTORIA STEBLOVSKAYA  Subtract the first equation from the second, taking into account that ˇf(ρi) = f(ρi), to get  ˇf(λ ∨ ρk+1) − f(λ ∨ ρk+1) ≤ ˇf(λ) − f(λ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Thus ˇf(λ′) < f(λ′) and ρk+1 ⪯ λ′, contradiction to the maximality of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='d  Claim 2: ˇf dominates f on Ω = [0, 1]m, namely ˇf(x) ≥ f(x) for all x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Proof: Since Ω = [0, 1]m is the convex hull of L = {0, 1}m, every x ∈ Ω is a convex combination  x =  �  λ∈L  tλ · λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Since ˇf is affine and f is convex, Claim 1 implies that  ˇf(x) = ˇf(  �  λ∈L  tλ · λ) =  �  λ∈L  tλ ˇf(λ) ≥  �  λ∈L  tλf(λ) ≥ f(  �  λ∈L  tλλ) = f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='d  Claim 2 implies that for any p ∈ M(Ω, b)  Ep(f) ≤ Ep( ˇf) = Ep(  m  �  i=0  αiℓi) =  m  �  i=0  αiEp(ℓi) = α0 +  m  �  i=1  αibi = Eq∗(f|L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  However, q∗ extends to �q∗ ∈ M(Ω, b) and clearly E �  q∗(f) = Eq∗(f|L) so we get  max  p∈M(Ω,b) Ep(f) = Eq∗(f|L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Notice that Eq∗(f|L) = �m  i=0 q∗(ρi) · f(ρi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='10 applied to q∗ ∈ M(Ω, b) and the  continuity of f easily imply that there exist λ ∈ M+(Ω, b) such that Eλ(f) is arbitrarily close  to Eq∗(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' If follows that supM+(Ω,b) Ep(f) = Eq∗(f|L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' (L, b)-stable probability measures  Main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' We consider n iterations of the random vector L = (ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ℓm) over Ω =  [0, 1]m from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' We obtain a sequence L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Ln of random vectors in Rm with sample  space Ωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' In fact  Lk : Ωn → Ω ⊆ Rm  is the projection to the kth factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' This is the universal process on Ωn, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Denote  by M+(Ω, n) the set of all non-degenerate probability measures on Ωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' We will identify these  with the set of pdf’s p: Ωn → R which are non-vanishing a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  With the notation and terminology of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='B we make the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Consider some b ∈ int(Ω) = (0, 1)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' A pdf p ∈ M+(Ω, n) is called (L, b)-  stable if for any 0 ≤ k ≤ n − 1  Ep(Lk+1|L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Lk) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  That is, the function Ep(Lk+1|L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Lk): Ωk → Rm is constant with value b a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' The set of  all (L, b)-stable p ∈ M+(Ω, n) is denoted  M+(Ω, n, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' A function f : Ωn → R is called fibrewise convex-supermodular if it is convex-  supermodular at each fibre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Namely, for any τ ∈ Ωk−1 and any θ ∈ Ωn−k the function  g: Ω → R defined by g: ω �→ f(τ, ω, θ) is convex-supermodular (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  EUROPEAN BASKETS IN DISCRETE-TIME CONTINUOUS-BINOMIAL MARKET MODELS  13  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' For any J = (j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , jn) ∈ Pn(m), see (5), let ℓJ : Ωn → R denote the function  ℓJ = ℓj1 ⊗ · · · ⊗ ℓjn, namely  ℓJ : (ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωn) �→ ℓj1(ω1) · · · ℓjn(ωn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Consider f : Ωn → R of the form f = h ◦ g where h: R → R is convex and g: Ωn → R is of  the form  g =  �  J∈Pn(m)  aJ · ℓJ  where aJ ≥ 0 for all J ̸= (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Then f is fibrewise convex-supermodular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Proof: It is clear that if τ ∈ Ωk−1 and θ ∈ Ωn−k then the restriction of ℓJ to the fibre  {τ} × Ω × {θ} ⊆ Ωn is equal to �n  i=0 biℓi where bi ≥ 0 for all i ≥ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' In fact bi is the sum of  all aJ in which the kth entry is equal to i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' The result follows from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='6(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  The main results of this section are the following two theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Let f : Ωn → R be a continuous function of the form f = h ◦ g in Example  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Assume that f ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Suppose that b ∈ int(Ω) = (0, 1)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Then  inf  p∈M+(Ω,n,b) Ep(f|L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Lk)(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk) = f(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, b, · · · , b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Recall the upper supermodular vertex q∗ : L → R from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' It extends to an  atomic probability measure on Ω supported on L which we abusively denote q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Let  q∗⊗k  be the obvious product probability measure on Lk as well as its extension to Ωk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Set qj = q∗(ρj) for all 0 ≤ j ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' For any J ∈ Pk(m) denote  qJ  =  �  j∈J  qj  ρJ  =  (ρj1, · · · , ρjk) ∈ Ωk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  If f : Ωn → R is measurable and (ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk) ∈ Ωk, we obtain a measurable function  g: Ωn−k → R by g(−) = f(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, −).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' If Q is a probability measure on Ωn−k we will  write EQ(f(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, −)) for EQ(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Let f : Ωn → R be a continuous fibrewise convex-supermodular, f ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Then  sup  p∈M+(Ω,n,b)  Ep(f|L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Lk)(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk) = Eq∗⊗n−k(f(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, −))  =  �  J∈Pn−k(m)  qJ · f(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, ρJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  In the remainder of this section we prove Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='4 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Throughout we fix b ∈ (0, 1)m  and assume that it is non-increasing, namely b1 ≥ · · · ≥ bm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Consider some p ∈ M+(Ω, n, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Suppose that 0 ≤ k ≤ n − 1 and consider  ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Set p′ = pLk+1|(L1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=',Lk)=(ω1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=',ωk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' See Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Then p′ ∈ M+(Ω, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' First, p′ is a pdf and p′ > 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=', see (9) in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Set I = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , k} and  J = {k + 1}, subsets of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Write LI for the random vector (L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Lk) and ωI =  14  JAREK KĘDRA, ASSAF LIBMAN, AND VICTORIA STEBLOVSKAYA  (ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk) ∈ Ωk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' We use Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='1 and the fact that p ∈ M+(Ω, n, b) to compute  Ep′(L) = Ep′(ω �→ Lk+1(ω)) = Ep′(ω �→ Ep(Lk+1|LI∪J)(ωI, ω))  = Ep(Lk+1|LI = ωI) = Ep(Lk+1|L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Lk)(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  By definition, then, p′ ∈ M+(Ω, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  □  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Let p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , pn ∈ M+(Ω, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Then p1 ⊗ · · · ⊗ pn ∈ M+(Ω, n, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' It is clear that p1 ⊗ · · · ⊗ pn is a non degenerate pdf on Ωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='3(3) and  since by definition Epk+1(L) = b,  Ep1⊗···⊗pn(Lk+1|L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Lk)(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk) = Epk+1(Lk+1) = Epk+1(L) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Since this holds for all 0 ≤ k ≤ n − 1, by definition p1 ⊗ · · · ⊗ pn ∈ M+(Ω, n, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  □  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='8 (n-fold Approximation Lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Let b ∈ (0, 1)n and consider q ∈ M(Ω, b) with  finite support {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , xr} and set qi = q({xi}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Let q⊗k denote the induced product measure  on Ωk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Let f : Ωn → R be continuous with f ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Then for any ǫ > 0 and any 0 ≤ k ≤ n  there exists P ∈ M+(Ω, n, b) such that  ���EP (f|L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Lk)(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk) − Eq⊗(n−k)(f(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, −))  ��� < ǫ  for all ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Since Ωn is compact, f is bounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='e ∥f∥∞ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Since f is uniformly continuous,  we choose δ > 0 suitable for  ǫ  3n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='10 with δ and with  ǫ  3n∥f∥∞ to obtain  p ∈ M+(Ω, b) and β <  ǫ  3n∥f∥∞ such that for any continuous g: Ω → R where g ≥ 0,  (12)  Ep(g) = β  �  Ω  g dµ +  r  �  i=1  (qi − β  r )g(ξi)  for some ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ξr ∈ Ω such that ∥xi − ξi∥∞ < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Set [r] = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' For any I = (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , in−k) ∈ [r]n−k set  qI = qi1 · · · qin−k  and  xI = (xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , xin−k) ∈ Ωn−k  and let fI : Ωk → R be the function  fI : (ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk) �→ f(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, xI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Observe that  (13)  Eq⊗(n−k)(f(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, −)) =  �  I∈[r]n−k  qI · fI(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  It is clear that ∥fI∥∞ ≤ ∥f∥∞ and that fI is uniformly continuous with the same δ suitable  for  ǫ  3n as that for f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Recall p ∈ M+(Ω, b) that we chose at the start of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Claim: Consider some 0 ≤ k < n and some I ∈ [r]n−k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Then for any ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk ∈ Ω  �����Ep  �  ω �→ fI(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, ω)  �  −  r  �  i=1  qi · fI(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, xi)  ����� < ǫ  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  EUROPEAN BASKETS IN DISCRETE-TIME CONTINUOUS-BINOMIAL MARKET MODELS  15  Proof: Set g(w) = fI(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Clearly g: Ω → R is continuous and ∥g∥∞ ≤ ∥f∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Moreover, it is uniformly continuous and clearly the same δ we chose for f suitable for  ǫ  3n  works for g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Since Eq(g) = �r  i=1 qig(xi) and since (12) holds  |Ep(g) − Eq(g)| ≤ β  �  Ω  g dµ + β  r  r  �  i=1  |g(ξi)| +  r  �  i=1  qi|g(ξi) − g(xi)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Since ∥g∥∞ ≤ ∥f∥∞ and β <  ǫ  3n∥f∥∞ the first and second terms in this sum are less than  ǫ  3n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Since ∥ξi − xi∥∞ < δ, the uniform continuity of g implies that the same is true for the last  term since �  i qi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' This completes the proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='d  Set P = p⊗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Then P ∈ M+(Ω, n, b) by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' In light of (13), we complete the  proof of the lemma by showing by downward induction on 0 ≤ k ≤ n that  (14)  ������  Ep⊗n(f|L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Lk)(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk) −  �  I∈[r]n−k  qI · fI(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk)  ������  ≤ (n − k) ǫ  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  The base of induction k = n is a triviality since  Ep⊗n(f|L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Ln)(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωn) = f(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωn)  and since fI = f and qI = 1 for the the only I ∈ [r]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Assume inductively that (14) holds for some 1 ≤ k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Fix some ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk−1 ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='3(2) imply that for any  Ep(τ �→ Ep⊗n(f|L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Lk)(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk−1, τ)) = Ep⊗n(f|L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Lk−1)(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Viewing the left hand side of (14) with ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk−1 fixed as a function of τ ∈ Ω, the linearity  of expectation Ep(−) implies  ������  Ep⊗n(f|L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Lk−1)(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' ωk−1) −  �  I∈[r]n−k  qIEp(fI(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk−1, −))  ������  < (n − k) ǫ  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Thanks to (13), in order to complete the induction step (to k − 1) it remains to show by that  ������  �  I∈[r]n−k  qI · Ep(fI(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk−1, −)) −  �  J∈[r]n−k+1  qJ · fJ(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk−1)  ������  < ǫ  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Given J = (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , in−k+1) ∈ [r]n−k+1 set I = (i2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , in−k+1) ∈ [r]n−k and observe that  qJ = qIqi1 and that fJ(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk−1) = fI(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk−1, xi1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' By the Claim above  ���  �  I∈[r]n−k  qIEp(fI(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk−1, −)) −  �  J∈[r]n−k+1  qJfJ(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk−1)  ��� =  =  ���  �  I∈[r]n−k  qIEp(fI(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk−1, −)) −  �  I∈[r]n−k  r  �  i=1  qIqifI(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk−1, xi)  ���  ≤  �  I∈[r]n−k  qI ·  ���Ep(fI(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk−1, −)) −  r  �  i=1  qifI(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk−1, xi)  ���  <  �  I∈[r]n−k−1  qI · ǫ  n = ǫ  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  This completes the induction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  □  16  JAREK KĘDRA, ASSAF LIBMAN, AND VICTORIA STEBLOVSKAYA  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' For any J = (j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , jn) ∈ Pn(m) and any ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , �  ωk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωn ∈ Ω  (meaning ωk is omitted) we have  ℓJ(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk−1, −, ωk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωn) =  �  i̸=k  ℓji(ωi) · ℓjk(−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Therefore, if λ ∈ M+(Ω, b) we get  Eλ(ℓJ(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk−1, −, ωk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωn)) = Eλ(ℓjk) ·  �  i̸=k  ℓji(ωi) = bjk ·  �  i̸=k  ℓji(ωi) =  ℓjk(b) ·  �  i̸=k  ℓji(ωi) = ℓJ(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk−1, b, ωk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  We use downward induction on 0 ≤ k ≤ n to show that for any p ∈ M+(Ω, n, b)  Ep(f|L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Lk)(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk) ≥ f(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , b)  almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' The base of induction k = n is a triviality (and in fact, equality holds a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Assume the inequality holds for k+1 ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Set p′ = pLk+1|L1=ω1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=',Lk=ωk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Then p′ ∈ M+(Ω, b)  by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='1 and the induction hypothesis imply  Ep(f|L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Lk)(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk) = Ep′(ω �→ Ep(f|L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Lk+1)(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, ω))  ≥ Ep′(ω �→ f(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, ω, b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Since f = h ◦ g with h convex and g as in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='3, Jensen’s inequality allows us to  continue the inequality  ≥ h(  �  J  aJEp′(ℓJ(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, −, b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , b))  = h(  �  J  aJℓJ(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , b))  = h(g(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , b))  = f(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  This completes the induction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  We deduce that in the statement of the theorem the right hand side is a lower bound  for the left hand side and it remains to show equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Let ν be the probability mea-  sure on Ω supported on {b}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='e ν({b}) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' It is clear that for any measurable function  g: Ωk → R we have Eν⊗k(g) = g(b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='8, for any ǫ > 0 there exists P ∈  M+(Ω, n, b) such that EP (f|L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Lk)(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk) is ǫ-close to Eν⊗(n−k)(f(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, −)) =  f(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  □  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' First, observe that  (15)  Eq∗⊗(n−k)(f(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, −)) =  �  I∈Pn−k(m)  qI · f(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, ρI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Next, we prove that for any p ∈ M+(Ω, n, b) and any 0 ≤ k ≤ n  (16)  Ep(f|L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Lk)(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk) ≤ E(q∗)⊗n−k(f(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, −)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Fix some p and use downward induction on k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' The base of induction k = n is a triviality  since Ep(f|L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Ln) = f a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Assume inductively that (16) holds for k + 1 ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Set  EUROPEAN BASKETS IN DISCRETE-TIME CONTINUOUS-BINOMIAL MARKET MODELS  17  p′ = pLk+1|(L1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=',Lk)=(ω1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=',ωk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='1 and the induction hypothesis together with (15)  imply that  Ep(f|L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Lk)(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk) = Ep′(ω �→ Ep(f|L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Lk+1)(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, ω))  ≤ Ep′(ω �→  �  I∈Pn−k−1  qIf(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, ω, ρI))  =  �  I∈Pn−k−1(m)  qI · Ep′(ω �→ f(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, ω, ρI)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  By the assumption on f, each function f(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, −, ρI) is convex-supermodular and con-  tinuous and.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='6 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='9 allow us to continue the estimate  ≤  �  I∈Pn−k−1(m)  qI ·  m  �  j=0  qj · f(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, ρj, ρI)) =  �  I∈Pn−k(m)  qI · f(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, ρI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Together with (15), this completes the induction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  We deduce that for any 0 ≤ k ≤ n the right hand side in the statement of the theorem  is an upper bound for the left hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='8 there exist P ∈ M+(Ω, n, b) such  that EP(f|L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Lk)(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk) are arbitrarily close to Eq∗⊗(n−k)(f(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, −).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' This  completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Proofs of the main results  In this section we prove the results in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' We start by setting up a formal framework  for the discrete-time continuous-binomial market model presented there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  We begin with the “one-step” process, namely description of the price jumps Ψi where  0 ≤ i ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' By definition Ψ0 = R and Ψi are chosen at random from the interval [Di, Ui].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' By  choosing a linear homeomorphisms [Di, Ui] ∼= [0, 1], a natural sample space for the probability  space underlying a single step is Ω = [0, 1]m and  Ψi(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , xm) = Di + (Ui − Di)xi,  Ψ0(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , xm) = R  With the notation of Section 3, for any 1 ≤ i ≤ m  Ψi = Diℓ0 + (Ui − Di)ℓi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  We will write Ψ: Ω → Rm+1 for the random vector  Ψ = (Ψ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Ψm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  The natural sample space for the n-step model is Ωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' We obtain a process Ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Ψn of  the price changes at time k:  Ψk : Ωn Lk  −→ Ω Ψ  −→ Rm+1  where Lk is the projection to the k-th factor and L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Lk form the universal process on  Ωn, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Thus,  Ψk  i = Di + (Ui − Di)Lk  i  where Lk  i is the ith component of Lk : Ωn → Ω ⊆ Rm+1 and we observe that (since ℓ0 =  1: Ω → R)  Lk  i = ℓ⊗(k−1)  0  ⊗ ℓi ⊗ ℓ⊗(n−k−1)  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Recall that we assume that 0 < Di < R < Ui so in particular Ψk  i > 0 for all i and all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  18  JAREK KĘDRA, ASSAF LIBMAN, AND VICTORIA STEBLOVSKAYA  The prices of the assets form an Rm+1-valued process  S0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Sn : Ωn → Rm+1  where Sk = (Sk  0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Sk  m) is the vector of prices of the assets at time k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' It is assumed by the  model that  Sk  i > 0  for all 0 ≤ i ≤ m and 0 ≤ k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  By construction of the model, the processes S0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Sn and Ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Ψn satisfy the relation  Sk  i = Sk−1  i  Ψk  i  (0 ≤ i ≤ m and 1 ≤ k ≤ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  It is therefore clear that for any 0 ≤ k ≤ n  Sk  i = S0  i · Ψ1  i · · · Ψk  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Recall the definition of Pk(m) from (5) in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  For J = (j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , jk) ∈ Pk(m) set  ℓJ = ℓj1 ⊗ · · · ⊗ ℓjk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' It follows that  Sk  i =  �  J∈Pk(m)  aJ · L1  j1 · · · Lk  jk =  �  J∈Pk(m)  aJ · ℓJ ⊗ 1 ⊗ · · · ⊗ 1  �  ��  �  n − k times  for some aJ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Comment: In Section 1 the processes Ψk and Sk were denoted Ψ(k) and S(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  The European basket in Section 1 is the function (random variable) F : Ωn → R  F =  � m  �  i=0  ci · Sn  i − K  �  ��  �  G  �+  where ci ≥ 0 for 1 ≤ i ≤ m and K > 0 is some number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Notice that  G =  �  J∈Pn(m)  aJℓJ  where aJ ≥ 0 for all J ̸= (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' It follows from Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='3 and since h(x) = x+ is  continuous, convex and non-negative that  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' F is continuous and fibrewise convex-supermodular and F ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Recall that a non-degenerate probability measure p on Ωn is called risk neutral if for any  k ≥ 0 and any j ≥ 1 such that k + j ≤ n  Ep(Sk+j|L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Lk)(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk) = Rj · Sk(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  We denote the set of these probability measures by RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' RN = M+(Ω, n, b) where b = (b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , bm) is defined in (2) in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Since Sk+j  i  = Sk  i · Ψk+1  i  · · Ψk+j  i  and since Sk  i > 0, it is clear that the condition for p  being a risk neutral measure is equivalent to the condition  Ep(Ψk+1  i  · · Ψk+j  i  |L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Lk) = Rj  (almost everywhere constant function Ωn−k → R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' It is easily verified using induction and  Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='1 that this condition (for any k, j such that k + j ≤ n) is equivalent to the  single step condition, namely  Ep(Ψk+1  i  |L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Lk) = R  EUROPEAN BASKETS IN DISCRETE-TIME CONTINUOUS-BINOMIAL MARKET MODELS  19  for all 1 ≤ k ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' But Ψk+1  i  = Di + (Ui − Di) · Lk+1  i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' So the condition above is equivalent  to  Di + (Ui − Di) · Ep(Lk+1  i  |L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Lk) = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Using the definition of b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , bm in (2), this is equivalent to Ep(Lk+1  i  |L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Lk) = bi, and  collecting these for all 1 ≤ i ≤ m we get  Ep(Lk+1|L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Lk) = b  which by definition is the condition for p ∈ M+(Ω, n, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  □  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='1 F is continuous convex-supermodular and F ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  The interval (Γmin(F, k) , Γmax(F, k)) of the rational values of F at some state of the world  (ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk) ∈ Ωk is known to be the collection of numbers  { Rk−n · Ep(F|L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Lk)(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk) }p∈RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='2 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='4 imply that  Rn−k · Γmin(F, k)(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk) = inf  p∈RN Ep(F|L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Lk)(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk) =  inf  p∈M+(Ω,n,b) Ep(F|L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Lk)(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk) = F(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  By definition of b, see (2), and since Ψi = Diℓ0 + (Ui − Di)ℓi,  Ψi(b) = R  for all 1 ≤ i ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  By definition, for any 1 ≤ i ≤ m  Sn  i (ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , b) = S0  i · Ψi(ω1) · · · ψi(ωk) · Ψi(b) · · · Ψi(b)  �  ��  �  n − k times  = Rn−kSk  i (ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Also, for i = 0 we clearly get Sn  0 = S0  0 · Rn = Rn−kSk  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' It follows that  F(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , b) =  �  Rn−k  m  �  i=0  ciSk  i − K  �+  (ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  This establishes the formula for Γmin(F, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  We note that the numbers qi defined in (3) and used in the statement of the theorem are  equal to q∗(ρi) of the upper supermodular vertex (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Since by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='1  the conditions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='5 hold, it follows that  Rn−k · Γmax(F, k)(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk) = sup  p∈RN  Ep(F|L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Lk)(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk)  =  sup  p∈M+(Ω,n,b)  Ep(F|L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Lk)(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk)  =  �  J∈Pn−k(m)  qJ · F(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, ρj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ρjn−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Since ℓi(ρj) = 1 if i ≤ j and ℓi(ρj) = 0 if i > j it follows that Ψi(ρj) = Di +(Ui −Di)ℓi(ρj) =  χi(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Therefore, for any J ∈ Pn−k(m),  Sn  i (ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, ρj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ρjn−k) = S0  i · Ψi(ω1) · · · Ψi(ωk) · Ψi(ρj1) · · · Ψi(ρjn−k) =  χi(J) · Sk  i (ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  20  JAREK KĘDRA, ASSAF LIBMAN, AND VICTORIA STEBLOVSKAYA  For i = 0 we get, of course, Sn  0 = S0  0 · Rn = Sk  0 · χ0(J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Substitution into the definition of F  we get  F(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk, ρj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ρjn−k) =  � m  �  i=0  ci · χi(J) · Sk  i − K  �+  (ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  This establishes the formula for Γmax(F, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  □  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Recall that Ψi(b) = Diℓ0 − (Ui − Di)ℓi(b) = Di + (Ui − Di)bi = R for  1 ≤ i ≤ m and that Ψ0 = R by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='1  Γmin(F, 0) = R−n · F(b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , b) = R−n(  m  �  i=0  ciS0  i · Ψi(b)n − K)+ = R−n ·  �  Rn  m  �  i=0  ciS0  i − K  �+  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  This is independent of Ui, Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Set bi as in (2) and denote by bi(s) the values of bi in our market model with parameters  ui(s) and di(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Notice that di(s) < R and that ui(s) > R for all 0 ≤ s < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Moreover,  di(0) = Di and ui(0) = Ui and limsր1 di(s) = R and limsր1 ui(s) = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' One checks that  bi(s) =  R − di(s)  ui(s) − di(s) = bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Therefore the values of qi = bi − bi+1 are independent of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' The values of χi(j) do depend on  s where χi(j)(s) = ui(s) if i ≤ j and χi(j)(s) = di(s) if i > j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Thus, χi(j)(s) is a polynomial  (of degree 1) in s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Moreover, limsր χi(j)(s) = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='1  ϕ(s) = Γmax(F, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' ui(s), di(s)) = R−n  �  J∈Pn(m)  qJ ·  \uf8eb  \uf8ed  m  �  i=0  ci · S0  i ·  �  j∈J  χi(j)(s)  \uf8f6  \uf8f8  +  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  So ϕ is a continuous function of s ∈ [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Now, ϕ(0) = Γmax(F, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Ui, Di) since di(0) = Di  and ui(0) = Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Since h: x �→ x+ is continuous and �m  j=0 qj = 1 we get  lim  sր1 ϕ(s) = R−n  �  J∈Pn(m)  qJ  \uf8eb  \uf8ed  m  �  i=0  ciS0  i · lim  sր1  �  j∈J  χi(j)(s) − K  \uf8f6  \uf8f8  +  = R−n  �  J∈Pn(m)  qJ  � m  �  i=0  ciS0  i · Rn − K  �+  = R−n  � m  �  i=0  ciS0  i · Rn − K  �+  = Γmin(F, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Ui, Di).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  The “intermediate value” result in the theorem follows from the continuity of ϕ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  □  In the next lemma we will consider processes on Ω, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Recall that the universal  process is L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Ln where Lk is the projection Ωn → Ω to the k-th factor followed by the  inclusion to Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Fix some n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Let q be a probability measure on Ω supported on {ρ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ρm}, see  (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Consider R-valued processes  r0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , rn−1 > 0  and  X0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Xn  EUROPEAN BASKETS IN DISCRETE-TIME CONTINUOUS-BINOMIAL MARKET MODELS  21  and Rm-valued processes  d0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , dn−1  and  ∆0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ∆n−1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Further, consider an Rm+1-valued processes  S0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Sn  and  Ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Ψn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  with components Sk = (Sk  0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Sk  m) and Ψk = (Ψk  0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Ψk  m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Assume that  (1) Sk  i = Sk−1  i  Ψk  i for all 1 ≤ k ≤ n and all 0 ≤ i ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  (2) Ψk+1  0  = rk and Ψk+1  i  = dk  i + ∆k  i · Lk+1  i  for any 0 ≤ k ≤ n − 1, and for any ωK =  (ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk) ∈ Ωk,  Eq(ω �→ Ψk+1  i  (ωK, ω)) = rk(ωK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  (3) Xk are fibrewise convex-supermodular, and for any 0 ≤ k ≤ n − 1 and any ωK ∈ Ωk  Eq(ω �→ Xk+1(ωK, ω)) = rk(ωK) · Xk(ωK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Then among all Rm+1-valued processes β0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , βn−1 there exists a process α0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , αn−1 which  minimises the R-valued random process  (17)  V k  β =  m  �  i=0  βk  i · Sk  i  subject to the condition  (18)  m  �  i=0  βk  i · Sk+1  i  ≥ Xk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Moreover, V k  α = Xk and the value of αk(ωK) at ωK ∈ Ωk can be computed by  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  αk  0(ωK)  αk  1(ωK)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  αk  m(ωK)  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb = T(ωK)−1 · M′(ωK)−1 · Q ·  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  Xk+1(ωKρ0)  Xk+1(ωKρ1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Xk+1(ωKρm)  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb  where Q is the matrix in the statement of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='2 and T, M′ : Ωk → Mat(m+1)×(m+1)(R)  are the (m + 1) × (m + 1) matrices  T =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  rk · Sk  0  Sk  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Sk  m  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb  M′ =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  1  dk  1  dk  2  · ·  dk  m  0  ∆k  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  0  ∆k  m  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' We prove the lemma in a sequence of claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Claim 1: If a process β0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , βn−1 satisfies (18) then V k  β ≥ Xk for all 0 ≤ k ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  22  JAREK KĘDRA, ASSAF LIBMAN, AND VICTORIA STEBLOVSKAYA  Proof: Choose some k and some ωK ∈ Ωk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Then (18) becomes the following system of  (infinitely many) inequalities in the unknowns βk  i (ωK)  (19)  m  �  i=0  βk  i (ωK) · Sk+1  i  (ωK, ω) ≥ Xk+1(ωK, ω),  (ω ∈ Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Let Φ(ω) denote the left hand side of (19) and Y (ω) denote its right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' These are  random variables with domain Ω so Eq(Φ) ≥ Eq(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' By the hypotheses on Xk  Eq(Y ) = Eq(ω �→ Xk+1(ωKω) = rk(ωK) · Xk(ωK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  By the hypotheses on Sk and Ψk  Eq(Φ) =  m  �  i=0  βk  i (ωK) · Sk  i (ωK) · Eq(ω �→ Ψk+1  i  (ωK, ω)) =  rk(ωK) ·  m  �  i=0  βk  i (ωK) · Sk  i (ωK) = rk(ωK) · V k  β (ωK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Since rk > 0 it follows that V k  β (ωK) ≥ Xk(ωK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='d  Consider some 0 ≤ k ≤ n − 1 and some ωK = (ω, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ωk) ∈ Ωk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' We will show in Claim 5  below that the system of m + 1 linear equations with m + 1 unknowns αk  0(ωK), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , αk  m(ωK)  (20)  m  �  i=0  Sk  i (ωK)Ψk+1  i  (ωK, ρj) · αk  i (ωK) = Xk+1(ωK, ρj),  0 ≤ j ≤ m  has a unique solution given by the matrices Q, M′, T as in the statement of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Since  Sk > 0 and dk and ∆k > 0 are measurable, we obtain measurable functions αk : Ωk → Rm+1  which form a process over Ω  α0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , αn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  We remark that (20) are the inequalities in (19) corresponding to ω = ρ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ρm with in-  equalities turned into equalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Claim 2: Consider some ωK ∈ Ωk where 0 ≤ k ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Then αk(ωK) solves the inequalities  (19) for all λ ∈ L ⊆ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Proof: As in Claim 1, write Φ(ω) for the left hand side of (19) and Y (ω) for the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' The  claim is that Φ(λ) ≥ Y (λ) for all λ ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Assume this is false, namely Φ(λ) < Y (λ) for some  λ ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Among all these λ’s choose one for which j is maximal with ρj ⪯ λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' see Definition  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='3 and the discussion below it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Clearly j < m because by definition of αk(ωK) we have  Φ(ρi) = Y (ρi) for all 0 ≤ i ≤ m and because ρm ∈ L is maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Set λ′ = λ ∨ ρj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' By  the choice of λ we get λ ∧ ρj+1 = ρj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Since Sk+1  i  (ωK, ω) = Sk  i (ωK) · Ψk+1  i  (ω) and since the  assignment ω �→ Ψk+1  i  (ωK, ω) is an affine function on Ω, it follows that Φ: Ω → R is affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Therefore  Φ(λ′) + Φ(ρj) = Φ(λ) + Φ(ρj+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  The assumption on Xk+1 implies that Y |L is supermodular, hence  Y (λ′) + Y (ρj) ≥ Y (λ) + Y (ρj+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Subtracting these inequalities, keeping in mind that by construction Φ(ρi) = Y (ρi), we get  Φ(λ′) − Y (λ′) ≤ Φ(λ) − Y (λ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Therefore Φ(λ′) < Y (λ′) and ρj+1 ⪯ λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' This contradicts the maximality of j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='d  Claim 3: αk(ωK) solves the inequalities (19) for all ω ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  EUROPEAN BASKETS IN DISCRETE-TIME CONTINUOUS-BINOMIAL MARKET MODELS  23  Proof: Since Ω is the convex hull of L, any ω ∈ Ω is a convex combination ω = �  λ∈L tλ · λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  The assumption on Xk+1 implies that Y : Ω → R is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Together with Claim 2 and since  Φ is affine  Φ(ω) = Φ(  �  λ∈L  tλλ) =  �  λ∈L  tλΦ(λ) ≥  �  λ∈L  tλY (λ) ≥ Y (  �  λ∈L  tλλ) = Y (ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='d  Claim 4: V k  α = Xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Proof: Denote qj = q(ρj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Consider some ωK ∈ Ωk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Equation (20) defining αk(ωK) yields  rk(ωK) · Xk(ωK) = Eq(ω �→ Xk+1(ωK, ω))  =  m  �  j=0  qjXk+1(ωK, ρj)  =  m  �  j=0  m  �  i=0  αk  i (ωK)Sk  i (ωK) · qjΨk+1  i  (ωK, ρj)  =  m  �  i=0  αk  i (ωK)Sk  i (ωK) · Eq(ω �→ Ψk+1  i  (ωK, ω))  = rk(ωK) · V k  α (ωK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Since rk > 0 it follows that V k  α (ωK) = Xk(ωK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='d  Claim 3 implies that α0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , αn−1 solve all the inequalities (19) and hence it solves the  constraints (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Claims 1 and 4 imply that V k  α ≤ V k  β for all k and all β0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , βn−1 that  satisfy (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' To complete the proof of the lemma it only remains to prove:  Claim 5: The system of equations (20) has a unique solution given by the matrices Q, T, M′  as in the statement of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Proof: Consider 0 ≤ k ≤ n−1 and ωK ∈ Ωk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' For 0 ≤ i, j ≤ m set χk+1  i  (j) = Ψk+1  i  (ωK, ρj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Notice that by the hypotheses  χk+1  0  (j) = rk(ωK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  For 1 ≤ i ≤ m observe that Lk+1  i  (ρj) = 1 if i ≤ j and Lk+1  i  (ρj) = 0 if i > j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Since  Ψk+1 = dk  i + ∆k  i (ωK)Lk+1  i  we get  χk  i (j) =  �  dk  i (ωK) + ∆k  i (ωK)  i ≤ j  dk  i (ωK)  i > j  Since Sk+1  i  (ωK, ρj) = Sk  i (ωK) · Ψk+1  i  (ωK, ρj), the matrix representing the system (20) is  M =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  Sk  0χk+1  0  (0)  Sk  1χk+1  1  (0)  · ·  Sk  mχk+1  m (0)  Sk  0χk+1  0  (1)  Sk  1χk+1  1  (1)  · ·  Sk  mχk+1  m (1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Sk  0χk+1  0  (m)  Sk  1χk+1  1  (m)  · ·  Sk  mχk+1  m (m)  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  1  χk+1  1  (0)  · ·  χk+1  m (0)  1  χk+1  1  (1)  · ·  χk+1  m (1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  1  χk+1  1  (m)  · ·  χk+1  m (m)  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb  �  ��  �  M′′ \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  rkSk  0  Sk  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Sk  m  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb  �  ��  �  T  24  JAREK KĘDRA, ASSAF LIBMAN, AND VICTORIA STEBLOVSKAYA  with all entries evaluated at ωK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Thus, M, M′′, T are functions Ωk → Mat(m+1)×(m+1)(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Oserve that for all 1 ≤ i ≤ m and 1 ≤ j ≤ m  χk+1  i  (j) − χk+1  i  (j − 1) =  �  ∆k+1  i  (ωK)  if i = j  0  if i ̸= j  It follows that  Q · M′′ =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  1  dk  1  dk  2  · ·  dk  m  0  ∆k  1  0  ∆k  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  0  ∆k  m  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  with these matrices evaluated at ωK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' We denote the latter matrix by M′ and notice that  it is invertible since ∆k  i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' In particular M(ωK) is invertible for any ωK ∈ Ωk so (20)  has a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Note that M−1 = T −1 · M′−1 · Q is a measurable function Ωn →  Mat(m+1)×(m+1)(R) and the solution of (20) is therefore the one given in the statement of the  lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  □  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' We apply Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='3 with the following data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' The probability measure  q is the upper supervertex q∗ : L → R (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='7) extended to a probability measure on  Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' The processes S0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Sn and Ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Ψn are the prices Sk = (Sk  0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Sk  m) of the assets  and their price jumps Ψk = (Ψk  0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Ψk  m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' The process r0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , rn−1 consists of the constant  functions with value R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' The processes d0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , dn−1 and ∆0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , ∆n−1 have components dk  i  and ∆k  i (1 ≤ i ≤ m) where dk  i is constant with value Di and ∆k  i is constant with value  Ui − Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' The process X0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , Xn are the upper bound of the option’s F price at time k,  namely Xk = Γmax(F, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  We need to show that the conditions of the lemma are fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  First, Sk  i > 0 for all  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  By construction of the model, Sk+1  i  = Sk  i · Ψk+1  i  and Ψk+1  0  = R = rk and Ψk+1  i  =  Di + (Ui − Di)Lk  i = dk  i + ∆k  i Lk+1  i  for all 0 ≤ k ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Also,  Eq(ω �→ Ψk+1  i  (ωK, ω)) = Eq(Di + (Ui − Di)ℓi(ω)) = Di + (Ui − Di)bi = R = rk(ωK)  by construction of q∗ (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='1  Xk(ωK) = Rk−n  �  J∈Pn−k(m)  qJ · F(ωK, ρJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Since F is fibrewise convex-supermodular by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='1, Xk is a linear combination with  non-negative coefficients of fibrewise convex-supermodular functions, hence it is one as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Finally, we check that  Eq∗(ω �→ Xk+1(ωK, ω)) = Eq∗  \uf8eb  \uf8edω �→ Rk+1−n  �  J∈Pn−k−1(m)  qJ · F(ωK, ω, ρJ)  \uf8f6  \uf8f8  = Rk+1−n  m  �  j=0  qj  �  J∈Pn−k−1(m)  qJF(ωK, ρj, ρJ)  = Rk+1−n  �  J∈Pn−k(m)  qJF(ωK, ρJ)  = R · Xk(ωK)  = rk(ωK) · Xk(ωK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  EUROPEAN BASKETS IN DISCRETE-TIME CONTINUOUS-BINOMIAL MARKET MODELS  25  All the conditions of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='3 are fulfilled and we obtain a process α0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' , αn−1 which  minimises  V k  α =  m  �  i=0  αk  i Sk  i  subject to the requirement that for all 0 ≤ k ≤ n − 1  m  �  i=0  αk  i Sk+1  i  ≥ Xk+1 = Γmax(F, k + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Thus, α(k) = αk is a minimum-cost maximal hedging strategy as required with the formulas  for its value given in the statement of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' It only remains to note that Yt(k) at the  state of the world ωK ∈ Ωk used in the statement of the theorem is precisely Xk+1(ωK, ρt)  because  Yt(k)(ωK) = Rk+1−n  �  J∈Pn−k−1(m)  qJ · (  m  �  i=0  ciχi(J)χi(t)Sk  i (ωK) − K)+  = Rk+1−n  �  J∈Pn−k−1(m)  qJ · (  m  �  i=0  ciχi(J)Ψi(ρt)Sk  i (ωK) − K)+  = Rk+1−n  �  J∈Pn−k−1(m)  qJ · (  m  �  i=0  ciχi(J)Sk  i (ωK, ρt) − K)+  = Γmax(F, k + 1)(ωK, ρt)  = Xk+1(ωK, ρt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  □  References  [1] William Feller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' An Introduction to Probability Theory and Its Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' John Wiley & Sons,  Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=', New York, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=', 1950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  [2] Jürgen Franke, Wolfgang Karl Härdle, and Christian Matthias Hafner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Statistics of financial markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  Universitext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Springer, Cham, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' An introduction, Fifth edition of [ MR2102757].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  [3] Allan Gut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Probability: a graduate course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Springer Texts in Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Springer, New York, second edition,  2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  [4] Paul R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Halmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Measure Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Van Nostrand Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=', Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=', New York, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=', 1950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  [5] Jarek Kędra, Assaf Libman, and Victoria Steblovskaya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Pricing and hedging contingent claims in a multi-  asset binomial market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='13283, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  [6] Jarek Kędra, Assaf Libman, and Victoria Steblovskaya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Hedging of european type contingent claims in  discrete time binomial market models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Preprint, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  [7] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Lovász.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Submodular functions and convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' In Mathematical programming: the state of the art (Bonn,  1982), pages 235–257.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Springer, Berlin, 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  [8] Stanley Pliska.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Introduction to Mathematical Finance - Discrete Time Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Blackwell Publishing, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  [9] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Royden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Real analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content=' Macmillan Publishing Company, New York, third edition, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='  University of Aberdeen and University of Szczecin  Email address: kedra@abdn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='uk  University of Aberdeen  Email address: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='libman@abdn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='uk  Bentley University  Email address: vsteblovskay@bentley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
+page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfSAzu/content/2301.04996v1.pdf'}
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+Factorial type I KMS states of Lie groups
+Tobias Simon
+January 10, 2023
+Abstract
+Motivated by the study of KMS conditions for C*- or W*-dynamical systems defined
+by covariant unitary representations of topological groups, we consider Gibbs states of a
+finite-dimensional Lie group G and prove that these are precisely the factorial type I KMS
+states. For an element X ∈ L(G) and an irreducible unitary representation ρ of G satisfying
+tr(ei∂ρ(X)) = 1, the corresponding Gibbs state is defined as ϕ(g) = tr(ρ(g)ei∂ρ(X)). We prove
+that under the mild assumption that ρ has discrete kernel, the condition tr(ei∂ρ(X)) < ∞
+implies that the generator X is an inner point of the set comp(g) of elliptic elements in
+g. This allows us to obtain a complete characterization of Lie algebras g, representations ρ
+with discrete kernel and generators X such that tr(ei∂ρ(X)) < ∞.
+Contents
+1
+Geometric and representation theoretic aspects of Gibbs states
+4
+1.1
+Ellipticity of Gibbs elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+1.2
+Characterization of representations of Gibbs type . . . . . . . . . . . . . . . . . .
+8
+2
+KMS states of topological groups
+12
+2.1
+Elementary properties of KMS states of topological groups . . . . . . . . . . . . .
+13
+2.2
+C*-methods for locally compact groups . . . . . . . . . . . . . . . . . . . . . . . .
+15
+2.3
+Semi-finite KMS states of topological groups
+. . . . . . . . . . . . . . . . . . . .
+19
+3
+Perspectives and open problems
+21
+3.1
+Extensions of KMS states of G to KMS states of G♭
+. . . . . . . . . . . . . . . .
+21
+3.2
+Factorial Type II KMS states of Lie groups . . . . . . . . . . . . . . . . . . . . .
+22
+A Appendix
+22
+A.1 Compactly embedded Cartan subalgebras and admissible Lie algebras
+. . . . . .
+22
+A.2 Convex geometric and representation-theoretic concepts . . . . . . . . . . . . . .
+24
+Notation and definitions
+• The group dynamical systems (G, α, R) considered in this paper consist of a topological
+group G and a group homomorphism α : R → Aut(G) with continuous orbit maps, i.e. α
+is strongly continuous. We call α a one-parameter group of automorphisms.
+• A covariant unitary representation of a group dynamical system (G, α, R) is a unitary
+representation of G♭ := G ⋊α R. These are precisely the unitary representations (π, H)
+1
+arXiv:2301.03444v1  [math.RT]  9 Jan 2023
+
+of G for which there exists a selfadjoint operator H on H which satisfies eitHπ(g)e−itH =
+π(αt(g)), for all t ∈ R, g ∈ G. We call H an implementing operator.
+• A state ϕ : G → C of a topological group G is a continuous function, for which the kernel
+Kϕ : G × G → C defined as Kϕ(x, y) := ϕ(x−1y) is positive definite and ϕ(e) = 1. Its
+GNS-representation is denoted by (πϕ, Hϕ, Ωϕ).
+• If G is a Lie group, we denote with L(G) or g its Lie algebra.
+• A von Neumann algebra M is uniquely decomposable as a direct sum of a type I, a type
+II1, a type II∞ and a type III von Neumann algebra (cf. [Tak02, Def. V.1.17], [Tak02,
+Thm. V.1.19]). It is called semi-finite if the type III summand is trivial. This is equivalent
+to the existence of a semi-finite faithful normal tracial weight τ on M (cf. [Tak03, Def.
+VII.1.1], [Tak02, Thm. V.2.15]).
+• We call a state of a topological group or C*-algebra semi-finite resp. factorial if the von
+Neumann algebra generated by its GNS-representation is semi-finite resp. is a factor.
+• An elliptic linear map endomorphism of a finite-dimensional vector space is a semi-simple
+linear map with purely imaginary eigenvalues.
+Introduction
+For C*- or W*-dynamical systems (M, τ, R), KMS states have been studied extensively, for
+instance in [BR96]. We are interested in KMS states of dynamical systems defined by group
+dynamical systems (G, α, R) and α-covariant representations π, i.e. we consider the von Neumann
+algebra M = π(G)′′ and the one-parameter group of *-automorphisms uniquely determined by
+τt(π(g)) = π(αt(g)). An example of such a dynamical system is the Weyl algebra W(E, σ) which
+is generated by the image of the Schrödinger representation of the Heisenberg group Heis(E, σ),
+where (E, σ) is a symplectic vector space (cf.
+[BR96, Thm.
+5.2.8]).
+In this example, the
+dynamics τ is implemented by Bogoliubov transformations corresponding to a one-parameter
+group of symplectic maps which is an example of the general situation described above. For such
+a dynamical system the (τ, β)-KMS condition of a state ω : M → C proves that the corresponding
+state ϕ : G → C, g �→ ω(π(g)) is an (α, β)-KMS in the sense of the following definition.
+Definition. Let (G, α, R) be a group dynamical system and ϕ : G → C be a state. For β > 0,
+we say that ϕ satisfies the (α, β)-KMS condition if the functions
+Fx,y : R → C,
+t �→ ϕ(xαt(y)),
+for
+x, y ∈ G,
+extend to continuous maps on the strip Sβ := {z ∈ C | 0 ≤ Im(z) ≤ β} that are holomorphic on
+the interior and satisfy the reflection property Fx,y(t + iβ) = ϕ(αt(y)x), for t ∈ R.
+If one conversely starts with an (α, β)-KMS state of a topological group and considers the cor-
+responding W*-dynamical system (W ∗(G), τ, R), where W ∗(G) is the group W*-algebra of G,
+one easily obtains a correspondence between (α, β)-KMS states of G and (τ, β)-KMS states of
+W ∗(G) (cf. Subsection 2.1). In the case, where the group G is locally compact, one has access to
+a C*-dynamical system given by group C*-algebra C∗(G) defined as the enveloping C*-algebra
+of L1(G) and a corresponding action. This makes KMS states of locally compact groups more
+accessible since there exist stronger decomposition results for C*-dynamical systems and their
+KMS states. In particular, for locally compact type I groups, where every factor state is a type I
+2
+
+state, every (α, β)-KMS state is an integral of extremal Gibbs states with respect to some Radon
+probability measure on the set of extremal (α, β)-KMS states (cf. Subsection 2.2).
+If an (α, β)-KMS state ϕ of a topological group G is semi-finite in the sense that πϕ(G)′′ is a
+semi-finite von Neumann algebra, there exists a semi-finite faithful normal trace τ on πϕ(G)′′.
+One can use the Radon-Nikodym theorem to prove that
+ϕ(g) = τ(π(g)e−βH)
+τ(e−βH)
+,
+for
+g ∈ G,
+(1)
+for a self-adjoint operator H affiliated to πϕ(G)′′ which satisfies eitHπϕ(g)e−itH = πϕ(αt(g)) and
+τ(e−βH) < ∞ (cf. Subsection 2.3). This applies in particular to the case, where πϕ(G)′′ is a
+type I factor, i.e. πϕ(G)′′ ∼= B(H), for some Hilbert space H. Thus one obtains in the manner of
+Equation (1) a one-to-one correspondence between factorial type I (α, β)-KMS states and unitary
+equivalence classes of irreducible α-covariant unitary representations π such that tr(e−βH) < ∞
+for the implementing operator H of α which satisfies eitHπ(g)e−itH = π(αt(g)).
+Suppose that G is a finite-dimensional Lie group and α is inner in the sense that there exists
+X ∈ L(G) such that αt is the conjugation by expG(tX). Then any unitary representation π is
+α-covariant since eit∂π(X)π(g)e−it∂π(X) = π(exp(tX)g exp(−tX)). This can always be achieved
+by considering, for not necessarily inner actions α, the group G♭ := G ⋊α R, for which
+(e, t)(g, s)(e, t)−1 = (αt(g), s),
+for
+g ∈ G, s, t ∈ R.
+For an action by inner automorphisms, the representations relevant in the study of factorial type
+I KMS states are the following:
+Definition. Let G be a finite dimensional Lie group and X ∈ L(G). We call a unitary repre-
+sentation (π, H) of G of Gibbs type with respect to X if ei∂π(X) is of trace class and elements in
+L(G), with respect to whom representations of Gibbs type exist, Gibbs elements. We denote the
+set of Gibbs elements of G with Gibbs(G, g) and for the universal covering group �G of G with
+Gibbs(g) := Gibbs( �G, g).
+Our main result is the following theorem, which turns out to be instrumental in characterizing
+representations of Gibbs type.
+Theorem. Let G be a finite dimensional Lie group and X ∈ L(G).
+If (π, H) is a unitary
+representation of G with discrete kernel such that ei∂π(X) is of trace class, then X is contained
+in the interior of the set comp(g) of elliptic elements in g (cf. Definition A.1.1).
+Irreducible unitary representations π of finite-dimensional Lie groups, for which there exists
+X ∈ comp(g)◦ such that the spectrum of i∂π(X) is bounded from above, where extensively
+studied in [Ne00]. Using these results, we obtain the following for irreducible representations of
+Gibbs type π with discrete kernel, the Lie algebras g, and Gibbs elements X:
+(i) g := L(G) is an admissible Lie algebra (cf. Definition A.1.5),
+(ii) there exists a compactly embedded Cartan subalgebra t ⊆ g containing X (cf. Definition
+A.1.1),
+(iii) there exists an admissible positive system of roots ∆+ ⊆ ∆(gC, tC) (cf. Definition 1.2.2)
+such that X is contained in the interior of the cone
+Cmax(∆+
+p ) := {Y ∈ t | iα(X0) ≥ 0 for all α ∈ ∆+
+p },
+where ∆+
+p is the set of non-compact positive roots (cf. Definition A.1.3),
+3
+
+(iv) π is a unitary highest weight representation with respect to the root system ∆+ (cf. Defi-
+nition A.2.9).
+Conversely, if one starts with an admissible Lie group G and a unitary highest weight repre-
+sentation π of G with respect to an admissible positive system ∆+, then [Ne00, Thm. IX.4.6]
+yields that ei∂π(X) is of trace class for all X ∈ Cmax(∆+
+p )◦. Hence we have characterized the Lie
+algebras, generators, and representations in question by geometric data. [Ne00, Thm. A.V.1]
+gives a complete classification of simple admissible Lie algebras, which are also called hermitian.
+For these, the admissible root systems are known and the cones Cmax(∆+
+p ) can be explicitly
+computed.
+The condition of semi-boundedness of the representations relevant to this paper is generalized in
+[Ni22] and shown to be a property of every KMS state of Lie groups. We also mention [LT18] as
+an example of trace class conditions arising in conformal field theory.
+Structure of this paper:
+This paper is divided into two sections. What motivated this paper is mostly contained in Sec-
+tion 2. In this section, most of what we do is take results already known for KMS states of C*-
+or W*-dynamical systems and connect these with the group dynamical picture. This leads us to
+unitary representations of Gibbs type which we characterize in Section 1. This characterization
+can be seen as a standalone result which is why we include Section 2 afterward as a motivation.
+Acknowledgements
+This research was supported by DFG-grant NE 413/10-1. I thank my supervisor Karl-Hermann
+Neeb for the many helpful corrections and Ricardo Correa da Silva for his advice on the theory
+of non-commutative integration.
+1
+Geometric and representation theoretic aspects of Gibbs
+states
+In this section, we study Gibbs elements and their corresponding unitary representations of Gibbs
+type. To do this, we make use of concepts introduced in Subsection A.1 and Subsection A.2 in
+the appendix. These subsections are meant as a quick introduction to concepts discussed in
+depth in the monograph [Ne00].
+Definition. We call a subalgebra a ⊆ g compactly embedded if ⟨eadg(a)⟩ ⊆ GL(g) is compact.
+We call an element x ∈ g elliptic if the subalgebra R · x ⊆ g is compactly embedded and denote
+with comp(g) the subset of elliptic elements in g.
+Under the assumption that π is an irreducible unitary representation of Gibbs type, we prove
+that the generator X is contained in the interior of the set of elliptic elements comp(g) in g.
+The condition comp(g)◦ ̸= ∅ is equivalent to the existence of a compactly embedded Cartan
+subalgebra t ⊆ g, and in this case comp(g) = Inn(g).t (cf. [Ne00, Thm. VII.1.8]). In Remark
+A.1.2, we discuss how, for a compactly embedded Cartan subalgebra t ⊆ g, one obtains a root
+system
+∆ := ∆(gC, tC) ⊆ it∗
+such that
+gC = tC ⊕
+�
+α∈∆
+gα
+C
+4
+
+is the simultaneous eigenspace decomposition with respect to t. In view of comp(g) = Inn(g).t
+and the Inn(g)-invariance of the set of Gibbs elements, it suffices to characterize the elements
+of Gibbs(g) ∩ t. We show that this characterization is possible by considering data associated
+to the root system ∆, namely we prove that for a unitary representation π of G with discrete
+kernel and an element X ∈ g satisfying ei∂π(X), one has the following:
+(i) g is an admissible Lie algebra, i.e. it contains an invariant generating closed convex subset
+containing no affine line.
+(ii) There exists a compactly embedded Cartan subalgebra t ⊆ g containing X.
+(iii) There exists an admissible positive system of roots ∆+ ⊆ ∆(gC, tC) (cf. Definition 1.2.2)
+such that X is contained in the interior of the cone Cmax(∆+
+p ) (cf. Definition A.1.4).
+(iv) π is a unitary highest weight representation with respect to the root system ∆+ (cf. Defi-
+nition A.2.9).
+In [Ne00, Thm. IX.4.6] it is shown that if π is a unitary highest weight representation with
+respect to an admissible positive system ∆+ and X ∈ Cmax(∆+
+p )◦, then ei∂π(X) is of trace class.
+If one now drops the assumption of the discrete kernel, one obtains a complete characterization of
+Gibbs representations of simply connected Lie groups and Gibbs elements by considering faithful
+unitary highest weight representations with respect to admissible positive systems of admissible
+quotients.
+1.1
+Ellipticity of Gibbs elements
+The goal of this subsection is to prove that, if π is a unitary representation of a finite-dimensional
+Lie group G with discrete kernel and X ∈ g := L(G) such that ei∂π(X) is of trace class, then
+X ∈ comp(g)◦. This is equivalent to gX := ker(ad(X)) ⊆ g being compactly embedded by [Ne00,
+Lemma VII.1.7(c)].
+The first crucial observation is that in this case, the eigenspaces of ei∂π(X) are all finite-dimensional
+and invariant under GX := ⟨exp(gX)⟩. Since π has discrete kernel, this implies that gX is a com-
+pact Lie algebra. In order to prove that it actually is compactly embedded, we consider the
+smallest ideal nY ⊴ g, for a fixed Y ∈ g, such that
+adg/nY : g/nY → g/nY , x + nY �→ [Y, x] + nY
+is an elliptic endomorphism. For Y ∈ gX, we prove that nY = {0} holds, which is equivalent to
+Y being elliptic. To do this, we proceed by first showing that nY is abelian and then use this to
+prove that nY is an at most one-dimensional central ideal.
+Lemma 1.1.1. Let (π, H) be a unitary representation of G with discrete kernel such that ei∂π(X)
+is of trace class. The ellipticity ideal nY is abelian, for every Y ∈ gX.
+Proof. Consider the unitary representation of G on the Hilbert-Schmidt operators B2(H) given
+by
+ρ : G → U(B2(H)),
+ρ(g)(A) = π(g)Aπ(g)∗.
+The eigenspace projections of ei∂π(X) have finite rank and thus are Hilbert-Schmidt opera-
+tors. Since π(exp(RY )) leaves the eigenspaces of ei∂π(X) invariant, the one-parameter groups
+5
+
+ρ(exp(RY )) fix the corresponding projections. Now Moore’s Theorem [Mo80, Thm. 1.1] implies
+that the normal subgroup NY := ⟨exp(nY )⟩ ⊴ G also fixes the projections onto the eigenspaces,
+which is equivalent to π(NY ) leaving the eigenspaces invariant. Hence the representation π|NY
+decomposes into finite-dimensional subrepresentations and has discrete kernel. Therefore the
+finite-dimensional unitary representations of nY separate the points which implies that nY is a
+compact Lie algebra. As a compact Lie algebra, nY is in particular reductive and decomposes
+as nY = znY ⊕ [nY , nY ]. We define a := [nY , nY ] ⊴ g and note that a is a semi-simple compact
+ideal. From [HN12, Cor. 5.6.14], one obtains that a is contained in a Levi complement s and
+thus for the ideal a ⊴ s there exists a complementary ideal a⊥ ⊴ s such that s = a ⊕ a⊥ since s
+is semi-simple. It is now easy to see that b := s⊥ ⊕ rad(g) is an ideal in g and satisfies g = a ⊕ b.
+This shows that g/b ∼= a is compact and thus adg/b(Y ) is elliptic which implies nY ⊆ b as nY is
+the smallest ideal with this property. Then a ⊆ nY and a ∩ b = {0} imply [nY , nY ] = a = {0}
+and thus the assertion follows.
+The group K := π(NY ) ⊆ U(H) on H leaves the finite dimensional eigenspaces of ei∂π(X)
+invariant, which implies that that K is compact and Lemma 1.1.1 shows that K is abelian. Since
+NY ⊴ G is a normal subgroup and π(NY ) ⊂ K is dense, it follows that π(g)Kπ(g)−1 ⊆ K and
+thus
+Φ : G → Aut(K),
+Φ(g)(k) = π(g)kπ(g)−1
+is a group homomorphism.
+Next we consider on Aut(K) a suitable topology1 for Φ to be
+continuous. We will use this topology to cite results that imply that a continuous action of a
+connected group acting on a compact abelian group by group homomorphisms is trivial.
+Lemma 1.1.2. Let (π, H) be a unitary representation of G with discrete kernel such that ei∂π(X)
+is of trace class. For Y ∈ gX and NY := ⟨exp(nY )⟩, the group K := π(NY ) is compact, abelian,
+and commutes with the identity component of G. In particular, nY is central in g.
+Proof. [HM06, Thm. 9.82] implies that the identity component of Aut(K) with respect to the
+topology introduced in the footnote above coincides with the inner automorphisms of K. Since
+K is abelian, it is trivial. As Φ is a group homomorphism, we only need to show that it is
+continuous with respect to O1 to prove its continuity with respect to O. By the exponential law
+for spaces of continuous functions (cf. [GN, Thm. 1.7.29]), this follows from the continuity of
+Φ∧ : G × K → K,
+(g, k) �→ Φ(g, k) = π(g)kπ(g)−1.
+This implies that the identity component of G acts trivially on K and therefore nY is central.
+The representation π was assumed to be irreducible, so that π(NY ) ⊆ π(G)′ = C1 and the
+discreteness of the kernel of π thus implies that nY ⊴ g is a central ideal.
+With the same
+argument, one obtains that z(g) is at most one-dimensional.
+It remains to prove that nY cannot be one-dimensional. In order to prove this, we will assume
+the opposite {0} ̸= nY ⊆ z := z(g). Then we are in the context of the following lemma which
+will allow us to use some results of the representation theory of the three-dimensional Heisenberg
+group to obtain a contradiction.
+1Consider the subspace topology O1 with respect to the topology of uniform convergence on C(K, K), which
+is the coarsest topology containing the sets U(V0, f0) := {f ∈ C(K, K) | f(k) ∈ V0f0(k) for all k ∈ K}, for
+f0 ∈ C(K, K) and identity neighbourhoods V0 in K.
+Let O2 be the unique topology on Aut(K) such that
+ι : Aut(K) → (Aut(K), O1), f �→ f−1 becomes a homeomorphism.
+If we now define O := O1 ∨ O2, then
+(Aut(K), O) defines a topological group (cf. [HM06, p. 257]).
+6
+
+Lemma 1.1.3. Let g be a real finite-dimensional Lie algebra with center z and suppose that
+y + z ∈ g/z is elliptic but y ∈ g is not. Then there exists x ∈ g such that [x, y] = z ∈ z \ {0}. In
+particular h := spanR{x, y, z} is isomorphic to the three dimensional Heisenberg algebra h3(R).
+Proof. We consider ad(y) as an endomorphism of gC with Jordan decomposition given by ad(y) =
+ad(y)s + ad(y)n. Since gC decomposes into the ad(y)n-invariant eigenspaces gλ
+C of ad(y)s, for
+λ ∈ C, and zC ⊆ g0
+C, it follows from the fact that y + z is elliptic that
+ad(y)n|gλ
+C ≡ 0,
+for
+λ ̸= 0,
+and
+ad(y)n(g0
+C) ⊆ zC.
+If ad(y)n were zero, then ad(y) = ad(y)s would be elliptic since zC ⊆ g0
+C. This proves that there
+exists x ∈ g0
+C such that ad(y)(x) = ad(y)n(x) ∈ zC \ {0}. If we write x = x0 + ix1 ∈ gC, then
+[x, y] ∈ zC \ {0} implies that either [x0, y] ∈ z \ {0} or [x1, y] ∈ z \ {0} hold. This proves the
+assertion.
+Remark 1.1.4. Consider the three-dimensional Heisenberg group defined as follows:
+Heis(R2) := R2 × R
+with multiplication
+(a, b, c)(x, y, z) := (a + x, b + y, c + z + ay),
+for a, b, c, x, y, z ∈ R. For irreducible unitary representations of Heis(R2) the center Z(Heis(R2)) ∼=
+R acts by scalars and thus, if the center acts non-trivially, defines a central character.
+The
+Stone-von-Neumann Theorem (cf.
+[DE09, Thm.
+10.2.1]) implies that this central character
+λ : R → T, x �→ eiλx uniquely determines the unitary representation and that the unitary
+representation is unitarily equivalent to the representation on L2(R) defined by
+πλ(a, b, c)f(x) := eiλ(bx+c)f(x + a)
+for
+a, b, c, x ∈ R, f ∈ L2(R).
+Consider the automorphism Φ : Heis(R2) → Heis(R2), (a, b, c) �→ (b, −a, c − ab), which fixes
+the center pointwise.
+We note that by the Stone-von-Neumann theorem πλ ∼= πλ ◦ Φ and
+Φ(0, t, 0) = (t, 0, 0) holds, for t ∈ R. This proves that the unitary one-parameter groups defined
+by
+πλ(t, 0, 0)f(x) = f(x + t)
+and
+πλ(0, t, 0)f(x) = eitλxf(x),
+for
+x, t ∈ R, f ∈ L2(R)
+are conjugate via U(H). In particular,πλ(R, 0, 0) and πλ(0, R, 0) do not have compact closure
+in U(H) since the identical representation of both groups on L2(R) clearly does not decompose
+into irreducible subrepresentations and the groups are conjugate.
+We now come to our first main result:
+Theorem 1.1.5. Let (π, H) be a unitary representation of G with discrete kernel such that
+ei∂π(X) is of trace class. Then π is a direct sum of irreducible representations and X is contained
+in comp(g)◦.
+Proof. Since ei∂π(X) ∈ π(G)′′ is a positive operator of trace class, it follows that π(G)′ leaves the
+finite-dimensional eigenspaces invariant. As finite dimensional C*-algebras are isomorphic to a
+direct sum of matrix algebras, this implies that there exists a maximal family of pairwise disjoint
+minimal projections in π(G)′ summing to the idenitfy in the strong operator topology. Therefore
+π is a direct sum of irreducible representations πj with discrete kernel satisfying tr(ei∂πj(X)) < ∞.
+Suppose now that π is irreducible and let Y ∈ gX. It follows from Lemma 1.1.2 that nY ⊆ z(g)
+and thus Y is either elliptic or 0 ̸= Y + z(g) is elliptic in g/z(g). In the latter case, Lemma
+1.1.3 implies that there exists x0 ∈ g such that [Y, x0] = z ∈ z(g) \ {0} and thus the subalgebra
+7
+
+h := spanR{Y, x0, z} is isomorphic to the three-dimensional Heisenberg algebra L(Heis(R2)) via
+the isomorphism defined by
+Y �→ (1, 0, 0), x0 �→ (0, 1, 0), z �→ (0, 0, 1).
+The simply connected Heisenberg group Heis(R2) is the universal covering group of ⟨exp(h)⟩ and
+thus the representation π|⟨exp(h)⟩ lifts to a representation ρ of Heis(R2). As π is irreducible, z(g)
+acts by scalars, i.e. there exists λ ∈ R such that
+∂ρ(0, 0, 1) = ∂π(z) = iλ1.
+One has λ ̸= 0, because ∂π(z) = 0 would imply nY ⊆ z(g) ⊆ ker(∂π) = {0}.
+The Stone-
+von-Neumann Theorem (cf. [Ne00, Thm. X.3.1]) implies that ρ is a multiple of πλ, where πλ
+is the unique irreducible unitary representation of the Heisenberg group with central character
+λ : R → T, x �→ eiλx. It follows from Remark 1.1.4, that the closure of
+ρ(exp(R(1, 0, 0))) = π(exp(RY ))
+is not compact.
+This is a contradiction since a closed subgroup of U(H) leaving the finite-
+dimensional eigenspaces of ei∂π(X) invariant is compact. Hence nY = {0} follows, which proves
+that Y is elliptic. As Y ∈ gX was arbitrary, we have shown that gX is contained in comp(g),
+which together with [Ne00, Lemma VII.1.5(b)] yields that gX is compactly embedded. From
+[Ne00, Lemma VII.1.7(c)], one obtains that X ∈ comp(g)◦.
+1.2
+Characterization of representations of Gibbs type
+The goal of this subsection is to summarize results from [Ne00] to prove that the irreducible
+unitary representation ρ of Gibbs type with discrete kernel coincide with faithful unitary high-
+est weight representations of a finite-dimensional connected Lie group G. The Lie algebras, for
+which such representations exist, are precisely the admissible Lie algebras, i.e. those containing
+a generating invariant closed convex subset containing no affine line.
+The Lie algebraic and
+representation theoretic concepts used in this subsection are introduced in Subsection A.1 and
+Subsection A.2.
+The convex momentum set Iπ ⊆ g∗ of a unitary representation (π, H) of a finite-dimensional Lie
+group G (cf. Definition A.2.3) connects convex geometry and representation theory. Its relevance
+for our study of representations of Gibbs type can be seen from the following result (cf. [Ne00,
+Lemma X.I.6]):
+B(Iπ) := {Y ∈ g | ⟨Iπ, Y ⟩ > −∞} = {Y ∈ g | sup spec(i∂π(Y )) < ∞}.
+If (π, H) is an irreducible unitary representation of G with discrete kernel for which there exists
+X ∈ g := L(G) such that ei∂π(X) is of trace class, then the spectrum of i∂π(X) is bounded
+from above and Theorem 1.1.5 implies X ∈ B(Iπ) ∩ comp(g)◦. Therefore [Ne00, Thm. X.3.8] is
+applicable and we obtain the following corollary:
+Corollary 1.2.1. Let G be a finite-dimensional Lie group and (π, H) be an irreducible unitary
+representation of G with discrete kernel for which there exists X ∈ g := L(G) such that ei∂π(X)
+is of trace class. Then there exist a compactly embedded Cartan subalgebra t ⊆ g containing X
+and a positive system of roots ∆+ ⊆ ∆(gC, tC), which is adapted (cf. Definition A.1.3), such that
+π is a unitary highest weight representation (cf. Definition A.2.9) with respect to ∆+ and X is
+contained in the interior of the cone
+Cmax := Cmax(∆+
+p ) = {X ∈ t | iα(X) ≥ 0 for all α ∈ ∆+
+p }.
+8
+
+For an element X ∈ g, the condition that there exists an adapted positive system ∆+ such
+that X ∈ Cmax(∆+
+p )◦ is not sufficient for X to be a Gibbs element. To ensure this, we need
+further restrictions on ∆+. The Gelfand-Rai’kov Theorem for Unitary Highest Weight Represen-
+tations [Ne00, Thm. X.4.8] states sufficient conditions for a positive root system ∆+ of a simply
+connected admissible Lie group such that unitary highest weight representations exist. In this
+theorem, one considers the convex cone
+Cmin(∆+
+p ) := cone({i[Zα, Z∗
+α] | Zα ∈ gα
+C, α ∈ ∆+
+p } ⊆ t,
+where ∗ : gC → gC, x + iy �→ −x + iy denotes the involution on gC discussed in Remark A.1.2.
+The condition stated in the following definition then ensures that the unitary highest weight
+representations separate the points.
+Definition 1.2.2. Let g be a Lie algebra and t ⊆ g be a compactly embedded Cartan subalgebra.
+We call an adapted positive system of roots ∆+ ⊆ ∆(gC, tC) such that Cmin(∆+
+p ) is pointed and
+contained in Cmax(∆+
+p ) admissible.
+Remark 1.2.3. The term admissible for a positive system of roots is not used in [Ne00]. We
+introduce it to state the conditions for positive systems of roots arising in our context more
+concisely.
+The condition of admissibility of a positive system arises quite naturally in the theory of unitary
+highest weight representations.
+In the Gelfand-Raik’kov theorem, it is stated as a sufficient
+condition for the existence of unitary highest weight representation of admissible Lie algebras.
+The following theorem, which is stated implicitly in [Ne00], shows that for faithful unitary highest
+weight representations the condition is also necessary:
+Theorem 1.2.4. ([Ne00, Thm. IX.2.17])
+Let G be a finite dimensional Lie group with Lie algebra g and (π, H) be an irreducible unitary
+highest representation of G with respect to a positive system ∆+ and suppose π has discrete
+kernel. Then g is admissible and ∆+ is an admissible positive system.
+Proof. We consider the space of smooth vectors H∞ which is a faithful, unitary highest weight
+module of gC. It follows from [Ne00, Prop. IX.1.14](iii) that, for the highest weight λ ∈ it∗ of
+π the gC-module H∞ is isomorphic to L(λ, ∆+), which is introduced there. Using [Ne00, Thm.
+IX.2.17], which is applicable to L(λ, ∆+), one obtains that g is admissible, ∆+ is adapted and
+−iλ ∈ t∗ is contained in the interior of
+Cmin(∆+
+p )⋆ := {f ∈ t∗ | f(Cmin(∆+
+p )) ⊆ [0, ∞)} ⊆ B(Cmin(∆+
+p )).
+Then [Ne00, Prop. V.1.15] implies that the convex cone Cmin(∆+
+p ) is pointed since B(Cmin(∆+
+p ))
+has inner points. To prove that Cmin(∆+
+p ) ⊆ Cmax(∆+
+p ) holds, we apply [Ne00, Prop. VIII.1.11]
+which is applicable if there exists f ∈ Cmin(∆+
+p )⋆ such that its orbit Of under the coadjoint
+action is closed, the convex hull of Of contains no affine lines and Of spans g∗.
+If one applies [Ne00, Thm. VIII.1.19] to the highest weight λ ∈ iCmin(∆+
+p )⋆ ⊆ it∗ of the unitary
+highest weight representation π with discrete kernel, one obtains that −iλ is admissible, i.e. its
+orbit O−iλ under the coadjoint action is closed and its convex hull contains no affine lines. Using
+[Ne00, Thm. X.4.1] which states that the momentum set of π is given by Iπ = conv(O−iλ) and
+(Iπ)⊥ = ker(dπ) = {0} (cf. [Ne00, Lemma X.1.6]), one obtains that Iπ ⊆ g∗ is generating and
+−iλ ∈ Cmin(∆+
+p )⋆
+satisfies
+spanRO−iλ = g∗.
+Hence [Ne00, Prop. VIII.1.11] is applicable which yields Cmin(∆+
+p ) ⊆ Cmax(∆+
+p ) and thus the
+positive system ∆+ is admissible.
+9
+
+Remark 1.2.5. Note that [Ne00, VIII.1.11] implies that the positive system ∆+ in the theorem
+above is uniquely determined by the condition that (π, H) is an irreducible unitary highest
+representation with respect to ∆+.
+In [Ne00], the highest weight modules L(λ, ∆+) are studied, for involutive Lie algebras with root
+decomposition (cf. [Ne00, Subsection IX.2]). For those modules, one has the following result:
+Theorem 1.2.6. ([Ne00, Thm. IX.4.6])
+Let g be an admissible Lie algebra, ∆+ an adapted positive system and L(λ, ∆+) a unitary highest
+weight module with respect to ∆+. If X ∈ Cmax(∆+
+p )◦ and ∂π denotes the representation of g on
+L(λ, ∆+), then tr(ei∂π(X)) < ∞.
+Remark 1.2.7. Let g be an admissible Lie algebra. Then [Ne00, Thm. VII.3.10] implies that
+there exists a compactly embedded Cartan subalgebra t in g and an admissible positive system
+of roots ∆+. Then [Ne00, Prop. VIII.3.7] associates to each admissible positive system ∆+ a
+closed generating invariant cone Wmax(∆+
+p ) ⊆ g such that Wmax ∩ t = Cmax. In the proof of
+[Ne00, Thm. VIII.3.10] it is also shown that
+Wmax(∆+
+p )◦ = Inn(g).Cmax(∆+
+p )◦ ⊆ comp(g)◦.
+If g is non-compact, the open invariant convex cone Wmax(∆+
+p )◦ is proper since comp(g)◦ ⊊ g.
+We are now able to present our second main result:
+Theorem 1.2.8. Let G be a finite-dimensional Lie group with Lie algebra g and (π, H) be an
+irreducible unitary highest representation of G with respect to a positive system ∆+. Then one
+has {X ∈ g | tr(ei∂π(X)) < ∞} = Wmax(∆+
+p )◦.
+Proof. This follows immediately from Corollary 1.2.1, Theorem 1.2.4, and Remark 1.2.5.
+As the Gelfand-Rai’kov Theorem (cf. [Ne00, Thm. X.4.8]) applies to simply connected admissible
+Lie groups, we may combine these results to obtain the following corollary:
+Corollary 1.2.9. Let G be a simply connected finite-dimensional Lie group with Lie algebra g
+and X ∈ g. Then there is a one-to-one correspondence between
+(i) irreducible unitary representation π of G such that ei∂π(X) is of trace class,
+(ii) irreducible unitary highest weight representations of admissible quotients q : g ↠ g1 with
+respect to an admissible positive system ∆+
+1 of g1 such that q(X) ∈ Wmax(∆+
+1,p)◦.
+We will now prove that an element X is contained in Gibbs(g) if and only if it is contained in
+an open proper invariant convex subset of g.
+Theorem 1.2.10.
+Geometric characterization of Gibbs elements
+Let g be a finite-dimensional Lie algebra. Then an element X is a Gibbs element if and only if
+it is contained in an open proper invariant convex subset of g.
+Proof. Suppose X ∈ Gibbs(g) and G is a simply connected Lie group with Lie algebra g. Then
+there exists a unitary representation π of G such that ei∂π(X) is of trace class. Then Corollary
+1.2.1 and Theorem 1.2.4 imply that g/ker(dπ) is admissible and there exists an admissible positive
+system ∆+ such that
+q(X) ∈ Cmax(∆+
+p )◦ ⊆ Wmax(∆+
+p )◦,
+10
+
+where q : g → g/ker(dπ) denotes the quotient map. If g/ker(dπ) is non-compact, the subset
+q−1(Wmax(∆+
+p )◦) ⊊ g
+is an open proper invariant convex subset containing X. If g′ := g/ker(dπ) is compact, choose
+an Inn(g′)-invariant metric on g′ and an open ball B ⊊ g′ with respect to this metric containing
+q(X). Then q−1(B) ⊊ g is an open proper invariant convex subset containing X.
+Suppose conversely that X is contained in an open proper invariant convex subset C ⊊ g. We
+adopt elements of the proof of [Ne00, Prop. VII.3.4]. We consider the set
+H(C) := {v ∈ g | v + C = C}.
+H(C) is a convex additive subgroup of the vector space g which implies that H(C) a linear
+subspace and the invariance of C yields that H(C) is an ideal.
+Consider the quotient map
+q : g → g′ := g/H(C) and observe that q(C) is an invariant open generating convex subset. It
+satisfies
+Hg′(q(C)) = {w ∈ g′ | w + q(C) = q(C)} = {q(v) ∈ g′ | v ∈ g, q(v + C) = q(C)} = {0}
+since q(v + C) = q(C) is equivalent to (v + C) + H(C) = C + H(C), and then C + H(C) = C
+implies v ∈ H(C). Therefore the closure of q(C) is an invariant generating closed convex subset
+containing no affine line and thus g′ is admissible. To apply Theorem 1.2.6 and Corollary 1.2.9,
+we show that there exists an admissible positive system ∆+ such that q(X) ∈ Cmax(∆+
+p )◦.
+The set q(C) is contained in the interior of its closure q(C) whose interior consists of elliptic
+elements (cf. [Ne00, Prop. VII.3.4(b)]) and thus q(X) is contained in the open invariant elliptic
+convex set q(C).
+For a compactly embedded Cartan subalgebra t ⊆ g′, one has comp(g′) = Inn(g′).t (cf. [Ne00,
+Thm. VII.1.8]), we may assume q(X) ∈ t. Then the Sandwich Theorem (cf. Theorem A.1.6)
+implies that there exists an adapted positive system ∆+ with respect to t such that
+q(C) ∩ t ⊆ Cmax(∆+
+p )
+and
+Cmin(∆+
+p ) ⊆ lim(q(C) ∩ t) := {v ∈ t | v + q(C) ∩ t ⊆ q(C) ∩ t}.
+As q(C) contains no affine line, the cone Cmin(∆+
+p ) is pointed and since Cmax is a cone and
+q(C) ∩ t ⊆ Cmax(∆+
+p ), it follows that
+Cmin(∆+
+p ) ⊆ lim(q(C) ∩ t) ⊆ Cmax(∆+
+p ).
+This proves that ∆+ is admissible and that q(X) ∈ Cmax(∆+
+p )◦. Hence the assertion follows.
+Example 1.2.11. Let (V, Ω) be a symplectic vector space, h(V, Ω) = R⊕V be the corresponding
+Heisenberg algebra and
+sp(V, Ω) := {A ∈ gl(V ) | Ω(v, Aw) + Ω(Av, w) = 0 for all v, w ∈ V }
+be the corresponding symplectic Lie algebra. The prototypical example of an admissible Lie
+algebra which is not semi-simple, is given by the Jacobi algebra g := hsp(V, Ω) := h(V, Ω) ⋊
+sp(V, Ω). The map
+Φ : hsp(V, Ω) → C∞(V ),
+Φ(s, v, A)(w) := 1
+2ΩA(w) + Ω(v, w) + s,
+11
+
+for s ∈ R, v, w ∈ V, A ∈ sp(V, Ω), defines an injective homomorphism of Lie algebras if one
+equips C∞(V ) with the Poisson-bracket corresponding to Ω (cf. [Ne00, Prop. A.IV.15]). The
+image of Φ coincides with the polynomials of degree ≤ 2 and it maps sp(V, Ω) to polynomials
+of homogeneous degree 2.
+Continuing onwards we identify hsp(V, Ω) ∼= im(Φ).
+Consider a
+symplectic basis
+{pj, qj | 1 ≤ j ≤ n := 1
+2dimRV }
+and define
+tl := spanR{p2
+i + q2
+i | 1 ≤ i ≤ n}.
+Then tl is a compactly embedded Cartan subalgebra of l := sp(V, Ω). If one computes the root
+system ∆ := ∆(lC, (tl)C) as in [Ne00, Ex. VII.2.30] and chooses the up to a sign unique positive
+admissible system ∆+, then one has
+Cmax(∆+
+p ) = cone({p2
+i + q2
+i | 1 ≤ i ≤ n}) = {ΩA ∈ tl | ΩA ≥ 0}.
+Since there is up to a sign only one admissible positive system of non-compact roots ∆+
+p in the
+simple hermitian Lie algebra sp(V, Ω) (cf. [Ne00, Lemma VII.2.16(iv)]) and ΩA > 0 if and only
+if A is elliptic and all its eigenvalues are contained in iR+, this proves that the set of Gibbs ele-
+ments consists of all elliptic symplectic endomorphisms for which the eigenvalues are contained
+in either iR+ or iR−.
+Then [Ne00, Ex. VII.2.30]) implies that t = z(g)⊕tl is a compactly embedded Cartan subalgebra
+of hsp(V, Ω) for the corresponding admissible positive systems of roots ∆±
+0 ⊆ ∆(gC, tC) defined
+by extending the elements of ∆± ⊆ ∆(lC, (tl)C) extended to t = z(g) ⊕ tl by zero on z(g). In
+particular, one has
+Wmax(∆±
+0,p)◦ = Inn(g).Cmax(∆±
+0,p)◦ = Inn(g).(z(g) ⊕ Cmax(∆±
+p )◦)
+which implies Wmax(∆±
+0,p)◦ = {(s, v, A) ∈ hsp(V, Ω) | ΩA > 0} which gets mapped to the set of
+strictly convex polynomials of degree ≤ 2 by Φ. This is the interior of the Inn(g)-invariant cone
+W := {f ∈ im(Φ) | inf f > ∞} =
+�
+Φ(s, v, A) | ΩA ≥ 0, v ∈ rad(ΩA)⊥Ω�
+of polynomials of degree ≤ 2 which are bounded from below. Note that this set is not closed as
+its boundary coincides with all polynomials of degree one.
+2
+KMS states of topological groups
+In this section, we discuss how results for KMS states of C*-algebras and W*-algebras can be
+transferred to topological groups.
+For general topological groups G, there exists a universal
+group W*-algebra W ∗(G). It has the property that, for a strongly continuous one-parameter
+group of automorphisms α : R → Aut(G), one obtains a σ-weakly continuous one-parameter
+group of *-automorphisms τ : R → Aut(W ∗(G)) such that a state ϕ of G is an (α, β)-KMS state
+if and only if its corresponding state ωϕ of W ∗(G) is a (τ, β)-KMS state (cf. Proposition 2.1.6).
+As is the case for W*-dynamical systems, one does not have good control over the set of KMS
+states in general. For locally compact groups this is not the case, as the availability of the group
+C*-algebra C∗(G) has the consequence that the set Kβ of (α, β)-KMS states of a locally compact
+topological group is a weak*-compact convex subset of L∞(G) ∼= L1(G)′ (cf. Theorem 2.2.4).
+Therefore the Krein-Millman Theorem implies that Kβ is the closed convex hull of its extremal
+points Ext(Kβ). As a KMS state is extremal if and only if it is a factor state, it suffices to study
+factorial KMS states. For general topological groups G, we characterize semi-finite KMS states
+12
+
+as generalized Gibbs states. This allows us to apply the results from Section 1 to factorial type I
+KMS states of finite-dimensional Lie groups G for one-parameter groups of inner automorphisms
+generated by X ∈ L(G), which correspond to irreducible unitary representations ρ such that
+tr(eiβ∂ρ(X)) < ∞.
+2.1
+Elementary properties of KMS states of topological groups
+The goal of this subsection is to relate the concepts (α, β)-KMS states of a topological group G
+and KMS states of W*-dynamical systems defined by α-covariant representations of the group G.
+In particular, we show that each (α, β)-KMS state of a topological group is α-invariant.
+Definition 2.1.1. (cf. [BR96, Def. 5.3.1], [BR96, Lemma 5.3.7]):
+Let ω be a state of a C*-algebra A (resp. normal state of a von Neumann algebra) and let τ : R →
+Aut(A) be a strongly (resp. σ-weakly) continuous one-parameter group of *-automorphisms. We
+call ω an (τ, β)-KMS state if the functions
+FA,B : R → C,
+t �→ ω(Aτt(B)),
+for
+A, B ∈ A,
+extend to continuous maps on the strip Sβ := {z ∈ C | 0 ≤ Im(z) ≤ β} that satisfy the reflection
+relation FA,B(t + iβ) = ω(τt(B)A), for t ∈ R, and are holomorphic on the interior.
+We mimic the equivalent definition of KMS states of C*-or W*-dynamical systems (A, τ, R) to
+define KMS states of topological groups (cf. [BR96, Lemma 5.3.7]).
+Definition 2.1.2. Let (G, α, R) be a group dynamical system and ϕ : G → C be a state. For
+β > 0, we say that ϕ satisfies the (α, β)-KMS condition if the functions
+Fx,y : R → C,
+t �→ ϕ(xαt(y)),
+for
+x, y ∈ G,
+extend to continuous maps on the strip Sβ := {z ∈ C | 0 ≤ Im(z) ≤ β} that are holomorphic on
+the interior and satisfy the reflection relation Fx,y(t + iβ) = ϕ(αt(y)x), for t ∈ R.
+We now define a W*-dynamical system (W ∗(G), τ, R) corresponding to a group dynamical system
+(G, α, R) and a state ωϕ of W ∗(G) corresponding to a state ϕ such that ϕ is an (α, β)-KMS state
+of G if and only if ωϕ is a (τ, β)-KMS state of W ∗(G).
+Definition 2.1.3. For a topological group G, we define the group W*-algebra as a pair of a von
+Neumann algebra W ∗(G) and a σ-weakly continuous group homomorphism π : G → U(W ∗(G))
+that has the universal property that for every strongly continuous unitary representation (ρ, H)
+of G there exists a unique *-homomorphism
+Φ : W ∗(G) → ρ(G)′′
+such that
+Φ(π(g)) = ρ(g),
+for all
+g ∈ G.
+For every topological group G, there exists such an algebra by [GN00, Prop. 4.2].
+Remark 2.1.4. Let G be a topological group.
+(i) By the universal property, for every α ∈ Aut(G) there exists a *-automorphism
+Φ : W ∗(G) → W ∗(G)
+such that
+Φ(π(g)) = π(α(g)),
+for all
+g ∈ G.
+For a strongly continuous one-parameter group of automorphisms α : R → Aut(G), the
+corresponding one-parameter group of *-automorphisms τ : R → Aut(W ∗(G)) is σ-weakly
+continuous since spanCπ(G) ⊆ W ∗(G) is σ-weakly dense and on U(H) the σ-weak and the
+strong operator topology coincide.
+13
+
+(ii) If ϕ is a state of G with GNS-representation (πϕ, Hϕ, Ωϕ), then the corresponding repre-
+sentation πϕ : W ∗(G) → πϕ(G)′′ defines the state
+ωϕ : W ∗(G) → C,
+A �→ ⟨Ωϕ, πϕ(A)Ωϕ⟩
+which satisfies ωϕ(π(g)) = ϕ(g), for all g ∈ G.
+If one adapts the proof of [BR96, Prop. 5.3.7] 2, one obtains the following lemma:
+Lemma 2.1.5. Suppose (A, τ, R) is a C*-dynamical system (resp. W*-dynamical system) and
+ω : A → C is a (resp. normal) state. If K ⊆ A is a norm-dense (resp. σ-weakly dense) subspace
+(resp. *-subalgebra) such that, for A, B ∈ K, the functions
+FA,B : R → C,
+t �→ ω(Aτt(B))
+extend to a continuous functions on the strip
+Sβ := {z ∈ C | 0 ≤ Im(z) ≤ β} that are
+holomorphic on the interior and satisfy FA,B(t + iβ) = ω(τt(A)B), for t ∈ R,, then ω is a
+(τ, β)-KMS state.
+Proposition 2.1.6. Let G be a topological group G, α : R → Aut(G) be a strongly continuous
+one-parameter group of automorphisms and ϕ : G → C a state. Then ϕ is an (α, β)-KMS state
+if and only if the corresponding state ωϕ : W ∗(G) → C is a (τ, β)-KMS state.
+Proof. Observe that ωϕ(π(x)τt(π(y))) = ωϕ(π(xαt(y))) = ϕ(xαt(y)) holds, for all x, y ∈ G.
+Then the assertion follows from Lemma 2.1.5 since spanCπ(G) is a σ-weakly dense *-subalgebra
+of W ∗(G) on which the KMS-condition is satisfied.
+Corollary 2.1.7. Let G be a topological group G, α : R → Aut(G) be a strongly continuous one-
+parameter group of automorphisms and ϕ : G → C an (α, β)-KMS state. Then ϕ is α-invariant.
+Proof. [BR96, Prop. 5.3.3] implies that ωϕ is τ-invariant. This proves the assertion.
+Let G be a topological group, α : R → Aut(G) be a strongly continuous one-parameter group of
+automorphisms and ϕ be an α-invariant state with GNS-representation (π, H, Ω). If (πϕ, Hϕ, Ωϕ)
+denotes the GNS-representation of ϕ, then there exists a strongly continuous unitary one-
+parameter group (Ut)t∈R on Hϕ such that
+Utπϕ(g)U−t = πϕ(αt(g))
+and
+UtΩϕ = Ωϕ,
+for
+t ∈ R, g ∈ G.
+Consider the normal state ωϕ : πϕ(G)′′ → C, A �→ ⟨Ωϕ, AΩϕ⟩ and the one-parameter group of
+*-automorphisms τ defined on πϕ(G)′′ by τt(A) := UtAU−t, for t ∈ R and A ∈ πϕ(G)′′, which is
+σ-weakly continuous.
+Proposition 2.1.8. A state ϕ of a group dynamical system (G, R, α) is an (α, β)-KMS state if
+and only if ωϕ is a (τ, β)-KMS state of the W*-dynamical system (πϕ(G)′′, τ, R).
+Proof. This follows immediately from the σ-weak density of spanCπϕ(G) ⊆ π(G)′′, the relation
+ω(πϕ(g)τt(πϕ(h))) = ϕ(gαt(h)),
+for
+g, h ∈ G, t ∈ R
+and Proposition 2.1.5.
+2In [BR96, Def. 5.3.1] KMS states of a C*-or W*-dynamical system are defined by a property for a suitably
+dense *-subalgebra B of τ-analytic elements to ensure that for A, B ∈ B the maps FA,B(t) := ω(Aτt(B)) extend
+to C and satisfy ω(Aτiβ(B)) = ω(BA). In the proof of [BR96, Prop. 5.3.7], it is only used that the *-subalgebra
+B consists of τ-analytic elements to ensure the extension property of the functions FA,B.
+14
+
+2.2
+C*-methods for locally compact groups
+Let G be a locally compact group.
+The group C∗-algebra of G is defined as the universal
+enveloping C∗-algebra C∗(G) := C∗(L1(G)) of the convolution algebra L1(G), i.e. the completion
+of L1(G) with respect to the maximal C∗-norm given on L1(G) by
+∥f∥C∗ := sup{∥π(f)∥ | π : L1(G) → B(H) non-degenerate *-representation}.
+Every non-degenerate *-representation π : L1(G) → B(H) extends uniquely to a continuous
+*-representation of C∗(G) and ∥π(f)∥ ≤ ∥f∥C∗ holds trivially for all f ∈ L1(G). We will usually
+also denote this extension by π. Note that non-degenerate *-representation are in one-to-one
+correspondence with unitary representations of G (cf. [DE09, Prop. 6.2.3]). This one-to-one
+correspondence yields the following one-to-one correspondence of states as they are uniquely
+determined by their cyclic GNS-representation and a fixed cyclic vector.
+Lemma 2.2.1. There is a one-to-one correspondence of states ϕ of G and states ω of C∗(G)
+established by Φ : S(G) → S(C∗(G)), ϕ �→ ωϕ, where ωϕ is uniquely determined by
+ωϕ(f) =
+�
+G
+f(x)ϕ(x)dx,
+for
+f ∈ L1(G).
+If one equips S(G) with the weak*-subspace topology from S(G) ⊆ L∞(G) ∼= L1(G)′ and S(C∗(G))
+with the weak*-topology of C∗(G)′, then Φ is affine linear and a weak*-continuous homeomor-
+phism.
+In order to get a one-to-one correspondence between KMS states of C∗(G) and KMS states of
+G, one needs to define a strongly continuous one-parameter group of automorphisms
+τ : R → Aut(C∗(G))
+corresponding to
+α : R → Aut(G)
+such that ωϕ is a (τ, β)-KMS state of C∗(G) if and only if ϕ is an (α, β)-KMS state of G. To do
+this, we note that the pullback (αt)∗µG of the Haar measure µG of G is a left-invariant Radon
+measure and thus there exists a unique scalar λt ∈ R+ satisfying
+µG ◦ α−t =: (αt)∗µG = λt · µG
+for all
+t ∈ R.
+It is an easy exercise to show that τ : R × L1(G) → L1(G), τt(f) := λt · (f ◦ α−t) extends to
+a strongly continuous one-parameter group of *-automorphisms of C∗(G) which we also denote
+by τ. For t ∈ R and f ∈ L1(G), one has
+π(τt(f)) =
+�
+G
+λt · f(α−t(x))π(x)dx =
+�
+G
+f(x)π(αt(x))dx = (π ◦ αt)(f).
+Lemma 2.2.2. Let G be a locally compact topological group and α : R → Aut(G) be a strongly
+continuous one-parameter group of automorphisms. For a state ϕ : G → C, one has
+ωϕ(f ∗ τt(g)) =
+�
+G
+�
+G
+f(y) g(x) ϕ(yαt(x))dxdy,
+for
+t ∈ R, f, g ∈ L1(G).
+Proof. Let (H, π, Ω) be the GNS-representation of ϕ and ωϕ.
+Using ϕ(x) = ⟨Ω, π(x)Ω⟩, we
+calculate ωϕ(f ∗ τt(g)) which is given by
+�
+G
+(f ∗ τt(g))(x)⟨Ω, π(x)Ω⟩dx =
+�
+G
+�
+G
+f(y) λt g(α−t(y−1x)) ϕ(x)dxdy
+=
+�
+G
+�
+G
+f(y) λt g(α−t(x)) ϕ(yx)dxdy =
+�
+G
+�
+G
+f(y) g(x) ϕ(yαt(x))dxdy.
+15
+
+Proposition 2.2.3. A state ϕ ∈ S(G) is an (α, β)-KMS state if and only if ωϕ ∈ S(C∗(G)) is
+a (τ, β)-KMS state in the sense of C∗-dynamical systems. In particular the map S(G) ∋ ϕ →
+ωϕ ∈ S(C∗(G)) restricts to an affine linear homeomorphism of the set of respective KMS states.
+Proof. Suppose ϕ is an (α, β)-KMS state of G. We first show that ωϕ is a (τ, β)-KMS state.
+[Prop. 5.3.12, BR96] implies that it suffices to show
+�
+R
+�ψ(t) ωϕ(f ∗ τt(g))dt =
+�
+R
+�ψ(t + iβ) ωϕ(τt(g) ∗ f)dt,
+for all ψ ∈ C∞
+c (R) and f, g ∈ L1(G).
+Fix ψ ∈ C∞
+c (R). The Paley-Wiener Theorem [BR96, Prop. 5.3.11] implies that, for R > 0 and
+supp(ψ) ⊆ [−R, R], the function �ψ is entire analytic and for every n ∈ N0 there exist constants
+Cn > 0 such that
+| �ψ(z)| ≤ Cn(1 + |z|)−n exp(R · |Im(z)|),
+for
+z ∈ C.
+This proves that the map t �→ �ψ(t)ϕ(yαt(x)) extends to continuous function F on the strip Sβ
+which is holomorphic on the interior. Choose, for r > 0, a piecewise smooth curve γr which
+parametrizes the rectangle of width depending on r and fixed height β depicted below:
+−r
+r
+iβ
+Applying [BR96, Prop. 5.3.5] to the continuous function F on Sβ which is holomorphic on the
+interior yields
+F(t + isβ) ≤ | �ψ(t + isβ)| · sup{|ϕ(xαt(y))|, |ϕ(αt(y)x)|} ≤ Cn(1 + β2 + t2)−n exp(βR).
+This proves that the restriction of F(z) to the right and left side of the rectangle parametrized
+above converges uniformly to zero for r → ∞ since it is given by |F(±r+isβ)|, for s ∈ [0, 1]. The
+length of the left and right side of the rectangle is finite and thus the integrals over the right and
+left sides tend to zero for r → ∞. The integrals of F(z) over the contractible curves γr vanish
+as F is holomorphic which implies
+0 = lim
+r→∞
+�
+γr
+F(z)dz =
+� ∞
+−∞
+�ψ(t)ϕ(yαt(x))
+�
+��
+�
+F (t)
+dt −
+� ∞
+−∞
+�ψ(t + iβ)ϕ(αt(x)y)
+�
+��
+�
+F (t+iβ)
+dt.
+To see that ωϕ is a (τ, β)-KMS state, we apply Lemma 2.2.2 and obtain
+�
+R
+�ψ(t) ωϕ(f ∗ τt(g))dt =
+�
+G
+�
+G
+f(y) g(x)
+�
+R
+�ψ(t) ϕ(yαt(x))dtdxdy
+=
+�
+G
+�
+G
+f(y) g(x)
+�
+R
+�ψ(t + iβ)ϕ(αt(x)y)dtdxdy =
+�
+R
+�ψ(t + iβ) ωϕ(τt(g) ∗ f)dt.
+16
+
+Suppose conversely that ωϕ is a (τ, β)-KMS state. Then [BR96, Prop. 5.3.4] implies that there
+exists a unique σ-weakly continuous one-parameter group of automorphisms �τ of πωϕ(C∗(G))′′
+such that �τt(πωϕ(A)) = πωϕ(τt(A))
+for all
+A ∈ C∗(G) and the state
+�ωϕ : πωϕ(C∗(G))′′ → C, A �→ ⟨Ωϕ, AΩϕ⟩
+is a (�τ, β)-KMS state in the sense of a W*-dynamical system. In particular
+�
+G
+f(x)�τt(πϕ(x))dx = (�τt ◦ πωϕ)(f) = πωϕ(τt(f)) =
+�
+G
+f(x)πϕ(αt(x))dx,
+implies that �τt(πϕ(x)) = πϕ(αt(x)), for all x ∈ G, as one can approximate with left shifts of a
+Dirac net. Therefore
+ϕ(xαt(y)) = ⟨Ωϕ, πϕ(x)πϕ(αt(y))Ωϕ⟩ = ⟨Ωϕ, πϕ(x)�τt(πϕ(y))Ωϕ⟩ = �ωϕ(πϕ(x)�τt(πϕ(y)))
+follows which proves that ϕ is an (α, β)-KMS state.
+Now we have translated all the necessary information from KMS states of groups to KMS states
+of a C*-algebra to prove the following theorem.
+Theorem 2.2.4. Let G be a locally compact topological group and α : R → Aut(G) be a strongly
+continuous one-parameter group of automorphisms. Then the set Kβ ⊆ S(G) of (α, β)-KMS
+states satisfies
+(i) Kβ is convex and weak-∗-compact with respect to the subspace topology of
+S(G) ⊆ L∞(G) ∼= L1(G)′.
+(ii) Kβ is a simplex (cf. Remark 2.2.5).
+(iii) ϕ ∈ Kβ is an extremal point of Kβ if and only if ϕ is a factor state.
+(iv) Kβ = Conv(Ext(Kβ)), where Ext(Kβ) are the extremal points in Kβ and the closure is
+with respect to the weak-∗-topology.
+Proof. Lemma 2.2.1 and Proposition 2.2.3 imply that we may view Kβ as the set of (τ, β)-KMS
+states of A := C∗(G). Then [BR96, p. 116] shows that the (τ, β)-KMS states are in one-to-one
+correspondence to the (τ+, β)-KMS states of the unitization A+ := A ⊕ C, where τ+ acts on A+
+via
+τ+ : R × A+ → A+,
+(t, (A, λ)) �→ (τt(A), λ).
+Then Theorem [BR87, Thm. 5.3.30] implies (i)-(iii) for the subset of (τ+, β)-KMS states. Since
+the correspondence was established via the affine linear homeomorphism with respect to the
+weak-∗-topology given by
+Kβ ∋ ω �→ ω+ : (A, λ) �→ ω(A) + λ,
+it follows that (i)-(iii) also hold for Kβ. Then (iv) is a direct consequence of (i) and the Krein-
+Milman Theorem.
+17
+
+Remark 2.2.5. For the compact convex subset Kβ of the locally convex topological vector space
+C∗(G)′ equipped with the weak-∗-topology [BR87, Subsection 4.1] provides a decomposition
+theory. For a given Radon probability measure µ on Kβ, [BR87, Prop. 4.1.1] proves that there
+exists a unique point b(µ) =
+�
+K ω dµ(ω) ∈ Kβ called the barycenter of µ, such that
+f(b(µ)) = µ(f) :=
+�
+K
+f(ω)dµ(ω),
+for all affine linear continuous functions f : Kβ → R. Then [BR87, Thm. 4.1.15] establishes the
+importance of Kβ being a simplex since it implies that every ω ∈ K is the barycenter of a unique
+normalized Radon measure µω maximal with respect to the partial order ≻ defined by
+µ ≻ ν
+:⇔
+µ(f) ≥ ν(f),
+for all f : K → R continuous and convex.
+For separable locally compact topological groups G the convolution algebra L1(G) is separa-
+ble which shows that C∗(G) is a separable C*-algebra.
+In particular, Kβ is metrizable and
+thus [BR87, Thm.
+4.1.11] implies that µω is supported on the extremal points of Kβ, i.e.
+µω(Ext(Kβ)) = 1. We thus obtain for each ω ∈ Kβ a unique decomposition
+ω =
+�
+Ext(Kβ)
+ω′dµω(ω′).
+For f ∈ L1(G), the function Ext(Kβ)×G ∋ (ω, x) �→ f(x)ϕω(x) is clearly integrable, the measure
+space Ext(Kβ) × G is σ-finite and thus from Fubini’s theorem, one obtains, for all ω0 ∈ Kβ, that
+ω0(f) =
+�
+Ext(Kβ)
+�
+G
+f(x)ϕω(x)dxdµω0(ω) =
+�
+G
+f(x)
+�
+Ext(Kβ)
+ϕω(x)dµω0(ω)dx.
+This implies that the state ϕω0 of G corresponding to ω0 is given by
+ϕω0(x) =
+�
+Ext(Kβ)
+ϕω(x)dµω0(ω),
+for
+x ∈ G.
+Since L1(G) ⊆ C∗(G) is dense, it follows the evaluation functionals
+δf : Kβ → C,
+ω �→ ω(f),
+for
+f ∈ L1(G),
+separate the points and thus the algebra generated by these functionals is dense in C(Kβ) with
+respect to the maximum norm by the Theorem of Stone-Weierstrass. This proves that µω0 is
+already uniquely determined by these functionals. As the set Kβ is convex and weak-∗-closed
+every such measure defines an (α, β)-KMS state. Hence we have proven the following proposition:
+Proposition 2.2.6. Let G be a separable locally compact topological group, α : R → Aut(G) be
+a strongly continuous one-parameter group and β ̸= 0. For each ϕ0 ∈ Kβ, there exists a Radon
+probability measure µ on Ext(Kβ) uniquely determined by
+ϕ0(x) =
+�
+Ext(Kβ)
+ϕ(x)dµ(ϕ),
+for all
+x ∈ G.
+Conversely every such measure defines an (α, β)-KMS state of G.
+18
+
+2.3
+Semi-finite KMS states of topological groups
+Consider an (α, β)-KMS state ϕ of a topological group G.
+If Mϕ = πϕ(G)′′ is a semifinite
+von Neumann algebra, then there exists a unique faithful semi-finite normal trace τ on Mϕ
+(cf. [Tak02, Thm. 2.15]). Since the faithful normal state ωϕ is invariant under the modular
+automorphism group στ
+t = idMϕ of the trace τ, it follows from the Radon-Nikodym Theorem
+[Tak03, Cor. 3.6] that there exists a unique non-singular positive self-adjoint operator h affiliated
+to Mϕ such that ωϕ(A) = τ(hA), for A ∈ Mϕ. Then [Tak03, Thm. 2.11] implies that the
+modular automorphism group of ωϕ is given by
+Ad(h−it/β) = Ad(eitH),
+for
+H = − 1
+β log(h).
+Furthermore one has τ(h) = τ(e−βH) = ωϕ(1) = 1 and ϕ is given by
+ϕ(g) = τ(πϕ(g)e−βH)
+τ(e−βH)
+,
+for
+g ∈ G.
+Suppose conversely that π is an α-covariant cyclic unitary representation such that π(G)′′ is semi-
+finite and for which there exists a non-singular implementing operator H of α affiliated to π(G)′′
+which satisfies τ(e−βH) = 1, for the unique faithful semi-finite normal trace τ on Mϕ = π(G)′′.
+We want to show that this defines an (α, β)-KMS state of G by constructing a (σ, β)-KMS state,
+where σt(A) := eitHAe−itH for A ∈ Mϕ. To do this, observe that [Tak02, Thm. 2.22] implies
+that the semi-finite von Neumann algebra Mϕ ⊗ Mop
+ϕ acts on L2(Mϕ, τ), which is defined as
+the completion of the complex pre-Hilbert space
+{A ∈ Mϕ | τ(|A|2) < ∞}
+with respect to the scalar product ⟨A, B⟩ := τ(A∗B), by left and right multiplication. If one
+denotes
+λ : Mϕ → B(L2(Mϕ, τ)),
+λ(a)x = ax,
+for
+τ(|x|2) < ∞
+and
+ρ : Mop
+ϕ → B(L2(Mϕ, τ)),
+ρ(a)x = xa,
+for
+τ(|x|2) < ∞,
+one has that λ(Mϕ)′ = ρ(Mϕ)′′ and ρ(Mϕ)′ = λ(Mϕ)′′ (cf. [Tak02, Thm. 2.22]).
+In particular if Ω = e− 1
+2 βH, then Ω ∈ L2(Mϕ, τ) is non-singular and self-adjoint. This implies
+that the vector Ω is cyclic and separating for Mϕ because by the Range-Kernel Lemma one has
+R(Ω) = ker(Ω∗) = ker(Ω) =
+{0} and therefore AΩ = 0 implies A = 0 and ΩB = 0 implies
+(ΩB)∗ = B∗Ω = 0, i.e. B∗ = 0 and thus B = 0, for A, B ∈ Mϕ.
+The corresponding modular objects (∆, J) are given on λ(M)Ω by J(AΩ) = ΩA∗ and ∆(AΩ) =
+ΩA, for A ∈ Ω and thus the corresponding modular automorphism group is given by
+σt : M → M,
+A �→ eitHAe−itH,
+for
+t ∈ R.
+This implies that the state
+ω : M → C,
+A �→ τ(Ae−βH)
+τ(e−βH) = ⟨Ω, AΩ⟩
+∥Ω∥2
+is a (σ, β)-KMS state. This proves that ϕ : G → C, g �→ ω(π(g)) is an (α, β)-KMS state. Hence
+one has the following proposition:
+19
+
+Proposition 2.3.1. Let G be a topological group,
+α : R → Aut(G) be a strongly continuous
+one-parameter group of automorphisms. There is a one-to-one correspondence between (α, β)-
+KMS states ϕ such that πϕ(G)′′ is a semi-finite von Neumann algebras and α-covariant cyclic
+unitary representation π such that πϕ(G)′′ is a semi-finite for which there exists an implementing
+operator H of α affiliated to π(G)′′ which satisfies τ(e−βH) = 1, for the unique faithful semi-finite
+normal trace τ on π(G)′′.
+Remark 2.3.2. If ϕ is a factorial type I (α, β)-KMS state, then Mϕ = πϕ(G)′′ is a type I
+von Neumann algebra and therefore there exists a Hilbert space H and an isomorphism of von
+Neumann algebras Φ : Mϕ → B(H). The unitary representation ρ := Φ ◦ πϕ is irreducible since
+the σ-weak closure of spanC(ρ(G)) is given by ρ(G)′′ = Φ(Mϕ) = B(H). This proves that there
+exists an operator H on H such that
+eitHρ(g)e−itH = ρ(αt(g))
+and
+tr(e−βH) < ∞
+since the GNS-representation of ϕ is α-covariant for the modular automorphism group, i.e.
+∆itπϕ(g)∆−it = πϕ(αt(g)) and Φ ◦ πϕ = ρ. In particular
+ϕ : G → C,
+tr(ρ(g)e−βH)
+tr(e−βH)
+,
+for
+g ∈ G,
+is a Gibbs state corresponding to the irreducible unitary representation ρ of G. Since the rep-
+resentation ρ is unique, the implementing operator H is unique up to the addition of a real
+scalar.
+Remark 2.3.3. By [Tak03, Thm. 3.14] a von Neumann algebra M is semi-finite if and only if
+for every faithful semi-finite normal weight ψ, the corresponding modular automorphism group
+σψ
+t is inner in the sense that there exists a self-adjoint operator H affiliated to M such that
+σψ
+t = Ad(eitH) as automorphisms of M. For KMS states of topological groups, this innerness of
+the modular automorphism group translates to the extension of the factorial KMS state to the
+larger group G♭ := G ⋊α R. Let ϕ be an (α, β)-KMS state such that πϕ(G)′′ is a semi-finite von
+Neumann algebra and H the corresponding implementing operator affiliated to Mϕ := πϕ(G)′′
+satisfying τ(e−βH) < ∞. It follows that the unitary representation
+π♭
+ϕ : G♭ = G ⋊α R → U(H),
+(g, t) �→ πϕ(g)eitH
+is a semi-finite representation since eitH ∈ Mϕ implies π♭
+ϕ(G ⋊α R)′′ = πϕ(G)′′ = Mϕ.
+In
+particular
+ϕ♭ : G♭ = G ⋊α R → C,
+(g, t) �→ τ(π♭
+ϕ(g, t)e−βH)
+τ(e−βH)
+is an (α♭, β)-KMS state of G♭ = G ⋊α R, where α♭ is the one-parameter group of inner automor-
+phisms of G♭ defined by conjugation with (e, t), i.e.
+α♭(g, s) = (e, t)(g, s)(e, t)−1 = (αt(g), s),
+for
+g ∈ G, s, t ∈ R.
+Note that the modular automorphism group which acts on Hϕ ∼= L2(Mϕ, τ) by conjugation with
+eitH and thus leaves the cyclic vector Ω =
+e
+− 1
+2 βH
+√
+tr(e−βH) invariant, but the left multiplication by
+eitH does not.
+20
+
+Theorem 2.3.4. Let β > 0 and suppose G is a finite dimensional Lie group and α : R → Aut(G)
+is given by αt(g) = exp(tX)g exp(−tX), for some fixed X ∈ g := L(G) and g ∈ G. Then the
+extremal type I (α, β)-KMS states are the extremal Gibbs states
+ϕ(g) = tr(ρ(g)eiβ∂ρ(X))
+tr(eiβ∂ρ(X))
+,
+where ρ is an irreducible representation of G such that eiβ∂ρ(X) is of trace class. In particular,
+if G is simply connected, then there is a one-to-one correspondence between
+(i) extremal type I (α, β)-KMS states
+(ii) equivalence classes of irreducible unitary highest weight representations of admissible quo-
+tients q : g ↠ g1 with respect to an admissible positive system ∆+
+1 of g1 such that q(X) is
+conjugate to Cmax(∆+
+1,p)◦.
+Proof. This follows immediately from Remark 2.3.2 since ρ(exp(tX)) = et∂ρ(X) = eit(−i∂ρ(X))
+implies that the corresponding implementing operator H, which is unique up to addition by a
+real scalar, is w.l.o.g. given by −i∂ρ(X). Therefore eiβ∂ρ(X) is of trace class and Corollary 1.2.9
+applies if G is simply connected.
+Remark 2.3.5. If G is a finite dimensional Lie group and the action α : R → Aut(G) is not
+inner, then one considers G♭ = G ⋊α R with Lie algebra g♭ := L(G♭). In order to classify all
+representations of Gibbs type, it suffices to classify all admissible quotients g′ of g♭ and check
+if the image of (0, 1) ∈ g♭ = g ⋊ R under the quotient map q : g♭ → g′ lies in the interior of
+Cmax(∆+
+p )◦, for an admissible positive system ∆+ of g′. If one then classifies the unitary highest
+weight representations of g′ with respect to ∆+, one obtains in the same manner as above a
+representation of Gibbs type of G, and all representations of Gibbs type are obtained in this
+manner.
+3
+Perspectives and open problems
+3.1
+Extensions of KMS states of G to KMS states of G♭
+To completely understand how the extension of a topological group G to G♭ affects the study
+of KMS states, it would be interesting to know if every (α, β)-KMS state of G extends to an
+(α♭, β)-KMS state of G♭ = G ⋊α R and if every KMS state of G♭ arises in this manner. We have
+seen in Remark 2.3.3 that semi-finite (α, β)-KMS states of G extend to (α♭, β)-KMS states of G♭.
+For a not necessarily semi-finite (α, β)-KMS state ϕ of G with GNS-representation (πϕ, Hϕ, Ωϕ)
+and M := πϕ(G)′′, it is not clear if such KMS extensions always exist. It might even be possible
+that this extension property already implies that the state is semi-finite.
+We considered the crossed product M⋊σR, where σ : R → Aut(M) is the modular automorphism
+group of the associated state ωϕ of M ⊆ B(Hϕ) (cf. [Tak03, Section X.1]). This crossed product
+is defined as the von Neumann algebra generated by the sets of bounded linear operators on
+L2(R, Hϕ) given by
+(πσ(A)f)(t) := σ−t(A)f(t)
+and
+(λ(s)f)(t) := f(t − s),
+for A ∈ M, s, t ∈ R, f ∈ L2(R, Hϕ).
+Then one can consider the associated dual weight
+�ωϕ of M ⋊σ R and [Tak03, Thm. X.1.17(ii)] implies that the modular automorphism group
+21
+
+�σ : R → Aut(M ⋊σ R) of the dual weight �ωϕ satisfies
+�σt(πσ(A)) = πσ(σt(A)),
+and
+�σt(λ(s)) = λ(s),
+for
+A ∈ M, s, t ∈ R.
+(2)
+For the unitary representation ρ of G♭ on L2(R, Hϕ) defined as ρ(g, t) := πσ(πϕ(g))λ(t), the
+relation πϕ(αt(g))) = σt(πϕ(g)) and (2) imply
+ρ(α♭
+t(g, s)) = ρ(αt(g), s) = �σt (ρ(g, s)) ,
+for
+g ∈ G, s, t ∈ R.
+But dual weight �ωϕ of a state does not define a state of the crossed product and thus one cannot
+naively define the extension of ϕ as ϕ♭(g, t) := �ωϕ(ρ(g, t)) for g ∈ G, t ∈ R. This does not work
+as the dual weight does not define a state. To see that, observe that [Tak03, Thm. X.1.17(i)]
+implies that, for a compactly supported M-valued continuous function f ∈ Cc(R, M) and for
+the representation �πσ of Cc(R, M) on L2(R, Hϕ) defined as
+�πσ(f) :=
+�
+R
+λ(s)πσ(f(s))ds,
+one has
+�ωϕ(�πσ(f)) = ωϕ(f(e)) = ⟨Ωϕ, f(e)Ωϕ⟩.
+As �πσ(Cc(R, M)) generates M ⋊σ R (cf. [Tak03, Lemma X.1.10(iii)]), this implies that the dual
+weight does not define a state of the cross product M⋊σ R since delta distributions do not define
+states as the passage from Cc(R, M) to L∞(R, M) suggests. Therefore this attempt does not
+seem to work but it illustrates a few key structures.
+3.2
+Factorial Type II KMS states of Lie groups
+We are not aware of how type II factor representations π of finite dimensional Lie groups G
+look like. For us, the more natural context of type II factor representation is given by inductive
+limits of Lie groups which then act for example an the hyperfinite II1 factor defined as a infinite
+product of two by two matrix algebras (cf.
+[Po67]).
+For KMS states of the inductive limit
+group U(∞) = lim
+−→n U(n) this has been done in [St78] whereas general factor representations of
+U(∞) are studied in [SV75]. A survey of these results is given in [SV82]. Nonetheless it would
+still be interesting to know what conditions does τ(eiβ∂π(X)) < ∞ implies for finite dimensional
+Lie groups G, representation π and generator X ∈ g? Clearly, the spectrum of H = −i∂π(X)
+does not need to be semi-bounded anymore as in type II factors there exists a sequence of
+non-zero mutually orthogonal projections {pn}n∈N such that 0 < τ(pn) → 0. Instead of the
+semi-boundedness of spec(H) on has that the spectrum decreases exponentially fast in the sense
+that
+�
+R e−βxdτ(PH(x)) < ∞ holds, i.e. τ(PH((−∞, t0])) ≤ eβt0τ(e−βH) goes to 0 exponentially
+as t → −∞.
+A
+Appendix
+A.1
+Compactly embedded Cartan subalgebras and admissible Lie al-
+gebras
+Definition A.1.1. We call a subalgebra a ⊆ g compactly embedded if ⟨eadg(a)⟩ ⊆ GL(g) is
+compact. We call an element x ∈ g elliptic if the subalgebra R · x ⊆ g is compactly embedded
+and denote with comp(g) the subset of elliptic elements in g.
+Remark A.1.2. (cf. [Ne00, Thm. VII.2.2]):
+If t is a compactly embedded Cartan subalgebra, then it is nilpotent and compact, i.e. abelian.
+22
+
+Therefore the subset adgC(tC) ⊆ gl(gC) is an abelian subalgebra consisting of diagonalizable
+elements since the compact group INNg(t) is a compact subgroup of the compact subgroup
+O(g, ⟨·, ·⟩) of gl(g) for some scalar product on g. Hence one obtains a root decomposition
+gC = tC ⊕
+�
+α∈∆
+gα
+C,
+where gα
+C := {X ∈ gC | [t, X] = α(t)X, for all t ∈ tC} and ∆ := {α ∈ t∗
+C | gα
+C ̸= {0}}. On the
+complex Lie algebra gC we define the involutive antihomomorphism
+(X + iY )∗ := −X + iY
+for
+X, Y ∈ g
+and observe that g = {X ∈ gC | X∗ = −X}. It follows from [Ne00, Thm. VII.2.2] that for
+Z ∈ gα
+C, one has Z∗ ∈ g−α
+C
+and [Z, Z∗] ∈ it. Define
+gC(Z) := spanC{Z, Z∗, [Z, Z∗]}
+and
+g(Z) := gC(Z) ∩ g
+For roots α ∈ ∆ , one now distinguishes between the following cases:
+(NS) There exists an element Z ∈ gα
+C such that α([Z, Z∗]) < 0 (the non-compact simple type).
+Then g(Z) ∼= sl(2, R).
+(CS) There exists an element Z ∈ gα
+C such that α([Z, Z∗]) > 0 (the compact simple type). Then
+g(Z) ∼= su(2).
+(N) α([Z, Z∗]) = 0 for all Z ∈ gα
+C and there exists Z ∈ gα
+C such that [Z, Z∗] ̸= 0 (the nilpotent
+type). Then g(Z) is isomorphic to the three-dimensional Heisenberg algebra h3(R).
+(A) [Z, Z∗] = 0 for all Z ∈ gα
+C (the abelian type). Then g(Z) ∼= R2.
+That only these four cases occur follows from [Ne00, Lemma VII.2.3] which implies dimCgα
+C = 1,
+for all α ∈ ∆ of simple type, and thus conditions (NS) and (CS) hold for all Z ∈ gα
+C if they hold
+for one.
+Definition A.1.3. (cf. [Ne00, Def. VII.2.4] and [Ne00, Def. VII.2.6]):
+Let t ⊆ g be a compactly embedded Cartan subalgebra.
+(i) We call a root α ∈ ∆ semisimple if it is of simple type and denote the set of semisimple
+roots by ∆s. The set of solvable roots is denoted by ∆r := ∆ \ ∆k.
+(ii) We call a root α ∈ ∆ compact if there exists Z ∈ gα
+C such that α([Z, Z∗]) < 0. All other
+roots are called non-compact and the set of all non-compact roots is denoted with ∆p
+and the set of compact roots as ∆k. We also define the non-compact semisimple roots
+∆p,s := ∆p ∩ ∆s.
+(iii) A subset ∆+ ⊆ ∆ is called a positive system if there exists a regular element X0 ∈ it such
+that
+∆+ = {α ∈ ∆ | α(X0) > 0}.
+If ∆+ is such a system of positive roots we define ∆+
+k , ∆+
+p , ∆+
+p,s and ∆+
+r as the intersection
+of the respective subset with ∆+.
+23
+
+(iv) We call a positive system ∆+ adapted if for all α ∈ ∆k and β ∈ ∆+
+p one has
+β(X0) > α(X0),
+for some X0 defining ∆+.
+Definition A.1.4. (cf. [Ne00, Def. VIII.2.6]):
+Let t ⊆ g be a compactly embedded Cartan subalgebra and ∆+ be a corresponding system of
+positive roots. We associate to the positive system of non-compact roots ∆+
+p the convex cones
+Cmin(∆+
+p ) := cone({i[Zα, Z∗
+α] | Zα ∈ gα
+C, α ∈ ∆+
+p } ⊆ t
+and
+Cmax(∆+
+p ) := (i∆+
+p )⋆ := {X ∈ t | iα(X) ≥ 0 for all α ∈ ∆+
+p }.
+If it is clear which system of positive non-compact roots ∆+
+p is considered, then we simply write
+Cmin, resp. Cmax, for Cmin(∆+
+p ), resp. Cmax(∆+
+p ).
+Definition A.1.5. We call a Lie algebra g admissible if g contains an invariant closed generating
+convex subset C which does not contain any affine line.
+Theorem A.1.6. (Sandwich Theorem [Ne00, Thm. VII.3.8])
+Let t ⊆ g be a compactly embedded Cartan subalgebra and ∅ ̸= C ⊆ g a closed generating invariant
+elliptic convex subset. Then there exists a uniquely determined adapted positive system ∆+
+p of
+non-compact roots such thatC ∩ t ⊆ Cmax(∆+
+p ). Moreover, we also have Cmin(∆+
+p ) ⊆ lim(C ∩ t).
+If, in addition, C is a cone, then this means that
+Cmin(∆+
+p ) ⊆ C ∩ t ⊆ Cmax(∆+
+p ).
+A.2
+Convex geometric and representation-theoretic concepts
+Definition A.2.1. (cf. [Ne00, Def. V.1.1] and [Ne00, Prop. V.1.6]):
+Let V be a finite-dimensional real vector space and W ⊆ V be a closed convex cone.
+(i) For a subset E ⊆ V , we put
+B(E) := {ω ∈ V ∗ | inf ω(E) > −∞}
+and
+E⋆ := {ω ∈ V ∗ | ω(E) ⊆ [0, ∞)}.
+(ii) We define, for a convex subset C ⊆ V , the recession cone lim(C) := {v ∈ V | v + C ⊆ C}.
+(iii) For I ⊆ V ∗, we define I⊥ := {v ∈ V | ω(v) = 0 for all ω ∈ I}.
+(iv) For a subset E ⊆ V , we define algint(E) as the interior of E with respect to the affine
+subspace spanned by E.
+Remark A.2.2. We remark the following:
+(i) E⋆ is always closed but B(E) is not closed in general.
+(ii) One has B
+�
+conv(E)
+�
+= B(E).
+(iii) B(E)⋆ = lim(E) for E convex (cf. [Ne00, Prop. V.1.14])
+24
+
+Definition A.2.3. (cf. [Ne00, Def. X.1.2]):
+Let (π, H) be a unitary representation of a Lie group G with Lie algebra L(G) = g on the
+Hilbert space H and let H∞ be the dense subspace of H of smooth vectors on which g acts as
+dπ(X)(v) :=
+d
+dt
+��
+t=0 π(expG(tX))v for X ∈ g,
+v ∈ H∞. Then we define the momentum map of
+the unitary representation π as
+Φπ : P(H∞) := (H∞ \ {0})/C× → g∗,
+Φπ([v])(X) := −i⟨v, dπ(X)v⟩
+⟨v, v⟩
+and the momentum set Iπ as the closed convex hull of the image of Φπ.
+Remark A.2.4. If (π, H) is a unitary representation of a Lie group G we consider the cone
+B(Iπ) = {X ∈ g | inf⟨Iπ, X⟩ > −∞}.
+This cone is invariant since Iπ is invariant under the coadjoint action (cf. [Ne00, Lemma X.1.3])
+and [Ne00, Lemma X.1.6] implies that
+B(Iπ) = {X ∈ g | sup(spec(i · dπ(X))) < ∞}.
+Definition A.2.5. We call a pair (g, ∗) an involutive Lie algebra if g is a complex Lie algebra
+and ∗ : g → g is an antilinear antiautomorphism, i.e. [x∗, y∗] = −[x, y]∗, holds for all x, y ∈ g.
+Definition A.2.6. We say that an involutive Lie algebra (g, ∗) has a root decomposition if there
+exists a ∗-invariant subalgebra h ⊆ g and a subset ∆ ⊆ g∗ such that α(h∗) = α(h) and
+g = h ⊕
+�
+α∈∆
+gα,
+for gα := {X ∈ g | ad(h)(X) = α(h)X for all h ∈ h} ̸= {0}.
+Definition A.2.7. Let (g, ∗) be an involutive Lie algebra. We call a g-module V unitary if there
+exists a positive definite hermitian form h : V × V → C that is also contravariant, i.e. it satisfies
+h(X.v, w) = h(v, X∗.w) for all X ∈ g and v, w ∈ V .
+Definition A.2.8. Let (g, ∗) be an involutive Lie algebra with root decomposition and ∆+ ⊆ ∆
+be a positive system. We call a g-module V a highest weight module with respect to ∆+ if there
+exists a linear functional
+λ : b := h ⊕
+�
+α∈∆+
+gα → C
+and a cyclic vector v ∈ V such that X.v = λ(X)v holds for all X ∈ b. We call λ the highest
+weight of V and v a highest weight vector.
+Definition A.2.9. An irreducible unitary representation (π, H) of a connected Lie group G is
+called a unitary highest weight representation if the Lie algebra g of G contains a compactly
+embedded Cartan subalgebra t ⊆ g and if k ⊆ g is the unique maximal compactly embedded
+subalgebra containing t (cf.
+[Ne00, Prop.
+VII.25]) and K := exp(k), then the space HK,∞
+of smooth K-finite vectors is a highest weight module for the involutive Lie algebra gC with
+involution (X + iY )∗ = −X + iY , for X, Y ∈ g.
+25
+
+References
+[BR87]
+Bratteli,O., and D. W. Robinson, “Operator Algebras and Quantum Statistical Me-
+chanics I,” 2nd ed., Texts and Monographs in Physics, Springer-Verlag, 1987.
+[BR96]
+Bratteli,O., and D. W. Robinson, “Operator Algebras and Quantum Statistical Me-
+chanics II,” 2nd ed., Texts and Monographs in Physics, Springer-Verlag, 1997.
+[DE09]
+Anton Deitmar, Siegfried Echterhoff, “Principles of Harmonic Analysis,” 1st ed.,
+Springer-Verlag 2009.
+[HM06]
+Karl H. Hofmann and Sidney A. Morris, “The Structure of Compact Groups,” A
+primer for the student—a handbook for the expert. Second revised and augmented
+edition. De Gruyter Studies in Mathematics, 25. Walter de Gruyter & Co., Berlin,
+2006.
+[HN12]
+Karl-Hermann Neeb, Joachim Hilgert, “Structure and Geometry of Lie Groups,” 1st
+ed., Springer Monographs in Mathematics. Springer, New York, 2012.
+[GN]
+Helge Glöckner, Karl-Hermann Neeb, “Infinite-dimensional Lie Groups; General
+Theory and Main Examples,” in preparation
+[GN00]
+Helge Glöckner, Karl-Hermann Neeb, “Minimally Almost Periodic Abelian Groups
+and Commutative W*-algebras,” Research & Exposition in Mathematics, Helder-
+mann Verlag Berlin, 2000
+[LT18]
+Roberto Longo, Yoh Tanimoto, “Rotational KMS states and type I conformal nets,”
+[Mo80]
+Calvin C. Moore, “The Mautner phenomenon for general unitary representations,”
+Pac. J. Math 86 (19 (1980), 155-169
+[Ne00]
+Karl-Hermann Neeb, “Holomorphy and convexity in lie theory,” De Gruyter Expo-
+sitions in Mathematics, 28. Walter de Gruyter & Co., Berlin, 2000.
+[Ni22]
+Milan Niestijl, “Generalized Positive Energy Representation of Groups of Jets,”
+arXiv:2211.10390 [math.RT]
+[Po67]
+Robert T. Powers, “Representations of Uniformly Hyperfinite Algebras and Their
+Associated von Neumann Rings,” Annals of Mathematics , Jul., 1967, Second Series,
+Vol. 86, No. 1 (Jul., 1967), pp. 138-171
+[St78]
+Stratila, S., “On a class of KMS states for the unitary group U(∞),” Math. Ann.
+235:1 (1978), 87-110
+[SV75]
+Stratila, S., and D. Voiculescu, “Representations of AF algebras and of the group
+U(∞),” Lecture Notes in Math. 486, Springer Verlag, Berlin, 1975
+[SV82]
+Stratila, S., and D. Voiculescu, “A survey on representation of the unitary group
+U(∞),” in “Spectral Theory”, Banach Center Publications 8, PWN-Polish Scientific
+Publishers, Warzaw, 1982; 415-434
+[Tak02]
+M. Takesaki, “Theory of operator algebras I,” (English summary) Reprint of the first
+(1979) edition. Encyclopedia of Mathematical Sciences, 124. Operator Algebras and
+Non-commutative Geometry, 5. Springer-Verlag, Berlin, 2002
+26
+
+[Tak03]
+M. Takesaki, “Theory of operator algebras II,” (English summary) Encyclopedia of
+Mathematical Sciences, 125. Operator Algebras and Non-commutative Geometry,
+6. Springer-Verlag, Berlin, 2003
+27
+
diff --git a/htE1T4oBgHgl3EQfzgUS/content/tmp_files/load_file.txt b/htE1T4oBgHgl3EQfzgUS/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..6ee23b2b10a713b9a43a063cecbc5bf97f762878
--- /dev/null
+++ b/htE1T4oBgHgl3EQfzgUS/content/tmp_files/load_file.txt
@@ -0,0 +1,1254 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf,len=1253
+page_content='Factorial type I KMS states of Lie groups  Tobias Simon  January 10, 2023  Abstract  Motivated by the study of KMS conditions for C*- or W*-dynamical systems defined  by covariant unitary representations of topological groups, we consider Gibbs states of a  finite-dimensional Lie group G and prove that these are precisely the factorial type I KMS  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' For an element X ∈ L(G) and an irreducible unitary representation ρ of G satisfying  tr(ei∂ρ(X)) = 1, the corresponding Gibbs state is defined as ϕ(g) = tr(ρ(g)ei∂ρ(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' We prove  that under the mild assumption that ρ has discrete kernel, the condition tr(ei∂ρ(X)) < ∞  implies that the generator X is an inner point of the set comp(g) of elliptic elements in  g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' This allows us to obtain a complete characterization of Lie algebras g, representations ρ  with discrete kernel and generators X such that tr(ei∂ρ(X)) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Contents  1  Geometric and representation theoretic aspects of Gibbs states  4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1  Ellipticity of Gibbs elements .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
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+page_content='  5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2  Characterization of representations of Gibbs type .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  13  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2  C*-methods for locally compact groups .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  15  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3  Semi-finite KMS states of topological groups  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  19  3  Perspectives and open problems  21  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1  Extensions of KMS states of G to KMS states of G♭  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  21  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2  Factorial Type II KMS states of Lie groups .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  22  A Appendix  22  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1 Compactly embedded Cartan subalgebras and admissible Lie algebras  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  22  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2 Convex geometric and representation-theoretic concepts .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  24  Notation and definitions  The group dynamical systems (G, α, R) considered in this paper consist of a topological  group G and a group homomorphism α : R → Aut(G) with continuous orbit maps, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' α  is strongly continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' We call α a one-parameter group of automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  A covariant unitary representation of a group dynamical system (G, α, R) is a unitary  representation of G♭ := G ⋊α R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' These are precisely the unitary representations (π, H)  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='03444v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='RT]  9 Jan 2023  of G for which there exists a selfadjoint operator H on H which satisfies eitHπ(g)e−itH =  π(αt(g)), for all t ∈ R, g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' We call H an implementing operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  A state ϕ : G → C of a topological group G is a continuous function, for which the kernel  Kϕ : G × G → C defined as Kϕ(x, y) := ϕ(x−1y) is positive definite and ϕ(e) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Its  GNS-representation is denoted by (πϕ, Hϕ, Ωϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  If G is a Lie group, we denote with L(G) or g its Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  A von Neumann algebra M is uniquely decomposable as a direct sum of a type I, a type  II1, a type II∞ and a type III von Neumann algebra (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' [Tak02, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='17], [Tak02,  Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' It is called semi-finite if the type III summand is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' This is equivalent  to the existence of a semi-finite faithful normal tracial weight τ on M (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' [Tak03, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1], [Tak02, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  We call a state of a topological group or C*-algebra semi-finite resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' factorial if the von  Neumann algebra generated by its GNS-representation is semi-finite resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' is a factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  An elliptic linear map endomorphism of a finite-dimensional vector space is a semi-simple  linear map with purely imaginary eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Introduction  For C*- or W*-dynamical systems (M, τ, R), KMS states have been studied extensively, for  instance in [BR96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' We are interested in KMS states of dynamical systems defined by group  dynamical systems (G, α, R) and α-covariant representations π, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' we consider the von Neumann  algebra M = π(G)′′ and the one-parameter group of *-automorphisms uniquely determined by  τt(π(g)) = π(αt(g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' An example of such a dynamical system is the Weyl algebra W(E, σ) which  is generated by the image of the Schrödinger representation of the Heisenberg group Heis(E, σ),  where (E, σ) is a symplectic vector space (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  [BR96, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  In this example, the  dynamics τ is implemented by Bogoliubov transformations corresponding to a one-parameter  group of symplectic maps which is an example of the general situation described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' For such  a dynamical system the (τ, β)-KMS condition of a state ω : M → C proves that the corresponding  state ϕ : G → C, g �→ ω(π(g)) is an (α, β)-KMS in the sense of the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Let (G, α, R) be a group dynamical system and ϕ : G → C be a state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' For β > 0,  we say that ϕ satisfies the (α, β)-KMS condition if the functions  Fx,y : R → C,  t �→ ϕ(xαt(y)),  for  x, y ∈ G,  extend to continuous maps on the strip Sβ := {z ∈ C | 0 ≤ Im(z) ≤ β} that are holomorphic on  the interior and satisfy the reflection property Fx,y(t + iβ) = ϕ(αt(y)x), for t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  If one conversely starts with an (α, β)-KMS state of a topological group and considers the cor-  responding W*-dynamical system (W ∗(G), τ, R), where W ∗(G) is the group W*-algebra of G,  one easily obtains a correspondence between (α, β)-KMS states of G and (τ, β)-KMS states of  W ∗(G) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' In the case, where the group G is locally compact, one has access to  a C*-dynamical system given by group C*-algebra C∗(G) defined as the enveloping C*-algebra  of L1(G) and a corresponding action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' This makes KMS states of locally compact groups more  accessible since there exist stronger decomposition results for C*-dynamical systems and their  KMS states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' In particular, for locally compact type I groups, where every factor state is a type I  2  state, every (α, β)-KMS state is an integral of extremal Gibbs states with respect to some Radon  probability measure on the set of extremal (α, β)-KMS states (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  If an (α, β)-KMS state ϕ of a topological group G is semi-finite in the sense that πϕ(G)′′ is a  semi-finite von Neumann algebra, there exists a semi-finite faithful normal trace τ on πϕ(G)′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  One can use the Radon-Nikodym theorem to prove that  ϕ(g) = τ(π(g)e−βH)  τ(e−βH)  ,  for  g ∈ G,  (1)  for a self-adjoint operator H affiliated to πϕ(G)′′ which satisfies eitHπϕ(g)e−itH = πϕ(αt(g)) and  τ(e−βH) < ∞ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' This applies in particular to the case, where πϕ(G)′′ is a  type I factor, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' πϕ(G)′′ ∼= B(H), for some Hilbert space H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Thus one obtains in the manner of  Equation (1) a one-to-one correspondence between factorial type I (α, β)-KMS states and unitary  equivalence classes of irreducible α-covariant unitary representations π such that tr(e−βH) < ∞  for the implementing operator H of α which satisfies eitHπ(g)e−itH = π(αt(g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Suppose that G is a finite-dimensional Lie group and α is inner in the sense that there exists  X ∈ L(G) such that αt is the conjugation by expG(tX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Then any unitary representation π is  α-covariant since eit∂π(X)π(g)e−it∂π(X) = π(exp(tX)g exp(−tX)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' This can always be achieved  by considering, for not necessarily inner actions α, the group G♭ := G ⋊α R, for which  (e, t)(g, s)(e, t)−1 = (αt(g), s),  for  g ∈ G, s, t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  For an action by inner automorphisms, the representations relevant in the study of factorial type  I KMS states are the following:  Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Let G be a finite dimensional Lie group and X ∈ L(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' We call a unitary repre-  sentation (π, H) of G of Gibbs type with respect to X if ei∂π(X) is of trace class and elements in  L(G), with respect to whom representations of Gibbs type exist, Gibbs elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' We denote the  set of Gibbs elements of G with Gibbs(G, g) and for the universal covering group �G of G with  Gibbs(g) := Gibbs( �G, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Our main result is the following theorem, which turns out to be instrumental in characterizing  representations of Gibbs type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Let G be a finite dimensional Lie group and X ∈ L(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  If (π, H) is a unitary  representation of G with discrete kernel such that ei∂π(X) is of trace class, then X is contained  in the interior of the set comp(g) of elliptic elements in g (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Irreducible unitary representations π of finite-dimensional Lie groups, for which there exists  X ∈ comp(g)◦ such that the spectrum of i∂π(X) is bounded from above, where extensively  studied in [Ne00].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Using these results, we obtain the following for irreducible representations of  Gibbs type π with discrete kernel, the Lie algebras g, and Gibbs elements X:  (i) g := L(G) is an admissible Lie algebra (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='5),  (ii) there exists a compactly embedded Cartan subalgebra t ⊆ g containing X (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Definition  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1),  (iii) there exists an admissible positive system of roots ∆+ ⊆ ∆(gC, tC) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2)  such that X is contained in the interior of the cone  Cmax(∆+  p ) := {Y ∈ t | iα(X0) ≥ 0 for all α ∈ ∆+  p },  where ∆+  p is the set of non-compact positive roots (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3),  3  (iv) π is a unitary highest weight representation with respect to the root system ∆+ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Defi-  nition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Conversely, if one starts with an admissible Lie group G and a unitary highest weight repre-  sentation π of G with respect to an admissible positive system ∆+, then [Ne00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='6]  yields that ei∂π(X) is of trace class for all X ∈ Cmax(∆+  p )◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Hence we have characterized the Lie  algebras, generators, and representations in question by geometric data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' [Ne00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1]  gives a complete classification of simple admissible Lie algebras, which are also called hermitian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  For these, the admissible root systems are known and the cones Cmax(∆+  p ) can be explicitly  computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  The condition of semi-boundedness of the representations relevant to this paper is generalized in  [Ni22] and shown to be a property of every KMS state of Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' We also mention [LT18] as  an example of trace class conditions arising in conformal field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Structure of this paper:  This paper is divided into two sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' What motivated this paper is mostly contained in Sec-  tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' In this section, most of what we do is take results already known for KMS states of C*-  or W*-dynamical systems and connect these with the group dynamical picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' This leads us to  unitary representations of Gibbs type which we characterize in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' This characterization  can be seen as a standalone result which is why we include Section 2 afterward as a motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Acknowledgements  This research was supported by DFG-grant NE 413/10-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' I thank my supervisor Karl-Hermann  Neeb for the many helpful corrections and Ricardo Correa da Silva for his advice on the theory  of non-commutative integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  1  Geometric and representation theoretic aspects of Gibbs  states  In this section, we study Gibbs elements and their corresponding unitary representations of Gibbs  type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' To do this, we make use of concepts introduced in Subsection A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1 and Subsection A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2 in  the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' These subsections are meant as a quick introduction to concepts discussed in  depth in the monograph [Ne00].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' We call a subalgebra a ⊆ g compactly embedded if ⟨eadg(a)⟩ ⊆ GL(g) is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  We call an element x ∈ g elliptic if the subalgebra R · x ⊆ g is compactly embedded and denote  with comp(g) the subset of elliptic elements in g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Under the assumption that π is an irreducible unitary representation of Gibbs type, we prove  that the generator X is contained in the interior of the set of elliptic elements comp(g) in g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  The condition comp(g)◦ ̸= ∅ is equivalent to the existence of a compactly embedded Cartan  subalgebra t ⊆ g, and in this case comp(g) = Inn(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='t (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' [Ne00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' In Remark  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2, we discuss how, for a compactly embedded Cartan subalgebra t ⊆ g, one obtains a root  system  ∆ := ∆(gC, tC) ⊆ it∗  such that  gC = tC ⊕  �  α∈∆  gα  C  4  is the simultaneous eigenspace decomposition with respect to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' In view of comp(g) = Inn(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='t  and the Inn(g)-invariance of the set of Gibbs elements, it suffices to characterize the elements  of Gibbs(g) ∩ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' We show that this characterization is possible by considering data associated  to the root system ∆, namely we prove that for a unitary representation π of G with discrete  kernel and an element X ∈ g satisfying ei∂π(X), one has the following:  (i) g is an admissible Lie algebra, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' it contains an invariant generating closed convex subset  containing no affine line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  (ii) There exists a compactly embedded Cartan subalgebra t ⊆ g containing X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  (iii) There exists an admissible positive system of roots ∆+ ⊆ ∆(gC, tC) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2)  such that X is contained in the interior of the cone Cmax(∆+  p ) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  (iv) π is a unitary highest weight representation with respect to the root system ∆+ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Defi-  nition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  In [Ne00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='6] it is shown that if π is a unitary highest weight representation with  respect to an admissible positive system ∆+ and X ∈ Cmax(∆+  p )◦, then ei∂π(X) is of trace class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  If one now drops the assumption of the discrete kernel, one obtains a complete characterization of  Gibbs representations of simply connected Lie groups and Gibbs elements by considering faithful  unitary highest weight representations with respect to admissible positive systems of admissible  quotients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1  Ellipticity of Gibbs elements  The goal of this subsection is to prove that, if π is a unitary representation of a finite-dimensional  Lie group G with discrete kernel and X ∈ g := L(G) such that ei∂π(X) is of trace class, then  X ∈ comp(g)◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' This is equivalent to gX := ker(ad(X)) ⊆ g being compactly embedded by [Ne00,  Lemma VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='7(c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  The first crucial observation is that in this case, the eigenspaces of ei∂π(X) are all finite-dimensional  and invariant under GX := ⟨exp(gX)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Since π has discrete kernel, this implies that gX is a com-  pact Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' In order to prove that it actually is compactly embedded, we consider the  smallest ideal nY ⊴ g, for a fixed Y ∈ g, such that  adg/nY : g/nY → g/nY , x + nY �→ [Y, x] + nY  is an elliptic endomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' For Y ∈ gX, we prove that nY = {0} holds, which is equivalent to  Y being elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' To do this, we proceed by first showing that nY is abelian and then use this to  prove that nY is an at most one-dimensional central ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Let (π, H) be a unitary representation of G with discrete kernel such that ei∂π(X)  is of trace class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' The ellipticity ideal nY is abelian, for every Y ∈ gX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Consider the unitary representation of G on the Hilbert-Schmidt operators B2(H) given  by  ρ : G → U(B2(H)),  ρ(g)(A) = π(g)Aπ(g)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  The eigenspace projections of ei∂π(X) have finite rank and thus are Hilbert-Schmidt opera-  tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Since π(exp(RY )) leaves the eigenspaces of ei∂π(X) invariant, the one-parameter groups  5  ρ(exp(RY )) fix the corresponding projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Now Moore’s Theorem [Mo80, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1] implies  that the normal subgroup NY := ⟨exp(nY )⟩ ⊴ G also fixes the projections onto the eigenspaces,  which is equivalent to π(NY ) leaving the eigenspaces invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Hence the representation π|NY  decomposes into finite-dimensional subrepresentations and has discrete kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Therefore the  finite-dimensional unitary representations of nY separate the points which implies that nY is a  compact Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' As a compact Lie algebra, nY is in particular reductive and decomposes  as nY = znY ⊕ [nY , nY ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' We define a := [nY , nY ] ⊴ g and note that a is a semi-simple compact  ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' From [HN12, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='14], one obtains that a is contained in a Levi complement s and  thus for the ideal a ⊴ s there exists a complementary ideal a⊥ ⊴ s such that s = a ⊕ a⊥ since s  is semi-simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' It is now easy to see that b := s⊥ ⊕ rad(g) is an ideal in g and satisfies g = a ⊕ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  This shows that g/b ∼= a is compact and thus adg/b(Y ) is elliptic which implies nY ⊆ b as nY is  the smallest ideal with this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Then a ⊆ nY and a ∩ b = {0} imply [nY , nY ] = a = {0}  and thus the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  The group K := π(NY ) ⊆ U(H) on H leaves the finite dimensional eigenspaces of ei∂π(X)  invariant, which implies that that K is compact and Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1 shows that K is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Since  NY ⊴ G is a normal subgroup and π(NY ) ⊂ K is dense, it follows that π(g)Kπ(g)−1 ⊆ K and  thus  Φ : G → Aut(K),  Φ(g)(k) = π(g)kπ(g)−1  is a group homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Next we consider on Aut(K) a suitable topology1 for Φ to be  continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' We will use this topology to cite results that imply that a continuous action of a  connected group acting on a compact abelian group by group homomorphisms is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Let (π, H) be a unitary representation of G with discrete kernel such that ei∂π(X)  is of trace class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' For Y ∈ gX and NY := ⟨exp(nY )⟩, the group K := π(NY ) is compact, abelian,  and commutes with the identity component of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' In particular, nY is central in g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' [HM06, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='82] implies that the identity component of Aut(K) with respect to the  topology introduced in the footnote above coincides with the inner automorphisms of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Since  K is abelian, it is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' As Φ is a group homomorphism, we only need to show that it is  continuous with respect to O1 to prove its continuity with respect to O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' By the exponential law  for spaces of continuous functions (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' [GN, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='29]), this follows from the continuity of  Φ∧ : G × K → K,  (g, k) �→ Φ(g, k) = π(g)kπ(g)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  This implies that the identity component of G acts trivially on K and therefore nY is central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  The representation π was assumed to be irreducible, so that π(NY ) ⊆ π(G)′ = C1 and the  discreteness of the kernel of π thus implies that nY ⊴ g is a central ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  With the same  argument, one obtains that z(g) is at most one-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  It remains to prove that nY cannot be one-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' In order to prove this, we will assume  the opposite {0} ̸= nY ⊆ z := z(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Then we are in the context of the following lemma which  will allow us to use some results of the representation theory of the three-dimensional Heisenberg  group to obtain a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  1Consider the subspace topology O1 with respect to the topology of uniform convergence on C(K, K), which  is the coarsest topology containing the sets U(V0, f0) := {f ∈ C(K, K) | f(k) ∈ V0f0(k) for all k ∈ K}, for  f0 ∈ C(K, K) and identity neighbourhoods V0 in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Let O2 be the unique topology on Aut(K) such that  ι : Aut(K) → (Aut(K), O1), f �→ f−1 becomes a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  If we now define O := O1 ∨ O2, then  (Aut(K), O) defines a topological group (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' [HM06, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' 257]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  6  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Let g be a real finite-dimensional Lie algebra with center z and suppose that  y + z ∈ g/z is elliptic but y ∈ g is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Then there exists x ∈ g such that [x, y] = z ∈ z \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' In  particular h := spanR{x, y, z} is isomorphic to the three dimensional Heisenberg algebra h3(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' We consider ad(y) as an endomorphism of gC with Jordan decomposition given by ad(y) =  ad(y)s + ad(y)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Since gC decomposes into the ad(y)n-invariant eigenspaces gλ  C of ad(y)s, for  λ ∈ C, and zC ⊆ g0  C, it follows from the fact that y + z is elliptic that  ad(y)n|gλ  C ≡ 0,  for  λ ̸= 0,  and  ad(y)n(g0  C) ⊆ zC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  If ad(y)n were zero, then ad(y) = ad(y)s would be elliptic since zC ⊆ g0  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' This proves that there  exists x ∈ g0  C such that ad(y)(x) = ad(y)n(x) ∈ zC \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' If we write x = x0 + ix1 ∈ gC, then  [x, y] ∈ zC \\ {0} implies that either [x0, y] ∈ z \\ {0} or [x1, y] ∈ z \\ {0} hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' This proves the  assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Consider the three-dimensional Heisenberg group defined as follows:  Heis(R2) := R2 × R  with multiplication  (a, b, c)(x, y, z) := (a + x, b + y, c + z + ay),  for a, b, c, x, y, z ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' For irreducible unitary representations of Heis(R2) the center Z(Heis(R2)) ∼=  R acts by scalars and thus, if the center acts non-trivially, defines a central character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  The  Stone-von-Neumann Theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  [DE09, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1]) implies that this central character  λ : R → T, x �→ eiλx uniquely determines the unitary representation and that the unitary  representation is unitarily equivalent to the representation on L2(R) defined by  πλ(a, b, c)f(x) := eiλ(bx+c)f(x + a)  for  a, b, c, x ∈ R, f ∈ L2(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Consider the automorphism Φ : Heis(R2) → Heis(R2), (a, b, c) �→ (b, −a, c − ab), which fixes  the center pointwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  We note that by the Stone-von-Neumann theorem πλ ∼= πλ ◦ Φ and  Φ(0, t, 0) = (t, 0, 0) holds, for t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' This proves that the unitary one-parameter groups defined  by  πλ(t, 0, 0)f(x) = f(x + t)  and  πλ(0, t, 0)f(x) = eitλxf(x),  for  x, t ∈ R, f ∈ L2(R)  are conjugate via U(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' In particular,πλ(R, 0, 0) and πλ(0, R, 0) do not have compact closure  in U(H) since the identical representation of both groups on L2(R) clearly does not decompose  into irreducible subrepresentations and the groups are conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  We now come to our first main result:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Let (π, H) be a unitary representation of G with discrete kernel such that  ei∂π(X) is of trace class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Then π is a direct sum of irreducible representations and X is contained  in comp(g)◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Since ei∂π(X) ∈ π(G)′′ is a positive operator of trace class, it follows that π(G)′ leaves the  finite-dimensional eigenspaces invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' As finite dimensional C*-algebras are isomorphic to a  direct sum of matrix algebras, this implies that there exists a maximal family of pairwise disjoint  minimal projections in π(G)′ summing to the idenitfy in the strong operator topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Therefore  π is a direct sum of irreducible representations πj with discrete kernel satisfying tr(ei∂πj(X)) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Suppose now that π is irreducible and let Y ∈ gX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' It follows from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2 that nY ⊆ z(g)  and thus Y is either elliptic or 0 ̸= Y + z(g) is elliptic in g/z(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' In the latter case, Lemma  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3 implies that there exists x0 ∈ g such that [Y, x0] = z ∈ z(g) \\ {0} and thus the subalgebra  7  h := spanR{Y, x0, z} is isomorphic to the three-dimensional Heisenberg algebra L(Heis(R2)) via  the isomorphism defined by  Y �→ (1, 0, 0), x0 �→ (0, 1, 0), z �→ (0, 0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  The simply connected Heisenberg group Heis(R2) is the universal covering group of ⟨exp(h)⟩ and  thus the representation π|⟨exp(h)⟩ lifts to a representation ρ of Heis(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' As π is irreducible, z(g)  acts by scalars, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' there exists λ ∈ R such that  ∂ρ(0, 0, 1) = ∂π(z) = iλ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  One has λ ̸= 0, because ∂π(z) = 0 would imply nY ⊆ z(g) ⊆ ker(∂π) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  The Stone-  von-Neumann Theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' [Ne00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1]) implies that ρ is a multiple of πλ, where πλ  is the unique irreducible unitary representation of the Heisenberg group with central character  λ : R → T, x �→ eiλx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' It follows from Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='4, that the closure of  ρ(exp(R(1, 0, 0))) = π(exp(RY ))  is not compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  This is a contradiction since a closed subgroup of U(H) leaving the finite-  dimensional eigenspaces of ei∂π(X) invariant is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Hence nY = {0} follows, which proves  that Y is elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' As Y ∈ gX was arbitrary, we have shown that gX is contained in comp(g),  which together with [Ne00, Lemma VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='5(b)] yields that gX is compactly embedded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' From  [Ne00, Lemma VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='7(c)], one obtains that X ∈ comp(g)◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2  Characterization of representations of Gibbs type  The goal of this subsection is to summarize results from [Ne00] to prove that the irreducible  unitary representation ρ of Gibbs type with discrete kernel coincide with faithful unitary high-  est weight representations of a finite-dimensional connected Lie group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' The Lie algebras, for  which such representations exist, are precisely the admissible Lie algebras, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' those containing  a generating invariant closed convex subset containing no affine line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  The Lie algebraic and  representation theoretic concepts used in this subsection are introduced in Subsection A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1 and  Subsection A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  The convex momentum set Iπ ⊆ g∗ of a unitary representation (π, H) of a finite-dimensional Lie  group G (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3) connects convex geometry and representation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Its relevance  for our study of representations of Gibbs type can be seen from the following result (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' [Ne00,  Lemma X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='6]):  B(Iπ) := {Y ∈ g | ⟨Iπ, Y ⟩ > −∞} = {Y ∈ g | sup spec(i∂π(Y )) < ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  If (π, H) is an irreducible unitary representation of G with discrete kernel for which there exists  X ∈ g := L(G) such that ei∂π(X) is of trace class, then the spectrum of i∂π(X) is bounded  from above and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='5 implies X ∈ B(Iπ) ∩ comp(g)◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Therefore [Ne00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='8] is  applicable and we obtain the following corollary:  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Let G be a finite-dimensional Lie group and (π, H) be an irreducible unitary  representation of G with discrete kernel for which there exists X ∈ g := L(G) such that ei∂π(X)  is of trace class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Then there exist a compactly embedded Cartan subalgebra t ⊆ g containing X  and a positive system of roots ∆+ ⊆ ∆(gC, tC), which is adapted (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3), such that  π is a unitary highest weight representation (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='9) with respect to ∆+ and X is  contained in the interior of the cone  Cmax := Cmax(∆+  p ) = {X ∈ t | iα(X) ≥ 0 for all α ∈ ∆+  p }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  8  For an element X ∈ g, the condition that there exists an adapted positive system ∆+ such  that X ∈ Cmax(∆+  p )◦ is not sufficient for X to be a Gibbs element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' To ensure this, we need  further restrictions on ∆+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' The Gelfand-Rai’kov Theorem for Unitary Highest Weight Represen-  tations [Ne00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='8] states sufficient conditions for a positive root system ∆+ of a simply  connected admissible Lie group such that unitary highest weight representations exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' In this  theorem, one considers the convex cone  Cmin(∆+  p ) := cone({i[Zα, Z∗  α] | Zα ∈ gα  C, α ∈ ∆+  p } ⊆ t,  where ∗ : gC → gC, x + iy �→ −x + iy denotes the involution on gC discussed in Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  The condition stated in the following definition then ensures that the unitary highest weight  representations separate the points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Let g be a Lie algebra and t ⊆ g be a compactly embedded Cartan subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  We call an adapted positive system of roots ∆+ ⊆ ∆(gC, tC) such that Cmin(∆+  p ) is pointed and  contained in Cmax(∆+  p ) admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' The term admissible for a positive system of roots is not used in [Ne00].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' We  introduce it to state the conditions for positive systems of roots arising in our context more  concisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  The condition of admissibility of a positive system arises quite naturally in the theory of unitary  highest weight representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  In the Gelfand-Raik’kov theorem, it is stated as a sufficient  condition for the existence of unitary highest weight representation of admissible Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  The following theorem, which is stated implicitly in [Ne00], shows that for faithful unitary highest  weight representations the condition is also necessary:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' ([Ne00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='17])  Let G be a finite dimensional Lie group with Lie algebra g and (π, H) be an irreducible unitary  highest representation of G with respect to a positive system ∆+ and suppose π has discrete  kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Then g is admissible and ∆+ is an admissible positive system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' We consider the space of smooth vectors H∞ which is a faithful, unitary highest weight  module of gC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' It follows from [Ne00, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='14](iii) that, for the highest weight λ ∈ it∗ of  π the gC-module H∞ is isomorphic to L(λ, ∆+), which is introduced there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Using [Ne00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='17], which is applicable to L(λ, ∆+), one obtains that g is admissible, ∆+ is adapted and  −iλ ∈ t∗ is contained in the interior of  Cmin(∆+  p )⋆ := {f ∈ t∗ | f(Cmin(∆+  p )) ⊆ [0, ∞)} ⊆ B(Cmin(∆+  p )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Then [Ne00, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='15] implies that the convex cone Cmin(∆+  p ) is pointed since B(Cmin(∆+  p ))  has inner points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' To prove that Cmin(∆+  p ) ⊆ Cmax(∆+  p ) holds, we apply [Ne00, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='11]  which is applicable if there exists f ∈ Cmin(∆+  p )⋆ such that its orbit Of under the coadjoint  action is closed, the convex hull of Of contains no affine lines and Of spans g∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  If one applies [Ne00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='19] to the highest weight λ ∈ iCmin(∆+  p )⋆ ⊆ it∗ of the unitary  highest weight representation π with discrete kernel, one obtains that −iλ is admissible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' its  orbit O−iλ under the coadjoint action is closed and its convex hull contains no affine lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Using  [Ne00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1] which states that the momentum set of π is given by Iπ = conv(O−iλ) and  (Iπ)⊥ = ker(dπ) = {0} (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' [Ne00, Lemma X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='6]), one obtains that Iπ ⊆ g∗ is generating and  −iλ ∈ Cmin(∆+  p )⋆  satisfies  spanRO−iλ = g∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Hence [Ne00, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='11] is applicable which yields Cmin(∆+  p ) ⊆ Cmax(∆+  p ) and thus the  positive system ∆+ is admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  9  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Note that [Ne00, VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='11] implies that the positive system ∆+ in the theorem  above is uniquely determined by the condition that (π, H) is an irreducible unitary highest  representation with respect to ∆+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  In [Ne00], the highest weight modules L(λ, ∆+) are studied, for involutive Lie algebras with root  decomposition (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' [Ne00, Subsection IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' For those modules, one has the following result:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' ([Ne00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='6])  Let g be an admissible Lie algebra, ∆+ an adapted positive system and L(λ, ∆+) a unitary highest  weight module with respect to ∆+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' If X ∈ Cmax(∆+  p )◦ and ∂π denotes the representation of g on  L(λ, ∆+), then tr(ei∂π(X)) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Let g be an admissible Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Then [Ne00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='10] implies that  there exists a compactly embedded Cartan subalgebra t in g and an admissible positive system  of roots ∆+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Then [Ne00, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='7] associates to each admissible positive system ∆+ a  closed generating invariant cone Wmax(∆+  p ) ⊆ g such that Wmax ∩ t = Cmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' In the proof of  [Ne00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='10] it is also shown that  Wmax(∆+  p )◦ = Inn(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='Cmax(∆+  p )◦ ⊆ comp(g)◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  If g is non-compact, the open invariant convex cone Wmax(∆+  p )◦ is proper since comp(g)◦ ⊊ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  We are now able to present our second main result:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Let G be a finite-dimensional Lie group with Lie algebra g and (π, H) be an  irreducible unitary highest representation of G with respect to a positive system ∆+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Then one  has {X ∈ g | tr(ei∂π(X)) < ∞} = Wmax(∆+  p )◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' This follows immediately from Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='4, and Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  As the Gelfand-Rai’kov Theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' [Ne00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='8]) applies to simply connected admissible  Lie groups, we may combine these results to obtain the following corollary:  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Let G be a simply connected finite-dimensional Lie group with Lie algebra g  and X ∈ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Then there is a one-to-one correspondence between  (i) irreducible unitary representation π of G such that ei∂π(X) is of trace class,  (ii) irreducible unitary highest weight representations of admissible quotients q : g ↠ g1 with  respect to an admissible positive system ∆+  1 of g1 such that q(X) ∈ Wmax(∆+  1,p)◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  We will now prove that an element X is contained in Gibbs(g) if and only if it is contained in  an open proper invariant convex subset of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Geometric characterization of Gibbs elements  Let g be a finite-dimensional Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Then an element X is a Gibbs element if and only if  it is contained in an open proper invariant convex subset of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Suppose X ∈ Gibbs(g) and G is a simply connected Lie group with Lie algebra g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Then  there exists a unitary representation π of G such that ei∂π(X) is of trace class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Then Corollary  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='4 imply that g/ker(dπ) is admissible and there exists an admissible positive  system ∆+ such that  q(X) ∈ Cmax(∆+  p )◦ ⊆ Wmax(∆+  p )◦,  10  where q : g → g/ker(dπ) denotes the quotient map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' If g/ker(dπ) is non-compact, the subset  q−1(Wmax(∆+  p )◦) ⊊ g  is an open proper invariant convex subset containing X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' If g′ := g/ker(dπ) is compact, choose  an Inn(g′)-invariant metric on g′ and an open ball B ⊊ g′ with respect to this metric containing  q(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Then q−1(B) ⊊ g is an open proper invariant convex subset containing X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Suppose conversely that X is contained in an open proper invariant convex subset C ⊊ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' We  adopt elements of the proof of [Ne00, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' We consider the set  H(C) := {v ∈ g | v + C = C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  H(C) is a convex additive subgroup of the vector space g which implies that H(C) a linear  subspace and the invariance of C yields that H(C) is an ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Consider the quotient map  q : g → g′ := g/H(C) and observe that q(C) is an invariant open generating convex subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' It  satisfies  Hg′(q(C)) = {w ∈ g′ | w + q(C) = q(C)} = {q(v) ∈ g′ | v ∈ g, q(v + C) = q(C)} = {0}  since q(v + C) = q(C) is equivalent to (v + C) + H(C) = C + H(C), and then C + H(C) = C  implies v ∈ H(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Therefore the closure of q(C) is an invariant generating closed convex subset  containing no affine line and thus g′ is admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' To apply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='6 and Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='9,  we show that there exists an admissible positive system ∆+ such that q(X) ∈ Cmax(∆+  p )◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  The set q(C) is contained in the interior of its closure q(C) whose interior consists of elliptic  elements (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' [Ne00, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='4(b)]) and thus q(X) is contained in the open invariant elliptic  convex set q(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  For a compactly embedded Cartan subalgebra t ⊆ g′, one has comp(g′) = Inn(g′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='t (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' [Ne00,  Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='8]), we may assume q(X) ∈ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Then the Sandwich Theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='6)  implies that there exists an adapted positive system ∆+ with respect to t such that  q(C) ∩ t ⊆ Cmax(∆+  p )  and  Cmin(∆+  p ) ⊆ lim(q(C) ∩ t) := {v ∈ t | v + q(C) ∩ t ⊆ q(C) ∩ t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  As q(C) contains no affine line, the cone Cmin(∆+  p ) is pointed and since Cmax is a cone and  q(C) ∩ t ⊆ Cmax(∆+  p ), it follows that  Cmin(∆+  p ) ⊆ lim(q(C) ∩ t) ⊆ Cmax(∆+  p ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  This proves that ∆+ is admissible and that q(X) ∈ Cmax(∆+  p )◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Hence the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Let (V, Ω) be a symplectic vector space, h(V, Ω) = R⊕V be the corresponding  Heisenberg algebra and  sp(V, Ω) := {A ∈ gl(V ) | Ω(v, Aw) + Ω(Av, w) = 0 for all v, w ∈ V }  be the corresponding symplectic Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' The prototypical example of an admissible Lie  algebra which is not semi-simple, is given by the Jacobi algebra g := hsp(V, Ω) := h(V, Ω) ⋊  sp(V, Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' The map  Φ : hsp(V, Ω) → C∞(V ),  Φ(s, v, A)(w) := 1  2ΩA(w) + Ω(v, w) + s,  11  for s ∈ R, v, w ∈ V, A ∈ sp(V, Ω), defines an injective homomorphism of Lie algebras if one  equips C∞(V ) with the Poisson-bracket corresponding to Ω (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' [Ne00, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' The  image of Φ coincides with the polynomials of degree ≤ 2 and it maps sp(V, Ω) to polynomials  of homogeneous degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Continuing onwards we identify hsp(V, Ω) ∼= im(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Consider a  symplectic basis  {pj, qj | 1 ≤ j ≤ n := 1  2dimRV }  and define  tl := spanR{p2  i + q2  i | 1 ≤ i ≤ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Then tl is a compactly embedded Cartan subalgebra of l := sp(V, Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' If one computes the root  system ∆ := ∆(lC, (tl)C) as in [Ne00, Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='30] and chooses the up to a sign unique positive  admissible system ∆+, then one has  Cmax(∆+  p ) = cone({p2  i + q2  i | 1 ≤ i ≤ n}) = {ΩA ∈ tl | ΩA ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Since there is up to a sign only one admissible positive system of non-compact roots ∆+  p in the  simple hermitian Lie algebra sp(V, Ω) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' [Ne00, Lemma VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='16(iv)]) and ΩA > 0 if and only  if A is elliptic and all its eigenvalues are contained in iR+, this proves that the set of Gibbs ele-  ments consists of all elliptic symplectic endomorphisms for which the eigenvalues are contained  in either iR+ or iR−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Then [Ne00, Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='30]) implies that t = z(g)⊕tl is a compactly embedded Cartan subalgebra  of hsp(V, Ω) for the corresponding admissible positive systems of roots ∆±  0 ⊆ ∆(gC, tC) defined  by extending the elements of ∆± ⊆ ∆(lC, (tl)C) extended to t = z(g) ⊕ tl by zero on z(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' In  particular, one has  Wmax(∆±  0,p)◦ = Inn(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='Cmax(∆±  0,p)◦ = Inn(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' (z(g) ⊕ Cmax(∆±  p )◦)  which implies Wmax(∆±  0,p)◦ = {(s, v, A) ∈ hsp(V, Ω) | ΩA > 0} which gets mapped to the set of  strictly convex polynomials of degree ≤ 2 by Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' This is the interior of the Inn(g)-invariant cone  W := {f ∈ im(Φ) | inf f > ∞} =  �  Φ(s, v, A) | ΩA ≥ 0, v ∈ rad(ΩA)⊥Ω�  of polynomials of degree ≤ 2 which are bounded from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Note that this set is not closed as  its boundary coincides with all polynomials of degree one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  2  KMS states of topological groups  In this section, we discuss how results for KMS states of C*-algebras and W*-algebras can be  transferred to topological groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  For general topological groups G, there exists a universal  group W*-algebra W ∗(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' It has the property that, for a strongly continuous one-parameter  group of automorphisms α : R → Aut(G), one obtains a σ-weakly continuous one-parameter  group of *-automorphisms τ : R → Aut(W ∗(G)) such that a state ϕ of G is an (α, β)-KMS state  if and only if its corresponding state ωϕ of W ∗(G) is a (τ, β)-KMS state (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  As is the case for W*-dynamical systems, one does not have good control over the set of KMS  states in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' For locally compact groups this is not the case, as the availability of the group  C*-algebra C∗(G) has the consequence that the set Kβ of (α, β)-KMS states of a locally compact  topological group is a weak*-compact convex subset of L∞(G) ∼= L1(G)′ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Therefore the Krein-Millman Theorem implies that Kβ is the closed convex hull of its extremal  points Ext(Kβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' As a KMS state is extremal if and only if it is a factor state, it suffices to study  factorial KMS states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' For general topological groups G, we characterize semi-finite KMS states  12  as generalized Gibbs states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' This allows us to apply the results from Section 1 to factorial type I  KMS states of finite-dimensional Lie groups G for one-parameter groups of inner automorphisms  generated by X ∈ L(G), which correspond to irreducible unitary representations ρ such that  tr(eiβ∂ρ(X)) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1  Elementary properties of KMS states of topological groups  The goal of this subsection is to relate the concepts (α, β)-KMS states of a topological group G  and KMS states of W*-dynamical systems defined by α-covariant representations of the group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  In particular, we show that each (α, β)-KMS state of a topological group is α-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' [BR96, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1], [BR96, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='7]):  Let ω be a state of a C*-algebra A (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' normal state of a von Neumann algebra) and let τ : R →  Aut(A) be a strongly (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' σ-weakly) continuous one-parameter group of *-automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' We  call ω an (τ, β)-KMS state if the functions  FA,B : R → C,  t �→ ω(Aτt(B)),  for  A, B ∈ A,  extend to continuous maps on the strip Sβ := {z ∈ C | 0 ≤ Im(z) ≤ β} that satisfy the reflection  relation FA,B(t + iβ) = ω(τt(B)A), for t ∈ R, and are holomorphic on the interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  We mimic the equivalent definition of KMS states of C*-or W*-dynamical systems (A, τ, R) to  define KMS states of topological groups (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' [BR96, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Let (G, α, R) be a group dynamical system and ϕ : G → C be a state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' For  β > 0, we say that ϕ satisfies the (α, β)-KMS condition if the functions  Fx,y : R → C,  t �→ ϕ(xαt(y)),  for  x, y ∈ G,  extend to continuous maps on the strip Sβ := {z ∈ C | 0 ≤ Im(z) ≤ β} that are holomorphic on  the interior and satisfy the reflection relation Fx,y(t + iβ) = ϕ(αt(y)x), for t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  We now define a W*-dynamical system (W ∗(G), τ, R) corresponding to a group dynamical system  (G, α, R) and a state ωϕ of W ∗(G) corresponding to a state ϕ such that ϕ is an (α, β)-KMS state  of G if and only if ωϕ is a (τ, β)-KMS state of W ∗(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' For a topological group G, we define the group W*-algebra as a pair of a von  Neumann algebra W ∗(G) and a σ-weakly continuous group homomorphism π : G → U(W ∗(G))  that has the universal property that for every strongly continuous unitary representation (ρ, H)  of G there exists a unique *-homomorphism  Φ : W ∗(G) → ρ(G)′′  such that  Φ(π(g)) = ρ(g),  for all  g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  For every topological group G, there exists such an algebra by [GN00, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Let G be a topological group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  (i) By the universal property, for every α ∈ Aut(G) there exists a *-automorphism  Φ : W ∗(G) → W ∗(G)  such that  Φ(π(g)) = π(α(g)),  for all  g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  For a strongly continuous one-parameter group of automorphisms α : R → Aut(G), the  corresponding one-parameter group of *-automorphisms τ : R → Aut(W ∗(G)) is σ-weakly  continuous since spanCπ(G) ⊆ W ∗(G) is σ-weakly dense and on U(H) the σ-weak and the  strong operator topology coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  13  (ii) If ϕ is a state of G with GNS-representation (πϕ, Hϕ, Ωϕ), then the corresponding repre-  sentation πϕ : W ∗(G) → πϕ(G)′′ defines the state  ωϕ : W ∗(G) → C,  A �→ ⟨Ωϕ, πϕ(A)Ωϕ⟩  which satisfies ωϕ(π(g)) = ϕ(g), for all g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  If one adapts the proof of [BR96, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='7] 2, one obtains the following lemma:  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Suppose (A, τ, R) is a C*-dynamical system (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' W*-dynamical system) and  ω : A → C is a (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' normal) state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' If K ⊆ A is a norm-dense (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' σ-weakly dense) subspace  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' *-subalgebra) such that, for A, B ∈ K, the functions  FA,B : R → C,  t �→ ω(Aτt(B))  extend to a continuous functions on the strip  Sβ := {z ∈ C | 0 ≤ Im(z) ≤ β} that are  holomorphic on the interior and satisfy FA,B(t + iβ) = ω(τt(A)B), for t ∈ R,, then ω is a  (τ, β)-KMS state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Let G be a topological group G, α : R → Aut(G) be a strongly continuous  one-parameter group of automorphisms and ϕ : G → C a state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Then ϕ is an (α, β)-KMS state  if and only if the corresponding state ωϕ : W ∗(G) → C is a (τ, β)-KMS state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Observe that ωϕ(π(x)τt(π(y))) = ωϕ(π(xαt(y))) = ϕ(xαt(y)) holds, for all x, y ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Then the assertion follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='5 since spanCπ(G) is a σ-weakly dense *-subalgebra  of W ∗(G) on which the KMS-condition is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Let G be a topological group G, α : R → Aut(G) be a strongly continuous one-  parameter group of automorphisms and ϕ : G → C an (α, β)-KMS state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Then ϕ is α-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' [BR96, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3] implies that ωϕ is τ-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' This proves the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Let G be a topological group, α : R → Aut(G) be a strongly continuous one-parameter group of  automorphisms and ϕ be an α-invariant state with GNS-representation (π, H, Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' If (πϕ, Hϕ, Ωϕ)  denotes the GNS-representation of ϕ, then there exists a strongly continuous unitary one-  parameter group (Ut)t∈R on Hϕ such that  Utπϕ(g)U−t = πϕ(αt(g))  and  UtΩϕ = Ωϕ,  for  t ∈ R, g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Consider the normal state ωϕ : πϕ(G)′′ → C, A �→ ⟨Ωϕ, AΩϕ⟩ and the one-parameter group of  *-automorphisms τ defined on πϕ(G)′′ by τt(A) := UtAU−t, for t ∈ R and A ∈ πϕ(G)′′, which is  σ-weakly continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' A state ϕ of a group dynamical system (G, R, α) is an (α, β)-KMS state if  and only if ωϕ is a (τ, β)-KMS state of the W*-dynamical system (πϕ(G)′′, τ, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' This follows immediately from the σ-weak density of spanCπϕ(G) ⊆ π(G)′′, the relation  ω(πϕ(g)τt(πϕ(h))) = ϕ(gαt(h)),  for  g, h ∈ G, t ∈ R  and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  2In [BR96, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1] KMS states of a C*-or W*-dynamical system are defined by a property for a suitably  dense *-subalgebra B of τ-analytic elements to ensure that for A, B ∈ B the maps FA,B(t) := ω(Aτt(B)) extend  to C and satisfy ω(Aτiβ(B)) = ω(BA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' In the proof of [BR96, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='7], it is only used that the *-subalgebra  B consists of τ-analytic elements to ensure the extension property of the functions FA,B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  14  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2  C*-methods for locally compact groups  Let G be a locally compact group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  The group C∗-algebra of G is defined as the universal  enveloping C∗-algebra C∗(G) := C∗(L1(G)) of the convolution algebra L1(G), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' the completion  of L1(G) with respect to the maximal C∗-norm given on L1(G) by  ∥f∥C∗ := sup{∥π(f)∥ | π : L1(G) → B(H) non-degenerate *-representation}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Every non-degenerate *-representation π : L1(G) → B(H) extends uniquely to a continuous  *-representation of C∗(G) and ∥π(f)∥ ≤ ∥f∥C∗ holds trivially for all f ∈ L1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' We will usually  also denote this extension by π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Note that non-degenerate *-representation are in one-to-one  correspondence with unitary representations of G (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' [DE09, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' This one-to-one  correspondence yields the following one-to-one correspondence of states as they are uniquely  determined by their cyclic GNS-representation and a fixed cyclic vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' There is a one-to-one correspondence of states ϕ of G and states ω of C∗(G)  established by Φ : S(G) → S(C∗(G)), ϕ �→ ωϕ, where ωϕ is uniquely determined by  ωϕ(f) =  �  G  f(x)ϕ(x)dx,  for  f ∈ L1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  If one equips S(G) with the weak*-subspace topology from S(G) ⊆ L∞(G) ∼= L1(G)′ and S(C∗(G))  with the weak*-topology of C∗(G)′, then Φ is affine linear and a weak*-continuous homeomor-  phism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  In order to get a one-to-one correspondence between KMS states of C∗(G) and KMS states of  G, one needs to define a strongly continuous one-parameter group of automorphisms  τ : R → Aut(C∗(G))  corresponding to  α : R → Aut(G)  such that ωϕ is a (τ, β)-KMS state of C∗(G) if and only if ϕ is an (α, β)-KMS state of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' To do  this, we note that the pullback (αt)∗µG of the Haar measure µG of G is a left-invariant Radon  measure and thus there exists a unique scalar λt ∈ R+ satisfying  µG ◦ α−t =: (αt)∗µG = λt · µG  for all  t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  It is an easy exercise to show that τ : R × L1(G) → L1(G), τt(f) := λt · (f ◦ α−t) extends to  a strongly continuous one-parameter group of *-automorphisms of C∗(G) which we also denote  by τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' For t ∈ R and f ∈ L1(G), one has  π(τt(f)) =  �  G  λt · f(α−t(x))π(x)dx =  �  G  f(x)π(αt(x))dx = (π ◦ αt)(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Let G be a locally compact topological group and α : R → Aut(G) be a strongly  continuous one-parameter group of automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' For a state ϕ : G → C, one has  ωϕ(f ∗ τt(g)) =  �  G  �  G  f(y) g(x) ϕ(yαt(x))dxdy,  for  t ∈ R, f, g ∈ L1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Let (H, π, Ω) be the GNS-representation of ϕ and ωϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Using ϕ(x) = ⟨Ω, π(x)Ω⟩, we  calculate ωϕ(f ∗ τt(g)) which is given by  �  G  (f ∗ τt(g))(x)⟨Ω, π(x)Ω⟩dx =  �  G  �  G  f(y) λt g(α−t(y−1x)) ϕ(x)dxdy  =  �  G  �  G  f(y) λt g(α−t(x)) ϕ(yx)dxdy =  �  G  �  G  f(y) g(x) ϕ(yαt(x))dxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  15  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' A state ϕ ∈ S(G) is an (α, β)-KMS state if and only if ωϕ ∈ S(C∗(G)) is  a (τ, β)-KMS state in the sense of C∗-dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' In particular the map S(G) ∋ ϕ →  ωϕ ∈ S(C∗(G)) restricts to an affine linear homeomorphism of the set of respective KMS states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Suppose ϕ is an (α, β)-KMS state of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' We first show that ωϕ is a (τ, β)-KMS state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  [Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='12, BR96] implies that it suffices to show  �  R  �ψ(t) ωϕ(f ∗ τt(g))dt =  �  R  �ψ(t + iβ) ωϕ(τt(g) ∗ f)dt,  for all ψ ∈ C∞  c (R) and f, g ∈ L1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Fix ψ ∈ C∞  c (R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' The Paley-Wiener Theorem [BR96, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='11] implies that, for R > 0 and  supp(ψ) ⊆ [−R, R], the function �ψ is entire analytic and for every n ∈ N0 there exist constants  Cn > 0 such that  | �ψ(z)| ≤ Cn(1 + |z|)−n exp(R · |Im(z)|),  for  z ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  This proves that the map t �→ �ψ(t)ϕ(yαt(x)) extends to continuous function F on the strip Sβ  which is holomorphic on the interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Choose, for r > 0, a piecewise smooth curve γr which  parametrizes the rectangle of width depending on r and fixed height β depicted below:  −r  r  iβ  Applying [BR96, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='5] to the continuous function F on Sβ which is holomorphic on the  interior yields  F(t + isβ) ≤ | �ψ(t + isβ)| · sup{|ϕ(xαt(y))|, |ϕ(αt(y)x)|} ≤ Cn(1 + β2 + t2)−n exp(βR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  This proves that the restriction of F(z) to the right and left side of the rectangle parametrized  above converges uniformly to zero for r → ∞ since it is given by |F(±r+isβ)|, for s ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' The  length of the left and right side of the rectangle is finite and thus the integrals over the right and  left sides tend to zero for r → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' The integrals of F(z) over the contractible curves γr vanish  as F is holomorphic which implies  0 = lim  r→∞  �  γr  F(z)dz =  � ∞  −∞  �ψ(t)ϕ(yαt(x))  �  ��  �  F (t)  dt −  � ∞  −∞  �ψ(t + iβ)ϕ(αt(x)y)  �  ��  �  F (t+iβ)  dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  To see that ωϕ is a (τ, β)-KMS state, we apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2 and obtain  �  R  �ψ(t) ωϕ(f ∗ τt(g))dt =  �  G  �  G  f(y) g(x)  �  R  �ψ(t) ϕ(yαt(x))dtdxdy  =  �  G  �  G  f(y) g(x)  �  R  �ψ(t + iβ)ϕ(αt(x)y)dtdxdy =  �  R  �ψ(t + iβ) ωϕ(τt(g) ∗ f)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  16  Suppose conversely that ωϕ is a (τ, β)-KMS state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Then [BR96, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='4] implies that there  exists a unique σ-weakly continuous one-parameter group of automorphisms �τ of πωϕ(C∗(G))′′  such that �τt(πωϕ(A)) = πωϕ(τt(A))  for all  A ∈ C∗(G) and the state  �ωϕ : πωϕ(C∗(G))′′ → C, A �→ ⟨Ωϕ, AΩϕ⟩  is a (�τ, β)-KMS state in the sense of a W*-dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' In particular  �  G  f(x)�τt(πϕ(x))dx = (�τt ◦ πωϕ)(f) = πωϕ(τt(f)) =  �  G  f(x)πϕ(αt(x))dx,  implies that �τt(πϕ(x)) = πϕ(αt(x)), for all x ∈ G, as one can approximate with left shifts of a  Dirac net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Therefore  ϕ(xαt(y)) = ⟨Ωϕ, πϕ(x)πϕ(αt(y))Ωϕ⟩ = ⟨Ωϕ, πϕ(x)�τt(πϕ(y))Ωϕ⟩ = �ωϕ(πϕ(x)�τt(πϕ(y)))  follows which proves that ϕ is an (α, β)-KMS state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Now we have translated all the necessary information from KMS states of groups to KMS states  of a C*-algebra to prove the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Let G be a locally compact topological group and α : R → Aut(G) be a strongly  continuous one-parameter group of automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Then the set Kβ ⊆ S(G) of (α, β)-KMS  states satisfies  (i) Kβ is convex and weak-∗-compact with respect to the subspace topology of  S(G) ⊆ L∞(G) ∼= L1(G)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  (ii) Kβ is a simplex (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  (iii) ϕ ∈ Kβ is an extremal point of Kβ if and only if ϕ is a factor state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  (iv) Kβ = Conv(Ext(Kβ)), where Ext(Kβ) are the extremal points in Kβ and the closure is  with respect to the weak-∗-topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3 imply that we may view Kβ as the set of (τ, β)-KMS  states of A := C∗(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Then [BR96, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' 116] shows that the (τ, β)-KMS states are in one-to-one  correspondence to the (τ+, β)-KMS states of the unitization A+ := A ⊕ C, where τ+ acts on A+  via  τ+ : R × A+ → A+,  (t, (A, λ)) �→ (τt(A), λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Then Theorem [BR87, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='30] implies (i)-(iii) for the subset of (τ+, β)-KMS states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Since  the correspondence was established via the affine linear homeomorphism with respect to the  weak-∗-topology given by  Kβ ∋ ω �→ ω+ : (A, λ) �→ ω(A) + λ,  it follows that (i)-(iii) also hold for Kβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Then (iv) is a direct consequence of (i) and the Krein-  Milman Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  17  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' For the compact convex subset Kβ of the locally convex topological vector space  C∗(G)′ equipped with the weak-∗-topology [BR87, Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1] provides a decomposition  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' For a given Radon probability measure µ on Kβ, [BR87, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1] proves that there  exists a unique point b(µ) =  �  K ω dµ(ω) ∈ Kβ called the barycenter of µ, such that  f(b(µ)) = µ(f) :=  �  K  f(ω)dµ(ω),  for all affine linear continuous functions f : Kβ → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Then [BR87, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='15] establishes the  importance of Kβ being a simplex since it implies that every ω ∈ K is the barycenter of a unique  normalized Radon measure µω maximal with respect to the partial order ≻ defined by  µ ≻ ν  :⇔  µ(f) ≥ ν(f),  for all f : K → R continuous and convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  For separable locally compact topological groups G the convolution algebra L1(G) is separa-  ble which shows that C∗(G) is a separable C*-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  In particular, Kβ is metrizable and  thus [BR87, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='11] implies that µω is supported on the extremal points of Kβ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  µω(Ext(Kβ)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' We thus obtain for each ω ∈ Kβ a unique decomposition  ω =  �  Ext(Kβ)  ω′dµω(ω′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  For f ∈ L1(G), the function Ext(Kβ)×G ∋ (ω, x) �→ f(x)ϕω(x) is clearly integrable, the measure  space Ext(Kβ) × G is σ-finite and thus from Fubini’s theorem, one obtains, for all ω0 ∈ Kβ, that  ω0(f) =  �  Ext(Kβ)  �  G  f(x)ϕω(x)dxdµω0(ω) =  �  G  f(x)  �  Ext(Kβ)  ϕω(x)dµω0(ω)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  This implies that the state ϕω0 of G corresponding to ω0 is given by  ϕω0(x) =  �  Ext(Kβ)  ϕω(x)dµω0(ω),  for  x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Since L1(G) ⊆ C∗(G) is dense, it follows the evaluation functionals  δf : Kβ → C,  ω �→ ω(f),  for  f ∈ L1(G),  separate the points and thus the algebra generated by these functionals is dense in C(Kβ) with  respect to the maximum norm by the Theorem of Stone-Weierstrass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' This proves that µω0 is  already uniquely determined by these functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' As the set Kβ is convex and weak-∗-closed  every such measure defines an (α, β)-KMS state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Hence we have proven the following proposition:  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Let G be a separable locally compact topological group, α : R → Aut(G) be  a strongly continuous one-parameter group and β ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' For each ϕ0 ∈ Kβ, there exists a Radon  probability measure µ on Ext(Kβ) uniquely determined by  ϕ0(x) =  �  Ext(Kβ)  ϕ(x)dµ(ϕ),  for all  x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Conversely every such measure defines an (α, β)-KMS state of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  18  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3  Semi-finite KMS states of topological groups  Consider an (α, β)-KMS state ϕ of a topological group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  If Mϕ = πϕ(G)′′ is a semifinite  von Neumann algebra, then there exists a unique faithful semi-finite normal trace τ on Mϕ  (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' [Tak02, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Since the faithful normal state ωϕ is invariant under the modular  automorphism group στ  t = idMϕ of the trace τ, it follows from the Radon-Nikodym Theorem  [Tak03, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='6] that there exists a unique non-singular positive self-adjoint operator h affiliated  to Mϕ such that ωϕ(A) = τ(hA), for A ∈ Mϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Then [Tak03, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='11] implies that the  modular automorphism group of ωϕ is given by  Ad(h−it/β) = Ad(eitH),  for  H = − 1  β log(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Furthermore one has τ(h) = τ(e−βH) = ωϕ(1) = 1 and ϕ is given by  ϕ(g) = τ(πϕ(g)e−βH)  τ(e−βH)  ,  for  g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Suppose conversely that π is an α-covariant cyclic unitary representation such that π(G)′′ is semi-  finite and for which there exists a non-singular implementing operator H of α affiliated to π(G)′′  which satisfies τ(e−βH) = 1, for the unique faithful semi-finite normal trace τ on Mϕ = π(G)′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  We want to show that this defines an (α, β)-KMS state of G by constructing a (σ, β)-KMS state,  where σt(A) := eitHAe−itH for A ∈ Mϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' To do this, observe that [Tak02, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='22] implies  that the semi-finite von Neumann algebra Mϕ ⊗ Mop  ϕ acts on L2(Mϕ, τ), which is defined as  the completion of the complex pre-Hilbert space  {A ∈ Mϕ | τ(|A|2) < ∞}  with respect to the scalar product ⟨A, B⟩ := τ(A∗B), by left and right multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' If one  denotes  λ : Mϕ → B(L2(Mϕ, τ)),  λ(a)x = ax,  for  τ(|x|2) < ∞  and  ρ : Mop  ϕ → B(L2(Mϕ, τ)),  ρ(a)x = xa,  for  τ(|x|2) < ∞,  one has that λ(Mϕ)′ = ρ(Mϕ)′′ and ρ(Mϕ)′ = λ(Mϕ)′′ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' [Tak02, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  In particular if Ω = e− 1  2 βH, then Ω ∈ L2(Mϕ, τ) is non-singular and self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' This implies  that the vector Ω is cyclic and separating for Mϕ because by the Range-Kernel Lemma one has  R(Ω) = ker(Ω∗) = ker(Ω) =  {0} and therefore AΩ = 0 implies A = 0 and ΩB = 0 implies  (ΩB)∗ = B∗Ω = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' B∗ = 0 and thus B = 0, for A, B ∈ Mϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  The corresponding modular objects (∆, J) are given on λ(M)Ω by J(AΩ) = ΩA∗ and ∆(AΩ) =  ΩA, for A ∈ Ω and thus the corresponding modular automorphism group is given by  σt : M → M,  A �→ eitHAe−itH,  for  t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  This implies that the state  ω : M → C,  A �→ τ(Ae−βH)  τ(e−βH) = ⟨Ω, AΩ⟩  ∥Ω∥2  is a (σ, β)-KMS state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' This proves that ϕ : G → C, g �→ ω(π(g)) is an (α, β)-KMS state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Hence  one has the following proposition:  19  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Let G be a topological group,  α : R → Aut(G) be a strongly continuous  one-parameter group of automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' There is a one-to-one correspondence between (α, β)-  KMS states ϕ such that πϕ(G)′′ is a semi-finite von Neumann algebras and α-covariant cyclic  unitary representation π such that πϕ(G)′′ is a semi-finite for which there exists an implementing  operator H of α affiliated to π(G)′′ which satisfies τ(e−βH) = 1, for the unique faithful semi-finite  normal trace τ on π(G)′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' If ϕ is a factorial type I (α, β)-KMS state, then Mϕ = πϕ(G)′′ is a type I  von Neumann algebra and therefore there exists a Hilbert space H and an isomorphism of von  Neumann algebras Φ : Mϕ → B(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' The unitary representation ρ := Φ ◦ πϕ is irreducible since  the σ-weak closure of spanC(ρ(G)) is given by ρ(G)′′ = Φ(Mϕ) = B(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' This proves that there  exists an operator H on H such that  eitHρ(g)e−itH = ρ(αt(g))  and  tr(e−βH) < ∞  since the GNS-representation of ϕ is α-covariant for the modular automorphism group, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  ∆itπϕ(g)∆−it = πϕ(αt(g)) and Φ ◦ πϕ = ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' In particular  ϕ : G → C,  tr(ρ(g)e−βH)  tr(e−βH)  ,  for  g ∈ G,  is a Gibbs state corresponding to the irreducible unitary representation ρ of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Since the rep-  resentation ρ is unique, the implementing operator H is unique up to the addition of a real  scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' By [Tak03, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='14] a von Neumann algebra M is semi-finite if and only if  for every faithful semi-finite normal weight ψ, the corresponding modular automorphism group  σψ  t is inner in the sense that there exists a self-adjoint operator H affiliated to M such that  σψ  t = Ad(eitH) as automorphisms of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' For KMS states of topological groups, this innerness of  the modular automorphism group translates to the extension of the factorial KMS state to the  larger group G♭ := G ⋊α R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Let ϕ be an (α, β)-KMS state such that πϕ(G)′′ is a semi-finite von  Neumann algebra and H the corresponding implementing operator affiliated to Mϕ := πϕ(G)′′  satisfying τ(e−βH) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' It follows that the unitary representation  π♭  ϕ : G♭ = G ⋊α R → U(H),  (g, t) �→ πϕ(g)eitH  is a semi-finite representation since eitH ∈ Mϕ implies π♭  ϕ(G ⋊α R)′′ = πϕ(G)′′ = Mϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  In  particular  ϕ♭ : G♭ = G ⋊α R → C,  (g, t) �→ τ(π♭  ϕ(g, t)e−βH)  τ(e−βH)  is an (α♭, β)-KMS state of G♭ = G ⋊α R, where α♭ is the one-parameter group of inner automor-  phisms of G♭ defined by conjugation with (e, t), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  α♭(g, s) = (e, t)(g, s)(e, t)−1 = (αt(g), s),  for  g ∈ G, s, t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Note that the modular automorphism group which acts on Hϕ ∼= L2(Mϕ, τ) by conjugation with  eitH and thus leaves the cyclic vector Ω =  e  − 1  2 βH  √  tr(e−βH) invariant, but the left multiplication by  eitH does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  20  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Let β > 0 and suppose G is a finite dimensional Lie group and α : R → Aut(G)  is given by αt(g) = exp(tX)g exp(−tX), for some fixed X ∈ g := L(G) and g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Then the  extremal type I (α, β)-KMS states are the extremal Gibbs states  ϕ(g) = tr(ρ(g)eiβ∂ρ(X))  tr(eiβ∂ρ(X))  ,  where ρ is an irreducible representation of G such that eiβ∂ρ(X) is of trace class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' In particular,  if G is simply connected, then there is a one-to-one correspondence between  (i) extremal type I (α, β)-KMS states  (ii) equivalence classes of irreducible unitary highest weight representations of admissible quo-  tients q : g ↠ g1 with respect to an admissible positive system ∆+  1 of g1 such that q(X) is  conjugate to Cmax(∆+  1,p)◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' This follows immediately from Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2 since ρ(exp(tX)) = et∂ρ(X) = eit(−i∂ρ(X))  implies that the corresponding implementing operator H, which is unique up to addition by a  real scalar, is w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' given by −i∂ρ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Therefore eiβ∂ρ(X) is of trace class and Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='9  applies if G is simply connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' If G is a finite dimensional Lie group and the action α : R → Aut(G) is not  inner, then one considers G♭ = G ⋊α R with Lie algebra g♭ := L(G♭).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' In order to classify all  representations of Gibbs type, it suffices to classify all admissible quotients g′ of g♭ and check  if the image of (0, 1) ∈ g♭ = g ⋊ R under the quotient map q : g♭ → g′ lies in the interior of  Cmax(∆+  p )◦, for an admissible positive system ∆+ of g′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' If one then classifies the unitary highest  weight representations of g′ with respect to ∆+, one obtains in the same manner as above a  representation of Gibbs type of G, and all representations of Gibbs type are obtained in this  manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  3  Perspectives and open problems  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1  Extensions of KMS states of G to KMS states of G♭  To completely understand how the extension of a topological group G to G♭ affects the study  of KMS states, it would be interesting to know if every (α, β)-KMS state of G extends to an  (α♭, β)-KMS state of G♭ = G ⋊α R and if every KMS state of G♭ arises in this manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' We have  seen in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3 that semi-finite (α, β)-KMS states of G extend to (α♭, β)-KMS states of G♭.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  For a not necessarily semi-finite (α, β)-KMS state ϕ of G with GNS-representation (πϕ, Hϕ, Ωϕ)  and M := πϕ(G)′′, it is not clear if such KMS extensions always exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' It might even be possible  that this extension property already implies that the state is semi-finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  We considered the crossed product M⋊σR, where σ : R → Aut(M) is the modular automorphism  group of the associated state ωϕ of M ⊆ B(Hϕ) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' [Tak03, Section X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' This crossed product  is defined as the von Neumann algebra generated by the sets of bounded linear operators on  L2(R, Hϕ) given by  (πσ(A)f)(t) := σ−t(A)f(t)  and  (λ(s)f)(t) := f(t − s),  for A ∈ M, s, t ∈ R, f ∈ L2(R, Hϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Then one can consider the associated dual weight  �ωϕ of M ⋊σ R and [Tak03, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='17(ii)] implies that the modular automorphism group  21  �σ : R → Aut(M ⋊σ R) of the dual weight �ωϕ satisfies  �σt(πσ(A)) = πσ(σt(A)),  and  �σt(λ(s)) = λ(s),  for  A ∈ M, s, t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  (2)  For the unitary representation ρ of G♭ on L2(R, Hϕ) defined as ρ(g, t) := πσ(πϕ(g))λ(t), the  relation πϕ(αt(g))) = σt(πϕ(g)) and (2) imply  ρ(α♭  t(g, s)) = ρ(αt(g), s) = �σt (ρ(g, s)) ,  for  g ∈ G, s, t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  But dual weight �ωϕ of a state does not define a state of the crossed product and thus one cannot  naively define the extension of ϕ as ϕ♭(g, t) := �ωϕ(ρ(g, t)) for g ∈ G, t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' This does not work  as the dual weight does not define a state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' To see that, observe that [Tak03, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='17(i)]  implies that, for a compactly supported M-valued continuous function f ∈ Cc(R, M) and for  the representation �πσ of Cc(R, M) on L2(R, Hϕ) defined as  �πσ(f) :=  �  R  λ(s)πσ(f(s))ds,  one has  �ωϕ(�πσ(f)) = ωϕ(f(e)) = ⟨Ωϕ, f(e)Ωϕ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  As �πσ(Cc(R, M)) generates M ⋊σ R (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' [Tak03, Lemma X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='10(iii)]), this implies that the dual  weight does not define a state of the cross product M⋊σ R since delta distributions do not define  states as the passage from Cc(R, M) to L∞(R, M) suggests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Therefore this attempt does not  seem to work but it illustrates a few key structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2  Factorial Type II KMS states of Lie groups  We are not aware of how type II factor representations π of finite dimensional Lie groups G  look like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' For us, the more natural context of type II factor representation is given by inductive  limits of Lie groups which then act for example an the hyperfinite II1 factor defined as a infinite  product of two by two matrix algebras (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  [Po67]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  For KMS states of the inductive limit  group U(∞) = lim  −→n U(n) this has been done in [St78] whereas general factor representations of  U(∞) are studied in [SV75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' A survey of these results is given in [SV82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Nonetheless it would  still be interesting to know what conditions does τ(eiβ∂π(X)) < ∞ implies for finite dimensional  Lie groups G, representation π and generator X ∈ g?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Clearly, the spectrum of H = −i∂π(X)  does not need to be semi-bounded anymore as in type II factors there exists a sequence of  non-zero mutually orthogonal projections {pn}n∈N such that 0 < τ(pn) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Instead of the  semi-boundedness of spec(H) on has that the spectrum decreases exponentially fast in the sense  that  �  R e−βxdτ(PH(x)) < ∞ holds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' τ(PH((−∞, t0])) ≤ eβt0τ(e−βH) goes to 0 exponentially  as t → −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  A  Appendix  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1  Compactly embedded Cartan subalgebras and admissible Lie al-  gebras  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' We call a subalgebra a ⊆ g compactly embedded if ⟨eadg(a)⟩ ⊆ GL(g) is  compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' We call an element x ∈ g elliptic if the subalgebra R · x ⊆ g is compactly embedded  and denote with comp(g) the subset of elliptic elements in g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' [Ne00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2]):  If t is a compactly embedded Cartan subalgebra, then it is nilpotent and compact, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  22  Therefore the subset adgC(tC) ⊆ gl(gC) is an abelian subalgebra consisting of diagonalizable  elements since the compact group INNg(t) is a compact subgroup of the compact subgroup  O(g, ⟨·, ·⟩) of gl(g) for some scalar product on g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Hence one obtains a root decomposition  gC = tC ⊕  �  α∈∆  gα  C,  where gα  C := {X ∈ gC | [t, X] = α(t)X, for all t ∈ tC} and ∆ := {α ∈ t∗  C | gα  C ̸= {0}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' On the  complex Lie algebra gC we define the involutive antihomomorphism  (X + iY )∗ := −X + iY  for  X, Y ∈ g  and observe that g = {X ∈ gC | X∗ = −X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' It follows from [Ne00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2] that for  Z ∈ gα  C, one has Z∗ ∈ g−α  C  and [Z, Z∗] ∈ it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Define  gC(Z) := spanC{Z, Z∗, [Z, Z∗]}  and  g(Z) := gC(Z) ∩ g  For roots α ∈ ∆ , one now distinguishes between the following cases:  (NS) There exists an element Z ∈ gα  C such that α([Z, Z∗]) < 0 (the non-compact simple type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Then g(Z) ∼= sl(2, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  (CS) There exists an element Z ∈ gα  C such that α([Z, Z∗]) > 0 (the compact simple type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Then  g(Z) ∼= su(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  (N) α([Z, Z∗]) = 0 for all Z ∈ gα  C and there exists Z ∈ gα  C such that [Z, Z∗] ̸= 0 (the nilpotent  type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Then g(Z) is isomorphic to the three-dimensional Heisenberg algebra h3(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  (A) [Z, Z∗] = 0 for all Z ∈ gα  C (the abelian type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Then g(Z) ∼= R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  That only these four cases occur follows from [Ne00, Lemma VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3] which implies dimCgα  C = 1,  for all α ∈ ∆ of simple type, and thus conditions (NS) and (CS) hold for all Z ∈ gα  C if they hold  for one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' [Ne00, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='4] and [Ne00, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='6]):  Let t ⊆ g be a compactly embedded Cartan subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  (i) We call a root α ∈ ∆ semisimple if it is of simple type and denote the set of semisimple  roots by ∆s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' The set of solvable roots is denoted by ∆r := ∆ \\ ∆k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  (ii) We call a root α ∈ ∆ compact if there exists Z ∈ gα  C such that α([Z, Z∗]) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' All other  roots are called non-compact and the set of all non-compact roots is denoted with ∆p  and the set of compact roots as ∆k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' We also define the non-compact semisimple roots  ∆p,s := ∆p ∩ ∆s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  (iii) A subset ∆+ ⊆ ∆ is called a positive system if there exists a regular element X0 ∈ it such  that  ∆+ = {α ∈ ∆ | α(X0) > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  If ∆+ is such a system of positive roots we define ∆+  k , ∆+  p , ∆+  p,s and ∆+  r as the intersection  of the respective subset with ∆+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  23  (iv) We call a positive system ∆+ adapted if for all α ∈ ∆k and β ∈ ∆+  p one has  β(X0) > α(X0),  for some X0 defining ∆+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' [Ne00, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='6]):  Let t ⊆ g be a compactly embedded Cartan subalgebra and ∆+ be a corresponding system of  positive roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' We associate to the positive system of non-compact roots ∆+  p the convex cones  Cmin(∆+  p ) := cone({i[Zα, Z∗  α] | Zα ∈ gα  C, α ∈ ∆+  p } ⊆ t  and  Cmax(∆+  p ) := (i∆+  p )⋆ := {X ∈ t | iα(X) ≥ 0 for all α ∈ ∆+  p }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  If it is clear which system of positive non-compact roots ∆+  p is considered, then we simply write  Cmin, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Cmax, for Cmin(∆+  p ), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Cmax(∆+  p ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' We call a Lie algebra g admissible if g contains an invariant closed generating  convex subset C which does not contain any affine line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' (Sandwich Theorem [Ne00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='8])  Let t ⊆ g be a compactly embedded Cartan subalgebra and ∅ ̸= C ⊆ g a closed generating invariant  elliptic convex subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Then there exists a uniquely determined adapted positive system ∆+  p of  non-compact roots such thatC ∩ t ⊆ Cmax(∆+  p ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Moreover, we also have Cmin(∆+  p ) ⊆ lim(C ∩ t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  If, in addition, C is a cone, then this means that  Cmin(∆+  p ) ⊆ C ∩ t ⊆ Cmax(∆+  p ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2  Convex geometric and representation-theoretic concepts  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' [Ne00, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1] and [Ne00, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='6]):  Let V be a finite-dimensional real vector space and W ⊆ V be a closed convex cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  (i) For a subset E ⊆ V , we put  B(E) := {ω ∈ V ∗ | inf ω(E) > −∞}  and  E⋆ := {ω ∈ V ∗ | ω(E) ⊆ [0, ∞)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  (ii) We define, for a convex subset C ⊆ V , the recession cone lim(C) := {v ∈ V | v + C ⊆ C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  (iii) For I ⊆ V ∗, we define I⊥ := {v ∈ V | ω(v) = 0 for all ω ∈ I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  (iv) For a subset E ⊆ V , we define algint(E) as the interior of E with respect to the affine  subspace spanned by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' We remark the following:  (i) E⋆ is always closed but B(E) is not closed in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  (ii) One has B  �  conv(E)  �  = B(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  (iii) B(E)⋆ = lim(E) for E convex (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' [Ne00, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='14])  24  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' [Ne00, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2]):  Let (π, H) be a unitary representation of a Lie group G with Lie algebra L(G) = g on the  Hilbert space H and let H∞ be the dense subspace of H of smooth vectors on which g acts as  dπ(X)(v) :=  d  dt  ��  t=0 π(expG(tX))v for X ∈ g,  v ∈ H∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Then we define the momentum map of  the unitary representation π as  Φπ : P(H∞) := (H∞ \\ {0})/C× → g∗,  Φπ([v])(X) := −i⟨v, dπ(X)v⟩  ⟨v, v⟩  and the momentum set Iπ as the closed convex hull of the image of Φπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' If (π, H) is a unitary representation of a Lie group G we consider the cone  B(Iπ) = {X ∈ g | inf⟨Iπ, X⟩ > −∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  This cone is invariant since Iπ is invariant under the coadjoint action (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' [Ne00, Lemma X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='3])  and [Ne00, Lemma X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='6] implies that  B(Iπ) = {X ∈ g | sup(spec(i · dπ(X))) < ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' We call a pair (g, ∗) an involutive Lie algebra if g is a complex Lie algebra  and ∗ : g → g is an antilinear antiautomorphism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' [x∗, y∗] = −[x, y]∗, holds for all x, y ∈ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' We say that an involutive Lie algebra (g, ∗) has a root decomposition if there  exists a ∗-invariant subalgebra h ⊆ g and a subset ∆ ⊆ g∗ such that α(h∗) = α(h) and  g = h ⊕  �  α∈∆  gα,  for gα := {X ∈ g | ad(h)(X) = α(h)X for all h ∈ h} ̸= {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Let (g, ∗) be an involutive Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' We call a g-module V unitary if there  exists a positive definite hermitian form h : V × V → C that is also contravariant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' it satisfies  h(X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='v, w) = h(v, X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='w) for all X ∈ g and v, w ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Let (g, ∗) be an involutive Lie algebra with root decomposition and ∆+ ⊆ ∆  be a positive system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' We call a g-module V a highest weight module with respect to ∆+ if there  exists a linear functional  λ : b := h ⊕  �  α∈∆+  gα → C  and a cyclic vector v ∈ V such that X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='v = λ(X)v holds for all X ∈ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' We call λ the highest  weight of V and v a highest weight vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' An irreducible unitary representation (π, H) of a connected Lie group G is  called a unitary highest weight representation if the Lie algebra g of G contains a compactly  embedded Cartan subalgebra t ⊆ g and if k ⊆ g is the unique maximal compactly embedded  subalgebra containing t (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  [Ne00, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='25]) and K := exp(k), then the space HK,∞  of smooth K-finite vectors is a highest weight module for the involutive Lie algebra gC with  involution (X + iY )∗ = −X + iY , for X, Y ∈ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  25  References  [BR87]  Bratteli,O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=', and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Robinson, “Operator Algebras and Quantum Statistical Me-  chanics I,” 2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=', Texts and Monographs in Physics, Springer-Verlag, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  [BR96]  Bratteli,O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=', and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Robinson, “Operator Algebras and Quantum Statistical Me-  chanics II,” 2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=', Texts and Monographs in Physics, Springer-Verlag, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  [DE09]  Anton Deitmar, Siegfried Echterhoff, “Principles of Harmonic Analysis,” 1st ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=',  Springer-Verlag 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  [HM06]  Karl H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Hofmann and Sidney A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Morris, “The Structure of Compact Groups,” A  primer for the student—a handbook for the expert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Second revised and augmented  edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' De Gruyter Studies in Mathematics, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Walter de Gruyter & Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=', Berlin,  2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  [HN12]  Karl-Hermann Neeb, Joachim Hilgert, “Structure and Geometry of Lie Groups,” 1st  ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=', Springer Monographs in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Springer, New York, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  [GN]  Helge Glöckner, Karl-Hermann Neeb, “Infinite-dimensional Lie Groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' General  Theory and Main Examples,” in preparation  [GN00]  Helge Glöckner, Karl-Hermann Neeb, “Minimally Almost Periodic Abelian Groups  and Commutative W*-algebras,” Research & Exposition in Mathematics, Helder-  mann Verlag Berlin, 2000  [LT18]  Roberto Longo, Yoh Tanimoto, “Rotational KMS states and type I conformal nets,”  [Mo80]  Calvin C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Moore, “The Mautner phenomenon for general unitary representations,”  Pac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Math 86 (19 (1980), 155-169  [Ne00]  Karl-Hermann Neeb, “Holomorphy and convexity in lie theory,” De Gruyter Expo-  sitions in Mathematics, 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Walter de Gruyter & Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=', Berlin, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  [Ni22]  Milan Niestijl, “Generalized Positive Energy Representation of Groups of Jets,”  arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='10390 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='RT]  [Po67]  Robert T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Powers, “Representations of Uniformly Hyperfinite Algebras and Their  Associated von Neumann Rings,” Annals of Mathematics , Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=', 1967, Second Series,  Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
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+page_content=', 1967), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' 138-171  [St78]  Stratila, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
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+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content='  235:1 (1978), 87-110  [SV75]  Stratila, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=', and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Voiculescu, “Representations of AF algebras and of the group  U(∞),” Lecture Notes in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' 486, Springer Verlag, Berlin, 1975  [SV82]  Stratila, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=', and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Voiculescu, “A survey on representation of the unitary group  U(∞),” in “Spectral Theory”, Banach Center Publications 8, PWN-Polish Scientific  Publishers, Warzaw, 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' 415-434  [Tak02]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Takesaki, “Theory of operator algebras I,” (English summary) Reprint of the first  (1979) edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Encyclopedia of Mathematical Sciences, 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Operator Algebras and  Non-commutative Geometry, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Springer-Verlag, Berlin, 2002  26  [Tak03]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Takesaki, “Theory of operator algebras II,” (English summary) Encyclopedia of  Mathematical Sciences, 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Operator Algebras and Non-commutative Geometry,  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
+page_content=' Springer-Verlag, Berlin, 2003  27' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfzgUS/content/2301.03444v1.pdf'}
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+Preprint 11 January 2023
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+The impact of human expert visual inspection on the discovery of strong
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+Melo,17 Guillaume Mahler,26,27 Ricardo L. C. Ogando,28 Frédéric Courbin,9 Alexander Fritz,29 Aniruddh
+Herle,17,30 Javier A. Acevedo Barroso,9 Raoul Cañameras,17 Claude Cornen,31 Birendra Dhanasingham,32
+Karl Glazebrook,33 Michael N. Martinez,13 Dan Ryczanowski,34 Elodie Savary,9 Filipe Góis-Silva,28 L.
+Arturo Ureña-López,35 Matthew P. Wiesner,36 Joshua Wilde,37 Gabriel Valim Calçada,28 Rémi Cabanac,38
+Yue Pan,6 Isaac Sierra,6 Giulia Despali,39 Micaele V. Cavalcante-Gomes,40,28 Christine Macmillan,30 Jacob
+Maresca,25 Aleksandra Grudskaia,17 Jackson H. O’Donnell,41,42 Eric Paic,9 Anna Niemiec,26,27 Lucia F.
+de la Bella,1 Jane Bromley,42 Devon M. Williams.43
+Affiliations are listed at the end of the paper
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+We investigate the ability of human ’expert’ classifiers to identify strong gravitational lens candidates in Dark Energy Survey
+like imaging. We recruited a total of 55 people that completed more than 25% of the project. During the classification task, we
+present to the participants 1489 images. The sample contains a variety of data including lens simulations, real lenses, non-lens
+examples, and unlabeled data. We find that experts are extremely good at finding bright, well-resolved Einstein rings, whilst
+arcs with g-band signal-to-noise less than ∼25 or Einstein radii less than ∼1.2 times the seeing are rarely recovered. Very few
+non-lenses are scored highly. There is substantial variation in the performance of individual classifiers, but they do not appear
+to depend on the classifier’s experience, confidence or academic position. These variations can be mitigated with a team of 6 or
+more independent classifiers. Our results give confidence that humans are a reliable pruning step for lens candidates, providing
+pure and quantifiably complete samples for follow-up studies.
+Key words: gravitational lensing: strong
+1 INTRODUCTION
+Strong gravitational lensing offers a rare opportunity to directly
+weight objects in the distant Universe. Mass warps space-time and
+in strong lens systems, the surface density is so great that multiple
+images of a background source are observed.
+Strong lenses enable a magnified view of the high redshift Uni-
+verse (Christensen et al. 2012; Stark et al. 2015; Shu et al. 2016;
+Ebeling et al. 2018; Shu et al. 2022), a direct probe of dark matter in
+galaxies, clusters and substructures (Oguri et al. 2002; Vegetti et al.
+2010; Jiménez-Vicente et al. 2015; Nierenberg et al. 2017; Gilman
+et al. 2020) and a geometrical probe of the cosmological parameters
+(Collett & Auger 2014; Bonvin et al. 2017; Wong et al. 2020). Almost
+all applications of strong lensing are limited by sample size, but the
+⋆ karina.rojas@port.ac.uk
+† thomas.collett@port.ac.uk
+era of deep wide area surveys offers an opportunity to grow strong
+lens samples a hundredfold (Collett 2015).
+As astronomy enters the era of billion object surveys, sophisti-
+cated methods for discovering strong lenses have been developed
+(e.g. Lanusse et al. 2018; Avestruz et al. 2019) and applied (Jacobs
+et al. 2017, 2019b,a; Rojas et al. 2022; Savary et al. 2022; Petrillo
+et al. 2017, 2019). These methods have been extremely successful at
+identifying candidate lenses, but the rarity of lenses means that even
+classifiers with 99.99% accuracy produce 100 false positives for ev-
+ery true lens. The gold standard for confirming a lens is spectroscopic
+confirmation of multiple redshifts, but reducing false positive rates
+is needed for a spectroscopic confirmation campaign to be viable. In
+most cases, a human expert step is used as the final stage filtering
+step to remove false positives.
+Expert human classification has been very successful at identify-
+ing the best strong lens candidates. For example, Tran et al. (2022)
+recently targeted 79 lens candidates, spectroscopically confirming 53
+© 2022 The Authors
+arXiv:2301.03670v1  [astro-ph.GA]  9 Jan 2023
+
+2
+K. Rojas et al.
+and definitely ruling out only 4 1. However, introducing a human into
+any classification task is likely to bring in selection biases: it is much
+easier to identify a bright arc that is well resolved from the lensing
+galaxy and significantly different in colour.
+The primary purpose of this work is to calibrate and understand
+how introducing human experts biases lens searches. In addition,
+we aim to understand how the choice of "experts" impacts the lens
+candidate sample selected and how search teams can mitigate the
+biases of their members. We set out to answer the following questions:
+• What are the properties of lensing systems that human experts
+identify reliably as lenses, and what do they miss?
+• Do human experts confuse non-lenses for lenses?
+• How does expert classification depend on the experience and
+confidence of the experts?
+• How reliable are individual classifications?
+• When ranking lens candidates, what do the scores of teams of
+experts mean?
+• How should lens searchers best build an expert team to classify
+their candidates?
+In Section 2 we will lay out our experiment, data and expert
+participants. In Section 3 we investigate how subsets of the lens
+candidates are scored by the ensemble of our users, enabling us
+to understand the selection biases of our experts. In Section 4 we
+investigate how individual users perform on the classification task.
+We summarize the conclusions of this work in Section 5.
+2 THE EXPERIMENT
+Our experiment is designed to understand human expert biases when
+performing visual inspection of strong lens candidates. With experts
+we refer to any person involved in strong lensing research, with
+an academic status from masters student to Professor (or similar).
+Additionally, we invited a small number of citizen scientists from
+outside the academic strong lensing community. Several of these are
+experienced users from the Spacewarps project. The details of our
+invitation to be part of this experiment can be found in Appendix A.
+We used the citizen science web portal Zoouniverse2 to serve a
+sample of 1489 images for classification.
+The users were asked to choose the best description for the object
+displayed from four options: (1) Certain lens (> 90%), (2) Probable
+lens (50 − 90%), (3) Probably not a lens (2 − 50%), and (4) Very
+unlikely (< 1%). We included percentages of confidence to be a lens
+to avoid semantic uncertainty 3.
+We designed our experiment to closely mimic a real strong lens
+classification task. In real lens searches, there are no tutorials, and
+there is limited prior knowledge of the completeness and purity of
+the sample to be classified. We therefore avoided offering further
+guidelines about the composition of the data sets to be classified. For
+the same reason, we did not offer a tutorial nor examples as would
+be usual in a citizen science project.
+1 20 of the remaining objects are likely strong lenses but redshifts weren’t
+obtained for both lens and source in 20 of the systems. 2 systems yielded no
+redshifts.
+2 https://www.zooniverse.org/projects/krojas26/
+experts-visual-inspection-experiment
+3 Formally, these percentages cannot be probabilities since we did not provide
+the users with prior information on the composition of the sample being
+classified.
+Default
+Blue
+Sqrt
+Figure 1. Example of a DES cutout of 50 × 50 pixels 13′′ × 13′′ of an object
+displayed in the three different scales presented to the users in the experiment.
+2.1 The data
+The image cutouts are from the Dark Energy Survey (DES). DES uses
+the Blanco 4-m telescope and the Dark Energy Camera (DECam,
+Honscheid & DePoy 2008; Flaugher et al. 2015) located at Cerro
+Tololo Inter-American Observatory (CTIO), Chile. The observations
+are performed in the optical grizY bands. We used gri-bands to
+produce colour composite 50 × 50 pixels (∼ 13′′ × 13′′) images
+centering the object of interest in the middle. The images have a
+typical 5σ depth of 23.72, 23.35, 22.88 in gri respectively.
+We simultaneously display three different colour scalings of each
+image to facilitate the recognition of the different features in the
+stamp (see Fig. 1). The three imaging scalings are: Default, blue and
+sqrt, and they are described in Appendix B.
+Since we aimed to assess the classification skills of experts, we
+required both a sufficiently large number of experts to participate
+and sufficient classifications per expert. This creates tension for ex-
+periment design since experts have difficulty engaging with the ex-
+periment if the classification task is too large. We decided that a
+sample size of around 1000 objects (around an hour of time assum-
+ing 2 seconds per classification) was small enough to get significant
+engagement and large enough to provide useful data.
+Given the rarity of real lenses on the sky, a random sample of
+∼1000 objects will not provide useful data. Instead, we designed a
+sample to have a broad variety of data including: simulated lenses;
+real lens candidates; non-lens examples; and unlabeled data. We also
+duplicated 105 cutouts, 15 cutouts from seven out of nine data sets
+excluding the eleven examples from SLACS and the four lenses from
+Rojas et al. (2022). The idea is to investigate the level of consistency
+of individuals when classifying. In total we had 1489 images to
+classify.
+Below we describe the different data sets, which are in 3 main
+categories: lens simulations and lens candidates that are labelled
+as lenses; negative examples that are labelled as non-lenses; and
+unlabeled data. In Tab. 1 we present the name, number of objects and
+category of each data set and in Fig. 2 we present as an example the
+image of four objects of each labelled data set, more details about
+these sets can be found in the following sections.
+2.1.1 Lens simulations and additional lens examples.
+We created two sets of simulations of strong lens systems. Each of
+them contains 160 images and both have a uniform Einstein radius
+distribution between 0.8′′ < θE < 3.0′′.
+The simulations were created using Lenstronomy4 (Birrer &
+Amara 2018; Birrer et al. 2021) and are based on real images for
+the source and the lens. The full procedure is described in Rojas
+4 https://github.com/lenstronomy/lenstronomy
+MNRAS 000, 1–15 (2022)
+
+Expert visual inspection of strong lenses
+3
+Table 1. Data sets presented in the experiment.
+Data set name
+number of objects
+category
+"Bright" simulations
+150
+Lens data set
+Default simulations
+150
+Lens data set
+SLACS
+11
+Lens data set
+R22 lenses
+4
+Lens data set
+LRGs
+150
+Non-lens data set
+CNNs = 0
+150
+Non-lens data set
+Non-lens simulations
+150
+Non-lens data set
+CNN-best
+300
+Unlabeled data
+Random
+300
+Unlabeled data
+et al. (2022) and can be summarized as follows: We pair Luminous
+Red Galaxies (LRGs) from the DES and source galaxies from the
+HST/HSC combined catalogue compiled by Cañameras et al. (2020),
+here the galaxies have the HST/ACS F814W high-resolution (Leau-
+thaud et al. 2007; Scoville et al. 2007; Koekemoer et al. 2007) and the
+colour information from Hyper Suprime Cam (HSC) ultra-deep stack
+images (Aihara et al. 2018). We modelled the mass of the systems
+as a Singular Isothermal Ellipsoid (SIE), which has the following
+parameters: the Einstein radius (θE), Position Angle, the axis ratio
+and the central position. The Einstein radius was calculated using
+the lens and source redshifts and the lens velocity dispersion. We
+inferred the lens galaxy redshift and velocity dispersion using a K-
+Nearest-Neighbors (KNN) algorithm, based on the assumption that
+galaxies with similar gri magnitudes will also have similar redshifts
+and velocity dispersion, this procedure is explained in detail in Rojas
+et al. (2022). The rest of the parameters are derived by fitting an
+elliptical Sérsic profile to the DES r-band image of the LRG. From
+this mass model we calculate the deflection of light, and trace rays
+back onto the source plane. This process is done on an 0.03′′pixel
+grid and then downsampled and PSF matched to the DES cutout of
+the LRG, and the flux scaled to the DES zero point. This simulated
+arc is then added to the DES LRG image to create a simulated lens.
+The first data set is the "Bright simulations". This sample contains
+a selection of simulations used to train the Convolutional Neural
+Network (CNN) in Rojas et al. (2022), with the addition of smaller
+Einstein radius simulations in the range 0.8′′ < θE < 1.2 that were
+not used to train the neural network in that work. In this dataset, the
+magnitude of the sources is boosted by one magnitude brighter than
+the observed HSC sources. This gives a population of bright lensing
+features designed for the CNN to easily learn the properties of strong
+lenses. These simulations should be the easiest for experts to identify
+as strong lenses.
+The second data is the "Default simulations". These were created
+with the same procedure as in Rojas et al. (2022), but without boosting
+the magnitude of the source. In this set of simulations we expect the
+systems to be harder to classify, since the signal-to-noise ratio (SNR)
+will be lower and the sources will stand out less brightly relative to
+the lensing galaxies.
+Additionally to these two simulated sets of lenses we added the
+eleven lenses from the Sloan Lens ACS Survey (SLACS, Auger
+et al. 2009) in the field of view of DES. These are spectroscopically
+confirmed, smaller Einstein radius systems, that are clear in Hubble
+Space Telescope imaging but represent a challenge to identify in
+ground-based resolution. Since small Einstein radius systems are
+expected to dominate in the real Universe (Collett 2015), we include
+the SLACS sample to see if there is any chance to identify these
+systems with visual inspection of ground-based telescopes. Finally,
+four lens candidates from Rojas et al. (2022) categorized with high
+"Bright"
+simulations
+"Bright"
+simulations
+"Bright"
+simulations
+"Bright"
+simulations
+Default
+simulations
+Default
+simulations
+Default
+simulations
+Default
+simulations
+SLACS
+SLACS
+SLACS
+SLACS
+R22 lenses
+R22 lenses
+R22 lenses
+R22 lenses
+LRGs
+LRGs
+LRGs
+LRGs
+CNNs=0,
+CNNs=0,
+CNNs=0,
+CNNs=0,
+Non-lens
+simulations
+Non-lens
+simulations
+Non-lens
+simulations
+Non-lens
+simulations
+Figure 2. Example of objects presented in the different labelled data
+sets."Bright" simulations, Default simulations, SLACS and R22 lenses from
+Rojas et al. (2022) are examples of objects labelled as lenses, while LRGs,
+CNNs=0 and Non-lens simulations are examples of objects labelled as non-
+lenses. The cutouts are 50 × 50 pixels and they are displayed in the default
+scale.
+scores in the "sure lens" list were also shown to the participant to see
+if our classifiers agreed with the authors of Rojas et al. (2022), we
+call this data set "R22 lenses".
+2.1.2 Negative examples.
+We included three data sets with 150 objects each that contained non-
+lens examples. These samples test the classifiers’ ability to reject non-
+lens systems. The first data set is a random sub-sample of the negative
+examples presented in the training set used to train the CNN in Rojas
+et al. (2022), we call this data set "Non-lenses training set" and
+contains LRGs that were not used to create strong lens simulations 5.
+The second data set consisted of a random selection of objects that
+were classified by the CNN with scores near zero. We called this data
+set "CNNs=0" and it contains stamps from the LRG selection that
+are highly improbable to contain any lens feature according to the
+5 it is not impossible that this sample contains a real lens, but it is statistically
+unlikely
+MNRAS 000, 1–15 (2022)
+
+4
+K. Rojas et al.
+CNN. The third data set is called "Non-lens simulations": we follow
+the same procedure as for strong lens simulations, but we disable the
+lensing deflections. Instead, we paint the source nearby to the LRG.
+This is designed to mimic a ’source in front of LRG’ alignment,
+which is a potential false positive for lens classification. This sample
+allows us to evaluate if classifiers are able to distinguish between real
+compact lenses and unlensed blue galaxies close to LRGs.
+2.1.3 Unlabeled data.
+We additionally included two data sets of unlabeled data, each set
+containing 300 objects. The first one contains the 300 best stamps
+graded by the CNN in Rojas et al. (2022), where 6 of them were
+flagged as "Maybe lens" in that work. The objective of this data
+set is to re-do the visual inspection and compare the classifications
+of the authors of Rojas et al. (2022) with the participants in our
+experiment. We call this data set unlabeled as we do not have a
+confirmed classification of the objects displayed, and although this
+data set was previously inspected by another group we do not use this
+information as prior. The second set was created by selecting random
+objects with CNN scores distributed between 0.1 and 0.9 from the
+sample analysed in Rojas et al. (2022). The objective of this is both
+to see if CNN grades and expert grades are correlated and to see if
+a population of high-quality candidates are likely to be missed by
+CNNs.
+2.2 Participants.
+We asked all the participants to complete a google form requesting
+some basic and confidential information that we used to have a more
+deep analysis of this experiment. We asked three multiple choice
+questions. These questions and their options are:
+(i) How many years have you worked in the field of gravitational
+lensing? a. Less than one year, b. Between 1-4 years, c. Between 4-8
+years, d. Between 8-12 years, e. More than 12 years.
+(ii) What is your research status? a. Master student, b. Ph.D. stu-
+dent, c.Postdoc, d.Professor/lecturer/similar, e. Amateur enthusiast6.
+(iii) How confident do you feel classifying lens systems? a.Very
+confident, b. Confident, c. A bit confident, d. Not confident.
+This information was asked so that we could search for correlations
+in the performance of the classifiers.
+A total of 80 people filled in the google form and a total of 69,592
+classifications were made in the project. Some of the users con-
+tributed only a small number of classifications and 15% of them
+did not perform any classification - we discarded the classifications
+of such users. We included all classifications made by users that
+analyzed more than 25% of the sample (370 objects). This cutoff
+leaves 55 classifiers, where 51% of them finished the whole project.
+Then we have a total of 66,835 classifications, with an average of 45
+classifications per object.
+The breakdown of the participants into the categories of academic
+position, years of experience, and confidence are listed in Table 2.
+A point of interest from the information compiled at this point is
+the correlation between confidence and experience. As is expected,
+classifiers with more experience (either a higher academic status or
+years working in the field) feel more confident performing the task
+as we can see in Appendix C.
+6 For better understanding we call this group "Citizen scientist" here.
+Table 2. Number of participants split in the three different categories re-
+quested at the beginning of this experiment.
+Academic status
+N. of participants
+Professors
+16
+Postdocs
+13
+Ph.D. Students
+15
+Master Students
+3
+Citizen Scientist
+8
+Years of experience in the field
+More than 12
+11
+8-12
+8
+4-8
+10
+1-4
+17
+Less than a year
+9
+Confidence
+Very confident
+16
+Confident
+22
+A bit confident
+14
+Not confident
+3
+3 RESULTS A: THE DISCOVERY OF LENSES WITH
+EXPERT VISUAL INSPECTION
+In this section we look at the performance of the ensemble of classi-
+fiers in identifying strong lenses.
+3.1 Scoring objects
+To compute a score for each object classified by our users, we translate
+the four different options into numbers as follows: "Certain lens" = 1,
+"Probable lens" = 2/3, "Probably not lens" = 1/3 and "Very unlikely" =
+0. The score for each object is the mean value of all its classifications.
+This gives every object a score between 0 and 1: objects scoring 1
+are universally considered to be a strong lens and objects scoring 0
+are universally considered non-lenses.
+In Fig. 3 we present the mean score histograms split into the
+various data subsets. Overall we find that the scores of non-lenses
+are low and for many lenses the scores are high.
+We investigated an alternative scoring system that up-weighted
+users with higher classification skill (Marshall et al. 2015) but this
+had no impact on our results (See Appendix D).
+3.2 Scores for images known to contain lenses.
+The left panel in Fig. 3 shows the distribution of scores for the four
+data sets of objects labelled as lenses. Both simulated data sets have
+scores spanning the full range. This is not unsurprising: some of the
+simulations are bright arcs in textbook configurations whereas others
+are extremely faint or are not easily resolved from the lensing galaxy.
+It is not a surprise that the "Bright" simulation set contains a higher
+number of objects classified as lenses than in the "Default" one: the
+brightest arcs stand out more from the lenses. In Fig. 4 we present the
+8 cutouts with the best scores for each of the lens simulation samples
+on the two top panels.
+The simulated lenses also give us an insight into the selection
+function of lens discovery with visual inspection. Since our sample
+is relatively small, we can only gain a coarse understanding of the
+selection function. In Fig. 5 and Fig. E1 we compare the recovery
+MNRAS 000, 1–15 (2022)
+
+Expert visual inspection of strong lenses
+5
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+20
+40
+60
+80
+100
+120
+140
+Number of subjects
+Lens data sets
+"Bright" simulations
+Default simulations
+SLACS
+R22 lenses
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Mean score
+Non-lens data sets
+LRGs
+CNNs=0
+Non-lens simulations
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Unlabeled data
+CNN-best
+Random
+Figure 3. Mean score per object separated in each of the data sets presented in this work: lens examples (left panel), non-lens examples (middle panel), and
+unlabeled data (right panel). All the histograms have the same binning with the expectation of SLACS and R22 lenses data sets where the bin size is double for
+visualization purposes.
+s = 1.0
+"Bright"
+simulations
+"Bright"
+simulations
+"Bright"
+simulations
+"Bright"
+simulations
+"Bright"
+simulations
+"Bright"
+simulations
+"Bright"
+simulations
+"Bright"
+simulations
+s = 1.0
+s = 1.0
+s = 1.0
+s = 0.99
+s = 0.99
+s = 0.98
+s = 0.98
+s = 1.0
+Default
+simulations
+Default
+simulations
+Default
+simulations
+Default
+simulations
+Default
+simulations
+Default
+simulations
+Default
+simulations
+Default
+simulations
+s = 1.0
+s = 0.99
+s = 0.99
+s = 0.98
+s = 0.97
+s = 0.97
+s = 0.97
+s = 0.2
+LRGs
+LRGs
+LRGs
+LRGs
+LRGs
+LRGs
+LRGs
+LRGs
+s = 0.19
+s = 0.19
+s = 0.19
+s = 0.18
+s = 0.17
+s = 0.16
+s = 0.16
+s = 0.31
+CNNs=0
+CNNs=0
+CNNs=0
+CNNs=0
+CNNs=0
+CNNs=0
+CNNs=0
+CNNs=0
+s = 0.23
+s = 0.18
+s = 0.17
+s = 0.17
+s = 0.17
+s = 0.16
+s = 0.15
+s = 0.31
+Non-lens
+simulations
+Non-lens
+simulations
+Non-lens
+simulations
+Non-lens
+simulations
+Non-lens
+simulations
+Non-lens
+simulations
+Non-lens
+simulations
+Non-lens
+simulations
+s = 0.3
+s = 0.3
+s = 0.29
+s = 0.28
+s = 0.28
+s = 0.27
+s = 0.26
+Figure 4. Examples of the eight objects with the highest mean score classified in the data sets: "Bright" simulations, Default simulations, LRGs, CNNs=0 and
+non-lens simulations. The mean score is displayed at the bottom of each cutout.
+fraction of simulated lenses and its standard deviation as a function
+of signal-to-noise ratio in the g-band, the magnitude of the Arc in the
+g-band and the Einstein radius of the lens. Since all of our images are
+simulations of the approximately uniform depth Dark Energy Survey,
+the first two quantities are tightly correlated. It is clear from the heat
+maps of Fig. 5 that there is a fairly sharp cutoff in recovery fraction
+for each quantity. Arcs fainter than ∼23rd magnitude (corresponding
+to a total SNR less than about 25), or with Einstein radius less than
+∼1.2 ′′are not discoverable by human eye in DES-like imaging. On
+the other hand, from the standard deviation of the scores we do not
+see any trend that allows us to get further information.
+In the same way, when we analyzed the SLACS sample we found
+that they were classified as "Non-lenses". The selection function
+of these systems drives them to have very bright lens galaxies and
+Einstein radii below 1.0 ′′ in most of the cases (Dobler et al. 2008).
+Given the results on similar simulated lenses, it is therefore not
+MNRAS 000, 1–15 (2022)
+
+6
+K. Rojas et al.
+2.2-2.6
+1.8-2.2
+1.4-1.8
+1.0-1.4
+0.6-1.0
+0.2-0.6
+log10(SNR g-band)
+"Bright" Simulations
+Default Simulations
+0.8-1.2
+1.2-1.6
+1.6-2.0
+2.0-2.4
+2.4-2.8
+2.8-3.2
+24.0-25.0
+23.0-24.0
+22.0-23.0
+21.0-22.0
+20.0-21.0
+19.0-20.0
+Arc g-band magnitude 
+0.8-1.2
+1.2-1.6
+1.6-2.0
+2.0-2.4
+2.4-2.8
+2.8-3.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Mean score
+Einstein Radii (arcsec)
+Figure 5. Heat maps with the average mean score per bins for the objects in the simulated lenses data sets. The left panels correspond to the "Bright" simulations,
+while the right panels to the "Default" simulations. In the top panels we display the logarithm of the SNR in g-band, while in the bottom panels we present the
+magnitude of the arc in the g-band. The color bar was selected with the purpose to more easily identify the bins where the simulations can be recognized as lens
+systems (in red) and where they are not identified (blue).
+surprising that the human experts struggled to classify these systems
+as lenses. This is almost certainly because of the challenge to visually
+deblend the lens and source in DES imaging. The eleven SLACS
+systems in DES are shown in Fig. 6, in the three different colour scales
+along with the score that they obtained from the visual inspection.
+On the other hand, the "R22 lenses" received scores between 0.6
+and 1.0, i.e. the classifiers consider them to probably be lenses. These
+R22 lenses were discovered using a visual inspection of the same DES
+data, so it is not surprising that these lenses remain discoverable for
+our classifiers. Although, as we can see in Fig. 7, the visual inspection
+scores obtained in Rojas et al. (2022) are somewhat different to those
+of our classifiers. We attribute this to human factors which we will
+discuss further in Sect. 4.3.
+3.3 Performance in negative examples.
+Our Non-lens examples are divided in three different data sets, the
+distribution of scores of their objects are shown in the middle panel
+of Fig. 3. Here we see that most of them were classified with scores
+between 0 and 0.3, meaning they are very unlikely to be lenses.
+The "Non-lens simulations" (mimicking a chance non-lensing align-
+ment) data set shows a broader distribution towards higher values
+these objects were potentially false positives, but they clearly do not
+particularly confuse our expert classifiers.
+Even though most of the objects are correctly identified as non-
+lenses, in Fig. 4 we present the eight cutouts of each sample with
+higher scores. Here we can see that most of the objects in the
+MNRAS 000, 1–15 (2022)
+
+Expert visual inspection of strong lenses
+7
+s = 0.21
+2MASXJ0216-0813
+Default scale
+Blue scale
+Sqrt scale
+s = 0.13
+2MASXJ0044+0113
+Default scale
+Blue scale
+Sqrt scale
+s = 0.12
+SDSSJ0008-0004
+s = 0.11
+2MASXJ2341+0000
+s = 0.11
+2MASSJ2111-0038
+s = 0.1
+SDSSJ0252+0039
+s = 0.1
+SDSSJ2347-0005
+s = 0.07
+SDSSJ0157-0056
+s = 0.06
+SDSSJ0029-0055
+s = 0.06
+SDSSJ2300+0022
+s = 0.04
+2MASXJ2141-0001
+Figure 6. Mosaic of the SLACS sample in the DES. We displayed the cutouts of each system in the same three different scales presented in the experiment:
+Default, Blue and Sqrt. On the cutout with the Default scale we added on the top the name of the system and on the bottom the visual inspection mean score
+obtained in this search.
+"LRGs" and "CNNs=0" data sets have little blue or red-ish com-
+panions around the central galaxy that could be mistaken for signs of
+lensing. In the same way, the cutouts of the "Non-lens simulations"
+set can be easily mistaken by very compact (low Einstein radii) lens
+systems, producing that some users gave a higher score to these ob-
+jects. Strictly speaking, the LRG and "CNNs=0" sets could contain
+a lens, though the probability of this is ≪ 1%.
+3.4 Performance in unlabeled data.
+We have two unlabeled data sets, "CNN-best" and "Random". Intu-
+itively we should hope that the CNN-selected sample should contain
+more lens candidates than the random sample. All of the CNN-best
+lenses have been inspected in Rojas et al. (2022). In the right panel
+of Fig. 3 we see that the CNN-best objects are distributed between
+0.0 and 0.8 (0.5 in the case of the Random sample), but the peak of
+these distributions are around 0.2, with most of the objects classified
+as Non-lenses and only a few of them have a score above 0.5.
+MNRAS 000, 1–15 (2022)
+
+8
+K. Rojas et al.
+s = 0.96 
+cnn=1.00, R22=1.00
+DES J060653-585843
+s = 0.96 
+cnn=1.00, R22=1.00
+DES J060653-585843
+s = 0.96 
+cnn=1.00, R22=1.00
+DES J060653-585843
+Default scale
+Blue scale
+Sqrt scale
+s = 0.84 
+cnn=1.00, R22=1.00
+DES J003727-413149
+s = 0.84 
+cnn=1.00, R22=1.00
+DES J003727-413149
+s = 0.84 
+cnn=1.00, R22=1.00
+DES J003727-413149
+s = 0.71 
+cnn=1.00, R22=0.86
+DES J033143-612315
+s = 0.71 
+cnn=1.00, R22=0.86
+DES J033143-612315
+s = 0.71 
+cnn=1.00, R22=0.86
+DES J033143-612315
+s = 0.62 
+cnn=1.00, R22=0.57
+DES J052648-375125
+s = 0.62 
+cnn=1.00, R22=0.57
+DES J052648-375125
+s = 0.62 
+cnn=1.00, R22=0.57
+DES J052648-375125
+Figure 7. Mosaic of the four systems in Rojas et al. (2022) classified in their
+category "Sure lens" displayed in the three different scales presented in this
+experiment. On the cutout with the Default scale we added on the top the
+name of the system, and on the bottom the mean score (s) from our visual
+inspection, the CNN (cnn) and visual inspection score (R22) from Rojas et al.
+(2022)
+In Fig. 8 we present the 16 cutouts with the highest scores for
+each of the samples. In the "CNN-best" data set we find that five
+of the objects were also recognized as candidates in Rojas et al.
+(2022), all of them classified in their "Maybe lens" catalogue. There
+are also seven objects with scores above 0.5 that were not classified
+as potential candidates in Rojas et al. (2022). In the case of the
+"Random" sample, none of the cutouts was classified as a potential
+lens candidate, although some get close to a score of 0.5. We can
+see some of them were highly graded by the CNN, this means that
+they went through the visual inspection steps in Rojas et al. (2022)
+but were not selected as lenses by those authors. Fig. 9 shows the
+scatter graph of aggregated expert scores against the CNN scores of
+Rojas et al. (2022). These two scores are essentially uncorrelated,
+indicating that the CNN and the experts are not responding to the
+same features. Since none of the objects in the Random data received
+a human score of 0.5 or more, we see no evidence of the CNN missing
+goodcandidates, however,thiscannotbeadefinitiveconclusion given
+the small sample size and the lack of correlation between the CNN
+and human scores. On the other hand in the non-lens datasets we see
+that most of the scores are below 0.5 and closer to 0.0, although a
+few non-lens simulations managed to confuse the CNN. In a similar
+way, the CNN fails to recognize a few lens simulations, some of
+them obtaining a score even lower than the one given by the human.
+Surprisingly some of the SLACS lenses obtained high scores by the
+CNN, this makes us wonder if the CNN is able to see some features
+that we humans can not. Nevertheless, giving an explanation of the
+CNN’s behaviour is out of the scope of this work.
+4 RESULTS B: UNDERSTANDING EXPERT
+CLASSIFICATION
+In this section, we investigate how individual users performed when
+executing the classification task.
+4.1 How accurately do individuals classify our sample?
+In order to evaluate the classification performed by each user we
+used the labeled data where we know the underlying truth: objects
+are either Lens or Non-lens. In our labeled category we find 58%
+of all the classifications (38,432 in total). Knowing the true label of
+each object and the classification given by each user we can compute
+confusion matrices to compare user performance when classifying
+the objects.
+Aggregating all of the classifiers and classifications, we computed
+a confusion matrix displaying the four different classification op-
+tions. In Fig. 10) we see the percentage of classifications that are
+in agreement or disagreement with the original label of the object.
+From here we can see that objects voted in the options "Certain lens",
+"Probable lens" and "Very unlikely" are in general well classified,
+achieving overall a 99%, 83%, and 80% of objects correctly labelled.
+On the other hand the classification "Probably not lens" is evenly
+split between labelled lenses and labelled non-lenses.
+To see in more detail the performance for each of the labelled
+data sets, we calculated a confusion matrix displaying the percentage
+of classifications according to the four different options for each
+data set. From Fig. 11 we can see that the three data sets labelled
+as "Non-lenses" obtained a very high percentage of votes in the
+option "Very Unlikely". However, there is confusion when classifying
+the simulated strong lens systems. SLACS lenses are mostly not
+recognized as lenses and only the four "R22 lenses" get a high amount
+of classifications in the categories "Certain lens" and "Probable lens".
+From the classifications of the labelled data, we compute confusion
+matrices for each user. In this section, we call objects Lenses if they
+are classified as either "Certain lens" or "Probable lens". Systems
+classified as "Probably not lens" or "Very unlikely" are considered
+Non-lenses. In that way we build a 2 × 2 confusion matrix that will
+tell us what it is the probability that a user classifies an object as
+"Lens" or "Non-lens" given that the true label is "Lens" or "Non-
+lens", this means the true positive and true negative rates in the
+confusion matrix.
+Following Marshall et al. (2015) we plotted these probabilities in
+Fig. 12, where we can see that most of the classifiers have very low
+false positive rates. This makes these classifiers extremely good at
+identifying easy lenses, but there is a significant range in their ability
+to identify challenging lenses. Some classifiers manage complete-
+ness of ∼ 60% at high purity, whilst more pessimistic classifiers are
+identifying half as many lenses.
+A handful of the classifiers are more optimistic, classifying more
+marginal systems as lenses. This comes at the cost of more false
+positives, which is not desirable in a real search where lenses are
+intrinsically rare.
+Additionally, we computed the same 2 × 2 confusion matrix join-
+ing all the classifications of the users among the different groups
+separated by academic status, years of experience and confidence in
+performing the classification. This gives a confusion matrix for each
+MNRAS 000, 1–15 (2022)
+
+Expert visual inspection of strong lenses
+9
+s = 0.84, 
+R22=1.00
+CNN Best
+CNN Best
+CNN Best
+CNN Best
+CNN Best
+CNN Best
+CNN Best
+CNN Best
+s = 0.74, 
+R22=0.57
+s = 0.72, 
+R22=0.43
+s = 0.7
+s = 0.57, 
+R22=0.71
+s = 0.56
+s = 0.55
+s = 0.54, 
+R22=0.71
+s = 0.54
+s = 0.52
+s = 0.51
+s = 0.5
+s = 0.49
+s = 0.47
+s = 0.46
+s = 0.45
+s = 0.49 ,
+cnn=0.89
+Random
+Random
+Random
+Random
+Random
+Random
+Random
+Random
+s = 0.48 ,
+cnn=0.18
+s = 0.44 ,
+cnn=0.84
+s = 0.43 ,
+cnn=0.87
+s = 0.36 ,
+cnn=0.79
+s = 0.36 ,
+cnn=0.41
+s = 0.36 ,
+cnn=0.41
+s = 0.35 ,
+cnn=0.41
+s = 0.32 ,
+cnn=0.58
+s = 0.32 ,
+cnn=0.35
+s = 0.32 ,
+cnn=0.14
+s = 0.31 ,
+cnn=0.90
+s = 0.31 ,
+cnn=0.41
+s = 0.31 ,
+cnn=0.15
+s = 0.31 ,
+cnn=0.34
+s = 0.3 ,
+cnn=0.41
+Figure 8. Examples of the 16 highly graded cutouts in the unlabeled data sets. The CNN-best data set is displayed in the two top panels, on the bottom of the
+cutouts we present the mean score (s) from this experiment, and visual inspection score (R22) from Rojas et al. (2022) in case they were part of the final catalogs
+presented in that work. The CNN score for all these objects is CNN 1.00. The Random sample is shown in the two bottom panels, on the bottom of each cutout
+we display the mean score (s) from this experiment, and the CNN score (cnn) from Rojas et al. (2022).
+0.010
+0.100
+0.500
+0.900
+0.990
+0.999
+Mean score
+0.000
+0.001
+0.100
+0.500
+0.900
+0.999
+CNN score
+Random
+LRGs
+sim-nl
+Bright sim
+Default sim
+SLACS
+Figure 9. Comparison between the human expert scores from our classifiers
+and the CNN scores given by the model in Rojas et al. (2022). The dashed
+black line is the one-to-one line. Grey triangles are from the unlabeled 300
+randomly drawn DES images. Squares are non-lens labelled data sets, in blue
+LRGs and in green the simulated non-lenses. Circles are simulated lenses, in
+yellow the bright data set and in red the default data set. Purple Stars are the
+SLACS lenses.
+grouping. To derive the error bars for these values, we use the stan-
+dard deviation of the individual true positive and true negative rates of
+each user in the group. Fig. 13 shows these results: overall the results
+are very similar. That is to say that, regardless of academic status,
+years of experience and confidence, each grouping produces a very
+similar average classification. There are very significant differences
+between individual users, but time in the field, academic position or
+reported confidence are not predictive of a user’s classification skill.
+Certain lens
+Probable lens
+Probably not lens
+Very unlikely
+Visual Inspection Values
+L
+NL
+Real Values 
+99 %
+83 %
+47 %
+20 %
+1 %
+17 %
+53 %
+80 %
+Figure 10. Confusion matrix of the four different classification options, "Cer-
+tain lens", "Probable lens", "Probably not lens" and "Very unlikely" contrasted
+with the real labels L: lens and NL: No lens. The percentages shown are the
+number of lenses (non-lenses) classified in a determined option.
+4.2 How reliable are individual classifications?
+To test the reliability of human classifications, we duplicated 105
+objects in the sample. We randomly selected 15 objects from each of
+the following data sets: CNN-best, LRGs,CNNs=0, Random, Default
+simulations, "Bright" simulations, and Non-lens simulations. The
+duplicate cutouts were shown at random points in the experiment 7.
+Overall, we had a total of 3797 duplicated classifications, with
+73% graded the same as before. On the other hand, 15% (10%)
+received an upgrade (downgrade) of one point in the classification,
+7 Because of the way Zooniverse serves images, some users finished the
+classification task but continued to classify a small number of randomly
+drawn images. We also took these into account in assessing the reliability of
+classifications.
+MNRAS 000, 1–15 (2022)
+
+10
+K. Rojas et al.
+"Bright"
+simulations
+Default
+simulations
+SLACS
+R22 lenses
+LRGs
+CNNs=0
+Non-lens
+simulations
+Data sets
+CL
+PL
+PNL
+VU
+Visual inspection values 
+32 %
+23 %
+0 %
+54 %
+0 %
+0 %
+0 %
+24 %
+24 %
+4 %
+29 %
+2 %
+2 %
+6 %
+22 %
+24 %
+22 %
+12 %
+16 %
+17 %
+25 %
+22 %
+28 %
+74 %
+5 %
+82 %
+81 %
+68 %
+Figure 11. Confusion matrix of the seven different labeled data sets analyzed
+in this experiment contrasted with the classification options CL: Certain
+lens, PL: Probable lens, PNL: Probably not lens and VU: Very unlikely. The
+percentages represent the amount of classifications made in each category.
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+Completeness
+0.6
+0.7
+0.8
+0.9
+1.0
+Purity
+Professor
+Postdoc
+Ph.D Student
+Master Student
+Citizen Scientist
+Figure 12. Completeness and purity percentages of each user when classify-
+ing our labelled data.
+this means that if for example an object was originally classified as
+"Probable lens" in the second time performing the classification the
+user classified the object as "Certain lens" in the upgrade case and
+as "Probably not lens" in the downgrade case. Only 1.3% (0.6%)
+of the classifications were upgraded (downgraded) by 2 points, and
+0.13% (0.03%) by 3 points, which means changing completely the
+classification from "Certain lens" to "Very unlikely" or vice versa.
+These very low percentages for extreme cases are a very good sign
+that the users are not obtuse classifiers and are somehow confident
+about their classifications. When we break these results down by self-
+reported confidence (Fig. 14), we see that "Very confident" users
+perform relatively consistently, with a ∼ 75% of reliability in the
+two extreme classification options. On the other hand, the users that
+signed as "Not confident" hesitate more at the time to use the "Certain
+lens" option, reaching only 40% of reliability in this class, but a 76%
+of reliability in the option "Probable lens" shows that they are more
+comfortable with this more ambiguous selection.
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+pr(C-LENS|T-LENS)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+pr(C-NO-L|T-NO-L)
+Obtuse
+Optimistic
+Astute
+Pessimistic
+Random
+Users
+Professor
+Postdoc
+Ph.D Student
+Master Student
+Citizen Scientist
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+pr(C-LENS|T-LENS)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+pr(C-NO-L|T-NO-L)
+Obtuse
+Optimistic
+Astute
+Pessimistic
+Random
+Users
+More than 12 years
+between 8-12 years
+between 4-8 years
+between 1-4 years
+Less than one year
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+pr(C-LENS|T-LENS)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+pr(C-NO-L|T-NO-L)
+Obtuse
+Optimistic
+Astute
+Pessimistic
+Random
+Users
+Very confident
+Confident
+A bit confident
+Not confident
+Figure 13. Probability that an object is classified as "Lens" given that it is
+a lens (true positive rate) vs. the probability that the object is classified as
+"Non-lens" given that it is not a lens (true negative rate). The values for each of
+the users are displayed with a black circle, whose size represents the amount
+of classification made by that user. The coloured stars represent the joint
+result of the different groups of classifiers presented in this work, separated
+by academic position (top panel), years of experience (middle panel) and
+confidence in performing the classification task (bottom panel).
+MNRAS 000, 1–15 (2022)
+
+Expert visual inspection of strong lenses
+11
+Very confident
+Confident
+A bit confident
+Not confident
+Certain lens
+Probable lens
+Probably not lens
+Very unlikely
+75 %
+68 %
+57 %
+40 %
+57 %
+59 %
+56 %
+76 %
+57 %
+51 %
+57 %
+61 %
+76 %
+74 %
+62 %
+53 %
+Percentage of reliability
+Figure 14. Mean percentage of reliability of users at the time to re-classify a
+object, according to their level of confidence.
+4
+7
+10
+13
+16
+19
+Team size
+0.04
+0.06
+0.08
+0.10
+0.12
+0.14
+0.16
+0.18
+0.20
+Classification error
+Figure 15. Standard deviation of scores when classified by a subset of users
+relative to the ’truth’ from all our users.
+4.3 How many classifiers are needed for a reliable score?
+The classification of an object into any of the options is a personal and
+subjective opinion. Even with clear guidelines and examples, there
+will be disagreement among users, as the visual inspection work in
+Rojas et al. (2022); Savary et al. (2022) showed. Typical strong lens
+searches have had a handful of expert classifiers (e.g. 3 for Jacobs
+et al. 2019b). Given the individual expert variation seen in Sect. 4.1
+and the lack of reliability seen in Sect. 4.2, it is to be expected that
+using a small number of experts can significantly bias final scores. To
+assess how significant this bias can be, we divide our classifications
+into random ’teams’ of n users. We computed new scores using only
+the classification of the team members. We did this for team sizes
+ranging from one to twenty participants randomly drawn from our
+users.
+We compare the team scores with the final score from the entirety
+of our classifiers. We created 200 teams of n random users to recal-
+culate the score for objects labelled as lenses. Fig. 15 shows how the
+team size affects the scoring of images. The standard deviation error
+is substantial for very small teams but decreases quickly up to a team
+size of ∼ 8, which has a classification error of 0.07 per object. Above
+8 users the classification error decreases much more slowly. Given
+that most objects are unambiguously classified as very unlikely (i.e.
+score of 0), these represent big differences in scores for the other
+objects: small teams are not very good at assessing the quality of a
+strong lens candidate.
+The results in this section go some way to explaining why the
+scores of Rojas et al. (2022) vary from those of our classifiers. There
+are not enough systems with overlapping data to draw firm conclu-
+sions (9 images), but the standard deviation is 0.08, as is expected
+for a team of 6 classifiers.
+Since teams will be most concerned about discovering marginal
+lenses, it is salient to focus on team accuracy when classifying lenses
+with marginal scores. Focusing on images with true scores between
+0.3 and 0.8 the classification error grows substantially: it is 0.17 for
+a team of 2 classifiers, 0.12 for a team of 4 and 0.07 for a team of 8.
+A team of 6 classifiers is required to achieve an expected accuracy
+better than 0.1, 15 are needed for an accuracy better than 0.05.
+5 CONCLUSIONS
+This work has investigated how strong lensing experts visually in-
+spect images of galaxy-scale strong lens candidates. We showed them
+a sample of 1489 mock and real images of the Dark Energy Survey
+and asked them to grade each image as either: Certain lens; Prob-
+able lens; Probably not a lens; or, Very unlikely to be a lens. With
+the resulting 66835 classifications we can now answer our initial
+questions.
+• What are the properties of lensing systems that human experts
+identify reliably as lenses, and what do they miss?
+Most gravitational lenses are reliably identified with an Einstein Ra-
+dius greater than 1.2′′and an arc g-band magnitude less than 23.
+This corresponds to roughly 1.2 times the seeing of the Dark Energy
+Survey and a g-band signal-to-noise of 25. Some lenses are discov-
+ered with fainter or smaller radius arcs. The Einstein radius cut-off is
+sharp with only a handful of very bright arcs discovered with Einstein
+radius of less than 1.2′′. The flux cut-off is smoother: roughly twenty
+percent of lenses are recovered even with an arc signal-to-noise of
+between 4 and 10.
+• Do human experts confuse non-lenses for lenses?
+For our labelled data, none of the non-lenses scored higher than
+0.3. Our simulations do not include face-on spirals, or ring galaxies,
+but the experts had no problem rejecting chance alignments of blue
+sources close to LRGs. Our labelled sample had an almost equal
+split of lenses and non-lenses. Unless robotic candidate selection
+improves substantially, real lens searches will have far more non-
+lenses than lenses, even so, it seems that human experts are very
+good at discarding the kinds of non-lenses shown here. Follow-up
+campaigns should be confident that highly scored candidates are
+almost certainly lenses.
+• How does expert classification depend on the experience and
+confidence of the experts?
+MNRAS 000, 1–15 (2022)
+
+12
+K. Rojas et al.
+We see substantial variation in the purity and completeness of indi-
+vidual classifiers, but there are no significant trends with experience,
+confidence, or academic position. None of these traits reliably predict
+the over-pessimism or over-optimism of some users.
+• How reliable are individual classifications?
+Classifications are not reliable when repeated. Even classifiers who
+self-report as ’very confident’ do not grade candidates consistently.
+When reclassifying the same images, certain and very unlikely lenses
+are scored the same roughly three-quarters of the time by confident
+and very confident classifiers, whereas probable and probably not
+lenses are only reproduced three-fifths of the time. Fewer than 2
+percent of reclassified targets changed by more than one classification
+step.
+• When ranking lens candidates, what do the scores of teams of
+experts mean? How should lens searchers best build an expert team
+to classify their candidates?
+Given the fact that classifications by a single expert are not reli-
+able when repeated, it is not surprising that small teams make for
+poor classifiers. On a 0-1 scoring system, teams of 6 classifiers will
+produce results within 0.1 of the ensemble average of all users when
+classifying marginal systems. Senior classifiers are, on the whole, not
+better than junior classifiers so teams should classify independently
+and not defer to the opinions of senior faculty members.
+We found no correlation between CNN and human scores sug-
+gesting that CNNs are not trained to recognise the same features
+as human experts. Bigger samples are needed to assess if this is a
+problem for lens finding in future surveys.
+A traditional search would use a small number of classifiers to
+grade a large number of images. To understand the human classifi-
+cation process, we have done the opposite. In the real Universe, real
+lenses are much rarer than our sample, so it is possible that our results
+do not perfectly scale to a search of a billion objects. However, if we
+assume that our discovery thresholds map onto searches of entire
+surveys our results suggest that previous forecasts are likely to be
+somewhat optimistic. The discovery signal-to-noise of our experts
+is broadly consistent with the assumptions of Collett (2015), how-
+ever our experts recovered few lenses with Einstein radius less than
+1.2′′, which represents ∼40 percent of the forecasted DES population
+in Collett (2015). Collett (2015) had assumed that users would be
+shown lens-subtracted images, such as in Sonnenfeld et al. (2018)
+and future work should investigate if expert inspection can recover
+even more lenses with such an approach.
+ACKNOWLEDGEMENTS
+This work has received funding from the European Research Council
+(ERC) under the European Union’s Horizon 2020 research and in-
+novation programme (LensEra: grant agreement No 945536). TC is
+funded by the Royal Society through a University Research Fellow-
+ship. For the purpose of open access, the authors have applied a Cre-
+ative Commons Attribution (CC BY) licence to any Author Accepted
+Manuscript version arising. DB is funded by a graduate studentship
+from UK Research and Innovation’s STFC and the University of
+Portsmouth. This work is also in part supported by the Swiss National
+Science Foundation (SNSF) and by the European Research Council
+(ERC) under the European UnionâĂŹs Horizon 2020 research and
+innovation program (COSMICLENS: grant agreement No 787886).
+JHHC aknowledge the generosity of Eric and Wendy Schmidt by
+recommendation of the Schmidt Futures program. FG acknowledges
+the support from grant PRIN MIUR 2017-20173ML3WW_001. RJ
+acknowledges the support from the research project grant "Under-
+standing the Dynamic Universe" funded by the Knut and Alice Wal-
+lenberg Foundation under Dnr KAW 2018.0067. S.S. thank the Max
+Planck Society for support through the Max Planck Research Group
+for SHS. This project has received funding from the European Re-
+search Council (ERC) under the European Unions Horizon 2020
+research and innovation programme (LENSNOVA: grant agreement
+No 771776). This research is supported in part by the Excellence
+Cluster ORIGINS which is funded by the Deutsche Forschungsge-
+meinschaft (DFG, German Research Foundation) under Germany’s
+Excellence Strategy – EXC-2094 – 390783311. TD Thanks the sup-
+port by an LSSTC Catalyst Fellowship awarded by LSST Corporation
+with funding from the John Templeton Foundation grant ID #62192.
+GM acknowledges funding from the European Union’s Horizon 2020
+research and innovation programme under the Marie SkÅĆodowska-
+Curie grant agreement No MARACHAS - DLV-896778. BD was
+supported in part by the NASA Astrophysics Theory Program under
+grant 80NSSC18K1014. F.G.S. and G.V.C. Coordenação de Aper-
+feiçoamento de Pessoal de Ensino Superior (CAPES) - Finance Code
+001 LAU-L thanks CONACyT México for support under grants No.
+A1-S-17899, No. 286897, No. 297771, No. 304001; and the Insti-
+tuto Avanzado de Cosmología Collaboration. JW was supported by
+the Science and Technology Facilities Council under grant number
+ST/P006760/1, the DISCnet Centre for Doctoral Training in Data-
+Intensive Science. JM acknowledges the support of the UK Science
+and Technology Facilities Council (STFC).
+DATA AVAILABILITY
+The inclusion of a Data Availability Statement is a requirement for
+articles published in MNRAS. Data Availability Statements provide
+a standardised format for readers to understand the availability of data
+underlying the research results described in the article. The statement
+may refer to original data generated in the course of the study or to
+third-party data analysed in the article. The statement should describe
+and provide means of access, where possible, by linking to the data or
+providing the required accession numbers for the relevant databases
+or DOIs.
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+AFFILIATIONS
+1 Institute of Cosmology and Gravitation, University of Portsmouth,
+Burnaby Rd, Portsmouth PO1 3FX, UK.
+2Kavli Institute for Particle Astrophysics and Cosmology and
+Department of Physics, Stanford University, Stanford, CA 94305,
+USA.
+3SLAC National Accelerator Laboratory, Menlo Park, CA, 94025.
+4Department of Physics and Astronomy, Stony Brook University,
+Stony Brook, NY 11794, USA.
+5Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL
+60510, USA.
+6Department of Astronomy and Astrophysics, University of Chicago,
+Chicago, IL 60637, USA.
+7Department of Physics and Astronomy, Lehman College of the
+CUNY, Bronx, NY 10468, USA.
+8Department of Astrophysics, American Museum of Natural History,
+Central Park West and 79th Street, NY 10024-5192, USA.
+9Institute of Physics, Laboratory of Astrophysics, Ecole Polytech-
+nique Fédérale de Lausanne (EPFL), Observatoire de Sauverny,
+1290 Versoix, Switzerland.
+10Instituto de FÃŋsica de Cantabria. (CSIC-UC). Avda Los Castros
+s/n 39095 Santander. Spain.
+11University of Bologna, Department of Physics and Astronomy
+(DIFA), Via Gobetti 93/2, I-40129, Bologna, Italy.
+12INAF – Osservatorio di Astrofisica e Scienza dello Spazio, via
+Gobetti 93/3 - 40129, Bologna - Italy.
+13Physics Department, University of Wisconsin-Madison, Madison,
+WI 53706, USA.
+14Oskar Klein Centre for Cosmoparticle Physics, Department of
+Physics, Stockholm University, Stockholm SE-106 91, Sweden.
+15Consejo Nacional de Ciencia y Tecnología, Av. Insurgentes Sur
+1582, Col. Crédito Constructor, Alc. Benito Juárez, CP 03940,
+México.
+16Mesoamerican Centre for Theoretical Physics, Universidad
+Autónoma de Chiapas, Carretera Zapata Km. 4, Real del Bosque,
+29040,
+Tuxtla Gutiérrez, Chiapas, México.
+17Max-Planck-Institut für Astrophysik, Karl-Schwarzschild Str. 1,
+85748 Garching, Germany.
+18Technische Universität München, Physik Department, James-
+Franck Str. 1, 85748 Garching, Germany.
+19INAF – Osservatorio Astronomico di Capodimonte, Salita
+Moiariello 16, 80131 - Napoli, Italy.
+20Instituto de Astronomía, Observatorio Astronómico Nacional,
+Universidad Nacional Autónoma de México, Apartado postal 106,
+C.P. 22800,
+Ensenada, B.C., México.
+21Sub-department of Astrophysics, Denys Wilkinson Building,
+University of Oxford, Keble Road, Oxford, OX1 3RH, UK.
+22Department of Astrophysical Sciences, Princeton University, 4 Ivy
+Lane, Princeton, NJ 08544.
+23LSSTC Catalyst Fellow.
+24Department of Physics and Astronomy, School of Natural Sci-
+ences, University of Manchester, Oxford Rd, Manchester M13 9PL.
+25School of Physics & Astronomy, University of Nottingham,
+University Park, Nottingham, NG7 2RD, UK.
+26Centre for Extragalactic Astronomy, Department of Physics,
+Durham University, Durham DH1 3LE, UK.
+27Institute for Computational Cosmology, Durham University, South
+Road, Durham DH1 3LE, UK.
+28Observatório Nacional, Rua Gal. José Cristino 77, Rio de Janeiro,
+RJ - 20921-400, Brazil.
+29OmegaLambdaTec GmbH, Lichtenbergstraße 8, 85748 Garching.
+30Universitäts-Sternwarte,
+Fakultät
+fÃijr
+Physik,
+Ludwig-
+Maximilians Universität München, Scheinerstr. 1, 81679 München,
+Germany.
+31Citizen
+Scientist,
+Zooniverse,
+Astrophysics
+Sub-department,
+University of Oxford, Keble Road, Oxford, OX1 3NP, UK.
+32Department of Physics and Astronomy, University of New Mexico,
+210 Yale Blvd NE, Albuquerque, NM 87106, USA.
+33Centre for Astrophysics and Supercomputing, Swinburne Univer-
+sity of Technology, PO Box 218, Hawthorn, VIC 3122, Australia.
+34University of Birmingham, School of Physics and Astronomy,
+Edgbaston, Birmingham, B15 2TT, UK.
+35Departamento de Física, DCI, Campus León, Universidad de
+Guanajuato, 37150, León, Guanajuato, México.
+36Department of Physics, Benedictine University, 5700 College
+Road, Lisle, Illinois, 60532.
+37School of Physical Sciences, The Open University, Walton Hall,
+Milton Keynes, MK7 6AA, UK.
+38IRAP, UniversitÃľ de Toulouse, UPS, CNRS.
+39Institut für Theoretische Astrophysik, Zentrum für Astronomie,
+Heidelberg Universität, Albert-Ueberle-Str. 2, 69120, Heidelberg,
+Germany.
+40Instituto Sciety Lab, São José dos Campos, SP, Brazil
+41Department of Physics, University of California, 1156 High St,
+Santa Cruz, CA 95064, USA.
+42Santa Cruz Institute of Particle Physics, University of California,
+1156 High St, Santa Cruz, CA, 95064 USA.
+43School of Computing & Communications, The Open University,
+Walton Hall, Milton Keynes MK7 6AA, UK.
+APPENDIX A: INVITATION
+We send an invitation to members of the strong lensing community,
+including LSST, DES, and Euclid strong lensing working groups and
+MNRAS 000, 1–15 (2022)
+
+14
+K. Rojas et al.
+we extended the invitation to Space Warp citizen scientists. They
+received the following invitation:
+"We would like to invite you to participate in a lens classification experiment
+with the goal of understanding how people in the field of gravitational lensing
+are performing when they do a visual classification task. The task will take
+about an hour of your time. Participants will be invited to coauthor the
+resulting paper.
+The motivation of this experiment comes from the explosion of new lens
+systems discovered by the use of CNN and subsequent validation through
+visual inspection performed for each team. There seem to be lots of differences
+in the expert validation and we want to see if this can be understood and
+calibrated.
+We expect that with this social experiment we can get key conclusions about
+our performance, and hope that in the future lens finders can benefit from this
+information. If you want to be part of this experiment please fill this quick
+google form first. The information requested here will help us to analyze the
+data, although your personal data (name, email and galaxy zoo username)
+will remain private.
+And now you are welcome to classify 1000 objects! If you follow this link8
+You will see DES gri-color composite images of each object, each stamp has
+a size of 50x50 pixels (13"x13"). The same object is displayed in 3 different
+color-scales to help the recognition of features. The task is simple: you have
+to click on the option that better represents the object(s) in the image and go
+to the next.
+You don’t have the obligation to complete the classification all at once, this
+might take you a couple of hours. Your progress is recorded and you can come
+back anytime you want ;). For a successful analysis we hope you can commit
+to completing the classification of the whole data set, or at least a big portion
+of it.
+Please share this among your group, postdocs, phd students, masters students,
+etc, that work in the field of strong lensing. All are encouraged to participate
+as we want to test a broad variety of expertise, but please do not share it
+among people outside of the field."
+APPENDIX B: IMAGE SCALINGS
+The "default" and "blue" composite images are scaled with an arcsinh
+stretch using (HumVI, Marshall et al. 2015). We tuned the rgb-scale
+parameters, the brightness (Q) and contrast (α) for default (blue)
+images as follows: r-scale = 1.0 (0.51), g-scale = 1.2 (0.68), b-scale=
+2.8 (3.12), Q = 1.0 (0.64), α = 0.03 (0.03). The third image is scaled
+using a square root image scaling. To ensure that the object of interest
+is not overshadowed by another brighter one we set minimum and
+maximum values for the pixels in the images. The minimum value is
+obtained using a square of 5×5 pixels in the corners of the gri-images
+and we select the minimum value among them. On the other hand,
+the maximum value is obtained by placing a box of 10 × 10 pixels in
+the middle of the three images and obtaining the mean among them.
+We use these two values to scale the three gri-images.
+APPENDIX C: EXPERIENCE AND CONFIDENCE
+In Fig. C1 we present the distribution of the levels of confidence
+compared to the academic status and years of experience in the field,
+according to the information provided by the classifiers when they
+subscribed to this experiment. From both plots, we can clearly see
+that at higher academic status or years of experience in the field,
+the classifiers feel more confident that they will perform a successful
+8 https://www.zooniverse.org/projects/krojas26/
+experts-visual-inspection-experiment/classify
+Very confident
+Confident
+A bit confident
+Not confident
+3
+6
+9
+Professor
+Postdoc
+PhD. Student
+Master Student
+Citizen Scientist
+Very confident
+Confident
+A bit confident
+Not confident
+3
+6
+9
+> 12 years
+8-12 years
+4-8 years
+1-4 years
+< 1 year
+Figure C1. The radar charts show the level of confidence that users have to
+perform the classification task separately in academic status (top panel) and
+years of experience (bottom panel).
+classification, while undergrad students and people with between 0-4
+years of experience feel "not" or only a "bit confident".
+APPENDIX D: SCORING SYSTEM
+In addition, we wanted to explore the impact of weighting the classi-
+fication of the users according to their percentage of correct classifi-
+cations in each of the 4 options given. To obtain this percentage we
+computed a 2 × 4 confusion matrix similar to Fig. 10 but this time
+for each user. Then we take the 4 relevant percentages: Classified as
+"Certain lens" or "Probable lens" given that it is a lens and "Probably
+not lens" or "Very unlikely" given that it is not a lens. Using these
+percentages as weight we then calculated a weighted mean score, this
+means that we multiply each score by the corresponding weight, sum
+them and divide by the sum of the weights. After weighting all the
+classifications we re-scale the mean score weighted between 0 and 1
+as we did with the previous score.
+We expected that weighting the mean score could provide a better
+score system because it will take into account the performance of the
+users, but most of the users performed in a pretty similar way. We
+can see in Fig. D1 that the mean score and the mean score weighted
+are highly correlated. For this reason, we are going to continue our
+MNRAS 000, 1–15 (2022)
+
+Expert visual inspection of strong lenses
+15
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Mean score
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Mean score weighted
+Figure D1. Scatter plot of the mean score compared with the mean score
+weighted by the corresponding percentage of successful classification in the
+options displayed.
+analysis using only the mean score and we are not going to explore
+the mean score weighted implications in further analysis.
+APPENDIX E: STANDARD DEVIATION
+We calculated the standard deviation of the scores given to each
+object in the experiment. With this information, we create a heat
+map (Fig. E1) to compare the standard error of the recovery fraction
+of the simulated lenses data sets in response to the signal-to-noise
+ratio in the g-band, the magnitude of the arc in the g-band and the
+Einstein radius of the lens. In contrast with the heat maps with the
+average scores (Fig. 5), we do not see any trend in the standard
+deviation.
+This paper has been typeset from a TEX/LATEX file prepared by the author.
+MNRAS 000, 1–15 (2022)
+
+16
+K. Rojas et al.
+2.2-2.6
+1.8-2.2
+1.4-1.8
+1.0-1.4
+0.6-1.0
+0.2-0.6
+log10(SNR g-band)
+"Bright" Simulations
+Default Simulations
+0.8-1.2
+1.2-1.6
+1.6-2.0
+2.0-2.4
+2.4-2.8
+2.8-3.2
+24.0-25.0
+23.0-24.0
+22.0-23.0
+21.0-22.0
+20.0-21.0
+19.0-20.0
+Arc g-band magnitude 
+0.8-1.2
+1.2-1.6
+1.6-2.0
+2.0-2.4
+2.4-2.8
+2.8-3.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Standard deviation
+Einstein Radii (arcsec)
+Figure E1. Heat maps with the average standard deviation of the mean score per bins for the objects in the simulated lenses data sets. The left panels correspond
+to the "Bright" simulations, while the right panels to the "Default" simulations. In the top panels we display the logarithm of the SNR in g-band, while in the
+bottom panels we present the magnitude of the arc in the g-band.
+MNRAS 000, 1–15 (2022)
+
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+page_content='0  The impact of human expert visual inspection on the discovery of strong  gravitational lenses  Karina Rojas,1⋆ Thomas E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Collett,1 † Daniel Ballard,1 Mark R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Magee,1 Simon Birrer,2,3,4 Elizabeth  Buckley-Geer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='5,6 James H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Chan,7 8 Benjamin Clément,9 José M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Diego,10 Fabrizio Gentile,11,12 Ji-  mena González,13 Rémy Joseph,14 Jorge Mastache,15,16 Stefan Schuldt,17,18 Crescenzo Tortora,19 Tomás  Verdugo,20 Aprajita Verma,21 Tansu Daylan,22,23 Martin Millon,9 Neal Jackson,24 Simon Dye,25 Alejandra  Melo,17 Guillaume Mahler,26,27 Ricardo L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Ogando,28 Frédéric Courbin,9 Alexander Fritz,29 Aniruddh  Herle,17,30 Javier A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Acevedo Barroso,9 Raoul Cañameras,17 Claude Cornen,31 Birendra Dhanasingham,32  Karl Glazebrook,33 Michael N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Martinez,13 Dan Ryczanowski,34 Elodie Savary,9 Filipe Góis-Silva,28 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  Arturo Ureña-López,35 Matthew P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Wiesner,36 Joshua Wilde,37 Gabriel Valim Calçada,28 Rémi Cabanac,38  Yue Pan,6 Isaac Sierra,6 Giulia Despali,39 Micaele V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Cavalcante-Gomes,40,28 Christine Macmillan,30 Jacob  Maresca,25 Aleksandra Grudskaia,17 Jackson H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' O’Donnell,41,42 Eric Paic,9 Anna Niemiec,26,27 Lucia F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  de la Bella,1 Jane Bromley,42 Devon M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Williams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='43  Affiliations are listed at the end of the paper  Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' in original form ZZZ  ABSTRACT  We investigate the ability of human ’expert’ classifiers to identify strong gravitational lens candidates in Dark Energy Survey  like imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' We recruited a total of 55 people that completed more than 25% of the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' During the classification task, we  present to the participants 1489 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The sample contains a variety of data including lens simulations, real lenses, non-lens  examples, and unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' We find that experts are extremely good at finding bright, well-resolved Einstein rings, whilst  arcs with g-band signal-to-noise less than ∼25 or Einstein radii less than ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='2 times the seeing are rarely recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Very few  non-lenses are scored highly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' There is substantial variation in the performance of individual classifiers, but they do not appear  to depend on the classifier’s experience, confidence or academic position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' These variations can be mitigated with a team of 6 or  more independent classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Our results give confidence that humans are a reliable pruning step for lens candidates, providing  pure and quantifiably complete samples for follow-up studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  Key words: gravitational lensing: strong  1 INTRODUCTION  Strong gravitational lensing offers a rare opportunity to directly  weight objects in the distant Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Mass warps space-time and  in strong lens systems, the surface density is so great that multiple  images of a background source are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  Strong lenses enable a magnified view of the high redshift Uni-  verse (Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Shu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  Ebeling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Shu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 2022), a direct probe of dark matter in  galaxies, clusters and substructures (Oguri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Vegetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Jiménez-Vicente et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Nierenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Gilman  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 2020) and a geometrical probe of the cosmological parameters  (Collett & Auger 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Bonvin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Wong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Almost  all applications of strong lensing are limited by sample size, but the  ⋆ karina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='rojas@port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='uk  † thomas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='collett@port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='uk  era of deep wide area surveys offers an opportunity to grow strong  lens samples a hundredfold (Collett 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  As astronomy enters the era of billion object surveys, sophisti-  cated methods for discovering strong lenses have been developed  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Lanusse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Avestruz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 2019) and applied (Jacobs  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 2017, 2019b,a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Savary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Petrillo  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 2017, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' These methods have been extremely successful at  identifying candidate lenses, but the rarity of lenses means that even  classifiers with 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='99% accuracy produce 100 false positives for ev-  ery true lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The gold standard for confirming a lens is spectroscopic  confirmation of multiple redshifts, but reducing false positive rates  is needed for a spectroscopic confirmation campaign to be viable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' In  most cases, a human expert step is used as the final stage filtering  step to remove false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  Expert human classification has been very successful at identify-  ing the best strong lens candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' For example, Tran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' (2022)  recently targeted 79 lens candidates, spectroscopically confirming 53  © 2022 The Authors  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='03670v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='GA]  9 Jan 2023  2  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  and definitely ruling out only 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' However, introducing a human into  any classification task is likely to bring in selection biases: it is much  easier to identify a bright arc that is well resolved from the lensing  galaxy and significantly different in colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  The primary purpose of this work is to calibrate and understand  how introducing human experts biases lens searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' In addition,  we aim to understand how the choice of "experts" impacts the lens  candidate sample selected and how search teams can mitigate the  biases of their members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' We set out to answer the following questions:  What are the properties of lensing systems that human experts  identify reliably as lenses, and what do they miss?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  Do human experts confuse non-lenses for lenses?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  How does expert classification depend on the experience and  confidence of the experts?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  How reliable are individual classifications?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  When ranking lens candidates, what do the scores of teams of  experts mean?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  How should lens searchers best build an expert team to classify  their candidates?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  In Section 2 we will lay out our experiment, data and expert  participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' In Section 3 we investigate how subsets of the lens  candidates are scored by the ensemble of our users, enabling us  to understand the selection biases of our experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' In Section 4 we  investigate how individual users perform on the classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  We summarize the conclusions of this work in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  2 THE EXPERIMENT  Our experiment is designed to understand human expert biases when  performing visual inspection of strong lens candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' With experts  we refer to any person involved in strong lensing research, with  an academic status from masters student to Professor (or similar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  Additionally, we invited a small number of citizen scientists from  outside the academic strong lensing community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Several of these are  experienced users from the Spacewarps project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The details of our  invitation to be part of this experiment can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  We used the citizen science web portal Zoouniverse2 to serve a  sample of 1489 images for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  The users were asked to choose the best description for the object  displayed from four options: (1) Certain lens (> 90%), (2) Probable  lens (50 − 90%), (3) Probably not a lens (2 − 50%), and (4) Very  unlikely (< 1%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' We included percentages of confidence to be a lens  to avoid semantic uncertainty 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  We designed our experiment to closely mimic a real strong lens  classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' In real lens searches, there are no tutorials, and  there is limited prior knowledge of the completeness and purity of  the sample to be classified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' We therefore avoided offering further  guidelines about the composition of the data sets to be classified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' For  the same reason, we did not offer a tutorial nor examples as would  be usual in a citizen science project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  1 20 of the remaining objects are likely strong lenses but redshifts weren’t  obtained for both lens and source in 20 of the systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 2 systems yielded no  redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  2 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='zooniverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='org/projects/krojas26/  experts-visual-inspection-experiment  3 Formally, these percentages cannot be probabilities since we did not provide  the users with prior information on the composition of the sample being  classified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  Default  Blue  Sqrt  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Example of a DES cutout of 50 × 50 pixels 13′′ × 13′′ of an object  displayed in the three different scales presented to the users in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='1 The data  The image cutouts are from the Dark Energy Survey (DES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' DES uses  the Blanco 4-m telescope and the Dark Energy Camera (DECam,  Honscheid & DePoy 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Flaugher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 2015) located at Cerro  Tololo Inter-American Observatory (CTIO), Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The observations  are performed in the optical grizY bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' We used gri-bands to  produce colour composite 50 × 50 pixels (∼ 13′′ × 13′′) images  centering the object of interest in the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The images have a  typical 5σ depth of 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='72, 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='35, 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='88 in gri respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  We simultaneously display three different colour scalings of each  image to facilitate the recognition of the different features in the  stamp (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The three imaging scalings are: Default, blue and  sqrt, and they are described in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  Since we aimed to assess the classification skills of experts, we  required both a sufficiently large number of experts to participate  and sufficient classifications per expert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' This creates tension for ex-  periment design since experts have difficulty engaging with the ex-  periment if the classification task is too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' We decided that a  sample size of around 1000 objects (around an hour of time assum-  ing 2 seconds per classification) was small enough to get significant  engagement and large enough to provide useful data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  Given the rarity of real lenses on the sky, a random sample of  ∼1000 objects will not provide useful data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Instead, we designed a  sample to have a broad variety of data including: simulated lenses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  real lens candidates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' non-lens examples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' and unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' We also  duplicated 105 cutouts, 15 cutouts from seven out of nine data sets  excluding the eleven examples from SLACS and the four lenses from  Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The idea is to investigate the level of consistency  of individuals when classifying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' In total we had 1489 images to  classify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  Below we describe the different data sets, which are in 3 main  categories: lens simulations and lens candidates that are labelled  as lenses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' negative examples that are labelled as non-lenses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' and  unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 1 we present the name, number of objects and  category of each data set and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 2 we present as an example the  image of four objects of each labelled data set, more details about  these sets can be found in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='1 Lens simulations and additional lens examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  We created two sets of simulations of strong lens systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Each of  them contains 160 images and both have a uniform Einstein radius  distribution between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='8′′ < θE < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  The simulations were created using Lenstronomy4 (Birrer &  Amara 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Birrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 2021) and are based on real images for  the source and the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The full procedure is described in Rojas  4 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='com/lenstronomy/lenstronomy  MNRAS 000, 1–15 (2022)  Expert visual inspection of strong lenses  3  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Data sets presented in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  Data set name  number of objects  category  "Bright" simulations  150  Lens data set  Default simulations  150  Lens data set  SLACS  11  Lens data set  R22 lenses  4  Lens data set  LRGs  150  Non-lens data set  CNNs = 0  150  Non-lens data set  Non-lens simulations  150  Non-lens data set  CNN-best  300  Unlabeled data  Random  300  Unlabeled data  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' (2022) and can be summarized as follows: We pair Luminous  Red Galaxies (LRGs) from the DES and source galaxies from the  HST/HSC combined catalogue compiled by Cañameras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' (2020),  here the galaxies have the HST/ACS F814W high-resolution (Leau-  thaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Scoville et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Koekemoer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 2007) and the  colour information from Hyper Suprime Cam (HSC) ultra-deep stack  images (Aihara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' We modelled the mass of the systems  as a Singular Isothermal Ellipsoid (SIE), which has the following  parameters: the Einstein radius (θE), Position Angle, the axis ratio  and the central position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The Einstein radius was calculated using  the lens and source redshifts and the lens velocity dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' We  inferred the lens galaxy redshift and velocity dispersion using a K-  Nearest-Neighbors (KNN) algorithm, based on the assumption that  galaxies with similar gri magnitudes will also have similar redshifts  and velocity dispersion, this procedure is explained in detail in Rojas  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The rest of the parameters are derived by fitting an  elliptical Sérsic profile to the DES r-band image of the LRG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' From  this mass model we calculate the deflection of light, and trace rays  back onto the source plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' This process is done on an 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='03′′pixel  grid and then downsampled and PSF matched to the DES cutout of  the LRG, and the flux scaled to the DES zero point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' This simulated  arc is then added to the DES LRG image to create a simulated lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  The first data set is the "Bright simulations".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' This sample contains  a selection of simulations used to train the Convolutional Neural  Network (CNN) in Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' (2022), with the addition of smaller  Einstein radius simulations in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='8′′ < θE < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='2 that were  not used to train the neural network in that work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' In this dataset, the  magnitude of the sources is boosted by one magnitude brighter than  the observed HSC sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' This gives a population of bright lensing  features designed for the CNN to easily learn the properties of strong  lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' These simulations should be the easiest for experts to identify  as strong lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  The second data is the "Default simulations".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' These were created  with the same procedure as in Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' (2022), but without boosting  the magnitude of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' In this set of simulations we expect the  systems to be harder to classify, since the signal-to-noise ratio (SNR)  will be lower and the sources will stand out less brightly relative to  the lensing galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  Additionally to these two simulated sets of lenses we added the  eleven lenses from the Sloan Lens ACS Survey (SLACS, Auger  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 2009) in the field of view of DES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' These are spectroscopically  confirmed, smaller Einstein radius systems, that are clear in Hubble  Space Telescope imaging but represent a challenge to identify in  ground-based resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Since small Einstein radius systems are  expected to dominate in the real Universe (Collett 2015), we include  the SLACS sample to see if there is any chance to identify these  systems with visual inspection of ground-based telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Finally,  four lens candidates from Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' (2022) categorized with high  "Bright"  simulations  "Bright"  simulations  "Bright"  simulations  "Bright"  simulations  Default  simulations  Default  simulations  Default  simulations  Default  simulations  SLACS  SLACS  SLACS  SLACS  R22 lenses  R22 lenses  R22 lenses  R22 lenses  LRGs  LRGs  LRGs  LRGs  CNNs=0,  CNNs=0,  CNNs=0,  CNNs=0,  Non-lens  simulations  Non-lens  simulations  Non-lens  simulations  Non-lens  simulations  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Example of objects presented in the different labelled data  sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' "Bright" simulations, Default simulations, SLACS and R22 lenses from  Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' (2022) are examples of objects labelled as lenses, while LRGs,  CNNs=0 and Non-lens simulations are examples of objects labelled as non-  lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The cutouts are 50 × 50 pixels and they are displayed in the default  scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  scores in the "sure lens" list were also shown to the participant to see  if our classifiers agreed with the authors of Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' (2022), we  call this data set "R22 lenses".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='2 Negative examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  We included three data sets with 150 objects each that contained non-  lens examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' These samples test the classifiers’ ability to reject non-  lens systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The first data set is a random sub-sample of the negative  examples presented in the training set used to train the CNN in Rojas  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' (2022), we call this data set "Non-lenses training set" and  contains LRGs that were not used to create strong lens simulations 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  The second data set consisted of a random selection of objects that  were classified by the CNN with scores near zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' We called this data  set "CNNs=0" and it contains stamps from the LRG selection that  are highly improbable to contain any lens feature according to the  5 it is not impossible that this sample contains a real lens, but it is statistically  unlikely  MNRAS 000, 1–15 (2022)  4  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The third data set is called "Non-lens simulations": we follow  the same procedure as for strong lens simulations, but we disable the  lensing deflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Instead, we paint the source nearby to the LRG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  This is designed to mimic a ’source in front of LRG’ alignment,  which is a potential false positive for lens classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' This sample  allows us to evaluate if classifiers are able to distinguish between real  compact lenses and unlensed blue galaxies close to LRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='3 Unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  We additionally included two data sets of unlabeled data, each set  containing 300 objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The first one contains the 300 best stamps  graded by the CNN in Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' (2022), where 6 of them were  flagged as "Maybe lens" in that work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The objective of this data  set is to re-do the visual inspection and compare the classifications  of the authors of Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' (2022) with the participants in our  experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' We call this data set unlabeled as we do not have a  confirmed classification of the objects displayed, and although this  data set was previously inspected by another group we do not use this  information as prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The second set was created by selecting random  objects with CNN scores distributed between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='9 from the  sample analysed in Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The objective of this is both  to see if CNN grades and expert grades are correlated and to see if  a population of high-quality candidates are likely to be missed by  CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='2 Participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  We asked all the participants to complete a google form requesting  some basic and confidential information that we used to have a more  deep analysis of this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' We asked three multiple choice  questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' These questions and their options are:  (i) How many years have you worked in the field of gravitational  lensing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Less than one year, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Between 1-4 years, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Between 4-8  years, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Between 8-12 years, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' More than 12 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  (ii) What is your research status?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Master student, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' stu-  dent, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='Postdoc, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='Professor/lecturer/similar, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Amateur enthusiast6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  (iii) How confident do you feel classifying lens systems?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='Very  confident, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Confident, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' A bit confident, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Not confident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  This information was asked so that we could search for correlations  in the performance of the classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  A total of 80 people filled in the google form and a total of 69,592  classifications were made in the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Some of the users con-  tributed only a small number of classifications and 15% of them  did not perform any classification - we discarded the classifications  of such users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' We included all classifications made by users that  analyzed more than 25% of the sample (370 objects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' This cutoff  leaves 55 classifiers, where 51% of them finished the whole project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  Then we have a total of 66,835 classifications, with an average of 45  classifications per object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  The breakdown of the participants into the categories of academic  position, years of experience, and confidence are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  A point of interest from the information compiled at this point is  the correlation between confidence and experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' As is expected,  classifiers with more experience (either a higher academic status or  years working in the field) feel more confident performing the task  as we can see in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  6 For better understanding we call this group "Citizen scientist" here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Number of participants split in the three different categories re-  quested at the beginning of this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  Academic status  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' of participants  Professors  16  Postdocs  13  Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Students  15  Master Students  3  Citizen Scientist  8  Years of experience in the field  More than 12  11  8-12  8  4-8  10  1-4  17  Less than a year  9  Confidence  Very confident  16  Confident  22  A bit confident  14  Not confident  3  3 RESULTS A: THE DISCOVERY OF LENSES WITH  EXPERT VISUAL INSPECTION  In this section we look at the performance of the ensemble of classi-  fiers in identifying strong lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='1 Scoring objects  To compute a score for each object classified by our users, we translate  the four different options into numbers as follows: "Certain lens" = 1,  "Probable lens" = 2/3, "Probably not lens" = 1/3 and "Very unlikely" =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The score for each object is the mean value of all its classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  This gives every object a score between 0 and 1: objects scoring 1  are universally considered to be a strong lens and objects scoring 0  are universally considered non-lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 3 we present the mean score histograms split into the  various data subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Overall we find that the scores of non-lenses  are low and for many lenses the scores are high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  We investigated an alternative scoring system that up-weighted  users with higher classification skill (Marshall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 2015) but this  had no impact on our results (See Appendix D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='2 Scores for images known to contain lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  The left panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 3 shows the distribution of scores for the four  data sets of objects labelled as lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Both simulated data sets have  scores spanning the full range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' This is not unsurprising: some of the  simulations are bright arcs in textbook configurations whereas others  are extremely faint or are not easily resolved from the lensing galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  It is not a surprise that the "Bright" simulation set contains a higher  number of objects classified as lenses than in the "Default" one: the  brightest arcs stand out more from the lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 4 we present the  8 cutouts with the best scores for each of the lens simulation samples  on the two top panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  The simulated lenses also give us an insight into the selection  function of lens discovery with visual inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Since our sample  is relatively small, we can only gain a coarse understanding of the  selection function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' E1 we compare the recovery  MNRAS 000, 1–15 (2022)  Expert visual inspection of strong lenses  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='0  0  20  40  60  80  100  120  140  Number of subjects  Lens data sets  "Bright" simulations  Default simulations  SLACS  R22 lenses  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='5  Mean score  Non-lens data sets  LRGs  CNNs=0  Non-lens simulations  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='0  Unlabeled data  CNN-best  Random  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Mean score per object separated in each of the data sets presented in this work: lens examples (left panel), non-lens examples (middle panel), and  unlabeled data (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' All the histograms have the same binning with the expectation of SLACS and R22 lenses data sets where the bin size is double for  visualization purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='0  "Bright"  simulations  "Bright"  simulations  "Bright"  simulations  "Bright"  simulations  "Bright"  simulations  "Bright"  simulations  "Bright"  simulations  "Bright"  simulations  s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='0  s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='0  s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='0  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='99  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='99  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='98  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='98  s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='0  Default  simulations  Default  simulations  Default  simulations  Default  simulations  Default  simulations  Default  simulations  Default  simulations  Default  simulations  s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='0  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='99  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='99  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='98  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='97  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='97  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='97  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='2  LRGs  LRGs  LRGs  LRGs  LRGs  LRGs  LRGs  LRGs  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='19  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='19  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='19  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='18  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='17  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='16  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='16  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='31  CNNs=0  CNNs=0  CNNs=0  CNNs=0  CNNs=0  CNNs=0  CNNs=0  CNNs=0  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='23  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='18  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='17  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='17  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='17  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='16  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='15  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='31  Non-lens  simulations  Non-lens  simulations  Non-lens  simulations  Non-lens  simulations  Non-lens  simulations  Non-lens  simulations  Non-lens  simulations  Non-lens  simulations  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='3  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='3  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='29  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='28  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='28  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='27  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='26  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Examples of the eight objects with the highest mean score classified in the data sets: "Bright" simulations, Default simulations, LRGs, CNNs=0 and  non-lens simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The mean score is displayed at the bottom of each cutout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  fraction of simulated lenses and its standard deviation as a function  of signal-to-noise ratio in the g-band, the magnitude of the Arc in the  g-band and the Einstein radius of the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Since all of our images are  simulations of the approximately uniform depth Dark Energy Survey,  the first two quantities are tightly correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' It is clear from the heat  maps of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 5 that there is a fairly sharp cutoff in recovery fraction  for each quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Arcs fainter than ∼23rd magnitude (corresponding  to a total SNR less than about 25), or with Einstein radius less than  ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='2 ′′are not discoverable by human eye in DES-like imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' On  the other hand, from the standard deviation of the scores we do not  see any trend that allows us to get further information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  In the same way, when we analyzed the SLACS sample we found  that they were classified as "Non-lenses".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The selection function  of these systems drives them to have very bright lens galaxies and  Einstein radii below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='0 ′′ in most of the cases (Dobler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  Given the results on similar simulated lenses, it is therefore not  MNRAS 000, 1–15 (2022)  6  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='2-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='8-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='4-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='0-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
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+page_content='6  log10(SNR g-band)  "Bright" Simulations  Default Simulations  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
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+page_content='2  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='0-25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='0  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
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+page_content='0-23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='0  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='0-22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
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+page_content='0-21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
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+page_content='0  Arc g-band magnitude  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='8-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
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+page_content='6-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
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+page_content='4-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
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+page_content='8-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
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+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='0  Mean score  Einstein Radii (arcsec)  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Heat maps with the average mean score per bins for the objects in the simulated lenses data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The left panels correspond to the "Bright" simulations,  while the right panels to the "Default" simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' In the top panels we display the logarithm of the SNR in g-band, while in the bottom panels we present the  magnitude of the arc in the g-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The color bar was selected with the purpose to more easily identify the bins where the simulations can be recognized as lens  systems (in red) and where they are not identified (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  surprising that the human experts struggled to classify these systems  as lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' This is almost certainly because of the challenge to visually  deblend the lens and source in DES imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The eleven SLACS  systems in DES are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 6, in the three different colour scales  along with the score that they obtained from the visual inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  On the other hand, the "R22 lenses" received scores between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='6  and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' the classifiers consider them to probably be lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' These  R22 lenses were discovered using a visual inspection of the same DES  data, so it is not surprising that these lenses remain discoverable for  our classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Although, as we can see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 7, the visual inspection  scores obtained in Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' (2022) are somewhat different to those  of our classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' We attribute this to human factors which we will  discuss further in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='3 Performance in negative examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  Our Non-lens examples are divided in three different data sets, the  distribution of scores of their objects are shown in the middle panel  of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Here we see that most of them were classified with scores  between 0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='3, meaning they are very unlikely to be lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  The "Non-lens simulations" (mimicking a chance non-lensing align-  ment) data set shows a broader distribution towards higher values  these objects were potentially false positives, but they clearly do not  particularly confuse our expert classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  Even though most of the objects are correctly identified as non-  lenses, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 4 we present the eight cutouts of each sample with  higher scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Here we can see that most of the objects in the  MNRAS 000, 1–15 (2022)  Expert visual inspection of strong lenses  7  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='21  2MASXJ0216-0813  Default scale  Blue scale  Sqrt scale  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='13  2MASXJ0044+0113  Default scale  Blue scale  Sqrt scale  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='12  SDSSJ0008-0004  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='11  2MASXJ2341+0000  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='11  2MASSJ2111-0038  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='1  SDSSJ0252+0039  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='1  SDSSJ2347-0005  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='07  SDSSJ0157-0056  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='06  SDSSJ0029-0055  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='06  SDSSJ2300+0022  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='04  2MASXJ2141-0001  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Mosaic of the SLACS sample in the DES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' We displayed the cutouts of each system in the same three different scales presented in the experiment:  Default, Blue and Sqrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' On the cutout with the Default scale we added on the top the name of the system and on the bottom the visual inspection mean score  obtained in this search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  "LRGs" and "CNNs=0" data sets have little blue or red-ish com-  panions around the central galaxy that could be mistaken for signs of  lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' In the same way, the cutouts of the "Non-lens simulations"  set can be easily mistaken by very compact (low Einstein radii) lens  systems, producing that some users gave a higher score to these ob-  jects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Strictly speaking, the LRG and "CNNs=0" sets could contain  a lens, though the probability of this is ≪ 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='4 Performance in unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  We have two unlabeled data sets, "CNN-best" and "Random".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Intu-  itively we should hope that the CNN-selected sample should contain  more lens candidates than the random sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' All of the CNN-best  lenses have been inspected in Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' In the right panel  of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 3 we see that the CNN-best objects are distributed between  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='8 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='5 in the case of the Random sample), but the peak of  these distributions are around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='2, with most of the objects classified  as Non-lenses and only a few of them have a score above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  MNRAS 000, 1–15 (2022)  8  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='96  cnn=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='00, R22=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='00  DES J060653-585843  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='96  cnn=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='00, R22=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='00  DES J060653-585843  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='96  cnn=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='00, R22=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='00  DES J060653-585843  Default scale  Blue scale  Sqrt scale  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='84  cnn=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='00, R22=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='00  DES J003727-413149  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='84  cnn=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='00, R22=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='00  DES J003727-413149  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='84  cnn=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='00, R22=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='00  DES J003727-413149  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='71  cnn=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='00, R22=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='86  DES J033143-612315  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='71  cnn=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='00, R22=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='86  DES J033143-612315  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='71  cnn=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='00, R22=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='86  DES J033143-612315  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='62  cnn=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='00, R22=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='57  DES J052648-375125  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='62  cnn=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='00, R22=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='57  DES J052648-375125  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='62  cnn=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='00, R22=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='57  DES J052648-375125  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Mosaic of the four systems in Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' (2022) classified in their  category "Sure lens" displayed in the three different scales presented in this  experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' On the cutout with the Default scale we added on the top the  name of the system, and on the bottom the mean score (s) from our visual  inspection, the CNN (cnn) and visual inspection score (R22) from Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  (2022)  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 8 we present the 16 cutouts with the highest scores for  each of the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' In the "CNN-best" data set we find that five  of the objects were also recognized as candidates in Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  (2022), all of them classified in their "Maybe lens" catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' There  are also seven objects with scores above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='5 that were not classified  as potential candidates in Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' In the case of the  "Random" sample, none of the cutouts was classified as a potential  lens candidate, although some get close to a score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' We can  see some of them were highly graded by the CNN, this means that  they went through the visual inspection steps in Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' (2022)  but were not selected as lenses by those authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 9 shows the  scatter graph of aggregated expert scores against the CNN scores of  Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' These two scores are essentially uncorrelated,  indicating that the CNN and the experts are not responding to the  same features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Since none of the objects in the Random data received  a human score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='5 or more, we see no evidence of the CNN missing  goodcandidates, however,thiscannotbeadefinitiveconclusion given  the small sample size and the lack of correlation between the CNN  and human scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' On the other hand in the non-lens datasets we see  that most of the scores are below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='5 and closer to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='0, although a  few non-lens simulations managed to confuse the CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' In a similar  way, the CNN fails to recognize a few lens simulations, some of  them obtaining a score even lower than the one given by the human.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  Surprisingly some of the SLACS lenses obtained high scores by the  CNN, this makes us wonder if the CNN is able to see some features  that we humans can not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Nevertheless, giving an explanation of the  CNN’s behaviour is out of the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  4 RESULTS B: UNDERSTANDING EXPERT  CLASSIFICATION  In this section, we investigate how individual users performed when  executing the classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='1 How accurately do individuals classify our sample?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  In order to evaluate the classification performed by each user we  used the labeled data where we know the underlying truth: objects  are either Lens or Non-lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' In our labeled category we find 58%  of all the classifications (38,432 in total).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Knowing the true label of  each object and the classification given by each user we can compute  confusion matrices to compare user performance when classifying  the objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  Aggregating all of the classifiers and classifications, we computed  a confusion matrix displaying the four different classification op-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 10) we see the percentage of classifications that are  in agreement or disagreement with the original label of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  From here we can see that objects voted in the options "Certain lens",  "Probable lens" and "Very unlikely" are in general well classified,  achieving overall a 99%, 83%, and 80% of objects correctly labelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  On the other hand the classification "Probably not lens" is evenly  split between labelled lenses and labelled non-lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  To see in more detail the performance for each of the labelled  data sets, we calculated a confusion matrix displaying the percentage  of classifications according to the four different options for each  data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 11 we can see that the three data sets labelled  as "Non-lenses" obtained a very high percentage of votes in the  option "Very Unlikely".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' However, there is confusion when classifying  the simulated strong lens systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' SLACS lenses are mostly not  recognized as lenses and only the four "R22 lenses" get a high amount  of classifications in the categories "Certain lens" and "Probable lens".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  From the classifications of the labelled data, we compute confusion  matrices for each user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' In this section, we call objects Lenses if they  are classified as either "Certain lens" or "Probable lens".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Systems  classified as "Probably not lens" or "Very unlikely" are considered  Non-lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' In that way we build a 2 × 2 confusion matrix that will  tell us what it is the probability that a user classifies an object as  "Lens" or "Non-lens" given that the true label is "Lens" or "Non-  lens", this means the true positive and true negative rates in the  confusion matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  Following Marshall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' (2015) we plotted these probabilities in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 12, where we can see that most of the classifiers have very low  false positive rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' This makes these classifiers extremely good at  identifying easy lenses, but there is a significant range in their ability  to identify challenging lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Some classifiers manage complete-  ness of ∼ 60% at high purity, whilst more pessimistic classifiers are  identifying half as many lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  A handful of the classifiers are more optimistic, classifying more  marginal systems as lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' This comes at the cost of more false  positives, which is not desirable in a real search where lenses are  intrinsically rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  Additionally, we computed the same 2 × 2 confusion matrix join-  ing all the classifications of the users among the different groups  separated by academic status, years of experience and confidence in  performing the classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' This gives a confusion matrix for each  MNRAS 000, 1–15 (2022)  Expert visual inspection of strong lenses  9  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='84,  R22=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='00  CNN Best  CNN Best  CNN Best  CNN Best  CNN Best  CNN Best  CNN Best  CNN Best  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='74,  R22=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='57  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='72,  R22=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='43  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='7  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='57,  R22=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='71  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='56  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='55  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='54,  R22=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='71  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='54  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='52  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='51  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='5  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='49  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='47  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='46  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='45  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='49 ,  cnn=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='89  Random  Random  Random  Random  Random  Random  Random  Random  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='48 ,  cnn=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='18  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='44 ,  cnn=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='84  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='43 ,  cnn=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='87  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='36 ,  cnn=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='79  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='36 ,  cnn=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='41  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='36 ,  cnn=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='41  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='35 ,  cnn=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='41  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='32 ,  cnn=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='58  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='32 ,  cnn=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='35  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='32 ,  cnn=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='14  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='31 ,  cnn=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='90  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='31 ,  cnn=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='41  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='31 ,  cnn=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='15  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='31 ,  cnn=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='34  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='3 ,  cnn=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='41  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Examples of the 16 highly graded cutouts in the unlabeled data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The CNN-best data set is displayed in the two top panels, on the bottom of the  cutouts we present the mean score (s) from this experiment, and visual inspection score (R22) from Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' (2022) in case they were part of the final catalogs  presented in that work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The CNN score for all these objects is CNN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The Random sample is shown in the two bottom panels, on the bottom of each cutout  we display the mean score (s) from this experiment, and the CNN score (cnn) from Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='900  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='990  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='999  Mean score  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='900  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='999  CNN score  Random  LRGs  sim-nl  Bright sim  Default sim  SLACS  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Comparison between the human expert scores from our classifiers  and the CNN scores given by the model in Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The dashed  black line is the one-to-one line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Grey triangles are from the unlabeled 300  randomly drawn DES images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Squares are non-lens labelled data sets, in blue  LRGs and in green the simulated non-lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Circles are simulated lenses, in  yellow the bright data set and in red the default data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Purple Stars are the  SLACS lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  grouping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' To derive the error bars for these values, we use the stan-  dard deviation of the individual true positive and true negative rates of  each user in the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 13 shows these results: overall the results  are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' That is to say that, regardless of academic status,  years of experience and confidence, each grouping produces a very  similar average classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' There are very significant differences  between individual users, but time in the field, academic position or  reported confidence are not predictive of a user’s classification skill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  Certain lens  Probable lens  Probably not lens  Very unlikely  Visual Inspection Values  L  NL  Real Values  99 %  83 %  47 %  20 %  1 %  17 %  53 %  80 %  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Confusion matrix of the four different classification options, "Cer-  tain lens", "Probable lens", "Probably not lens" and "Very unlikely" contrasted  with the real labels L: lens and NL: No lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The percentages shown are the  number of lenses (non-lenses) classified in a determined option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='2 How reliable are individual classifications?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  To test the reliability of human classifications, we duplicated 105  objects in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' We randomly selected 15 objects from each of  the following data sets: CNN-best, LRGs,CNNs=0, Random, Default  simulations, "Bright" simulations, and Non-lens simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The  duplicate cutouts were shown at random points in the experiment 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  Overall, we had a total of 3797 duplicated classifications, with  73% graded the same as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' On the other hand, 15% (10%)  received an upgrade (downgrade) of one point in the classification,  7 Because of the way Zooniverse serves images, some users finished the  classification task but continued to classify a small number of randomly  drawn images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' We also took these into account in assessing the reliability of  classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  MNRAS 000, 1–15 (2022)  10  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  "Bright"  simulations  Default  simulations  SLACS  R22 lenses  LRGs  CNNs=0  Non-lens  simulations  Data sets  CL  PL  PNL  VU  Visual inspection values  32 %  23 %  0 %  54 %  0 %  0 %  0 %  24 %  24 %  4 %  29 %  2 %  2 %  6 %  22 %  24 %  22 %  12 %  16 %  17 %  25 %  22 %  28 %  74 %  5 %  82 %  81 %  68 %  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Confusion matrix of the seven different labeled data sets analyzed  in this experiment contrasted with the classification options CL: Certain  lens, PL: Probable lens, PNL: Probably not lens and VU: Very unlikely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The  percentages represent the amount of classifications made in each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
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+page_content='8  Completeness  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
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+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='0  Purity  Professor  Postdoc  Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='D Student  Master Student  Citizen Scientist  Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Completeness and purity percentages of each user when classify-  ing our labelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  this means that if for example an object was originally classified as  "Probable lens" in the second time performing the classification the  user classified the object as "Certain lens" in the upgrade case and  as "Probably not lens" in the downgrade case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='3% (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='6%)  of the classifications were upgraded (downgraded) by 2 points, and  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='13% (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='03%) by 3 points, which means changing completely the  classification from "Certain lens" to "Very unlikely" or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  These very low percentages for extreme cases are a very good sign  that the users are not obtuse classifiers and are somehow confident  about their classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' When we break these results down by self-  reported confidence (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 14), we see that "Very confident" users  perform relatively consistently, with a ∼ 75% of reliability in the  two extreme classification options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' On the other hand, the users that  signed as "Not confident" hesitate more at the time to use the "Certain  lens" option, reaching only 40% of reliability in this class, but a 76%  of reliability in the option "Probable lens" shows that they are more  comfortable with this more ambiguous selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
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+page_content='0  pr(C-LENS|T-LENS)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
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+page_content='0  pr(C-NO-L|T-NO-L)  Obtuse  Optimistic  Astute  Pessimistic  Random  Users  Professor  Postdoc  Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='D Student  Master Student  Citizen Scientist  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
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+page_content='0  pr(C-NO-L|T-NO-L)  Obtuse  Optimistic  Astute  Pessimistic  Random  Users  More than 12 years  between 8-12 years  between 4-8 years  between 1-4 years  Less than one year  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
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+page_content='0  pr(C-NO-L|T-NO-L)  Obtuse  Optimistic  Astute  Pessimistic  Random  Users  Very confident  Confident  A bit confident  Not confident  Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Probability that an object is classified as "Lens" given that it is  a lens (true positive rate) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' the probability that the object is classified as  "Non-lens" given that it is not a lens (true negative rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The values for each of  the users are displayed with a black circle, whose size represents the amount  of classification made by that user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The coloured stars represent the joint  result of the different groups of classifiers presented in this work, separated  by academic position (top panel), years of experience (middle panel) and  confidence in performing the classification task (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  MNRAS 000, 1–15 (2022)  Expert visual inspection of strong lenses  11  Very confident  Confident  A bit confident  Not confident  Certain lens  Probable lens  Probably not lens  Very unlikely  75 %  68 %  57 %  40 %  57 %  59 %  56 %  76 %  57 %  51 %  57 %  61 %  76 %  74 %  62 %  53 %  Percentage of reliability  Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Mean percentage of reliability of users at the time to re-classify a  object, according to their level of confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  4  7  10  13  16  19  Team size  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='20  Classification error  Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Standard deviation of scores when classified by a subset of users  relative to the ’truth’ from all our users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='3 How many classifiers are needed for a reliable score?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  The classification of an object into any of the options is a personal and  subjective opinion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Even with clear guidelines and examples, there  will be disagreement among users, as the visual inspection work in  Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Savary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' (2022) showed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Typical strong lens  searches have had a handful of expert classifiers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 3 for Jacobs  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Given the individual expert variation seen in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='1  and the lack of reliability seen in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='2, it is to be expected that  using a small number of experts can significantly bias final scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' To  assess how significant this bias can be, we divide our classifications  into random ’teams’ of n users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' We computed new scores using only  the classification of the team members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' We did this for team sizes  ranging from one to twenty participants randomly drawn from our  users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  We compare the team scores with the final score from the entirety  of our classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' We created 200 teams of n random users to recal-  culate the score for objects labelled as lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 15 shows how the  team size affects the scoring of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The standard deviation error  is substantial for very small teams but decreases quickly up to a team  size of ∼ 8, which has a classification error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='07 per object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Above  8 users the classification error decreases much more slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Given  that most objects are unambiguously classified as very unlikely (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  score of 0), these represent big differences in scores for the other  objects: small teams are not very good at assessing the quality of a  strong lens candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  The results in this section go some way to explaining why the  scores of Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' (2022) vary from those of our classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' There  are not enough systems with overlapping data to draw firm conclu-  sions (9 images), but the standard deviation is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='08, as is expected  for a team of 6 classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  Since teams will be most concerned about discovering marginal  lenses, it is salient to focus on team accuracy when classifying lenses  with marginal scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Focusing on images with true scores between  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='3 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='8 the classification error grows substantially: it is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='17 for  a team of 2 classifiers, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='12 for a team of 4 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='07 for a team of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  A team of 6 classifiers is required to achieve an expected accuracy  better than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='1, 15 are needed for an accuracy better than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  5 CONCLUSIONS  This work has investigated how strong lensing experts visually in-  spect images of galaxy-scale strong lens candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' We showed them  a sample of 1489 mock and real images of the Dark Energy Survey  and asked them to grade each image as either: Certain lens;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Prob-  able lens;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Probably not a lens;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' or, Very unlikely to be a lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' With  the resulting 66835 classifications we can now answer our initial  questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  What are the properties of lensing systems that human experts  identify reliably as lenses, and what do they miss?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  Most gravitational lenses are reliably identified with an Einstein Ra-  dius greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='2′′and an arc g-band magnitude less than 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  This corresponds to roughly 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='2 times the seeing of the Dark Energy  Survey and a g-band signal-to-noise of 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Some lenses are discov-  ered with fainter or smaller radius arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The Einstein radius cut-off is  sharp with only a handful of very bright arcs discovered with Einstein  radius of less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='2′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The flux cut-off is smoother: roughly twenty  percent of lenses are recovered even with an arc signal-to-noise of  between 4 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  Do human experts confuse non-lenses for lenses?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  For our labelled data, none of the non-lenses scored higher than  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Our simulations do not include face-on spirals, or ring galaxies,  but the experts had no problem rejecting chance alignments of blue  sources close to LRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Our labelled sample had an almost equal  split of lenses and non-lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Unless robotic candidate selection  improves substantially, real lens searches will have far more non-  lenses than lenses, even so, it seems that human experts are very  good at discarding the kinds of non-lenses shown here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Follow-up  campaigns should be confident that highly scored candidates are  almost certainly lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  How does expert classification depend on the experience and  confidence of the experts?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  MNRAS 000, 1–15 (2022)  12  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  We see substantial variation in the purity and completeness of indi-  vidual classifiers, but there are no significant trends with experience,  confidence, or academic position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' None of these traits reliably predict  the over-pessimism or over-optimism of some users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  How reliable are individual classifications?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  Classifications are not reliable when repeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Even classifiers who  self-report as ’very confident’ do not grade candidates consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  When reclassifying the same images, certain and very unlikely lenses  are scored the same roughly three-quarters of the time by confident  and very confident classifiers, whereas probable and probably not  lenses are only reproduced three-fifths of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Fewer than 2  percent of reclassified targets changed by more than one classification  step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  When ranking lens candidates, what do the scores of teams of  experts mean?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' How should lens searchers best build an expert team  to classify their candidates?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  Given the fact that classifications by a single expert are not reli-  able when repeated, it is not surprising that small teams make for  poor classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' On a 0-1 scoring system, teams of 6 classifiers will  produce results within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='1 of the ensemble average of all users when  classifying marginal systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Senior classifiers are, on the whole, not  better than junior classifiers so teams should classify independently  and not defer to the opinions of senior faculty members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  We found no correlation between CNN and human scores sug-  gesting that CNNs are not trained to recognise the same features  as human experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Bigger samples are needed to assess if this is a  problem for lens finding in future surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  A traditional search would use a small number of classifiers to  grade a large number of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' To understand the human classifi-  cation process, we have done the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' In the real Universe, real  lenses are much rarer than our sample, so it is possible that our results  do not perfectly scale to a search of a billion objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' However, if we  assume that our discovery thresholds map onto searches of entire  surveys our results suggest that previous forecasts are likely to be  somewhat optimistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The discovery signal-to-noise of our experts  is broadly consistent with the assumptions of Collett (2015), how-  ever our experts recovered few lenses with Einstein radius less than  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='2′′, which represents ∼40 percent of the forecasted DES population  in Collett (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Collett (2015) had assumed that users would be  shown lens-subtracted images, such as in Sonnenfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' (2018)  and future work should investigate if expert inspection can recover  even more lenses with such an approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  This work has received funding from the European Research Council  (ERC) under the European Union’s Horizon 2020 research and in-  novation programme (LensEra: grant agreement No 945536).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' TC is  funded by the Royal Society through a University Research Fellow-  ship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' For the purpose of open access, the authors have applied a Cre-  ative Commons Attribution (CC BY) licence to any Author Accepted  Manuscript version arising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' DB is funded by a graduate studentship  from UK Research and Innovation’s STFC and the University of  Portsmouth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' This work is also in part supported by the Swiss National  Science Foundation (SNSF) and by the European Research Council  (ERC) under the European UnionâĂŹs Horizon 2020 research and  innovation program (COSMICLENS: grant agreement No 787886).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  JHHC aknowledge the generosity of Eric and Wendy Schmidt by  recommendation of the Schmidt Futures program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' FG acknowledges  the support from grant PRIN MIUR 2017-20173ML3WW_001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' RJ  acknowledges the support from the research project grant "Under-  standing the Dynamic Universe" funded by the Knut and Alice Wal-  lenberg Foundation under Dnr KAW 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='0067.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' thank the Max  Planck Society for support through the Max Planck Research Group  for SHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' This project has received funding from the European Re-  search Council (ERC) under the European Unions Horizon 2020  research and innovation programme (LENSNOVA: grant agreement  No 771776).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' This research is supported in part by the Excellence  Cluster ORIGINS which is funded by the Deutsche Forschungsge-  meinschaft (DFG, German Research Foundation) under Germany’s  Excellence Strategy – EXC-2094 – 390783311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' TD Thanks the sup-  port by an LSSTC Catalyst Fellowship awarded by LSST Corporation  with funding from the John Templeton Foundation grant ID #62192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  GM acknowledges funding from the European Union’s Horizon 2020  research and innovation programme under the Marie SkÅĆodowska-  Curie grant agreement No MARACHAS - DLV-896778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' BD was  supported in part by the NASA Astrophysics Theory Program under  grant 80NSSC18K1014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Coordenação de Aper-  feiçoamento de Pessoal de Ensino Superior (CAPES) - Finance Code  001 LAU-L thanks CONACyT México for support under grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  A1-S-17899, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 286897, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 297771, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 304001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' and the Insti-  tuto Avanzado de Cosmología Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' JW was supported by  the Science and Technology Facilities Council under grant number  ST/P006760/1, the DISCnet Centre for Doctoral Training in Data-  Intensive Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' JM acknowledges the support of the UK Science  and Technology Facilities Council (STFC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  DATA AVAILABILITY  The inclusion of a Data Availability Statement is a requirement for  articles published in MNRAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Data Availability Statements provide  a standardised format for readers to understand the availability of data  underlying the research results described in the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The statement  may refer to original data generated in the course of the study or to  third-party data analysed in the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The statement should describe  and provide means of access, where possible, by linking to the data or  providing the required accession numbers for the relevant databases  or DOIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
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+page_content=', 2022, A&A, 666, A1  Scoville N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
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+page_content=', 2007, ApJS, 172, 38  Shu Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
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+page_content=', 2016, ApJ, 833, 264  Shu X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
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+page_content=', 2022, ApJ, 926, 155  Sonnenfeld A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
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+page_content=', 2018, PASJ, 70, S29  Stark D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
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+page_content=', 2015, MNRAS, 454, 1393  Tran K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
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+page_content=' arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='05307  Vegetti S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=', Koopmans L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
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+page_content=', Bolton A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=', Treu T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=', Gavazzi R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=', 2010, MNRAS,  408, 1969  Wong K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=', 2020, MNRAS, 498, 1420  AFFILIATIONS  1 Institute of Cosmology and Gravitation, University of Portsmouth,  Burnaby Rd, Portsmouth PO1 3FX, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  2Kavli Institute for Particle Astrophysics and Cosmology and  Department of Physics, Stanford University, Stanford, CA 94305,  USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  3SLAC National Accelerator Laboratory, Menlo Park, CA, 94025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  4Department of Physics and Astronomy, Stony Brook University,  Stony Brook, NY 11794, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  5Fermi National Accelerator Laboratory, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Box 500, Batavia, IL  60510, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  6Department of Astronomy and Astrophysics, University of Chicago,  Chicago, IL 60637, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  7Department of Physics and Astronomy, Lehman College of the  CUNY, Bronx, NY 10468, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  8Department of Astrophysics, American Museum of Natural History,  Central Park West and 79th Street, NY 10024-5192, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  9Institute of Physics, Laboratory of Astrophysics, Ecole Polytech-  nique Fédérale de Lausanne (EPFL), Observatoire de Sauverny,  1290 Versoix, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  10Instituto de FÃŋsica de Cantabria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' (CSIC-UC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Avda Los Castros  s/n 39095 Santander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  11University of Bologna, Department of Physics and Astronomy  (DIFA), Via Gobetti 93/2, I-40129, Bologna, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  12INAF – Osservatorio di Astrofisica e Scienza dello Spazio, via  Gobetti 93/3 - 40129, Bologna - Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  13Physics Department, University of Wisconsin-Madison, Madison,  WI 53706, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  14Oskar Klein Centre for Cosmoparticle Physics, Department of  Physics, Stockholm University, Stockholm SE-106 91, Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  15Consejo Nacional de Ciencia y Tecnología, Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Insurgentes Sur  1582, Col.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Crédito Constructor, Alc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Benito Juárez, CP 03940,  México.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  16Mesoamerican Centre for Theoretical Physics, Universidad  Autónoma de Chiapas, Carretera Zapata Km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 4, Real del Bosque,  29040,  Tuxtla Gutiérrez, Chiapas, México.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  17Max-Planck-Institut für Astrophysik, Karl-Schwarzschild Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 1,  85748 Garching, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  18Technische Universität München, Physik Department, James-  Franck Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 1, 85748 Garching, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  19INAF – Osservatorio Astronomico di Capodimonte, Salita  Moiariello 16, 80131 - Napoli, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  20Instituto de Astronomía, Observatorio Astronómico Nacional,  Universidad Nacional Autónoma de México, Apartado postal 106,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 22800,  Ensenada, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=', México.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  21Sub-department of Astrophysics, Denys Wilkinson Building,  University of Oxford, Keble Road, Oxford, OX1 3RH, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  22Department of Astrophysical Sciences, Princeton University, 4 Ivy  Lane, Princeton, NJ 08544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  23LSSTC Catalyst Fellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  24Department of Physics and Astronomy, School of Natural Sci-  ences, University of Manchester, Oxford Rd, Manchester M13 9PL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  25School of Physics & Astronomy, University of Nottingham,  University Park, Nottingham, NG7 2RD, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  26Centre for Extragalactic Astronomy, Department of Physics,  Durham University, Durham DH1 3LE, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  27Institute for Computational Cosmology, Durham University, South  Road, Durham DH1 3LE, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  28Observatório Nacional, Rua Gal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' José Cristino 77, Rio de Janeiro,  RJ - 20921-400, Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  29OmegaLambdaTec GmbH, Lichtenbergstraße 8, 85748 Garching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  30Universitäts-Sternwarte,  Fakultät  fÃijr  Physik,  Ludwig-  Maximilians Universität München, Scheinerstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 1, 81679 München,  Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  31Citizen  Scientist,  Zooniverse,  Astrophysics  Sub-department,  University of Oxford, Keble Road, Oxford, OX1 3NP, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  32Department of Physics and Astronomy, University of New Mexico,  210 Yale Blvd NE, Albuquerque, NM 87106, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  33Centre for Astrophysics and Supercomputing, Swinburne Univer-  sity of Technology, PO Box 218, Hawthorn, VIC 3122, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  34University of Birmingham, School of Physics and Astronomy,  Edgbaston, Birmingham, B15 2TT, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  35Departamento de Física, DCI, Campus León, Universidad de  Guanajuato, 37150, León, Guanajuato, México.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  36Department of Physics, Benedictine University, 5700 College  Road, Lisle, Illinois, 60532.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  37School of Physical Sciences, The Open University, Walton Hall,  Milton Keynes, MK7 6AA, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  38IRAP, UniversitÃľ de Toulouse, UPS, CNRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  39Institut für Theoretische Astrophysik, Zentrum für Astronomie,  Heidelberg Universität, Albert-Ueberle-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 2, 69120, Heidelberg,  Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  40Instituto Sciety Lab, São José dos Campos, SP, Brazil  41Department of Physics, University of California, 1156 High St,  Santa Cruz, CA 95064, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  42Santa Cruz Institute of Particle Physics, University of California,  1156 High St, Santa Cruz, CA, 95064 USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  43School of Computing & Communications, The Open University,  Walton Hall, Milton Keynes MK7 6AA, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  APPENDIX A: INVITATION  We send an invitation to members of the strong lensing community,  including LSST, DES, and Euclid strong lensing working groups and  MNRAS 000, 1–15 (2022)  14  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  we extended the invitation to Space Warp citizen scientists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' They  received the following invitation:  "We would like to invite you to participate in a lens classification experiment  with the goal of understanding how people in the field of gravitational lensing  are performing when they do a visual classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The task will take  about an hour of your time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Participants will be invited to coauthor the  resulting paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  The motivation of this experiment comes from the explosion of new lens  systems discovered by the use of CNN and subsequent validation through  visual inspection performed for each team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' There seem to be lots of differences  in the expert validation and we want to see if this can be understood and  calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  We expect that with this social experiment we can get key conclusions about  our performance, and hope that in the future lens finders can benefit from this  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' If you want to be part of this experiment please fill this quick  google form first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The information requested here will help us to analyze the  data, although your personal data (name, email and galaxy zoo username)  will remain private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  And now you are welcome to classify 1000 objects!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' If you follow this link8  You will see DES gri-color composite images of each object, each stamp has  a size of 50x50 pixels (13"x13").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The same object is displayed in 3 different  color-scales to help the recognition of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The task is simple: you have  to click on the option that better represents the object(s) in the image and go  to the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  You don’t have the obligation to complete the classification all at once, this  might take you a couple of hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Your progress is recorded and you can come  back anytime you want ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' For a successful analysis we hope you can commit  to completing the classification of the whole data set, or at least a big portion  of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  Please share this among your group, postdocs, phd students, masters students,  etc, that work in the field of strong lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' All are encouraged to participate  as we want to test a broad variety of expertise, but please do not share it  among people outside of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='"  APPENDIX B: IMAGE SCALINGS  The "default" and "blue" composite images are scaled with an arcsinh  stretch using (HumVI, Marshall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' We tuned the rgb-scale  parameters, the brightness (Q) and contrast (α) for default (blue)  images as follows: r-scale = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='51), g-scale = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='68), b-scale=  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='8 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='12), Q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='64), α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='03 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='03).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The third image is scaled  using a square root image scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' To ensure that the object of interest  is not overshadowed by another brighter one we set minimum and  maximum values for the pixels in the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The minimum value is  obtained using a square of 5×5 pixels in the corners of the gri-images  and we select the minimum value among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' On the other hand,  the maximum value is obtained by placing a box of 10 × 10 pixels in  the middle of the three images and obtaining the mean among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  We use these two values to scale the three gri-images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  APPENDIX C: EXPERIENCE AND CONFIDENCE  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' C1 we present the distribution of the levels of confidence  compared to the academic status and years of experience in the field,  according to the information provided by the classifiers when they  subscribed to this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' From both plots, we can clearly see  that at higher academic status or years of experience in the field,  the classifiers feel more confident that they will perform a successful  8 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='zooniverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='org/projects/krojas26/  experts-visual-inspection-experiment/classify  Very confident  Confident  A bit confident  Not confident  3  6  9  Professor  Postdoc  PhD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Student  Master Student  Citizen Scientist  Very confident  Confident  A bit confident  Not confident  3  6  9  > 12 years  8-12 years  4-8 years  1-4 years  < 1 year  Figure C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The radar charts show the level of confidence that users have to  perform the classification task separately in academic status (top panel) and  years of experience (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  classification, while undergrad students and people with between 0-4  years of experience feel "not" or only a "bit confident".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  APPENDIX D: SCORING SYSTEM  In addition, we wanted to explore the impact of weighting the classi-  fication of the users according to their percentage of correct classifi-  cations in each of the 4 options given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' To obtain this percentage we  computed a 2 × 4 confusion matrix similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 10 but this time  for each user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Then we take the 4 relevant percentages: Classified as  "Certain lens" or "Probable lens" given that it is a lens and "Probably  not lens" or "Very unlikely" given that it is not a lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Using these  percentages as weight we then calculated a weighted mean score, this  means that we multiply each score by the corresponding weight, sum  them and divide by the sum of the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' After weighting all the  classifications we re-scale the mean score weighted between 0 and 1  as we did with the previous score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  We expected that weighting the mean score could provide a better  score system because it will take into account the performance of the  users, but most of the users performed in a pretty similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' We  can see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' D1 that the mean score and the mean score weighted  are highly correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' For this reason, we are going to continue our  MNRAS 000, 1–15 (2022)  Expert visual inspection of strong lenses  15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='0  Mean score  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='0  Mean score weighted  Figure D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Scatter plot of the mean score compared with the mean score  weighted by the corresponding percentage of successful classification in the  options displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  analysis using only the mean score and we are not going to explore  the mean score weighted implications in further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  APPENDIX E: STANDARD DEVIATION  We calculated the standard deviation of the scores given to each  object in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' With this information, we create a heat  map (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' E1) to compare the standard error of the recovery fraction  of the simulated lenses data sets in response to the signal-to-noise  ratio in the g-band, the magnitude of the arc in the g-band and the  Einstein radius of the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' In contrast with the heat maps with the  average scores (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' 5), we do not see any trend in the standard  deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  MNRAS 000, 1–15 (2022)  16  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
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+page_content='6  log10(SNR g-band)  "Bright" Simulations  Default Simulations  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
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+page_content='8-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='0  Standard deviation  Einstein Radii (arcsec)  Figure E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' Heat maps with the average standard deviation of the mean score per bins for the objects in the simulated lenses data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' The left panels correspond  to the "Bright" simulations, while the right panels to the "Default" simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content=' In the top panels we display the logarithm of the SNR in g-band, while in the  bottom panels we present the magnitude of the arc in the g-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
+page_content='  MNRAS 000, 1–15 (2022)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE2T4oBgHgl3EQfHgbc/content/2301.03670v1.pdf'}
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+Version January 30, 2023
+Preprint typeset using LATEX style openjournal v. 09/06/15
+COSMOLOGY WITH 6 PARAMETERS IN THE STAGE-IV ERA: EFFICIENT MARGINALISATION OVER NUISANCE
+PARAMETERS
+B. Hadzhyiska1,2,∗, K. Wolz3,4, S. Azzoni5,6, D. Alonso5, C. García-García5, J. Ruiz-Zapatero5, and A. Slosar7
+1Miller Institute for Basic Research in Science, University of California, Berkeley, CA, 94720, USA
+2Physics Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720
+3International School for Advanced Studies (SISSA), Via Bonomea 265, 34136 Trieste, Italy
+4National Institute for Nuclear Physics (INFN) – Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy
+5Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, United Kingdom
+6Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU, WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan and
+7Physics Department, Brookhaven National Laboratory, Upton NY 11973, USA
+(Dated: January 30, 2023)
+Version January 30, 2023
+ABSTRACT
+The analysis of photometric large-scale structure data is often complicated by the need to account for many
+observational and astrophysical systematics. The elaborate models needed to describe them often introduce
+many “nuisance parameters”, which can be a major inhibitor of an efficient parameter inference. In this paper,
+we introduce an approximate method to analytically marginalise over a large number of nuisance parameters
+based on the Laplace approximation. We discuss the mathematics of the method, its relation to concepts such
+as volume effects and profile likelihood, and show that it can be further simplified for calibratable systematics
+by linearising the dependence of the theory on the associated parameters. We quantify the accuracy of this
+approach by comparing it with traditional sampling methods in the context of existing data from the Dark
+Energy Survey, as well as futuristic Stage-IV photometric data. The linearised version of the method is able to
+obtain parameter constraints that are virtually equivalent to those found by exploring the full parameter space
+for a large number of calibratable nuisance parameters, while reducing the computation time by a factor 3-10.
+Furthermore, the non-linearised approach is able to analytically marginalise over a large number of parameters,
+returning constraints that are virtually indistinguishable from the brute-force method in most cases, accurately
+reproducing both the marginalised uncertainty on cosmological parameters, and the impact of volume effects
+associated with this marginalisation. We provide simple recipes to diagnose when the approximations made by
+the method fail, and one should thus resort to traditional methods. The gains in sampling efficiency associated
+with this method enable the joint analysis of multiple surveys, typically hindered by the large number of nuisance
+parameters needed to describe them.
+1. INTRODUCTION
+The Λ Cold Dark Matter (ΛCDM) model of cosmology
+offers a compelling explanation for a wide variety of obser-
+vations despite the fact that the nature of its dominant com-
+ponents, dark matter and dark energy, remains unknown. As
+the statistical power of cosmological experiments grows in the
+next decade, our ability to stress test the ΛCDM model of
+the Universe will improve immensely and provide powerful
+constraints on its dark ingredients. In addition, these new ex-
+periments may shed light over discrepancies between early-
+and late-Universe-derived cosmological constraints that have
+recently emerged.
+Of highest relevance to this study is the tension affect-
+ing the measurement of the amplitude of matter fluctuations,
+𝑆8 ≡ 𝜎8(Ω𝑚/0.3)0.5, where 𝜎8 is the root mean square of
+matter fluctuations on an 8 Mpc/ℎ scale, and Ω𝑚 is the frac-
+tional energy density in non-relativistic matter. In particular,
+the latest prediction from Planck CMB data finds its value
+(Planck Collaboration et al. 2020) to be ∼2-3𝜎 higher than
+the analogous measurement from photometric surveys such as
+KiDS-1000 (KiDS), the Dark Energy Survey (DES) and Hyper
+Suprime-Cam (HSC) (Heymans et al. 2021; DES Collabora-
+tion 2022, 2018; MacCrann et al. 2015; Hamana et al. 2020).
+As data from current and future cosmological surveys such
+∗boryanah@berkeley.edu
+as the Legacy Survey of Space and Time (LSST), at the Vera
+Rubin Observatory (LSST Dark Energy Science Collabora-
+tion 2012), the Nancy Grace Roman Space Telescope (Spergel
+et al. 2015), or the Euclid satellite (Amendola et al. 2018)
+starts to trickle in, it is of paramount importance for cosmol-
+ogists to devise robust tests of their analysis pipelines, and to
+construct rigorous theoretical frameworks to better understand
+these tensions and extract maximal cosmological information.
+Photometric surveys offer a pathway to resolving many is-
+sues of the standard paradigm by providing measurements of
+the clustering and weak gravitational lensing of millions and
+soon billions of galaxies on the sky.
+In particular, the so-
+called “3×2pt” analysis (the joint analysis of galaxy cluster-
+ing and cosmic shear in tomographic bins) offers a powerful
+tool to cosmologists with the potential to break degenera-
+cies between cosmological and astrophysical parameters and
+yields stringent constraints (Heymans et al. 2021). Such tomo-
+graphic analyses have a leverage on the 𝑆8 and Ω𝑚 tensions and
+can yield ∼1-10%-level constraints (e.g., Nicola et al. 2020;
+Hadzhiyska et al. 2021; García-García et al. 2021). Nonethe-
+less, it is still unclear whether this tension is driven by system-
+atic and modeling errors in the analysis or by new physics.
+Large-scale structure surveys in general are affected by a
+large number of observational systematic uncertainties, as well
+as uncertainties in the physical relation between the astrophys-
+ical systems that form the basis of their observables, and the
+arXiv:2301.11895v1  [astro-ph.CO]  27 Jan 2023
+
+2
+underlying cosmological quantities we wish to constrain (com-
+monly labelled “astrophysical systematics”). With increasing
+statistical power, new sources of systematic uncertainty be-
+come more relevant, and previously known systematics require
+more detailed models. This inevitably leads to an inflation in
+the number of nuisance parameters relative to the number of
+the physically meaningful cosmological parameters. Exam-
+ples of these systematics are: uncertainties in the redshift dis-
+tribution of the samples, errors in the measured galaxy shapes
+(relevant for weak lensing studies), the link between the abun-
+dance of galaxies and the underlying matter overdensities, and
+the intrinsic alignments between galaxy orientations and the
+local structures. Moreover, since the future of cosmology lies
+in the combination of multiple observational probes, the simul-
+taneous modeling of these effects across different datasets, and
+the efficient sampling of the resulting (typically large) param-
+eter space is bound to become a top priority in the next few
+years.
+To illustrate this, the latest analysis of galaxy clustering and
+weak gravitational lensing carried out by the Dark Energy
+Survey (DES) (DES Collaboration 2022) included 6 cosmo-
+logical parameters (ΛCDM and the total neutrino mass) and
+25 nuisance parameters. Due to the curse of dimensionality,
+this increase in the number of parameters leads to strong ineffi-
+ciencies in the standard rejection sampling algorithms used to
+explore the parameter space. The resulting parameter chains
+take long times to converge (e.g. days or weeks for state-of-
+the-art datasets), even though we are ultimately only interested
+in the marginal posterior distribution of a much smaller pa-
+rameter space.
+This problem can be addressed through various approaches.
+On one hand, the use of gradient-based sampling algorithms,
+such as Hamiltonian Monte-Carlo approaches (MacKay 2002;
+Hoffman & Gelman 2011), or other methods that are well-
+suited to multiple-parameter problems, can significantly speed
+up the sampler convergence time and provide constraints in a
+reasonable amount of time.
+Recently, Ruiz-Zapatero et al.
+(2023) validated the analytical marginalisation of redshift cal-
+ibration models with up to ∼ 100 parameters using self-tuning
+Hamiltonian Monte-Carlo (a problem that has also been ad-
+dressed by e.g. Stölzner et al. (2021); Zhang et al. (2023) using
+other methods). However, the performance of the marginali-
+sation method depends on the choice of nuisance parameters,
+on one hand through their effect on the theory prediction, and
+on the other hand through their priors.
+For Gaussian data with parameters affecting the theory pre-
+dictions to linear order, this can be effectively done by modify-
+ing the covariance matrix and fixing the marginalised param-
+eters (Rybicki & Press 1992). For more complex parameters
+that appear as higher-order terms in the model such as the
+galaxy bias parameters, exact analytic marginalisation is not
+generally possible. One may resort to Gibbs sampling-like
+schemes (Geman & Geman 1984), where this marginalisa-
+tion is done numerically on the fly, and which can potentially
+lead to significant speed-ups in the sampling process. How-
+ever, the efficiency of this approach depends on the effective
+acceptance rate of the marginalisation step, and on the degen-
+eracy between nuisance and cosmological parameters. Here,
+we propose a general technique that allows for approximate
+analytical marginalisation over both linear and non-linear nui-
+sance parameters in an efficient manner.
+Similar methods
+have been put forward in the past (e.g., Taylor & Kitching
+2010), with a variety of applications in mind (Bridle et al.
+2002; Stölzner et al. 2021). Here, we will quantify the validity
+of this method in the context of photometric 3×2pt analyses
+with current data from DES, and futuristic Stage-IV-like data
+mimicking experiments such as LSST.
+This paper is organised as follows. In Section 2, we pro-
+vide a general introduction to our analytical marginalisation
+method and discuss the interpretation of the various terms
+we define and their relevance to volume effects and param-
+eter priors.
+In Section 3, we introduce relevant aspects of
+the analysis of cosmological photometric surveys and explore
+possible ways in which our method can be used to marginalise
+over linearisable and non-linearisable nuisance parameters in
+the model. In Section 4, we showcase the effect of employing
+our method both in terms of deriving unbiased constraints on
+the cosmological parameters and also in terms of the conver-
+gence time and performance. We summarise our findings and
+comment on their implications for the future of photometric
+survey analysis in Section 5.
+2. LIKELIHOODS AND NUISANCE PARAMETERS
+Our aim is to explore the posterior distribution 𝑝( �𝜃|d),
+where �𝜃 is a set of model parameters, and d is the data. Using
+Bayes’ theorem, the posterior can be written in terms of the
+likelihood 𝑝(d| �𝜃) and prior 𝑝( �𝜃) as
+𝑝( �𝜃|d) ∝ 𝑝(d| �𝜃)𝑝( �𝜃).
+(1)
+We will decompose the model parameters as �𝜃 = { �Ω, �𝑛},
+where �Ω is a set of parameters we care about (e.g. fundamen-
+tal cosmological parameters), and �𝑛 is a vector of nuisance
+parameters, describing various observational or theoretical
+uncertainties, which are largely irrelevant to the fundamen-
+tal question being explored. In other words, the distribution
+we aim to obtain is the marginalised posterior
+𝑝( �Ω|d) =
+∫
+𝑑�𝑛 𝑝( �Ω, �𝑛|d).
+(2)
+For simplicity, in what follows, for any probability distribu-
+tion 𝑝( �𝜃), we will define the “chi-squared”, 𝜒2, as
+𝜒2( �𝜃) ≡ −2 log 𝑝( �𝜃) + 𝐾,
+(3)
+where 𝐾 is an arbitrary constant that does not depend on the
+random variables �𝜃. Note that it is more common to define
+𝜒2 in terms of the likelihood, but in this paper, we will always
+apply it to the posterior, generalising to the presence of priors.
+2.1. Profiling and analytical marginalisation
+In order to approximate the marginal distribution in Eq. (2),
+let us start by considering the best-fit value of the nuisance
+parameters having fixed �Ω. That is, we define �𝑛∗( �Ω) as
+�𝑛∗( �Ω) ≡ arg max�𝑛𝑝( �Ω, �𝑛|d).
+(4)
+Assuming that the distribution is differentiable at all points,
+�𝑛∗ then satisfies
+𝜕𝜒2
+𝜕�𝑛
+����
+�𝑛∗
+= 0.
+(5)
+Following Taylor & Kitching (2010), we can then approx-
+imate the distribution at each value of �Ω by expanding 𝜒2 to
+second order in �𝑛 around �𝑛∗, i.e.:
+𝜒2( �Ω, �𝑛) ≃ 𝜒2
+∗ ( �Ω) + Δ�𝑛𝑇 F∗Δ�𝑛,
+(6)
+
+3
+p(Ω0, n|d)
+Exact
+Profile
+Profile + Laplace
+χ2(Ω0, n|d)
+n
+Ω
+p(Ω, n|d)
+n∗(Ω)
+Ω = Ω0
+p(Ω|d)
+Fig. 1.— Joint posterior distribution on two parameters, Ω and 𝑛, with an approximate degeneracy of the form 𝑛Ω1.2 ∼ const. The large bottom left panel shows
+the joint distribution as red contours, with the position of the best-fit value of 𝑛 as a function of Ω in solid black. The central panel shows the 𝜒2 for a fixed value
+of Ω = Ω0 (shown as the dotted line in the first panel). The red line shows the true 𝜒2, while the dashed black line shows the Laplace approximation. The top
+panel shows the probability distribution along Ω = Ω0 (given by 𝑝 ∝ exp(−𝜒2/2)), with the exact distribution and its Laplace approximation in red and dashed
+black respectively. The bottom right panel shows the distribution of Ω marginalised over 𝑛. The exact result is shown in red, with its Laplace approximation
+shown in dashed black. The blue line shows the marginalised profile likelihood obtained by simply maximising the joint likelihood over 𝑛 for each Ω. The Laplace
+approximation provides an excellent description of the marginalised distribution, while the profile likelihood returns a distribution with very similar width but
+centered, by construction, on the best-fit value of Ω, avoiding volume effects.
+where 𝜒2
+∗ ( �Ω) ≡ 𝜒2( �Ω, �𝑛∗), Δ�𝑛 ≡ �𝑛 − �𝑛∗, and we have defined
+the matrix
+F∗,𝑖 𝑗 = 1
+2
+𝜕2𝜒2
+𝜕𝑛𝑖𝜕𝑛 𝑗
+����
+�𝑛∗
+.
+(7)
+This is the so-called Laplace approximation (Kass et al. 1990).
+In this limit, the distribution is locally (i.e.
+at each �Ω) a
+multivariate normal distribution in �𝑛, and thus the integral in
+Eq. (2) can be solved analytically. The resulting marginalised
+likelihood has a 𝜒2
+𝑚( �Ω) ≡ −2 log 𝑝( �Ω|d) given by
+𝜒2
+𝑚( �Ω) ≃ 𝜒2
+∗ ( �Ω) + log
+�
+det
+�
+F∗( �Ω)
+��
++ const..
+(8)
+In what follows, we will label the two contributions in
+Eq. (8), 𝜒2
+∗ and log det F∗, as the profile and Laplace terms
+respectively:
+1. The profile term is related to the “profile likelihood”
+(Cole et al. 2013), defined as
+𝑝prof( �Ω|d) ∝ 𝑝( �Ω, �𝑛∗|d).
+(9)
+The profile likelihood is a tool commonly used in fre-
+quentist parameter inference (Cousins 1995). The ad-
+vantage of the profile likelihood is that its maximum is,
+by definition, the global maximum of the joint distri-
+bution. Understanding this maximum as an estimator
+for �Ω given the data, constraints on �Ω can be obtained
+by calculating this maximum for random simulated re-
+alisations of the data. Alternatively, if the distribution
+is sufficiently close to a Gaussian, these constraints can
+be simply obtained in terms of thresholds of the associ-
+ated residual 𝜒2, Δ𝜒2( �Ω) = 𝜒2( �Ω) − 𝜒2
+min (Feldman &
+Cousins 1998). Additionally, the posterior profile likeli-
+hood is, by construction, centered on the best-fit param-
+eters, and is thus free from volume effects associated
+with the choice of the nuisance parameters (Hamann
+et al. 2007; Herold et al. 2022; Campeti & Komatsu
+2022) (see Section 2.3).
+2. The Laplace term, sometimes referred to as Occam’s
+razor term, is associated with the quadratic contribution
+to the Laplace approximation (Eq. (8)), and accounts,
+to first order, for the volume in the space of nuisance
+parameters �𝑛 that has been integrated over for fixed �Ω
+(i.e. the local curvature of the joint distribution at each
+
+4
+�Ω)1. As we will see in Section 2.3, the Laplace term is
+associated with volume effects, and is subdominant with
+respect to the profile term for sufficiently constraining
+data.
+The role of the profile and Laplace terms is illustrated in Fig. 1.
+The figure shows a bivariate distribution for two parameters
+with an approximate degeneracy of the form 𝑛Ω1.2 ∼ const..
+The contour levels of the true distribution are shown in red
+in the bottom left panel, while the black solid line shows the
+best-fit value of 𝑛 as a function of Ω. The middle panel shows
+the exact 𝜒2 of the distribution as a function of 𝑛 for a fixed
+Ω = Ω0 (for convenience, we chose Ω0 to be the maximum
+of the distribution, also shown as a dotted line in the bottom
+panel). The black dashed line shows the quadratic Laplace
+approximation to the red curve, with the position of the best-
+fit 𝑛 (for Ω = Ω0) marked by the blue line. The top panel
+shows the distribution along the Ω = Ω0 line. The exact distri-
+bution is again shown in red, and the Laplace approximation
+to it is shown in dashed black. The “profile likelihood” ap-
+proximation, which fixes 𝑛 to its best-fit value, is shown in
+blue. Finally, the bottom right panel shows the distribution
+marginalised over 𝑛, 𝑝(Ω). The true marginal is shown in
+red. The result of analytically marginalising over 𝑛 using the
+Laplace approximation is shown in dashed black, and recov-
+ers the true marginal almost exactly. Finally, the conditionally
+maximised distribution accounting only for the profile term in
+Eq. (8) is shown in blue. As mentioned above, the profile-only
+approximation recovers a distribution that is centered at the
+best-fit value of Ω (marked by the dotted line). The shift in
+the peak of the true marginal observed is caused by volume
+effects, which we discuss in more detail in Section 2.3.
+Two qualitative results should be borne in mind in what
+follows.
+First, the Laplace approximation provides a rea-
+sonably accurate prediction for the marginal for sufficiently
+well-behaved distributions. Secondly, keeping only the profile
+term recovers a distribution that has approximately the same
+width but is, by construction, centered on the maximum of the
+distribution.
+It is worth noting that including the Laplace term in Eq. (8)
+should come at virtually no additional computational cost.
+Finding �𝑛∗( �Ω) requires solving for 𝜕�𝑛𝜒2 = 0, which can be
+done efficiently using gradient descent methods. Finding the
+optimal step size in these algorithms often requires evaluating
+the Hessian of the function being minimised, and therefore the
+matrix F∗ entering the Laplace term, is already a product of
+the minimisation algorithm. For instance, the iteration in the
+case of the Newton-Raphson algorithm is given by
+�𝑛∗,𝑖+1 = �𝑛∗,𝑖 −
+�
+[∇𝑛∇𝑇
+𝑛 𝜒2]−1 · ∇𝑛𝜒2�
+𝑖 ,
+(10)
+where ∇𝑛𝜒2 is the gradient of 𝜒2 with respect to �𝑛, and
+∇𝑛∇𝑇
+𝑛 𝜒2 ≡ 2F is its Hessian matrix. In the applications we
+will explore here, when Eq. (5) cannot be solved analytically,
+we will make use of a modified version of the Newton-Raphson
+algorithm, which we describe in the next section.
+2.2. Gaussian likelihoods
+Let us now apply the method described in the previous
+section to the case of Gaussian likelihoods. In this case we
+1 In the frequentist context, Barndorff-Nielsen (1983) introduced the for-
+mula in Eq. (8) under the name of “modified profile likelihood”.
+assume that the posterior distribution takes the form:
+− 2 log 𝑝( �Ω, �𝑛|d) = (d − t)𝑇 C−1(d − t) + 𝜒2
+𝑝,Ω( �Ω) + 𝜒2
+𝑝,𝑛(�𝑛).
+(11)
+Here t( �Ω, �𝑛) is the theory vector, which depends on the model
+parameters, C is the covariance matrix of the data, which we
+assume to be model-independent, and 𝜒2
+𝑝,Ω and 𝜒2
+𝑝,𝑛 are the
+parameter priors. Although the methodology described below
+is straightforward to generalise to the case of arbitrary priors,
+for simplicity we will assume that the nuisance parameters
+have Gaussian priors, and therefore
+𝜒2
+𝑝,𝑛(�𝑛) = (�𝑛 − �𝑛𝑝)𝑇 C−1
+𝑛 (�𝑛 − �𝑛𝑝),
+(12)
+where C𝑛 is the prior covariance.
+In order to find �𝑛∗ and F , we need the first and second
+derivatives of the 𝜒2 with respect to �𝑛. In this case, these are
+given by:
+𝜕𝜒2
+𝜕𝑛𝑖
+= −2𝜕𝑖t𝑇 C−1(d − t) + 2
+∑︁
+𝑗
+�
+C−1
+𝑛
+�
+𝑖 𝑗 (𝑛 𝑗 − 𝑛𝑝, 𝑗), (13)
+F𝑖 𝑗 = 𝐹𝑖 𝑗 + ΔF𝑖 𝑗,
+(14)
+where we have used the shorthand 𝜕𝑖 ≡ 𝜕/𝜕𝑛𝑖, and we have
+defined
+𝐹𝑖 𝑗 ≡ 𝜕𝑖t𝑇 C−1 𝜕𝑗t +
+�
+C−1
+𝑛
+�
+𝑖 𝑗 ,
+(15)
+ΔF𝑖 𝑗 ≡ 𝜕𝑖𝜕𝑗t𝑇 C−1(t − d).
+(16)
+On the one hand, the first contribution to F , 𝐹, has three
+interesting properties:
+• It is positive-definite, and therefore invertible.
+• It is independent of the data d.
+• It coincides with the Fisher matrix of the Gaussian like-
+lihood of Eq. (11) with respect to the nuisance parame-
+ters.
+On the other hand, when evaluated on the hypersurface �𝑛 =
+�𝑛∗( �Ω), t is close to d, and therefore the contribution from ΔF
+is usually smaller than 𝐹. These properties will be important
+when discussing volume effects in the next section. For now,
+we will use the positive-definiteness of 𝐹 to define a modified
+Newton-Raphson iteration, by swapping the Hessian matrix
+2F with 2𝐹:
+�𝑛∗,𝑖+1 = �𝑛∗,𝑖 −
+�1
+2𝐹−1 ∇𝑛𝜒2
+�
+𝑖
+.
+(17)
+This results in the Gauss-Newton method, which improves
+on the Newton-Raphson iteration in two aspects: first, 𝐹 is
+invertible and likely more stable than the Hessian matrix,
+which need not be positive-definite. Secondly, 𝐹 does not
+require calculating second derivatives of the theory vector
+(although this is not a computationally challenging problem
+for the cases considered here).
+Note that replacing Hes-
+sian matrix with Fisher matrix in Newton-Raphson iteration
+often appears in various problem, most notably in optimal
+quadratic estimators (Tegmark 1997), see Madhavacheril et al.
+(2015) for further discussion. Other approaches, such as the
+Levenberg-Marquardt algorithm (Levenberg 1944; Marquardt
+1963), build on this method by improving the numerical sta-
+bility of 𝐹−1 through regularisation. The problems addressed
+here do not require us to resort to these.
+
+5
+2.3. Volume effects and priors
+Consider now the case of a data vector with a Gaussian like-
+lihood and poor constraining power (e.g. noise-dominated).
+In this limit, we would naïvely expect the marginalised poste-
+rior distribution to return largely unconstrained �Ω. To verify if
+this is the case, consider first the profile contribution in Eq. (8),
+averaged over realisations of the data:
+⟨𝜒2
+∗⟩( �Ω) = Tr
+��
+(d − t( �Ω, �𝑛∗))(d − t( �Ω, �𝑛∗))𝑇 �
+C−1�
+. (18)
+If the data are noise-dominated, the fluctuations in d−t( �Ω, �𝑛∗)
+are dominated by noise, rather than the changes in �Ω. In that
+case ⟨(d − t)𝑇 (d − t)⟩ ≃ C, and therefore
+⟨𝜒2
+∗⟩(Ω) ≃ 𝑁𝑑,
+(19)
+where 𝑁𝑑 is the number of data points. Thus, as we expected,
+this contribution tends to a constant that does not favour any
+region of parameter space.
+Consider now the Laplace contribution.
+As we argued,
+the contribution ΔF∗ is normally small compared to 𝐹∗ (both
+evaluated at �𝑛∗), and therefore
+log det F∗ = log det(𝐹∗ + ΔF∗) ≃ log det 𝐹∗
+(20)
+which, as we discussed before, is independent of the data.
+The Laplace contribution to the marginal distribution is thus
+a parameter-dependent function that does not depend on the
+data, and which would favour particular regions of parameter
+space even in the absence of data!
+This is an example of a “volume effect”: the process of
+marginalisation favours regions of parameter space that cover
+a larger volume of the probability density in the direction of in-
+tegration, causing a shift in the maximum of the marginalised
+distribution with respect to the maximum of the full distribu-
+tion. Volume effects depend on the definition of the nuisance
+parameters and, if informative, the associated priors. A classi-
+cal example is trying to fit noisy data to a power-law model of
+the form 𝑛 𝑥Ω, where 𝑥 is an independent variable taking values
+𝑥 > 1, and (𝑛, Ω) are free parameters. If the data are noise-
+dominated, scattering around zero, a very negative value of the
+power law index Ω is able to obtain a reasonable fit to the data
+for a wider range of amplitudes 𝑛, and thus the marginalised
+distribution will favour values of Ω that are significantly dif-
+ferent from the best-fit.
+This often happens for non-linear
+parameters like Ω when marginalising over amplitude-like pa-
+rameters such as 𝑛. The most pernicious aspect of volume
+effects is that their impact on the marginalised distribution
+depends on the specific parametrisation used to define the nui-
+sance parameters. In the example above, redefining the model
+to 𝑛 (𝑥/𝑥0)Ω, where 𝑥0 is a fixed quantity, larger than any value
+of 𝑥, would result in marginalised posteriors that favour large
+and positive values of Ω for noisy data (instead of negative).
+A common choice to eliminate the dependence on model
+parametrisation, and thus to partially mitigate the impact of
+volume effects is to use the well-known Jeffreys prior:
+𝑝J( �𝜃) =
+√
+det 𝐹,
+(21)
+where 𝐹 is the Fisher information matrix, which we introduced
+in the previous section (Eq. (15) for Gaussian distributions).
+Comparing this with Eq. (8), we can see that the inclusion of a
+Jeffreys prior for the nuisance parameters has the effect of par-
+tially cancelling the contribution from the Laplace term (since,
+as we argued, ΔF is generally smaller than 𝐹2). This may not
+be entirely surprising since, as we saw, the Laplace term is
+in essence the type of volume effect that the Jeffreys prior is
+meant to address. However, this detour leads us to an interest-
+ing result: for Gaussian data with negligible parameter depen-
+dence in the covariance matrix, the profile likelihood will in
+general be a reasonable approximation to the true marginalised
+posterior in the presence of a Jeffreys prior. In this sense, max-
+imisation and marginalisation are approximately equivalent as
+long as we avoid volume effects.
+2.4. Linear parameters
+Let us now consider the case of linear parameters.
+I.e.
+consider a Gaussian likelihood in the form of Eq. (11) where
+all parameters live in the theory prediction, which has the form
+t = t0 + T�𝑛,
+(22)
+where t0 and T are a vector and a matrix independent of �𝑛, but
+potentially dependent on �Ω. For simplicity, we will assume
+that the prior on �𝑛 is centered at zero (�𝑛𝑝 = 0). This can always
+be achieved by simply redefining �𝑛′ ≡ �𝑛 − �𝑛𝑝, and adding the
+contribution T�𝑛𝑝 to t0.
+The case of linear parameters is particularly interesting,
+because the 𝜒2 is quadratic in �𝑛 by construction, and the
+Laplace approximation is exact. Since the second derivatives
+of the theory vector are zero, ΔF = 0, and the Fisher matrix
+is independent of �𝑛 and given by
+F = 𝐹 = T𝑇 C−1T + C−1
+𝑛 .
+(23)
+Furthermore, the best-fit parameters can be found analytically:
+�𝑛∗ = 𝐹−1 T𝑇 C−1r,
+(24)
+where r ≡ d − t0 is the data rescaled by the �𝑛-independent
+component of the theory. Using �𝑛∗ to compute the 𝜒2, and
+using Eq. (23), we obtain the marginalised 𝜒2
+𝑚 of Eq. (8) which,
+as we said, is exact in this case.
+The first thing worth noting is that, if the matrix T is indepen-
+dent of �Ω, then 𝐹 is constant, and so is the Laplace term. Up
+to an irrelevant overall constant, the marginalised 𝜒2 is then
+equivalent to 𝜒2
+∗, obtained by substituting the best-fit value
+of �𝑛. Thus, the approximate relation between marginalisation
+and maximisation we outlined in the previous section becomes
+an equivalence when the data is Gaussian with a linear model
+in �𝑛 since, in this case, there are no volume effects. All volume
+effects resulting from a dependence of T on �Ω are otherwise
+incorporated exactly in the Laplace term. If the priors on �𝑛
+are sufficiently tight, the second term in Eq. (23) dominates,
+and these volume effects become negligible.
+Let us now focus on the profile term. Substituting �𝑛∗ from
+Eq. (24), we obtain
+𝜒2
+∗ = (Wr)𝑇 C−1(Wr) + r𝑇 C−1T𝐹−1C−1
+𝑛 𝐹−1T𝑇 C−1r,
+(25)
+where the second term comes from the prior on �𝑛, and we have
+defined the matrix
+W ≡ I − T𝐹−1T𝑇 C−1,
+(26)
+with I the identity.
+2 In addition to this, whereas 𝐹 is evaluated along �𝑛∗ in the Laplace
+approximation, the Jeffreys prior is evaluated at every point in parameter
+space.
+
+6
+First, consider the limit of no external prior (i.e. C−1
+𝑛 = 0).
+In this case, we can ignore the second term in Eq. (25), and
+the Fisher matrix is 𝐹 = T𝑇 C−1T. We can then see that the
+matrix W projects r onto the subspace that is orthogonal to all
+the columns of T (with orthogonality defined using the inverse
+covariance of the data C−1 as a dot product). Marginalising
+over linear parameters is therefore equivalent in this limit to
+deprojecting all modes of the data that live in the subspace
+spanned by the columns of T (Rybicki & Press 1992).
+Secondly, Eq. (25) can be simplified significantly into
+𝜒2
+∗ = r𝑇 ˜C−1r,
+(27)
+where ˜C is a modified covariance given by
+˜C = C + TC𝑛T𝑇 .
+(28)
+To obtain this beautifully simple result, one only needs to ex-
+pand first term in Eq. (25), simplify the result, and make use of
+the Woodbury matrix identity (Woodbury 1950). It is worth
+stressing again that this result is an exact expression for both
+the marginal posterior (using a Jeffreys prior) and the con-
+ditionally maximised posterior, as explained in the previous
+section. Maximising and marginalising over �𝑛 therefore result
+in the same Gaussian likelihood with the theory vector evalu-
+ated at �𝑛 = 0 (or at its prior mean if non-zero), and a modified
+covariance ˜C, obtained by simply assigning additional vari-
+ance in quadrature to the modes of the data that align with the
+columns of T (with this extra variance given by the associated
+�𝑛 parameter priors).
+Note that while this approach is algorithmically the fastest, it
+does not produce a best-fit value of a given nuisance parameter.
+This is occasionally useful even for nuisance parameters. A
+canonical example is the shot noise contribution to the galaxy
+power spectrum, the value of which can tell us about the halos
+in which the given tracer galaxies reside.
+To summarise: in the case of Gaussian data, negligible pa-
+rameter dependence of the covariance matrix, and a theory
+model that is linear in the nuisance parameters, the Laplace
+approximation is exact. In this case, there is a mathematical
+equivalence between marginalisation (using a Jeffreys prior),
+𝜒2 minimisation, deprojection, and simply adding in quadra-
+ture the prior uncertainty on the marginalised parameters at
+the data level. If the modes associated with �𝑛 (i.e. the columns
+of T) depend on the other parameters of the model, the associ-
+ated volume effects are captured exactly by the Laplace term,
+which is simply given by the log-determinant of Eq. (23). Im-
+portantly, in this scenario, volume effects become negligible
+if the constraints on nuisance parameters are either dominated
+by data, or by the priors.
+3. COSMOLOGY FROM TOMOGRAPHIC
+LARGE-SCALE STRUCTURE DATA
+To explore the validity of the Laplace approximation in-
+troduced in the previous section to marginalise over nuisance
+parameters in the context of cosmology, we will study the case
+of 3×2pt analyses, combining photometric galaxy clustering
+and weak lensing.
+3.1. Galaxy clustering and weak lensing
+Photometric redshift surveys make use of two main cosmo-
+logical probes: cosmic shear (i.e. the distortion in the shape
+of galaxies caused by weak gravitational lensing) and galaxy
+clustering. In both cases, the data are typically split into pho-
+tometric redshift bins, and the data vector is constructed by
+combining various angular auto- and cross-correlations be-
+tween pairs of such bins. The galaxy overdensity 𝛿𝛼
+𝑔 ( ˆn) and
+the weak lensing shear 𝛾𝛼
+𝐺( ˆn) for galaxies in redshift bin 𝛼
+can be related to the three-dimensional fluctuations in the
+galaxy number density Δ𝑔(x) and the matter density Δ𝑚(x)
+via (Bartelmann & Schneider 2001; Krause et al. 2017)
+𝛿𝛼
+𝑔 ( ˆn) =
+∫
+𝜒𝐻
+0
+𝑑𝜒 𝑞𝛼
+𝑔 (𝜒) Δ𝑔(𝜒(𝑧) ˆn, 𝑧),
+𝛾𝛼
+𝐺( ˆn) =
+∫
+𝜒𝐻
+0
+𝑑𝜒 𝑞𝛼
+𝐺(𝜒)
+�
+−𝜒−2ð2∇−2Δ𝑚(𝜒ˆn, 𝑧)
+�
+,
+(29)
+where ˆn is the sky direction, 𝜒 is the comoving radial distance
+at redshift 𝑧, 𝜒𝐻 is the distance to the horizon, 𝑞𝛼
+𝑔 and 𝑞𝛼
+𝛾
+are the radial kernels for galaxy clustering and cosmic shear,
+respectively, and ð is the spin-raising differential operator,
+acting on a spin-𝑠 quantity as:
+ð 𝑠 𝑓 (𝜃, 𝜑) = −(sin 𝜃)𝑠
+� 𝜕
+𝜕𝜃 +
+𝑖
+sin 𝜃
+𝜕
+𝜕𝜑
+�
+(sin 𝜃)−𝑠 𝑠 𝑓
+(30)
+and turning it into a spin-(𝑠 + 1) quantity.
+The radial kernels in both cases are given by
+𝑞𝛼
+𝑔 (𝜒) ≡ 𝐻(𝑧)
+𝑐
+𝑝𝛼(𝑧),
+𝑞𝛼
+𝐺(𝜒) ≡ 3
+2𝐻2
+0Ω𝑚
+𝜒
+𝑎(𝜒)
+∫ ∞
+𝑧(𝜒)
+𝑑𝑧′𝑝𝛼(𝑧′) 𝜒(𝑧′) − 𝜒
+𝜒(𝑧′)
+,
+(31)
+where 𝑐 is the speed of light, 𝐻(𝑧) is the Hubble expansion
+rate, 𝐻0 ≡ 𝐻(𝑧 = 0), Ω𝑚 is the matter density parameter
+today, and 𝑝𝛼(𝑧) is the redshift distribution in bin 𝛼,
+Cosmic shear observations are sensitive not only to the weak
+lensing distortion of galaxy shapes, but also to the intrinsic
+alignments (IAs) in the orientation of galaxies due to local
+interactions. The total observed cosmic shear signal is thus
+𝛾𝛼 = 𝛾𝛼
+𝐺 + 𝐴IA𝛾𝛼
+𝐼 ,
+(32)
+where 𝛾𝛼
+𝐺 is the lensing signal, given by Eq. (29), and 𝐴IA𝛾𝛼
+𝐼 is
+the intrinsic alignment component. Using the linear non-linear
+alignment model, the latter is given by
+𝛾𝛼
+𝐼 ( ˆn) = −
+∫
+𝜒𝐻
+0
+𝑑𝜒 𝑞𝛼
+𝐼 (𝜒)[−𝜒−2ð2∇−2Δ𝑚(𝜒ˆn, 𝑧)],
+(33)
+and 𝐴IA is an unknown amplitude parameter describing the
+strength of the local alignment signal. The IA radial kernel is
+the same as the galaxy clustering kernel
+𝑞𝛼
+𝐼 (𝜒) = 𝐻(𝑧)
+𝑐
+𝑝𝛼(𝑧).
+(34)
+The relation between the galaxy overdensity Δ𝑔 and the mat-
+ter overdensity Δ𝑚 is complex in detail (non-linear, non-local
+and stochastic). Over mildly non-linear scales, one can how-
+ever use a perturbative approach, relating Δ𝑔 with scalar com-
+binations of the Hessian of the gravitational potential 𝜕𝑖𝜕𝑗Φ
+(where Φ is normalised so that ∇2Φ ≡ Δ𝑚). Following Mc-
+Donald & Roy (2009); Abidi & Baldauf (2018), here we will
+expand this bias relation to second order, including non-local
+contributions, so that:
+Δ𝑔 = 𝑏1Δ𝑚+ 𝑏2
+2! (Δ𝑚−⟨Δ2
+𝑚⟩)+ 𝑏𝑠
+2! (𝑠2−⟨𝑠2⟩)+𝑏∇∇2Δ𝑚. (35)
+Here 𝑠2 ≡ 𝑠𝑖 𝑗𝑠𝑖 𝑗 is the trace of the squared tidal tensor, where
+𝑠𝑖 𝑗 ≡ 𝜕𝑖𝜕𝑗Φ − ∇2Φ/3. The quantities 𝑏1, 𝑏2, 𝑏𝑠, and 𝑏∇ are
+
+7
+the so-called “linear”, “quadratic”, “tidal”, and “non-local”
+bias parameters, which characterise the response of the galaxy
+overdensity to the corresponding terms in perturbation theory,
+including the impact of non-local effects on scales compara-
+ble with the Lagrangian halo size. Within this formalism, and
+assuming that the bias parameters are constant within each
+redshift bin, the projected galaxy overdensity can then be ex-
+pressed as a sum over the projected version of the different
+operators in the previous equation:
+𝛿𝛼
+𝑔 ( ˆn) =
+∑︁
+𝑘
+𝑏𝛼,𝑘, 𝛿𝛼
+𝑘 ( ˆn),
+(36)
+where 𝑘 runs over the set {1, 2, 𝑠, ∇}, corresponding to the
+various operators, dependent only on the matter overdensity
+and tidal tensor, that contribute to the total galaxy overdensity.
+𝑏𝛼,𝑘 is the value of each associated bias parameter in bin 𝛼 is
+the value of each associated bias parameter in bin 𝛼, and
+𝛿𝛼
+𝑘 ( ˆn) ≡
+∫
+𝜒𝐻
+0
+𝑑𝜒 𝑞𝛼
+𝑔 (𝜒)Δ𝑘 ( ˆn),
+(37)
+with
+Δ1 ≡ Δ𝑚, Δ2 ≡ Δ2
+𝑚 − ⟨Δ2
+𝑚⟩,
+Δ𝑠 ≡ 𝑠2 − ⟨𝑠2⟩, Δ∇ ≡ ∇2Δ𝑚.
+(38)
+For some of the results shown in Section 4 we will consider
+only linear bias, setting 𝑏2 = 𝑏𝑠 = 𝑏∇ = 0. Either of these
+bias schemes is only valid on sufficiently large scales, and
+therefore we will limit our analysis to multipoles ℓ < 𝑘max ¯𝜒,
+where ¯𝜒 is the comoving distance to the mean redshift of
+the galaxy tracer under analysis. The maximum wavenumber
+used will be 𝑘max = 0.15 Mpc−1 when using linear bias, and
+𝑘max = 0.3 Mpc−1 when using the perturbative approach above
+(Pandey et al. 2020).
+Comparing Eq. (32) and Eq. (36), we see that we can de-
+scribe both tomographic galaxy clustering and cosmic shear
+as a projected tracer 𝑢𝛼( ˆn) with the generic form
+𝑢𝛼( ˆn) = 𝜖𝑢 𝑢𝛼
+𝑀 ( ˆn) +
+∑︁
+𝑘
+𝑏𝑢
+𝛼,𝑘 𝑢𝛼
+𝑘 ( ˆn) ,
+(39)
+where 𝑏𝑢
+𝛼,𝑘 are bias parameters (specifying the tracer type 𝑢),
+𝑢𝛼
+𝑀 and 𝑢𝛼
+𝑘 are projected quantities that depend only on cosmo-
+logical observables (matter overdensities, comoving distances
+etc.) and the radial kernels, and 𝜖𝑢 is a Boolean variable that is
+either 1 if the tracer contains an unbiased contribution (as is the
+case of cosmic shear), and 0 otherwise (as is the case of galaxy
+clustering). The index 𝛼 in Eq. (39) runs over the redshift bins,
+which allows for the general case of having redshift-dependent
+bias functions. If this is not the case, 𝑏𝑢
+𝛼,𝑘 ≡ 𝑏𝑢
+𝑘 can optionally
+be assumed to not vary across redshift bins.
+We can relate the power spectrum, 𝑃𝑋𝑌 (𝑘, 𝑧), of two three-
+dimensional quantities 𝑋 and 𝑌 (e.g., Δ𝑘) to the angular cross-
+power spectrum of their associated projected tracers in redshift
+bins 𝛼 and 𝛽 (e.g. 𝛿𝛼
+𝑘 ) via:
+𝐶 (𝑋,𝛼),(𝑌 ,𝛽)
+ℓ
+=
+∫
+𝑑𝜒
+𝜒2 𝑞𝛼
+𝑋 (𝜒) 𝑞𝛽
+𝑌 (𝜒) 𝑃𝑋𝑌
+�
+𝑘 = ℓ + 1/2
+𝜒
+, 𝑧(𝜒)
+�
+,
+(40)
+where we have assumed the Limber approximation (Limber
+1953; Afshordi et al. 2004), appropriate for the wide radial ker-
+nels considered in this work. Note that in principle the lensing
+kernel should be multiplied by an ℓ-dependent prefactor
+𝐺ℓ ≡
+√︄
+(ℓ + 2)!
+(ℓ − 2)!
+1
+(ℓ + 1/2)2
+(41)
+to account for the difference between angular and three-
+dimensional derivatives in Eq. (29) (i.e. 𝜒2ð2∇−2 � 1).
+To calculate the matter power spectrum we will use the
+Halofit fitting function Smith et al. (2003) with revisions
+from Takahashi et al. (2012a). To calculate the power spectra
+between the different perturbative matter fields involved in the
+bias expansion (Eq. (38)) we will make use of Fast-PT 3. The
+procedure is described in McEwen et al. (2016) and we refer
+readers to that paper for further details.
+The cosmological analysis of tomographic weak lensing and
+galaxy clustering two-point functions requires accounting for,
+and propagating uncertainties in some of the ingredients of the
+corresponding theoretical predictions. This is usually done by
+defining a model that describes the impact of the associated
+effects, and marginalising over the nuisance parameters of that
+model. The nature of these parameters can be classified into
+two main types:
+• Calibratable parameters: these are parameters on which
+relatively tight priors can be placed using external data
+(i.e. they can be calibrated). These are normally associ-
+ated with observational effects, such as shape measure-
+ment or photometric redshift errors.
+• Non-calibratable parameters: these are parameters for
+which no reliable prior information exists, and which
+must therefore be measured with our own data. These
+are normally associated with astrophysical uncertainties
+specific to the sample under study, such as galaxy bias
+or intrinsic alignment parameters.
+The next two sections describe the strategies we will use to
+marginalise over both types of nuisance parameters.
+3.2. Calibratable systematics: linearisation
+In the presence of tight priors, which we will further assume
+to be Gaussian, the nuisance parameters may not stray far from
+their prior mean. In that case, we can Taylor-expand the theory
+prediction as in Eq. (22)
+t( �Ω, �𝑛) = t0( �Ω) + T (�𝑛 − �𝑛𝑝),
+(42)
+where �𝑛𝑝 is the prior mean, and
+t0 ≡ t( �Ω, �𝑛𝑝),
+T ≡ 𝜕t
+𝜕�𝑛
+����
+�𝑛𝑝
+.
+(43)
+Since now the theory is linear with respect to �𝑛 − �𝑛𝑝, we
+can then follow the procedure in Section 2.4 to analytically
+marginalise over those parameters. As we discussed, we sim-
+ply modify the covariance of the data vector (in this case, a
+collection of power spectra) as in Eq. (28), and then sample the
+resulting Gaussian likelihood evaluating t at the prior mean of
+�𝑛. Two important things should be noted. First, in doing this,
+we have neglected the parameter dependence of the Laplacian
+term, given by
+log
+�
+det
+�
+T𝑇 C−1T + C−1
+𝑛
+��
+.
+(44)
+3 https://github.com/JoeMcEwen/FAST-PT
+
+8
+If the prior is sufficiently tight, this term is dominated by the
+constant C−1
+𝑛 contribution, so the approximation is reasonable.
+Secondly, since the modified covariance matrix (Eq. (28)) now
+involves a term of the form TC𝑛T𝑇 , in principle it depends
+on �Ω through T.
+Calculating the covariance at each point
+in the likelihood may be computationally costly, depending
+on the size of T. Instead, we will simply evaluate T at the
+best-fit value of �Ω, and ignore all parameter dependence of
+the covariance. It was shown in Kodwani et al. (2019) that
+the parameter dependence of the covariance can generally be
+neglected and, furthermore, for a sufficiently tight prior the
+TC𝑛T𝑇 contribution should be subdominant. Hadzhiyska et al.
+(2020) also showed that the choice of fiducial �Ω does not affect
+the final results as long as they are close to the center of the
+posterior distribution. Adopting these two approximations (in
+addition to the Taylor expansion of t) is therefore well justified
+and, as we will show in Section 4.1, leads to accurate results.
+One of the most important calibratable systematics in pho-
+tometric surveys comes from the uncertainties in the redshift
+distribution of the source and lens samples. To account for
+these, it is common to parametrise the potential deviations
+from a fiducial redshift distribution, and to calibrate those
+parameters through external data and simulations. In a sim-
+plified model, errors in photo-𝑧’s cause shifts in the means and
+changes in the width of the derived redshift distribution for a
+population of galaxies (Bonnett et al. 2016). The resulting
+model for the redshift distribution is
+𝑝𝛼(𝑧) ∝ ˆ𝑝𝛼(𝑧𝛼
+𝑐 + 𝑤𝛼
+𝑧 (𝑧 − 𝑧𝛼
+𝑐 ) + Δ𝑧𝛼),
+(45)
+where Δ𝑧𝛼 and 𝑤𝛼
+𝑧 parametrise deviations in the mean and
+width of the fiducial distribution ˆ𝑝𝛼, and 𝑧𝛼
+𝑐 is the redshift at
+which this fiducial distribution attains its maximum. We will
+label these the “shift” and “width” parameters respectively in
+what follows. Note that the normalisation of the distribution
+changes with both of these parameters, and it must be renor-
+malised to unit area before using it to compute any power
+spectra.
+In the rest of this paper, we will add one shift and one
+width parameter for each galaxy clustering bin, and one shift
+parameter for each cosmic shear bin, fixing their width. On the
+one hand, mean shifts have a strong impact on the weak lensing
+kernel, since it depends on a cumulative integral of the redshift
+distribution, and may also affect galaxy clustering by changing
+the growth factor. On the other hand, mild (∼ 10%) changes
+to the width have a very strong impact on the amplitude of the
+clustering auto-correlations, but leave the weak lensing kernel
+almost unchanged (Ruiz-Zapatero et al. 2023).
+Including these free parameters amounts to burdening the
+MCMC sampler with tens of extra parameters, which in-
+evitably leads to a substantial slowdown of its convergence.
+Moreover, there is no guarantee that all the uncertainty in
+the 𝑝(𝑧)s can be captured by these parameters, and several
+alternative procedures have been developed to ensure this. Al-
+though we will focus here on the shift-width parametrisation,
+Hadzhiyska et al. (2020) and Ruiz-Zapatero et al. (2023) ex-
+plore the most general case of treating the 𝑝(𝑧) as a step-wise
+function, with the function values at each step treated as free
+parameters, and show that the approximate marginalisation
+described here leads to accurate results for both clustering
+and shear. We will therefore focus here only on the simpler
+shift-width parametrisation, to exemplify the performance of
+the Laplace approximation in the case of calibratable nuisance
+parameters.
+We stress that we can apply the same procedure to any other
+parameter that appears to behave linearly in the theoretical
+model.
+This is the case for any other nuisance parameter
+with a sufficiently tight prior, and in fact this procedure is
+routinely used for multiplicative bias parameters in cosmic
+shear analyses (Hildebrandt et al. 2020). The procedure is also
+exact in the case of truly linear parameters, regardless of their
+priors. A good example of this is the amplitude of shot noise or
+stochastic contributions to galaxy clustering auto-correlations
+(García-García et al. 2021).
+3.3. Non-calibratable systematics: bias parameters
+Most sources of astrophysical uncertainty cannot be well-
+constrained from external data, and thus must be con-
+strained at the same time as the cosmological parameters, and
+marginalised over. In this case, the linearisation described in
+the previous section is not appropriate, and we must resort to
+numerical methods in order to obtain �𝑛∗ and the Laplace con-
+tribution in Eq. (8). In the model introduced in Section 3.1,
+these astrophysical uncertainties are described by the bias and
+intrinsic alignment parameters. In this formalism, all tracers
+can be expressed generically as in Eq. (39), where 𝑢𝛼
+𝑀 and 𝑢𝛼
+𝑘
+are projected maps depending purely cosmological fields (i.e.
+depending only on the matter overdensity and the tidal field),
+and all astrophysical uncertainties are incorporated in the 𝑏𝑢
+𝛼,𝑘
+parameters.
+Although the bias/IA description used here covers a wide
+range of state-of-the-art physical models used in current 3×2pt
+analyses, it is mathematically exceptionally simple.
+From
+Eq. (39) we see that the cross-correlation between any two
+such tracers (𝑢𝛼, 𝑤𝛽) is a simple quadratic function of the bias
+parameters:
+𝐶𝑢𝛼,𝑤 𝛽
+ℓ
+= 𝜖𝑢𝜖𝑤𝐶
+𝑢𝛼
+𝑀 ,𝑤 𝛽
+𝑀
+ℓ
++
+∑︁
+𝑖
+𝑏𝑢
+𝛼,𝑖𝜖𝑤𝐶
+𝑢𝛼
+𝑖 ,𝑤 𝛽
+𝑀
+ℓ
++
+∑︁
+𝑗
+𝜖𝑢𝑏𝑤
+𝛽, 𝑗𝐶
+𝑢𝛼
+𝑀 ,𝑤 𝛽
+𝑗
+ℓ
++
+∑︁
+𝑖, 𝑗
+𝑏𝑢
+𝛼,𝑖𝑏𝑤
+𝛽, 𝑗𝐶
+𝑢𝛼
+𝑖 ,𝑤 𝛽
+𝑗
+ℓ
+.
+(46)
+Here, 𝐶
+𝑢𝛼
+𝑀/𝑖,𝑤 𝛽
+𝑀/𝑗
+ℓ
+are the power spectra between the cosmo-
+logical projected fields 𝑢𝛼
+𝑀/𝑖 and 𝑤𝛽
+𝑀/𝑗, defined in Eq. (39),
+and the sums run over the associated bias terms as introduced
+in Eq. (36).
+Since these only involve radial projections of
+purely cosmological quantities, they can be treated as tem-
+plates that only depend on the cosmological parameters. Note
+that, in principle, these templates also depend on the calibrat-
+able nuisance parameters described in the previous section
+(e.g.
+through the modification in the radial kernels due to
+𝑝(𝑧) uncertainties). However, we assume that we have been
+able to marginalise over these analytically as we described
+above, and therefore they can be treated as fixed for all intents
+and purposes.
+The first derivative of the power spectrum with respect to
+
+9
+the bias parameters is thus a linear polynomial:
+𝜕𝐶𝑢𝛼,𝑤 𝛽
+ℓ
+𝜕𝑏𝑟
+𝛾,𝑘
+= 𝛿𝐾
+𝛼,𝛾𝛿𝐾
+𝑢,𝑟
+�
+𝜖𝑤𝐶
+𝑢𝛼
+𝑘 ,𝑤 𝛽
+𝑀
+ℓ
++
+∑︁
+𝑗
+𝑏𝑤
+𝛽, 𝑗𝐶
+𝑢𝛼
+𝑘 ,𝑤 𝛽
+𝑗
+ℓ
+�
++ 𝛿𝐾
+𝛽,𝛾𝛿𝐾
+𝑤,𝑟
+�
+𝜖𝑢𝐶
+𝑢𝛼
+𝑀 ,𝑤 𝛽
+𝑘
+ℓ
++
+∑︁
+𝑖
+𝑏𝑢
+𝛼,𝑖𝐶
+𝑢𝛼
+𝑖 ,𝑤 𝛽
+𝑘
+ℓ
+�
+,
+(47)
+where 𝛿𝐾 is the Kronecker delta, and 𝑏𝑟
+𝛾,𝑘 denotes the 𝑘th
+bias term of tracer type 𝑟 (i.e., galaxy overdensity or shear) in
+redshift bin 𝛾. Finally, the Hessian is constant with respect to
+the bias parameters
+𝜕2𝐶𝑢𝛼,𝑤 𝛽
+ℓ
+𝜕𝑏𝑟
+𝛾,𝑘𝜕𝑏𝑠𝜎,𝑚
+=
+(48)
+�
+𝛿𝐾
+𝛾,𝛼𝛿𝐾
+𝜎,𝛽𝛿𝐾
+𝑟,𝑢𝛿𝐾
+𝑠,𝑤 + 𝛿𝐾
+𝛾,𝛽𝛿𝐾
+𝜎,𝛼𝛿𝐾
+𝑟,𝑤𝛿𝐾
+𝑠,𝑢
+�
+𝐶
+𝑟 𝛾
+𝑘 ,𝑠𝜎
+𝑚
+ℓ
+.
+These expressions are remarkably simple and fast to evalu-
+ate, and thus computing the 𝜒2 and its derivatives (needed for
+minimisation, and to calculate the Laplace contribution) can
+be done extremely efficiently. As we will see, in practice we
+find that finding the minimum of the 𝜒2 takes 𝑂(10 − 100)
+Gauss-Newton iterations, each of which is orders of magnitude
+faster than recomputing the power spectrum templates when
+changing cosmological parameters. Computing the Laplace
+approximation to the marginalised posterior at each sample
+of the cosmological parameters is therefore virtually equiva-
+lent to evaluating the joint posterior for new cosmological +
+nuisance parameters if using brute-force marginalisation.
+3.4. Example Stage-III and Stage-IV datasets
+To test the validity of the methodology described above in
+practice, we will apply it to two different datasets:
+• Real data from the Dark Energy Survey Year-1 (DES-
+Y1) data release (DES Collaboration 2018).
+• A simulated 3×2-point data vector mimicking the char-
+acteristic of a Stage-IV survey such as LSST.
+These datasets are representative of both the current and future
+generation and thus span the plausible accuracy range for the
+foreseable future. We describe them next.
+3.4.1. The DES-Y1 data
+We use the galaxy-galaxy, galaxy-shear, and shear-shear
+power spectra and covariance matrix provided in García-
+García et al. (2021), constructed from the DES-Y1 data. DES
+is a five-year photometric survey which has observed 5000
+deg2 of the sky using five different filter bands (grizY) from
+the 4m Blanco Telescope at the Cerro Tololo Inter-American
+Observatory (CTIO), in Chile. García-García et al. (2021)
+employed the publicly available key Y1KP catalogs4, covering
+1786 deg2 before masking (DES Collaboration 2018; Drlica-
+Wagner et al. 2018).
+The galaxy clustering sample consists of luminous red
+galaxies (LRGs) selected with the redMaGiC algorithm, di-
+vided into 5 redshift bins, defined in Elvin-Poole et al. (2018).
+4
+https://des.ncsa.illinois.edu/releases/y1a1/
+key-catalogs
+In this analysis, we will make use of the fiducial redshift distri-
+butions released by DES to model the angular power spectra.
+We will also adopt the same galaxy weights to correct for sky
+systematics (see Elvin-Poole et al. 2018, for more details).
+The shear sample is the official source sample used in the
+DES-Y1 analysis (Zuntz et al. 2018), including all cuts and
+tomographic bin definitions. Galaxy shapes were determined
+using the Metacalibration algorithm (Huff & Mandelbaum
+2017; Sheldon & Huff 2017). The sample is divided into four
+tomographic bins, for which we use the official redshift distri-
+butions provided with the Y1 release (Hoyle et al. 2018). See
+Nicola et al. (2021) for further details regarding the estimation
+of the shear power spectra and covariance.
+Following the official DES-Y1 analysis, we use all cross-
+correlations between different shear bins and between shear
+and clustering bins, but only the auto-correlations between
+clustering bins.
+3.4.2. Synthetic Stage-IV data
+We consider a futuristic, idealised data set that resembles the
+characteristics of LSST. It is important to test our method in the
+low-noise regime, where the inferred posterior is even more
+sensitive to redshift distribution uncertainties or, in general,
+degeneracies between cosmological and nuisance parameters,
+and where the final error budget is more dominated by these
+effects.
+To define the clustering and shear samples we follow the
+same procedure outlined in Nicola et al. (2023). The shear
+sample is defined following the LSST Science Requirements
+Document (The LSST Dark Energy Science Collaboration
+et al. 2018) (see Appendices D1 and D2).
+We divide this
+redshift distribution into 5 bins in photometric redshift space,
+each containing the same number of sources. We assume a
+Gaussian photometric redshift uncertainty with standard de-
+viation 𝜎𝑧 = 0.05(1 + 𝑧), which thus defines the true-redshift
+tails of the distribution in each tomographic bin. The sample
+has an overall angular number density of 27 gals. arcmin−2.
+For galaxy clustering, we define a sample extending out to
+𝑧 ∼ 1.5 with a total density of 4 gals. arcmin−2 (as would
+be expected of an LRG-like sample for LSST). This number
+density and the associated redshift distribution were estimated
+using measurements of the luminosity function for red galax-
+ies as described in Alonso et al. (2015).
+The sample was
+divided into 6 redshift bins equi-spaced in photometric red-
+shift space, and assuming a photometric redshift uncertainty
+of 𝜎𝑧 = 0.02(1 + 𝑧). To simplify the analysis, we assume a
+constant linear galaxy bias 𝑏1 = 1, and set all higher-order bias
+coefficients to zero. The results obtained in the next section
+should be largely insensitive to this choice.
+For simplicity, we use a Gaussian covariance to describe the
+uncertainties of the resulting data vector, calculated assuming
+a sky fraction 𝑓sky = 0.4. The LSST data vector was generated
+assuming a true cosmology with parameters
+(Ω𝑚, Ω𝑏, ℎ, 𝑛𝑠, 𝜎8) = (0.3, 0.05, 0.7, 0.96, 0.8),
+(49)
+where Ω𝑚 and Ω𝑏 are the total matter and baryon fractions, ℎ is
+the reduced Hubble parameter, 𝑛𝑠 is the scalar spectral index,
+and 𝜎8 is the standard deviation of linear density perturbations
+smoothed on spheres of radius 8 Mpc ℎ−1 at redshift 𝑧 = 0.
+3.5. Likelihood
+To obtain constraints on cosmological parameters from the
+real and synthetic data described in the previous section, we
+
+10
+Parameter priors
+Parameter
+Prior
+Parameter
+Prior
+Cosmology
+Redshift calibration
+Ωm
+𝑈 (0.07, 0.8)
+Δ𝑧1s
+N(Δ𝑧1𝑠,∗, 0.016)
+Ωb
+𝑈 (0.03, 0.07)
+Δ𝑧2s
+N(Δ𝑧2𝑠,∗, 0.013)
+ℎ
+𝑈 (0.55, 0.91)
+Δ𝑧3s
+N(Δ𝑧3𝑠,∗, 0.011)
+𝑛s
+𝑈 (0.87, 1.07)
+Δ𝑧4,5
+s
+N(Δ𝑧4,5
+𝑠,∗, 0.022)
+𝜎8
+𝑈 (0.5, 1.1)
+Δ𝑧1g
+N(Δ𝑧1𝑔,∗, 0.007)
+Bias parameters
+Δ𝑧2g
+N(Δ𝑧2𝑔,∗, 0.007)
+𝑏𝑖
+1
+N(1.5, 100)
+Δ𝑧3g
+N(Δ𝑧3𝑔,∗, 0.006)
+𝑏𝑖
+2,𝑠,∇
+N(0, 100)
+Δ𝑧4,5,6
+g
+N(Δ𝑧4,5,6
+𝑔,∗ , 0.01)
+𝐴IA,0
+N(0, 100)
+𝑤𝑖𝑧,g
+N(1.00, 0.08)
+TABLE 1
+Prior distributions for the nuisance parameters entering our
+“3×2pt” analysis for each tracer. 𝑈 (𝑎, 𝑏) and N(𝜇, 𝜎) describe a
+uniform distribution with boundaries (𝑎, 𝑏) and a Gaussian
+distribution with mean 𝜇 and variance 𝜎, respectively. The index 𝑖
+in 𝑏𝑖g and 𝑚𝑖 runs over the different redshift bins. Δ𝑧∗ denotes the
+deviation from zero of the central/best-fit value of each redshift
+uncertainty parameter.
+will assume that the data vector (i.e. the clustering and shear
+power spectra), follows a Gaussian likelihood as in Eq. (11),
+with a parameter-independent covariance. The model will be
+described by 5 cosmological parameters, listed in Eq. (49),
+one or four linear parameters for each clustering redshift bins
+(for linear and PT bias respectively), one intrinsic alignment
+amplitude, one redshift shift parameter for each clustering and
+cosmic shear bin, and one redshift distribution width parame-
+ter for each clustering bin. The priors used for all parameters
+are provided in Table 1. The priors on cosmological param-
+eters are roughly based on the choices made for the DES-Y1
+analysis, except we sample over 𝜎8 instead of the scalar spec-
+trum amplitude 𝐴𝑠. The priors on the redshift shift parameters
+are based on the calibration of the DES-Y1 data (Hoyle et al.
+2018), and thus represent achievable calibration levels. The
+priors on the redshift width parameters are commensurate with
+those used in the DES Year-3 analysis (DES Collaboration
+2022) (the DES-Y1 analysis did not introduce width param-
+eters). Finally, we place an uninformative Gaussian prior on
+all the bias parameters, centered at zero and with a standard
+deviation of 100. The choice of using a very broad Gaussian
+prior as opposed to simply a flat prior is intended to enforce
+a smooth distribution as a function of these parameters, and
+to potentially aid the Gauss-Newton iterator when minimising
+the 𝜒2.
+We employ the cobaya MCMC sampler (Torrado & Lewis
+2019, 2021) with a convergence condition that the Gelman-
+Rubin diagnostic, 𝑅, ought to satisfy 𝑅 − 1 < 0.01. When
+using the Laplace approximation, we minimise the 𝜒2 over
+the nuisance parameters using a Gauss-Newton iterator, using
+the analytical derivatives with respect to bias parameters as
+described in Section 3.3, and modify the log-probability to
+be sampled by cobaya to be that of Eq. (8). We find that, in
+order to reduce the number of steps taken by the Gauss-Newton
+iterator, it is useful to determine a well educated global best-fit
+for the full parameter space before taking any samples, and to
+start the iterator from the corresponding best-fit value of the
+nuisance parameters.
+Throughout, we made use of the fitting formula of Eisenstein
+& Hu (1998) to calculate the linear matter power spectrum. We
+do this to speed up the calculations, and we have verified that
+the results obtained on the DES-Y1 data are insensitive to this
+choice compared to using a Boltzmann solver such as CLASS
+0.75
+0.80
+S8
+0.3
+0.4
+Ωm
+0.3
+0.4
+Ωm
+DES-Y1, p(z) marg. (x14 params.)
+Brute-force
+Analytical
+Fixed p(z)
+Fig. 2.— Contours comparing brute-force (silver) with analytic marginalisa-
+tion over photo-𝑧 uncertainties (red; see Section 3.2). Results are shown for
+the DES-Y1 data. We find that the contours are virtually unchanged, demon-
+strating the benefit of using an efficient analytic marginalisation scheme. We
+also show for posterity the result of assuming negligible error on the photo-𝑧
+distributions (blue). In reality, this assumption does not hold and potentially
+leads to a bias in the inferred cosmology.
+(Blas et al. 2011). The non-linear matter power spectrum is
+then computed using HALOFIT (Takahashi et al. 2012b).
+4. RESULTS
+4.1. Calibratable nuisance parameters
+We begin by focusing on calibratable systematics, for which
+we will follow the procedure described in Section 3.2: we will
+linearise the dependence of the theoretical prediction with re-
+spect to these parameters, for which a relatively tight prior can
+be obtained, and simply modify the covariance matrix as in
+Eq. (28), with T evaluated at a fixed set of parameters (fix-
+ing all cosmological parameters to the best-fit values found
+by Planck (Planck Collaboration et al. 2020) and all bias pa-
+rameters to their best-fit values). At this stage, we will thus
+only consider the nuisance parameters describing the uncer-
+tainty in the redshift distributions of the tracers under study,
+described in the right column of Table 1. All other parameters
+(cosmological, bias, and intrinsic alignment parameters) will
+for now be marginalised “brute-force” (i.e. treating them as
+free parameters in the MCMC chains). For simplicity, we will
+consider only linear bias, using scales 𝑘 < 0.15 Mpc−1, as
+described in Section 3.5.
+We first compare the performance of the method when ap-
+plied to the 3×2pt analysis of DES-Y1 data (Section 3.4.1).
+The results are shown in Fig. 2 and summarised in Table 2,
+with the exact results shown as black contour lines, and the
+results of the analytical marginalisation shown in red.
+We
+find that using the analytic marginalisation technique not only
+yields contours that are almost indistinguishable from those
+obtained by the traditional approach, but also does so signif-
+icantly faster (by a factor ∼ 10) with many fewer parameters.
+The blue contours in the figure show the constraints found by
+
+11
+0.795
+0.805
+S8
+0.6
+0.7
+h
+0.94
+0.96
+0.98
+ns
+0.04
+0.05
+0.06
+Ωb
+0.29
+0.30
+0.31
+Ωm
+0.29
+0.31
+Ωm
+0.05
+0.06
+Ωb
+0.94
+0.98
+ns
+0.6
+0.7
+h
+LSST, p(z) marg. (x17 params.)
+Brute-force
+Analytical
+Fixed p(z)
+Fig. 3.— Same as Fig. 2, but results are shown for a futuristic data vector with minimal noise meant to mimic the expected precision of LSST. The redshift
+distribution is less noisy compared with DES-Y1 and is split into 6 tomographic bins. We find that our analytic marginalisation recipe yields virtually equivalent
+results to the standard method for marginalising over redshift uncertainties by sampling directly the shift and width redshift parameters.
+0.79
+0.81
+S8
+0.28
+0.30
+0.32
+Ωm
+0.27
+0.30
+0.33
+Ωm
+LSST, p(z) marg. (x17 params., larger errors)
+Brute-force
+Analytical
+Fixed p(z)
+Fig. 4.— Same as Fig. 3 except for the mean redshift distribution is the same
+as that of DES-Y1, but the errors on the nuisance parameters are quadrupled.
+As before, we find that our method of analytic marginalisation works well
+despite the larger errors on the 𝑝(𝑧) measurements. Thus, we assert that the
+approximations in Section 3.2 hold even in the conservative scenario of large
+photometric uncertainties in futuristic data.
+fixing the nuisance parameters to their prior means, instead of
+marginalising over them. In this case, we observe that the red-
+shift distribution uncertainties cause only a mild broadening
+of the marginalised contours, which is not very challenging
+for the analytical approximate marginalisation to reproduce.
+In order to explore a more challenging scenario, we now
+move on to the case of an LSST-like 3×2pt dataset, as defined
+in Section 3.4.2. We will assume the same prior uncertainties
+used in the analysis of the DES-Y1 data. This will allow us to
+quantify the validity of the analytical marginalisation approach
+in a conservative scenario, in which, in spite of the much
+higher sensitivity of Stage-IV data, the precision with which
+we are able to calibrate redshift distributions has not improved
+with respect to the performance achieved with current data.
+Furthermore, we will explore an additional case in which the
+prior uncertainties are 4 times larger than the DES-Y1 ones.
+The reasoning behind this second test is two-fold: on one hand,
+our marginalisation method is an approximation that works in
+the regime where photometric uncertainties are linearisable
+and testing when that assumption breaks is essential; on the
+other hand, it is likely that, at the highest redshifts, and for
+the faintest samples, the 𝑝(𝑧) uncertainties will be somewhat
+larger for LSST than those of current surveys, especially at
+high redshifts. Hence, quadrupling the errors is an important
+worst-case scenario to consider.
+It is important to note that the results in this section are not
+meant to be interpreted as forecasts on the constraining power
+of LSST on cosmological parameters, but only as a quantifica-
+tion of our ability to analytically marginalise over photometric
+uncertainties when inferring the underlying cosmology.
+A
+
+12
+more thorough analysis would include a realistic treatment of
+the LSST true redshift distribution and noise, and a more care-
+ful treatment of galaxy bias and intrinsic alignments. As such,
+the results presented here give us a conservative estimate of the
+effect of analytic marginalisation on cosmology constraints.
+We present the results of the first scenario (i.e., 𝑝(𝑧) errors
+matching DES) in Fig. 3. The marginalised constraints on
+all cosmological parameters are virtually unchanged when we
+switch from the brute-force to the analytical marginalisation.
+This latter method is therefore successful at recovering ac-
+curate marginalised constraints on cosmological parameters.
+The figure shows the constraints on all cosmological param-
+eters, to highlight that the result extends even to the param-
+eters that 3×2pt datasets are less sensitive to: Ω𝑏, 𝑛𝑠 and ℎ.
+Also shown in blue are the constraints found assuming perfect
+knowledge of the redshift distributions (i.e. fixing all 𝑝(𝑧)
+parameters). In this case, the uncertainties on the redshift dis-
+tributions have a much larger effects than for the DES-Y1 data,
+inflating the uncertainties on 𝑆8 by a factor ∼ 2.6. Thus, even
+though marginalising over the redshift distribution parameters
+has an outsized effect on the final constraints, the analytical
+approximation approach is able to capture it almost exactly.
+Similarly reassuring is our result of quadrupling the uncer-
+tainty in the redshift nuisance parameters, shown in Fig. 4 in
+the (Ω𝑚, 𝑆8) plane. In this case, the increased redshift distribu-
+tion uncertainties broaden the final constraints on 𝑆8 by up to a
+factor ∼10. In spite of this, we find that the analytic marginal-
+isation method not only yields virtually the same constraints
+on the cosmological parameters, but does so 3-10 times faster
+than the traditional approach (see Table 2). This implicitly
+validates the approximation that a first-order expansion of the
+theory data vector with respect to a change in redshift distribu-
+tion is sufficient, even for prior uncertainties on 𝑝(𝑧) that are
+substantially worse than those achieved by current datasets.
+4.2. Bias parameters: tight posteriors
+Let us now focus on the bias parameters. In this case, the
+absence of an informative prior prevents us from linearising
+the dependence of the theory on these parameters, and we must
+therefore calculate the profile and Laplace terms numerically.
+Moreover, since the dependence of the 𝜒2 on these param-
+eters is not quadratic, marginalising over them may lead to
+significant volume effects in the marginalised posterior for the
+cosmological parameters. We will however start by exploring
+two situations in which the data is sensitive enough to measure
+these bias parameters accurately, in which case, as we saw in
+Section 2, the Laplace term and volume effects are small.
+In the first case, we make use of the DES-Y1 data, marginal-
+ising only over a single linear galaxy bias parameter per clus-
+tering bin, as well as an intrinsic alignment amplitude (i.e. a
+total of 6 nuisance bias parameters). This roughly coincides
+with the analysis choices made for the official DES-Y1 anal-
+ysis. The results are shown in the left panel of Fig. 5. The
+exact marginalised constrains (solid black contours) are accu-
+rately recovered by the Laplace approximation (red contours).
+While the former are obtained by running an MCMC with 11
+free parameters (5 cosmological, 6 nuisance 𝑝(𝑧) parameters
+marginalised as described in the previous section), the latter in-
+volve only a 5-dimensional parameter space, which is therefore
+significantly simpler to explore. Concretely, the 5-parameter
+chain converged 3 times faster than the 11-parameter chain
+(see Table 2). The blue contours in the same figure shows
+the constraints obtained after fixing the bias parameters to
+the best-fit values found by DES (Elvin-Poole et al. 2018).
+Fixing the galaxy bias shifts the cosmological constraining
+power from the cosmic shear data to the higher signal-to-noise
+clustering data, thus significantly reducing the uncertainties.
+Note that, although the red contours show the result of the full
+Laplace approximation (i.e. profile + Laplace contributions),
+the Laplace contribution is negligible, and the profile term is
+enough to recover the marginalised posterior.
+In order to explore the performance of the method with a
+significantly larger number of nuisance parameters, while still
+remaining in the regime where these parameters can be well
+constrained by the data, we now move to the LSST-like syn-
+thetic dataset, making use of the second-order perturbative
+expansion of Eq. (35) to describe galaxy bias. In this case,
+we include 4 free bias parameter in each of the 6 clustering
+redshift bins, adding up to a total of 25 nuisance parameters
+when combined with the intrinsic alignment amplitude. The
+results are shown in the right panel of Fig. 5, again as black
+lines for the brute-force marginalisation, i.e. considering all
+30 free parameters, and as red contours for the 5-parameter
+Laplace approximation. As before, we find that the approxi-
+mation is able to recover the marginalised constraints almost
+exactly.
+The impact of volume effects is also heavily sup-
+pressed, shifting the marginalised contours by less than 0.3𝜎
+(not shown in the figure).
+Note that, in both of these cases, besides the bias param-
+eters, we have also marginalised over a number of redshift
+distribution uncertainty parameters (14 for DES-Y1, 17 for
+LSST), thus reducing the model dimensionality from 37 and
+24 for LSST and DES-Y1, respectively, to only 5 cosmological
+parameters. Obtaining converged MCMC chains for these 5
+cosmological parameters takes approximately 2 hours for the
+LSST-like dataset, a factor ∼ 6 faster than the chains with only
+the 𝑝(𝑧) parameters analytically marginalised over, and a fac-
+tor ∼ 30 times faster than the full brute-force chains. The mag-
+nitude of these speed gains, however impressive, must be taken
+with a pinch of salt. The performance of MCMC sampling
+may depend significantly upon the design of the likelihood
+code, and whether it allows the sampler to decompose the
+space between “fast” and “slow” parameters, over-sampling
+the former, and making use of “dragging” techniques (Lewis
+2013; Neal 2005). The fast-slow split allows one to effectively
+marginalise over the fast subspace, and becomes particularly
+powerful in the presence of a large number of nuisance pa-
+rameters on which the likelihood has a computationally sim-
+ple dependence (as is the case of the bias parameters in our
+model). To provide a fairer assessment of the computational
+gains obtained using the Laplace approximation, we wrote
+an optimised version of the 3×2-point likelihood that allows
+cobaya to exploit the fast nature of the bias parameters as ef-
+ficiently as possible (assuming all 𝑝(𝑧) parameters) are fixed.
+In this case, the speed-up factor for the cases explored here
+ranged between ∼ 1.5 and ∼ 11, with the highest performance
+improvement achieved for the LSST 2×2pt data discussed in
+the next section. Ultimately the performance difference de-
+pends on the design of the likelihood code, the complexity
+in the parameter dependence of the likelihood, and the effi-
+ciency of the 𝜒2 minimisation method used to calculate �𝑛∗.
+On this latter point, we find that the Gauss-Newton method
+used here typically achieves convergence after only 10-20 it-
+erations. Thus finding the best-fit bias parameters is signifi-
+cantly faster than calculating the cosmology-dependent power
+spectrum templates of Eq. (46).
+
+13
+0.75
+0.80
+S8
+0.2
+0.3
+0.4
+Ωm
+0.2
+0.3
+0.4
+Ωm
+DES-Y1, linear bias marg. (x6 params.)
+Brute-force
+Analytical
+Fixed bias
+0.795
+0.805
+S8
+0.29
+0.30
+0.31
+Ωm
+0.29
+0.31
+Ωm
+LSST, PT bias marg. (x25 params.)
+Brute-force
+Analytical
+Fig. 5.— On the left, same as Fig. 2, but here we marginalise over the bias parameters using the Laplace approximation. This reduces the parameter space to
+only the cosmological parameters without degrading the accuracy of the constraints. On the right, as Fig. 3, but here we marginalise over the perturbation theory
+bias parameters via the Laplace approximation. The agreement with the full numerical marginalisation (black) is almost perfect.
+0.78
+0.80
+0.82
+S8
+0.28
+0.30
+0.32
+Ωm
+0.28
+0.30
+0.32
+Ωm
+LSST 2x2pt, PT bias marg. (x25 params.)
+Brute-force
+Brute-force (Jeffreys)
+Analytical
+Analytical (profile-only)
+0.75
+0.80
+S8
+0.2
+0.3
+0.4
+Ωm
+0.2
+0.3
+0.4
+Ωm
+DES-Y1, PT bias marg. (x21 params.)
+Brute-force
+Brute-force (Jeffreys)
+Analytical
+Analytical (profile-only)
+Fig. 6.— Demonstration of volume effects in the case of using less constraining data. In order to recover the best-fit parameter values (denoted with a star), when
+applying brute-force marginalisation, one needs to make use of Jeffreys prior (compare solid with dashed black line). Similarly, when analytically marginalising
+over the PT biases, one needs to remove the Laplace term to avoid biasing the constraints (compare blue and red contours). The plot on the left shows the
+constraints for a 2x2pt analysis with a LSST-like dataset, while on the right we show constraints using the DES-Y1 data. In the latter case, the volume effects
+become more pronounced, as the data has less constraining power.
+
+14
+4.3. Bias parameters: loose posteriors and volume effects
+In the cases explored in the previous sections, the data was
+able to constrain the nuisance parameters sufficiently well, and
+all volume effects associated with the choice of parametrisa-
+tion were subdominant. Let us now study the ability of the
+Laplace approximation to capture these volume effects when
+they become more prominent.
+First, let us consider the case of a “2×2pt” combination
+of the LSST data, consisting only of the galaxy-galaxy and
+galaxy-shear power spectra, excluding the shear-shear auto-
+correlation, and analysed under the perturbative bias expan-
+sion of Eq. (35). The logic for focusing on this scenario is that,
+in this case, the model relies completely on galaxy clustering
+to constrain cosmological parameters, and is therefore more
+sensitive to the complexity of the galaxy bias parametrisation
+(which dominates the dimensionality of the nuisance parame-
+ter space). The resulting constraints on Ω𝑚 and 𝑆8 are shown
+in the left panel of Fig. 6. The solid and dashed black contours
+show the results using brute-force marginalisation, assuming
+no prior and a Jeffreys prior respectively. The red contours
+then show the same constraints using the Laplace approxima-
+tion (including both profile and Laplace terms), while the blue
+contour shows the result of including only the profile like-
+lihood contribution. The position of the best-fit parameters
+is indicated by the orange star. We find that the profile-only
+approximation is able to recover well the exact marginalised
+constraints assuming a Jeffreys prior, and the full Laplace ap-
+proximation recovers the constraints found in the absence of
+this prior. In this case we see a clear ∼ 0.8𝜎 shift in the cosmo-
+logical constraints due to volume effects associated with the
+large number of bias parameters, although the Laplace term is
+able to capture this with high accuracy.
+As a second example, we explore the possibility of using
+the second-order perturbative bias expansion of Eq. (35) to
+analyse the DES-Y1 data. The lower sensitivity of this data
+compared to LSST should reduce its ability to constrain the
+higher-order bias parameters, and increase the impact of vol-
+ume effects. The results are shown in the right panel of Fig. 6,
+using the same color scheme as the last figure. In this case we
+can see that volume effects are a lot more significant, and can
+shift the confidence contours of the cosmological parameters
+by almost 3𝜎 with respect to their global best-fit value, marked
+by the orange star. We can also see that, in this more extreme
+case, the Laplace approximation starts to fail, manifesting mild
+shifts of ∼ 0.3 − 0.5𝜎 with respect to the exact marginalised
+constraints. This provides us with a useful “rule of thumb”
+to determine the reliability of the Laplace approximation: if
+a significant (> 1𝜎) shift is found between the marginalised
+contours obtained using the profile likelihood and those ob-
+tained accounting also for the Laplace term, the approxima-
+tion may start to fail, and a brute-force marginalisation over
+the nuisance parameters using a Jeffreys prior is needed to
+obtain accurate results. This test can be done without running
+the MCMC twice: one can run MCMC chain using Laplace
+approximation while also saving the profile likelihood values
+at each MCMC step and employing importance sampling to
+derive the profile likelihood posteriors. Nevertheless, even in
+these cases, we find that the Laplace approximation is able to
+recover the region of parameter space preferred by the data
+reasonably well, and can thus be used as a fast way to charac-
+terise this region that can then be refined (e.g. via importance
+sampling).
+In an extreme case, such as the one we just discussed, it
+is worth asking which of the constraints in Fig.
+6 are the
+“correct” ones. The presence of such large volume effects,
+combined with the arbitrariness of using flat priors for the
+nuisance parameters, would imply that the use of a Jeffreys
+prior should produce the most correct results. However, as
+we noted in Section 2.3, the Jeffreys prior is not guaranteed
+to cancel out all volume effects. One might thus argue that
+the profile likelihood, since it is by construction centered on
+the best-fit parameters, could provide an equally reasonable
+representation of the favoured confidence region. One might,
+however, view this choice critically due to the excessive gain
+in precision over the marginal likelihood that is expected in
+the case of many poorly constrained nuisance parameters (see
+e.g. Fig. 4). Finally, we could adopt a completely frequentist
+approach, determining the confidence intervals of all param-
+eters by extracting the maximum-likelihood estimate for the
+data and for a suite of simulated datasets compatible with it.
+5. CONCLUSIONS
+Forecasts of the next decade in cosmology predict that mean-
+ingful constraints on fundamental unknowns such as the mass
+of neutrinos and the nature of dark matter and dark energy
+will come from multi-scale, multi-tracer efforts, encompass-
+ing a wide range of probes and redshifts. It is for this reason
+that the efficient analysis of joint data sets combining low-
+and high-redshift probes in an accurate manner is of great
+importance to the field of large-scale structure analysis. How-
+ever, this comes at the cost of adding a colossal number of
+nuisance parameters characterising the observational and the-
+oretical systematic uncertainties of all probes involved, which
+can noticeably slow down the sampling of the parameter space.
+In galaxy clustering and weak lensing joint studies, the most
+significant obstacles to overcome are the accurate modeling of
+the redshift distribution, the galaxy bias relation, and intrin-
+sic alignments. Not doing so correctly can bias the inferred
+cosmological parameters, but also accounting for these effects
+via more elaborate models is a major inhibitor of efficient
+sampling.
+In this paper, we introduce a formal approach for speeding up
+the sampling process in the presence of a large number of nui-
+sance parameters, and investigate the accuracy of our method
+when applying it to photometric survey data. In particular, we
+study the current DES-Y1 data set as well as a synthetic data
+vector from an LSST-like survey to validate whether the fast
+analytic method proposed in this work is capable of reproduc-
+ing the posterior contours and constraints one arrives at when
+adopting the traditional method of diligently varying tens of
+nuisance parameters.
+In Section 2, we describe the general methodology behind
+analytically marginalising over any nuisance parameter by ap-
+proximating the marginal distribution, Eq. (8), as consisting
+of a “profile” term, centered on the best-fit nuisance param-
+eters, and a “Laplace” term, associated with the quadratic
+contribution. We then consider the special case of a Gaussian
+likelihood, showing that the Laplace term can be associated
+with volume effects, which the profile term alone is free from,
+being approximately equivalent to imposing a Jeffreys prior.
+This argument becomes exact when studying the case of nui-
+sance parameters that contribute linearly to the theory vector.
+In Section 3, we introduce the cosmological analysis of pho-
+tometric survey analysis and the specific problems associated
+with it.
+In particular, we introduce the relevant summary
+statistics utilised when performing “3×2pt” analysis and the
+two dominant sets of nuisance parameters associated with that
+
+15
+analysis: redshift distribution (𝑝(𝑧)) calibration parameters
+and bias/IA model parameters. We then provide details of
+how our model enables a fast marginalisation over these and
+how it can be applied to current and future data sets.
+We summarise our results in Section 4. The first half of that
+section deals with the 𝑝(𝑧) parameters. Since these parame-
+ters have external priors, the dependence of the theory vector
+on them can be linearised around the center of that prior, and
+the Laplace term can be ignored. Their impact can then be
+incorporated in a pre-sampling step by simply modifying the
+covariance matrix (see Eq. (28)). We showed (Figs. 2 and
+3), that this method is able to obtain constraints on cosmo-
+logical parameters that are indistinguishable from those found
+via brute-force marginalisation, while performing 5-10 times
+better in terms of computational time, even when considering
+calibration priors that are significantly worse than those that
+can be achieved with current data.
+We next focused on the marginalisation over bias parame-
+ters, using the full Laplace approximation, finding the maxi-
+mum of the conditional posterior distribution on the fly. We
+find that, when the data is able to place significant constraints
+on all model parameters, the Laplace approximation is an ex-
+cellent representation of the marginal distribution, and that the
+Laplace term becomes subdominant with respect to the profile
+likelihood. In the presence of less constraining data, volume
+effects associated with the choice of nuisance parametrisation
+become relevant and, when mild, can be accurately captured
+by the Laplace term. In these cases, we also find that the profile
+likelihood is almost indistinguishable from the marginal distri-
+bution when adopting a Jeffreys prior, cancelling these volume
+effects. When volume effects become more relevant, signifi-
+cantly shifting the parameter region favoured by the marginal
+distribution (e.g. by more than ∼ 1𝜎), the Laplace approxi-
+mation begins to fail, although the position and extent of the
+favoured region of parameter space are still qualitatively well
+described by it (both with or without a Jeffreys prior). Thus,
+even in this case, the method can be used to identify this region
+and then refine it with a standard MCMC.
+Through this paper we have assumed that the likelihood in
+the nuisance parameters is unimodal. Obviously, for multi-
+modal likelihoods with comparable posterior volumes in each
+mode, the approximations we use will fail. Since the likeli-
+hood is in general quartic in bias parameters, the likelihood
+can in principle have up to three local maxima in each bias
+direction making multimodal likelihoods a possibility. This
+has been observed “in the wild” in fits to galaxy auto-power
+spectrum using higher order bias parameters (e.g., Goldstein
+et al. 2022), but typically disappears with sufficiently aggres-
+sive scale cuts and when employing the full 3×2pt data vector.
+These problems can be detected by initialising the Newton-
+Raphson optimiser at different starting points and noting exis-
+tence of multiple maxima.
+The elimination of nuisance parameters is a nontrivial task.
+Caution must be taken when departing from our assumption of
+Gaussian data with a model-independent covariance matrix.
+While in this specific application, we find the profile likeli-
+hood to give unbiased parameter constraints and the marginal
+distribution to be biased, this does not hold in general. Com-
+mon tasks in data analysis (such as the estimation of the vari-
+ance from Gaussian data) have an opposite outcome, with the
+marginal distribution peaking at the unbiased parameter esti-
+mate, while in other cases neither method seems to perform
+optimally (Berger et al. 1999). It is therefore crucial to con-
+sider the role of volume effects in every specific likelihood
+model.
+The reduction in the dimensionality of the parameter space
+afforded by the Laplace approximation is accompanied by a
+boost in the speed with which convergence in the associated
+MCMC chains is achieved, and we find improvements by a
+factor ∼ 2 − 15. Achieving this performance improvement is
+greatly aided by the simplicity with which the theory predic-
+tion depends on the bias and IA nuisance parameters, which
+allows us to write down its derivatives analytically (see Sec-
+tion 3.3), and to evaluate them quickly (in 𝑂(10−3 s)). There
+may be more sophisticated astrophysical models that do not
+conform to this simple structure (e.g. physically motivated
+models for the redshift dependence of bias terms, halo-based
+models, etc.), and for which computing derivatives analyti-
+cally is impractical or unfeasible. A more general application
+of this method in these situations would thus require the use of
+automatic differentiation techniques (Margossian 2018), able
+to efficiently calculate these gradients regardless of how the
+model depends on the nuisance parameters. The Laplace ap-
+proximation is by far not the only application that benefits
+from having efficient access to derivatives of the likelihood
+with respect to some parameters.
+More general sampling
+methods including HMC (MacKay 2002) or variational in-
+ference (Blei et al. 2016) often require the use of likelihood
+gradients, as do efficient minimisation methods, or the calcu-
+lation of Jeffreys priors. Additionally, although here we have
+focused on the volume effects associated with the marginali-
+sation over nuisance parameters, the non-linear way in which
+all cosmological parameters enter the likelihood implies that
+degeneracies between them can also give rise to biasing vol-
+ume effects within the subspace of cosmological parameters.
+A Jeffreys prior (or any other form of correction for volume
+effects) should therefore be used routinely in cosmological pa-
+rameter inference, although this is normally prevented by the
+need to estimate derivatives efficiently. The development of
+fully automatically-differentiable cosmological theory predic-
+tions is therefore of paramount importance in the context of
+current and future experiments.
+In conclusion, our method produces satisfactory results in
+virtually all of the explored regimes. Thanks to the huge gain
+in sampling efficiency this approach offers, it can be used to
+enable the joint analysis of multiple galaxy surveys, which is
+typically hindered by the gigantic number of nuisance param-
+eters that need to be sampled. Fast marginalisation methods
+such as the one proposed in this work can help us address some
+of the inconsistencies in the present analysis of cosmological
+data. For example, current constraints find that both the ampli-
+tude of clustering (𝑆8) and the energy density of matter (Ω𝑚)
+inferred from the low-redshift analysis of galaxy and weak
+lensing surveys are systemically lower than those obtained
+from high-redshift probes such as the CMB (for a review, see
+Perivolaropoulos & Skara 2022). In the near term, we plan
+to adopt the method proposed in this paper to speed up the
+sampling process and analyze jointly the currently (publicly)
+available photometric survey data in an effort to tackle some
+of the obstacles standing in the way of gaining a fundamental
+understanding of our Universe.
+ACKNOWLEDGMENTS
+We would like to thank Deaglan Bartlett, Harry Desmond,
+Simone Ferraro, Arrykrishna Mootoovaloo, and Martin White
+for useful discussions. BH is supported by the Miller Institute
+for Basic Research in Science and the Chamberlain Fellow-
+
+16
+ship and Lawrence Berkeley National Lab. KW is funded by
+a SISSA Ph.D. fellowship. SA is funded by a Kavli/IPMU
+doctoral studentship.
+DA is supported by the Science and
+Technology Facilities Council through an Ernest Rutherford
+Fellowship, grant reference ST/P004474. CGG is supported
+by European Research Council Grant No: 693024 and the
+Beecroft Trust. JRZ is supported by an STFC doctoral stu-
+dentship.
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+APPENDIX
+A. SUMMARY OF RESULTS
+Table 2 lists the constraints on cosmological parameters,
+and the time needed for the corresponding MCMC chains to
+converge for the different cases explored in Section 4.1, where
+we marginalise over calibratable redshift distribution system-
+atics using the linearisation technique described in Section 3.2.
+Table 3 shows the same information for the cases explored in
+Sections 4.2 and 4.3, in which we make use of the full Laplace
+approximation to marginalise over galaxy bias and intrinsic
+alignment parameters.
+This paper was built using the Open Journal of Astrophysics
+LATEX template. The OJA is a journal which provides fast and
+easy peer review for new papers in the astro-ph section of the
+arXiv, making the reviewing process simpler for authors and
+referees alike. Learn more at http://astro.theoj.org.
+
+17
+Method
+𝑆8
+Ω𝑚
+Ω𝑏
+𝑛𝑠
+ℎ
+Converge time
+DES 3×2pt: brute-force
+0.784 ± 0.017
+0.304+0.029
+−0.041
+–
+–
+–
+338.34 Hrs
+DES 3×2pt: analytic
+0.783 ± 0.017
+0.305+0.030
+−0.042
+–
+–
+–
+31.74 Hrs
+DES 3×2pt: fixed 𝑧
+0.780 ± 0.015
+0.306+0.029
+−0.040
+–
+–
+–
+30.22 Hrs
+LSST 3×2pt: brute-force
+0.7999 ± 0.0021
+0.3001 ± 0.0057
+0.0501 ± 0.0036
+0.9600 ± 0.0090
+0.672+0.024
+−0.029
+58.24 Hrs
+LSST 3×2pt: analytic
+0.8000 ± 0.0021
+0.3001 ± 0.0057
+0.0502 ± 0.0037
+0.9597 ± 0.0089
+0.673+0.026
+−0.028
+13.00 Hrs
+LSST 3×2pt (4x): brute-force
+0.7996 ± 0.0056
+0.300 ± 0.010
+0.0501 ± 0.0038
+0.9599 ± 0.0097
+0.673+0.028
+−0.035
+23.19 Hrs
+LSST 3×2pt (4x): analytic
+0.7999 ± 0.0054
+0.300 ± 0.010
+0.0503 ± 0.0037
+0.9598 ± 0.0099
+0.673+0.029
+−0.035
+8.36 Hrs
+LSST 3×2pt: fixed 𝑧
+0.79998 ± 0.00078
+0.3000 ± 0.0026
+0.0501 ± 0.0036
+0.9599 ± 0.0068
+0.672+0.020
+−0.023
+12.39 Hrs
+TABLE 2
+Constraints on the cosmological parameters from our 3×2pt analysis applied to a current (DES-Y1) and future (LSST-like) data set, comparing
+the performance of two marginalisation approaches: analytic and brute-force (described in Section 3.2). “Analytic” refers to the new method
+proposed in this work, in which one accounts for photometric uncertainties prior to sampling the parameter space, whereas “brute-force”
+refers to the standard method of introducing ∼10 “shift” and “width” redshift calibration parameters (see Table 1 and Table 1). “Fixed 𝑧”
+considers the unrealistic scenario of completely ignoring photometric uncertainties.
+Method
+𝑆8
+Ω𝑚
+Ω𝑏
+𝑛𝑠
+ℎ
+Converge time
+DES-Y1 linear bias: brute-force
+0.785 ± 0.013
+0.303+0.030
+−0.043
+–
+–
+–
+7.1 Hrs
+DES-Y1 linear bias: analytical
+0.785 ± 0.012
+0.302+0.030
+−0.041
+–
+–
+–
+2.5 Hrs
+DES-Y1 linear bias: fixed 𝑏
+0.7810 ± 0.0090
+0.2691 ± 0.0094
+–
+–
+–
+2.1 Hrs
+DES-Y1 PT bias: brute-force, no J.P.
+0.780 ± 0.012
+0.347 ± 0.031
+–
+–
+–
+16.5 Hrs
+DES-Y1 PT bias: analytical, full Laplace
+0.775 ± 0.011
+0.362 ± 0.030
+–
+–
+–
+6.5 Hrs
+DES-Y1 PT bias: brute-force, with J.P.
+0.795 ± 0.011
+0.256+0.022
+−0.027
+–
+–
+–
+45.5 Hrs
+DES-Y1 PT bias: analytical, profile only
+0.792 ± 0.012
+0.273+0.027
+−0.034
+–
+–
+–
+8.1 Hrs
+LSST 3×2pt: brute-force
+0.7999 ± 0.0020
+0.2996 ± 0.0055
+0.0505 ± 0.0036
+0.9580 ± 0.0089
+0.676+0.025
+−0.030
+5.1 Hrs
+LSST 3×2pt: analytical
+0.8000 ± 0.0020
+0.3001 ± 0.0057
+0.0501 ± 0.0037
+0.9595 ± 0.0089
+0.672+0.025
+−0.030
+2.2 Hrs
+LSST 2×2pt: brute-force, no J.P.
+0.7946 ± 0.0086
+0.3055 ± 0.0081
+0.0482 ± 0.0044
+0.955 ± 0.037
+0.655+0.035
+−0.044
+35.1 Hrs
+LSST 2×2pt: analytical, full Laplace
+0.7949 ± 0.0088
+0.3059 ± 0.0081
+0.0483 ± 0.0043
+0.955 ± 0.038
+0.654+0.037
+−0.042
+2.1 Hrs
+LSST 2×2pt: brute-force, with J.P.
+0.7997 ± 0.0086
+0.2997+0.0072
+−0.0081
+0.0504 ± 0.0044
+0.954 ± 0.039
+0.680+0.040
+−0.049
+17.2 Hrs
+LSST 2×2pt: analytical, profile only
+0.7993 ± 0.0083
+0.2998+0.0075
+−0.0087
+0.0506 ± 0.0043
+0.953 ± 0.040
+0.682+0.041
+−0.049
+6.4 Hrs
+TABLE 3
+Constraints on the cosmological parameters from our 3×2pt analysis applied to a current (DES-Y1) and future (LSST-like) data set, comparing
+the performance of two marginalisation approaches: analytic and brute-force (described in Section 3.2). “Analytic” refers to the new method
+proposed in this work, in which one accounts for photometric uncertainties prior to sampling the parameter space, whereas “brute-force”
+refers to the standard method of introducing ∼10 “shift” and “width” redshift calibration parameters (see Table 1 and Table 1). “Fixed 𝑧”
+considers the unrealistic scenario of completely ignoring photometric uncertainties.
+
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+page_content=' University of Oxford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
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+page_content=' United Kingdom  6Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
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+page_content=' Kashiwa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
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+page_content=' Brookhaven National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Upton NY 11973,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' USA  (Dated: January 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2023)  Version January 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2023  ABSTRACT  The analysis of photometric large-scale structure data is often complicated by the need to account for many  observational and astrophysical systematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The elaborate models needed to describe them often introduce  many “nuisance parameters”, which can be a major inhibitor of an efficient parameter inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In this paper,  we introduce an approximate method to analytically marginalise over a large number of nuisance parameters  based on the Laplace approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' We discuss the mathematics of the method, its relation to concepts such  as volume effects and profile likelihood, and show that it can be further simplified for calibratable systematics  by linearising the dependence of the theory on the associated parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' We quantify the accuracy of this  approach by comparing it with traditional sampling methods in the context of existing data from the Dark  Energy Survey, as well as futuristic Stage-IV photometric data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The linearised version of the method is able to  obtain parameter constraints that are virtually equivalent to those found by exploring the full parameter space  for a large number of calibratable nuisance parameters, while reducing the computation time by a factor 3-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Furthermore, the non-linearised approach is able to analytically marginalise over a large number of parameters,  returning constraints that are virtually indistinguishable from the brute-force method in most cases, accurately  reproducing both the marginalised uncertainty on cosmological parameters, and the impact of volume effects  associated with this marginalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' We provide simple recipes to diagnose when the approximations made by  the method fail, and one should thus resort to traditional methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The gains in sampling efficiency associated  with this method enable the joint analysis of multiple surveys, typically hindered by the large number of nuisance  parameters needed to describe them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' INTRODUCTION  The Λ Cold Dark Matter (ΛCDM) model of cosmology  offers a compelling explanation for a wide variety of obser-  vations despite the fact that the nature of its dominant com-  ponents, dark matter and dark energy, remains unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' As  the statistical power of cosmological experiments grows in the  next decade, our ability to stress test the ΛCDM model of  the Universe will improve immensely and provide powerful  constraints on its dark ingredients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In addition, these new ex-  periments may shed light over discrepancies between early-  and late-Universe-derived cosmological constraints that have  recently emerged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Of highest relevance to this study is the tension affect-  ing the measurement of the amplitude of matter fluctuations,  𝑆8 ≡ 𝜎8(Ω𝑚/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='3)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='5, where 𝜎8 is the root mean square of  matter fluctuations on an 8 Mpc/ℎ scale, and Ω𝑚 is the frac-  tional energy density in non-relativistic matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In particular,  the latest prediction from Planck CMB data finds its value  (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2020) to be ∼2-3𝜎 higher than  the analogous measurement from photometric surveys such as  KiDS-1000 (KiDS), the Dark Energy Survey (DES) and Hyper  Suprime-Cam (HSC) (Heymans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' DES Collabora-  tion 2022, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' MacCrann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Hamana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  As data from current and future cosmological surveys such  ∗boryanah@berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='edu  as the Legacy Survey of Space and Time (LSST), at the Vera  Rubin Observatory (LSST Dark Energy Science Collabora-  tion 2012), the Nancy Grace Roman Space Telescope (Spergel  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2015), or the Euclid satellite (Amendola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2018)  starts to trickle in, it is of paramount importance for cosmol-  ogists to devise robust tests of their analysis pipelines, and to  construct rigorous theoretical frameworks to better understand  these tensions and extract maximal cosmological information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Photometric surveys offer a pathway to resolving many is-  sues of the standard paradigm by providing measurements of  the clustering and weak gravitational lensing of millions and  soon billions of galaxies on the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  In particular, the so-  called “3×2pt” analysis (the joint analysis of galaxy cluster-  ing and cosmic shear in tomographic bins) offers a powerful  tool to cosmologists with the potential to break degenera-  cies between cosmological and astrophysical parameters and  yields stringent constraints (Heymans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Such tomo-  graphic analyses have a leverage on the 𝑆8 and Ω𝑚 tensions and  can yield ∼1-10%-level constraints (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=', Nicola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Hadzhiyska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' García-García et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Nonethe-  less, it is still unclear whether this tension is driven by system-  atic and modeling errors in the analysis or by new physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Large-scale structure surveys in general are affected by a  large number of observational systematic uncertainties, as well  as uncertainties in the physical relation between the astrophys-  ical systems that form the basis of their observables, and the  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='11895v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='CO]  27 Jan 2023  2  underlying cosmological quantities we wish to constrain (com-  monly labelled “astrophysical systematics”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' With increasing  statistical power, new sources of systematic uncertainty be-  come more relevant, and previously known systematics require  more detailed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' This inevitably leads to an inflation in  the number of nuisance parameters relative to the number of  the physically meaningful cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Exam-  ples of these systematics are: uncertainties in the redshift dis-  tribution of the samples, errors in the measured galaxy shapes  (relevant for weak lensing studies), the link between the abun-  dance of galaxies and the underlying matter overdensities, and  the intrinsic alignments between galaxy orientations and the  local structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Moreover, since the future of cosmology lies  in the combination of multiple observational probes, the simul-  taneous modeling of these effects across different datasets, and  the efficient sampling of the resulting (typically large) param-  eter space is bound to become a top priority in the next few  years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  To illustrate this, the latest analysis of galaxy clustering and  weak gravitational lensing carried out by the Dark Energy  Survey (DES) (DES Collaboration 2022) included 6 cosmo-  logical parameters (ΛCDM and the total neutrino mass) and  25 nuisance parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Due to the curse of dimensionality,  this increase in the number of parameters leads to strong ineffi-  ciencies in the standard rejection sampling algorithms used to  explore the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The resulting parameter chains  take long times to converge (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' days or weeks for state-of-  the-art datasets), even though we are ultimately only interested  in the marginal posterior distribution of a much smaller pa-  rameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  This problem can be addressed through various approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  On one hand, the use of gradient-based sampling algorithms,  such as Hamiltonian Monte-Carlo approaches (MacKay 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Hoffman & Gelman 2011), or other methods that are well-  suited to multiple-parameter problems, can significantly speed  up the sampler convergence time and provide constraints in a  reasonable amount of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Recently, Ruiz-Zapatero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  (2023) validated the analytical marginalisation of redshift cal-  ibration models with up to ∼ 100 parameters using self-tuning  Hamiltonian Monte-Carlo (a problem that has also been ad-  dressed by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Stölzner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (2023) using  other methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' However, the performance of the marginali-  sation method depends on the choice of nuisance parameters,  on one hand through their effect on the theory prediction, and  on the other hand through their priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  For Gaussian data with parameters affecting the theory pre-  dictions to linear order, this can be effectively done by modify-  ing the covariance matrix and fixing the marginalised param-  eters (Rybicki & Press 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' For more complex parameters  that appear as higher-order terms in the model such as the  galaxy bias parameters, exact analytic marginalisation is not  generally possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' One may resort to Gibbs sampling-like  schemes (Geman & Geman 1984), where this marginalisa-  tion is done numerically on the fly, and which can potentially  lead to significant speed-ups in the sampling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' How-  ever, the efficiency of this approach depends on the effective  acceptance rate of the marginalisation step, and on the degen-  eracy between nuisance and cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Here,  we propose a general technique that allows for approximate  analytical marginalisation over both linear and non-linear nui-  sance parameters in an efficient manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Similar methods  have been put forward in the past (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=', Taylor & Kitching  2010), with a variety of applications in mind (Bridle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Stölzner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Here, we will quantify the validity  of this method in the context of photometric 3×2pt analyses  with current data from DES, and futuristic Stage-IV-like data  mimicking experiments such as LSST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  This paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In Section 2, we pro-  vide a general introduction to our analytical marginalisation  method and discuss the interpretation of the various terms  we define and their relevance to volume effects and param-  eter priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  In Section 3, we introduce relevant aspects of  the analysis of cosmological photometric surveys and explore  possible ways in which our method can be used to marginalise  over linearisable and non-linearisable nuisance parameters in  the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In Section 4, we showcase the effect of employing  our method both in terms of deriving unbiased constraints on  the cosmological parameters and also in terms of the conver-  gence time and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' We summarise our findings and  comment on their implications for the future of photometric  survey analysis in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' LIKELIHOODS AND NUISANCE PARAMETERS  Our aim is to explore the posterior distribution 𝑝( �𝜃|d),  where �𝜃 is a set of model parameters, and d is the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Using  Bayes’ theorem, the posterior can be written in terms of the  likelihood 𝑝(d| �𝜃) and prior 𝑝( �𝜃) as  𝑝( �𝜃|d) ∝ 𝑝(d| �𝜃)𝑝( �𝜃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  (1)  We will decompose the model parameters as �𝜃 = { �Ω, �𝑛},  where �Ω is a set of parameters we care about (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' fundamen-  tal cosmological parameters), and �𝑛 is a vector of nuisance  parameters, describing various observational or theoretical  uncertainties, which are largely irrelevant to the fundamen-  tal question being explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In other words, the distribution  we aim to obtain is the marginalised posterior  𝑝( �Ω|d) =  ∫  𝑑�𝑛 𝑝( �Ω, �𝑛|d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  (2)  For simplicity, in what follows, for any probability distribu-  tion 𝑝( �𝜃), we will define the “chi-squared”, 𝜒2, as  𝜒2( �𝜃) ≡ −2 log 𝑝( �𝜃) + 𝐾,  (3)  where 𝐾 is an arbitrary constant that does not depend on the  random variables �𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Note that it is more common to define  𝜒2 in terms of the likelihood, but in this paper, we will always  apply it to the posterior, generalising to the presence of priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Profiling and analytical marginalisation  In order to approximate the marginal distribution in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (2),  let us start by considering the best-fit value of the nuisance  parameters having fixed �Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' That is, we define �𝑛∗( �Ω) as  �𝑛∗( �Ω) ≡ arg max�𝑛𝑝( �Ω, �𝑛|d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  (4)  Assuming that the distribution is differentiable at all points,  �𝑛∗ then satisfies  𝜕𝜒2  𝜕�𝑛  ����  �𝑛∗  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  (5)  Following Taylor & Kitching (2010), we can then approx-  imate the distribution at each value of �Ω by expanding 𝜒2 to  second order in �𝑛 around �𝑛∗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' :  𝜒2( �Ω, �𝑛) ≃ 𝜒2  ∗ ( �Ω) + Δ�𝑛𝑇 F∗Δ�𝑛,  (6)  3  p(Ω0, n|d)  Exact  Profile  Profile + Laplace  χ2(Ω0, n|d)  n  Ω  p(Ω, n|d)  n∗(Ω)  Ω = Ω0  p(Ω|d)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='— Joint posterior distribution on two parameters, Ω and 𝑛, with an approximate degeneracy of the form 𝑛Ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='2 ∼ const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The large bottom left panel shows  the joint distribution as red contours, with the position of the best-fit value of 𝑛 as a function of Ω in solid black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The central panel shows the 𝜒2 for a fixed value  of Ω = Ω0 (shown as the dotted line in the first panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The red line shows the true 𝜒2, while the dashed black line shows the Laplace approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The top  panel shows the probability distribution along Ω = Ω0 (given by 𝑝 ∝ exp(−𝜒2/2)), with the exact distribution and its Laplace approximation in red and dashed  black respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The bottom right panel shows the distribution of Ω marginalised over 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The exact result is shown in red, with its Laplace approximation  shown in dashed black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The blue line shows the marginalised profile likelihood obtained by simply maximising the joint likelihood over 𝑛 for each Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The Laplace  approximation provides an excellent description of the marginalised distribution, while the profile likelihood returns a distribution with very similar width but  centered, by construction, on the best-fit value of Ω, avoiding volume effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  where 𝜒2  ∗ ( �Ω) ≡ 𝜒2( �Ω, �𝑛∗), Δ�𝑛 ≡ �𝑛 − �𝑛∗, and we have defined  the matrix  F∗,𝑖 𝑗 = 1  2  𝜕2𝜒2  𝜕𝑛𝑖𝜕𝑛 𝑗  ����  �𝑛∗  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  (7)  This is the so-called Laplace approximation (Kass et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  In this limit, the distribution is locally (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  at each �Ω) a  multivariate normal distribution in �𝑛, and thus the integral in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (2) can be solved analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The resulting marginalised  likelihood has a 𝜒2  𝑚( �Ω) ≡ −2 log 𝑝( �Ω|d) given by  𝜒2  𝑚( �Ω) ≃ 𝜒2  ∗ ( �Ω) + log  �  det  �  F∗( �Ω)  ��  + const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='.  (8)  In what follows, we will label the two contributions in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (8), 𝜒2  ∗ and log det F∗, as the profile and Laplace terms  respectively:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The profile term is related to the “profile likelihood”  (Cole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2013), defined as  𝑝prof( �Ω|d) ∝ 𝑝( �Ω, �𝑛∗|d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  (9)  The profile likelihood is a tool commonly used in fre-  quentist parameter inference (Cousins 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The ad-  vantage of the profile likelihood is that its maximum is,  by definition, the global maximum of the joint distri-  bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Understanding this maximum as an estimator  for �Ω given the data, constraints on �Ω can be obtained  by calculating this maximum for random simulated re-  alisations of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Alternatively, if the distribution  is sufficiently close to a Gaussian, these constraints can  be simply obtained in terms of thresholds of the associ-  ated residual 𝜒2, Δ𝜒2( �Ω) = 𝜒2( �Ω) − 𝜒2  min (Feldman &  Cousins 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Additionally, the posterior profile likeli-  hood is, by construction, centered on the best-fit param-  eters, and is thus free from volume effects associated  with the choice of the nuisance parameters (Hamann  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Herold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Campeti & Komatsu  2022) (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The Laplace term, sometimes referred to as Occam’s  razor term, is associated with the quadratic contribution  to the Laplace approximation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (8)), and accounts,  to first order, for the volume in the space of nuisance  parameters �𝑛 that has been integrated over for fixed �Ω  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' the local curvature of the joint distribution at each  4  �Ω)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' As we will see in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='3, the Laplace term is  associated with volume effects, and is subdominant with  respect to the profile term for sufficiently constraining  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  The role of the profile and Laplace terms is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  The figure shows a bivariate distribution for two parameters  with an approximate degeneracy of the form 𝑛Ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='2 ∼ const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='.  The contour levels of the true distribution are shown in red  in the bottom left panel, while the black solid line shows the  best-fit value of 𝑛 as a function of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The middle panel shows  the exact 𝜒2 of the distribution as a function of 𝑛 for a fixed  Ω = Ω0 (for convenience, we chose Ω0 to be the maximum  of the distribution, also shown as a dotted line in the bottom  panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The black dashed line shows the quadratic Laplace  approximation to the red curve, with the position of the best-  fit 𝑛 (for Ω = Ω0) marked by the blue line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The top panel  shows the distribution along the Ω = Ω0 line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The exact distri-  bution is again shown in red, and the Laplace approximation  to it is shown in dashed black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The “profile likelihood” ap-  proximation, which fixes 𝑛 to its best-fit value, is shown in  blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Finally, the bottom right panel shows the distribution  marginalised over 𝑛, 𝑝(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The true marginal is shown in  red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The result of analytically marginalising over 𝑛 using the  Laplace approximation is shown in dashed black, and recov-  ers the true marginal almost exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Finally, the conditionally  maximised distribution accounting only for the profile term in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (8) is shown in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' As mentioned above, the profile-only  approximation recovers a distribution that is centered at the  best-fit value of Ω (marked by the dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The shift in  the peak of the true marginal observed is caused by volume  effects, which we discuss in more detail in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Two qualitative results should be borne in mind in what  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  First, the Laplace approximation provides a rea-  sonably accurate prediction for the marginal for sufficiently  well-behaved distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Secondly, keeping only the profile  term recovers a distribution that has approximately the same  width but is, by construction, centered on the maximum of the  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  It is worth noting that including the Laplace term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (8)  should come at virtually no additional computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Finding �𝑛∗( �Ω) requires solving for 𝜕�𝑛𝜒2 = 0, which can be  done efficiently using gradient descent methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Finding the  optimal step size in these algorithms often requires evaluating  the Hessian of the function being minimised, and therefore the  matrix F∗ entering the Laplace term, is already a product of  the minimisation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' For instance, the iteration in the  case of the Newton-Raphson algorithm is given by  �𝑛∗,𝑖+1 = �𝑛∗,𝑖 −  �  [∇𝑛∇𝑇  𝑛 𝜒2]−1 · ∇𝑛𝜒2�  𝑖 ,  (10)  where ∇𝑛𝜒2 is the gradient of 𝜒2 with respect to �𝑛, and  ∇𝑛∇𝑇  𝑛 𝜒2 ≡ 2F is its Hessian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In the applications we  will explore here, when Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (5) cannot be solved analytically,  we will make use of a modified version of the Newton-Raphson  algorithm, which we describe in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Gaussian likelihoods  Let us now apply the method described in the previous  section to the case of Gaussian likelihoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In this case we  1 In the frequentist context, Barndorff-Nielsen (1983) introduced the for-  mula in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (8) under the name of “modified profile likelihood”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  assume that the posterior distribution takes the form:  − 2 log 𝑝( �Ω, �𝑛|d) = (d − t)𝑇 C−1(d − t) + 𝜒2  𝑝,Ω( �Ω) + 𝜒2  𝑝,𝑛(�𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  (11)  Here t( �Ω, �𝑛) is the theory vector, which depends on the model  parameters, C is the covariance matrix of the data, which we  assume to be model-independent, and 𝜒2  𝑝,Ω and 𝜒2  𝑝,𝑛 are the  parameter priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Although the methodology described below  is straightforward to generalise to the case of arbitrary priors,  for simplicity we will assume that the nuisance parameters  have Gaussian priors, and therefore  𝜒2  𝑝,𝑛(�𝑛) = (�𝑛 − �𝑛𝑝)𝑇 C−1  𝑛 (�𝑛 − �𝑛𝑝),  (12)  where C𝑛 is the prior covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  In order to find �𝑛∗ and F , we need the first and second  derivatives of the 𝜒2 with respect to �𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In this case, these are  given by:  𝜕𝜒2  𝜕𝑛𝑖  = −2𝜕𝑖t𝑇 C−1(d − t) + 2  ∑︁  𝑗  �  C−1  𝑛  �  𝑖 𝑗 (𝑛 𝑗 − 𝑛𝑝, 𝑗), (13)  F𝑖 𝑗 = 𝐹𝑖 𝑗 + ΔF𝑖 𝑗,  (14)  where we have used the shorthand 𝜕𝑖 ≡ 𝜕/𝜕𝑛𝑖, and we have  defined  𝐹𝑖 𝑗 ≡ 𝜕𝑖t𝑇 C−1 𝜕𝑗t +  �  C−1  𝑛  �  𝑖 𝑗 ,  (15)  ΔF𝑖 𝑗 ≡ 𝜕𝑖𝜕𝑗t𝑇 C−1(t − d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  (16)  On the one hand, the first contribution to F , 𝐹, has three  interesting properties:  It is positive-definite, and therefore invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  It is independent of the data d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  It coincides with the Fisher matrix of the Gaussian like-  lihood of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (11) with respect to the nuisance parame-  ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  On the other hand, when evaluated on the hypersurface �𝑛 =  �𝑛∗( �Ω), t is close to d, and therefore the contribution from ΔF  is usually smaller than 𝐹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' These properties will be important  when discussing volume effects in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' For now,  we will use the positive-definiteness of 𝐹 to define a modified  Newton-Raphson iteration, by swapping the Hessian matrix  2F with 2𝐹:  �𝑛∗,𝑖+1 = �𝑛∗,𝑖 −  �1  2𝐹−1 ∇𝑛𝜒2  �  𝑖  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  (17)  This results in the Gauss-Newton method, which improves  on the Newton-Raphson iteration in two aspects: first, 𝐹 is  invertible and likely more stable than the Hessian matrix,  which need not be positive-definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Secondly, 𝐹 does not  require calculating second derivatives of the theory vector  (although this is not a computationally challenging problem  for the cases considered here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Note that replacing Hes-  sian matrix with Fisher matrix in Newton-Raphson iteration  often appears in various problem, most notably in optimal  quadratic estimators (Tegmark 1997), see Madhavacheril et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  (2015) for further discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Other approaches, such as the  Levenberg-Marquardt algorithm (Levenberg 1944;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Marquardt  1963), build on this method by improving the numerical sta-  bility of 𝐹−1 through regularisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The problems addressed  here do not require us to resort to these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Volume effects and priors  Consider now the case of a data vector with a Gaussian like-  lihood and poor constraining power (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' noise-dominated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  In this limit, we would naïvely expect the marginalised poste-  rior distribution to return largely unconstrained �Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' To verify if  this is the case, consider first the profile contribution in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (8),  averaged over realisations of the data:  ⟨𝜒2  ∗⟩( �Ω) = Tr  ��  (d − t( �Ω, �𝑛∗))(d − t( �Ω, �𝑛∗))𝑇 �  C−1�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (18)  If the data are noise-dominated, the fluctuations in d−t( �Ω, �𝑛∗)  are dominated by noise, rather than the changes in �Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In that  case ⟨(d − t)𝑇 (d − t)⟩ ≃ C, and therefore  ⟨𝜒2  ∗⟩(Ω) ≃ 𝑁𝑑,  (19)  where 𝑁𝑑 is the number of data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Thus, as we expected,  this contribution tends to a constant that does not favour any  region of parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Consider now the Laplace contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  As we argued,  the contribution ΔF∗ is normally small compared to 𝐹∗ (both  evaluated at �𝑛∗), and therefore  log det F∗ = log det(𝐹∗ + ΔF∗) ≃ log det 𝐹∗  (20)  which, as we discussed before, is independent of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  The Laplace contribution to the marginal distribution is thus  a parameter-dependent function that does not depend on the  data, and which would favour particular regions of parameter  space even in the absence of data!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  This is an example of a “volume effect”: the process of  marginalisation favours regions of parameter space that cover  a larger volume of the probability density in the direction of in-  tegration, causing a shift in the maximum of the marginalised  distribution with respect to the maximum of the full distribu-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Volume effects depend on the definition of the nuisance  parameters and, if informative, the associated priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' A classi-  cal example is trying to fit noisy data to a power-law model of  the form 𝑛 𝑥Ω, where 𝑥 is an independent variable taking values  𝑥 > 1, and (𝑛, Ω) are free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' If the data are noise-  dominated, scattering around zero, a very negative value of the  power law index Ω is able to obtain a reasonable fit to the data  for a wider range of amplitudes 𝑛, and thus the marginalised  distribution will favour values of Ω that are significantly dif-  ferent from the best-fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  This often happens for non-linear  parameters like Ω when marginalising over amplitude-like pa-  rameters such as 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The most pernicious aspect of volume  effects is that their impact on the marginalised distribution  depends on the specific parametrisation used to define the nui-  sance parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In the example above, redefining the model  to 𝑛 (𝑥/𝑥0)Ω, where 𝑥0 is a fixed quantity, larger than any value  of 𝑥, would result in marginalised posteriors that favour large  and positive values of Ω for noisy data (instead of negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  A common choice to eliminate the dependence on model  parametrisation, and thus to partially mitigate the impact of  volume effects is to use the well-known Jeffreys prior:  𝑝J( �𝜃) =  √  det 𝐹,  (21)  where 𝐹 is the Fisher information matrix, which we introduced  in the previous section (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (15) for Gaussian distributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Comparing this with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (8), we can see that the inclusion of a  Jeffreys prior for the nuisance parameters has the effect of par-  tially cancelling the contribution from the Laplace term (since,  as we argued, ΔF is generally smaller than 𝐹2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' This may not  be entirely surprising since, as we saw, the Laplace term is  in essence the type of volume effect that the Jeffreys prior is  meant to address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' However, this detour leads us to an interest-  ing result: for Gaussian data with negligible parameter depen-  dence in the covariance matrix, the profile likelihood will in  general be a reasonable approximation to the true marginalised  posterior in the presence of a Jeffreys prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In this sense, max-  imisation and marginalisation are approximately equivalent as  long as we avoid volume effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Linear parameters  Let us now consider the case of linear parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  consider a Gaussian likelihood in the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (11) where  all parameters live in the theory prediction, which has the form  t = t0 + T�𝑛,  (22)  where t0 and T are a vector and a matrix independent of �𝑛, but  potentially dependent on �Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' For simplicity, we will assume  that the prior on �𝑛 is centered at zero (�𝑛𝑝 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' This can always  be achieved by simply redefining �𝑛′ ≡ �𝑛 − �𝑛𝑝, and adding the  contribution T�𝑛𝑝 to t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  The case of linear parameters is particularly interesting,  because the 𝜒2 is quadratic in �𝑛 by construction, and the  Laplace approximation is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Since the second derivatives  of the theory vector are zero, ΔF = 0, and the Fisher matrix  is independent of �𝑛 and given by  F = 𝐹 = T𝑇 C−1T + C−1  𝑛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  (23)  Furthermore, the best-fit parameters can be found analytically:  �𝑛∗ = 𝐹−1 T𝑇 C−1r,  (24)  where r ≡ d − t0 is the data rescaled by the �𝑛-independent  component of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Using �𝑛∗ to compute the 𝜒2, and  using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (23), we obtain the marginalised 𝜒2  𝑚 of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (8) which,  as we said, is exact in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  The first thing worth noting is that, if the matrix T is indepen-  dent of �Ω, then 𝐹 is constant, and so is the Laplace term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Up  to an irrelevant overall constant, the marginalised 𝜒2 is then  equivalent to 𝜒2  ∗, obtained by substituting the best-fit value  of �𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Thus, the approximate relation between marginalisation  and maximisation we outlined in the previous section becomes  an equivalence when the data is Gaussian with a linear model  in �𝑛 since, in this case, there are no volume effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' All volume  effects resulting from a dependence of T on �Ω are otherwise  incorporated exactly in the Laplace term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' If the priors on �𝑛  are sufficiently tight, the second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (23) dominates,  and these volume effects become negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Let us now focus on the profile term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Substituting �𝑛∗ from  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (24), we obtain  𝜒2  ∗ = (Wr)𝑇 C−1(Wr) + r𝑇 C−1T𝐹−1C−1  𝑛 𝐹−1T𝑇 C−1r,  (25)  where the second term comes from the prior on �𝑛, and we have  defined the matrix  W ≡ I − T𝐹−1T𝑇 C−1,  (26)  with I the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  2 In addition to this, whereas 𝐹 is evaluated along �𝑛∗ in the Laplace  approximation, the Jeffreys prior is evaluated at every point in parameter  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  6  First, consider the limit of no external prior (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' C−1  𝑛 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  In this case, we can ignore the second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (25), and  the Fisher matrix is 𝐹 = T𝑇 C−1T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' We can then see that the  matrix W projects r onto the subspace that is orthogonal to all  the columns of T (with orthogonality defined using the inverse  covariance of the data C−1 as a dot product).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Marginalising  over linear parameters is therefore equivalent in this limit to  deprojecting all modes of the data that live in the subspace  spanned by the columns of T (Rybicki & Press 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Secondly, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (25) can be simplified significantly into  𝜒2  ∗ = r𝑇 ˜C−1r,  (27)  where ˜C is a modified covariance given by  ˜C = C + TC𝑛T𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  (28)  To obtain this beautifully simple result, one only needs to ex-  pand first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (25), simplify the result, and make use of  the Woodbury matrix identity (Woodbury 1950).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' It is worth  stressing again that this result is an exact expression for both  the marginal posterior (using a Jeffreys prior) and the con-  ditionally maximised posterior, as explained in the previous  section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Maximising and marginalising over �𝑛 therefore result  in the same Gaussian likelihood with the theory vector evalu-  ated at �𝑛 = 0 (or at its prior mean if non-zero), and a modified  covariance ˜C, obtained by simply assigning additional vari-  ance in quadrature to the modes of the data that align with the  columns of T (with this extra variance given by the associated  �𝑛 parameter priors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Note that while this approach is algorithmically the fastest, it  does not produce a best-fit value of a given nuisance parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  This is occasionally useful even for nuisance parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' A  canonical example is the shot noise contribution to the galaxy  power spectrum, the value of which can tell us about the halos  in which the given tracer galaxies reside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  To summarise: in the case of Gaussian data, negligible pa-  rameter dependence of the covariance matrix, and a theory  model that is linear in the nuisance parameters, the Laplace  approximation is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In this case, there is a mathematical  equivalence between marginalisation (using a Jeffreys prior),  𝜒2 minimisation, deprojection, and simply adding in quadra-  ture the prior uncertainty on the marginalised parameters at  the data level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' If the modes associated with �𝑛 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' the columns  of T) depend on the other parameters of the model, the associ-  ated volume effects are captured exactly by the Laplace term,  which is simply given by the log-determinant of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Im-  portantly, in this scenario, volume effects become negligible  if the constraints on nuisance parameters are either dominated  by data, or by the priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' COSMOLOGY FROM TOMOGRAPHIC  LARGE-SCALE STRUCTURE DATA  To explore the validity of the Laplace approximation in-  troduced in the previous section to marginalise over nuisance  parameters in the context of cosmology, we will study the case  of 3×2pt analyses, combining photometric galaxy clustering  and weak lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Galaxy clustering and weak lensing  Photometric redshift surveys make use of two main cosmo-  logical probes: cosmic shear (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' the distortion in the shape  of galaxies caused by weak gravitational lensing) and galaxy  clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In both cases, the data are typically split into pho-  tometric redshift bins, and the data vector is constructed by  combining various angular auto- and cross-correlations be-  tween pairs of such bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The galaxy overdensity 𝛿𝛼  𝑔 ( ˆn) and  the weak lensing shear 𝛾𝛼  𝐺( ˆn) for galaxies in redshift bin 𝛼  can be related to the three-dimensional fluctuations in the  galaxy number density Δ𝑔(x) and the matter density Δ𝑚(x)  via (Bartelmann & Schneider 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Krause et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2017)  𝛿𝛼  𝑔 ( ˆn) =  ∫  𝜒𝐻  0  𝑑𝜒 𝑞𝛼  𝑔 (𝜒) Δ𝑔(𝜒(𝑧) ˆn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 𝑧),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  𝛾𝛼  𝐺( ˆn) =  ∫  𝜒𝐻  0  𝑑𝜒 𝑞𝛼  𝐺(𝜒)  �  −𝜒−2ð2∇−2Δ𝑚(𝜒ˆn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 𝑧)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  (29)  where ˆn is the sky direction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 𝜒 is the comoving radial distance  at redshift 𝑧,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 𝜒𝐻 is the distance to the horizon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 𝑞𝛼  𝑔 and 𝑞𝛼  𝛾  are the radial kernels for galaxy clustering and cosmic shear,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' and ð is the spin-raising differential operator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  acting on a spin-𝑠 quantity as:  ð 𝑠 𝑓 (𝜃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 𝜑) = −(sin 𝜃)𝑠  � 𝜕  𝜕𝜃 +  𝑖  sin 𝜃  𝜕  𝜕𝜑  �  (sin 𝜃)−𝑠 𝑠 𝑓  (30)  and turning it into a spin-(𝑠 + 1) quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  The radial kernels in both cases are given by  𝑞𝛼  𝑔 (𝜒) ≡ 𝐻(𝑧)  𝑐  𝑝𝛼(𝑧),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  𝑞𝛼  𝐺(𝜒) ≡ 3  2𝐻2  0Ω𝑚  𝜒  𝑎(𝜒)  ∫ ∞  𝑧(𝜒)  𝑑𝑧′𝑝𝛼(𝑧′) 𝜒(𝑧′) − 𝜒  𝜒(𝑧′)  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  (31)  where 𝑐 is the speed of light,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 𝐻(𝑧) is the Hubble expansion  rate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 𝐻0 ≡ 𝐻(𝑧 = 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Ω𝑚 is the matter density parameter  today,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' and 𝑝𝛼(𝑧) is the redshift distribution in bin 𝛼,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Cosmic shear observations are sensitive not only to the weak  lensing distortion of galaxy shapes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' but also to the intrinsic  alignments (IAs) in the orientation of galaxies due to local  interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The total observed cosmic shear signal is thus  𝛾𝛼 = 𝛾𝛼  𝐺 + 𝐴IA𝛾𝛼  𝐼 ,  (32)  where 𝛾𝛼  𝐺 is the lensing signal, given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (29), and 𝐴IA𝛾𝛼  𝐼 is  the intrinsic alignment component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Using the linear non-linear  alignment model, the latter is given by  𝛾𝛼  𝐼 ( ˆn) = −  ∫  𝜒𝐻  0  𝑑𝜒 𝑞𝛼  𝐼 (𝜒)[−𝜒−2ð2∇−2Δ𝑚(𝜒ˆn, 𝑧)],  (33)  and 𝐴IA is an unknown amplitude parameter describing the  strength of the local alignment signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The IA radial kernel is  the same as the galaxy clustering kernel  𝑞𝛼  𝐼 (𝜒) = 𝐻(𝑧)  𝑐  𝑝𝛼(𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  (34)  The relation between the galaxy overdensity Δ𝑔 and the mat-  ter overdensity Δ𝑚 is complex in detail (non-linear, non-local  and stochastic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Over mildly non-linear scales, one can how-  ever use a perturbative approach, relating Δ𝑔 with scalar com-  binations of the Hessian of the gravitational potential 𝜕𝑖𝜕𝑗Φ  (where Φ is normalised so that ∇2Φ ≡ Δ𝑚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Following Mc-  Donald & Roy (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Abidi & Baldauf (2018), here we will  expand this bias relation to second order, including non-local  contributions, so that:  Δ𝑔 = 𝑏1Δ𝑚+ 𝑏2  2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (Δ𝑚−⟨Δ2  𝑚⟩)+ 𝑏𝑠  2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (𝑠2−⟨𝑠2⟩)+𝑏∇∇2Δ𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (35)  Here 𝑠2 ≡ 𝑠𝑖 𝑗𝑠𝑖 𝑗 is the trace of the squared tidal tensor, where  𝑠𝑖 𝑗 ≡ 𝜕𝑖𝜕𝑗Φ − ∇2Φ/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The quantities 𝑏1, 𝑏2, 𝑏𝑠, and 𝑏∇ are  7  the so-called “linear”, “quadratic”, “tidal”, and “non-local”  bias parameters, which characterise the response of the galaxy  overdensity to the corresponding terms in perturbation theory,  including the impact of non-local effects on scales compara-  ble with the Lagrangian halo size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Within this formalism, and  assuming that the bias parameters are constant within each  redshift bin, the projected galaxy overdensity can then be ex-  pressed as a sum over the projected version of the different  operators in the previous equation:  𝛿𝛼  𝑔 ( ˆn) =  ∑︁  𝑘  𝑏𝛼,𝑘, 𝛿𝛼  𝑘 ( ˆn),  (36)  where 𝑘 runs over the set {1, 2, 𝑠, ∇}, corresponding to the  various operators, dependent only on the matter overdensity  and tidal tensor, that contribute to the total galaxy overdensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  𝑏𝛼,𝑘 is the value of each associated bias parameter in bin 𝛼 is  the value of each associated bias parameter in bin 𝛼, and  𝛿𝛼  𝑘 ( ˆn) ≡  ∫  𝜒𝐻  0  𝑑𝜒 𝑞𝛼  𝑔 (𝜒)Δ𝑘 ( ˆn),  (37)  with  Δ1 ≡ Δ𝑚, Δ2 ≡ Δ2  𝑚 − ⟨Δ2  𝑚⟩,  Δ𝑠 ≡ 𝑠2 − ⟨𝑠2⟩, Δ∇ ≡ ∇2Δ𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  (38)  For some of the results shown in Section 4 we will consider  only linear bias, setting 𝑏2 = 𝑏𝑠 = 𝑏∇ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Either of these  bias schemes is only valid on sufficiently large scales, and  therefore we will limit our analysis to multipoles ℓ < 𝑘max ¯𝜒,  where ¯𝜒 is the comoving distance to the mean redshift of  the galaxy tracer under analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The maximum wavenumber  used will be 𝑘max = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='15 Mpc−1 when using linear bias, and  𝑘max = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='3 Mpc−1 when using the perturbative approach above  (Pandey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Comparing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (32) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (36), we see that we can de-  scribe both tomographic galaxy clustering and cosmic shear  as a projected tracer 𝑢𝛼( ˆn) with the generic form  𝑢𝛼( ˆn) = 𝜖𝑢 𝑢𝛼  𝑀 ( ˆn) +  ∑︁  𝑘  𝑏𝑢  𝛼,𝑘 𝑢𝛼  𝑘 ( ˆn) ,  (39)  where 𝑏𝑢  𝛼,𝑘 are bias parameters (specifying the tracer type 𝑢),  𝑢𝛼  𝑀 and 𝑢𝛼  𝑘 are projected quantities that depend only on cosmo-  logical observables (matter overdensities, comoving distances  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=') and the radial kernels, and 𝜖𝑢 is a Boolean variable that is  either 1 if the tracer contains an unbiased contribution (as is the  case of cosmic shear), and 0 otherwise (as is the case of galaxy  clustering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The index 𝛼 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (39) runs over the redshift bins,  which allows for the general case of having redshift-dependent  bias functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' If this is not the case, 𝑏𝑢  𝛼,𝑘 ≡ 𝑏𝑢  𝑘 can optionally  be assumed to not vary across redshift bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  We can relate the power spectrum, 𝑃𝑋𝑌 (𝑘, 𝑧), of two three-  dimensional quantities 𝑋 and 𝑌 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=', Δ𝑘) to the angular cross-  power spectrum of their associated projected tracers in redshift  bins 𝛼 and 𝛽 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 𝛿𝛼  𝑘 ) via:  𝐶 (𝑋,𝛼),(𝑌 ,𝛽)  ℓ  =  ∫  𝑑𝜒  𝜒2 𝑞𝛼  𝑋 (𝜒) 𝑞𝛽  𝑌 (𝜒) 𝑃𝑋𝑌  �  𝑘 = ℓ + 1/2  𝜒  , 𝑧(𝜒)  �  ,  (40)  where we have assumed the Limber approximation (Limber  1953;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Afshordi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2004), appropriate for the wide radial ker-  nels considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Note that in principle the lensing  kernel should be multiplied by an ℓ-dependent prefactor  𝐺ℓ ≡  √︄  (ℓ + 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  (ℓ − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  1  (ℓ + 1/2)2  (41)  to account for the difference between angular and three-  dimensional derivatives in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (29) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 𝜒2ð2∇−2 � 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  To calculate the matter power spectrum we will use the  Halofit fitting function Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (2003) with revisions  from Takahashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (2012a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' To calculate the power spectra  between the different perturbative matter fields involved in the  bias expansion (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (38)) we will make use of Fast-PT 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The  procedure is described in McEwen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (2016) and we refer  readers to that paper for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  The cosmological analysis of tomographic weak lensing and  galaxy clustering two-point functions requires accounting for,  and propagating uncertainties in some of the ingredients of the  corresponding theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' This is usually done by  defining a model that describes the impact of the associated  effects, and marginalising over the nuisance parameters of that  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The nature of these parameters can be classified into  two main types:  Calibratable parameters: these are parameters on which  relatively tight priors can be placed using external data  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' they can be calibrated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' These are normally associ-  ated with observational effects, such as shape measure-  ment or photometric redshift errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Non-calibratable parameters: these are parameters for  which no reliable prior information exists, and which  must therefore be measured with our own data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' These  are normally associated with astrophysical uncertainties  specific to the sample under study, such as galaxy bias  or intrinsic alignment parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  The next two sections describe the strategies we will use to  marginalise over both types of nuisance parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Calibratable systematics: linearisation  In the presence of tight priors, which we will further assume  to be Gaussian, the nuisance parameters may not stray far from  their prior mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In that case, we can Taylor-expand the theory  prediction as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (22)  t( �Ω, �𝑛) = t0( �Ω) + T (�𝑛 − �𝑛𝑝),  (42)  where �𝑛𝑝 is the prior mean, and  t0 ≡ t( �Ω, �𝑛𝑝),  T ≡ 𝜕t  𝜕�𝑛  ����  �𝑛𝑝  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  (43)  Since now the theory is linear with respect to �𝑛 − �𝑛𝑝, we  can then follow the procedure in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='4 to analytically  marginalise over those parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' As we discussed, we sim-  ply modify the covariance of the data vector (in this case, a  collection of power spectra) as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (28), and then sample the  resulting Gaussian likelihood evaluating t at the prior mean of  �𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Two important things should be noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' First, in doing this,  we have neglected the parameter dependence of the Laplacian  term, given by  log  �  det  �  T𝑇 C−1T + C−1  𝑛  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  (44)  3 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='com/JoeMcEwen/FAST-PT  8  If the prior is sufficiently tight, this term is dominated by the  constant C−1  𝑛 contribution, so the approximation is reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Secondly, since the modified covariance matrix (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (28)) now  involves a term of the form TC𝑛T𝑇 , in principle it depends  on �Ω through T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Calculating the covariance at each point  in the likelihood may be computationally costly, depending  on the size of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Instead, we will simply evaluate T at the  best-fit value of �Ω, and ignore all parameter dependence of  the covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' It was shown in Kodwani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (2019) that  the parameter dependence of the covariance can generally be  neglected and, furthermore, for a sufficiently tight prior the  TC𝑛T𝑇 contribution should be subdominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Hadzhiyska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  (2020) also showed that the choice of fiducial �Ω does not affect  the final results as long as they are close to the center of the  posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Adopting these two approximations (in  addition to the Taylor expansion of t) is therefore well justified  and, as we will show in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='1, leads to accurate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  One of the most important calibratable systematics in pho-  tometric surveys comes from the uncertainties in the redshift  distribution of the source and lens samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' To account for  these, it is common to parametrise the potential deviations  from a fiducial redshift distribution, and to calibrate those  parameters through external data and simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In a sim-  plified model, errors in photo-𝑧’s cause shifts in the means and  changes in the width of the derived redshift distribution for a  population of galaxies (Bonnett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The resulting  model for the redshift distribution is  𝑝𝛼(𝑧) ∝ ˆ𝑝𝛼(𝑧𝛼  𝑐 + 𝑤𝛼  𝑧 (𝑧 − 𝑧𝛼  𝑐 ) + Δ𝑧𝛼),  (45)  where Δ𝑧𝛼 and 𝑤𝛼  𝑧 parametrise deviations in the mean and  width of the fiducial distribution ˆ𝑝𝛼, and 𝑧𝛼  𝑐 is the redshift at  which this fiducial distribution attains its maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' We will  label these the “shift” and “width” parameters respectively in  what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Note that the normalisation of the distribution  changes with both of these parameters, and it must be renor-  malised to unit area before using it to compute any power  spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  In the rest of this paper, we will add one shift and one  width parameter for each galaxy clustering bin, and one shift  parameter for each cosmic shear bin, fixing their width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' On the  one hand, mean shifts have a strong impact on the weak lensing  kernel, since it depends on a cumulative integral of the redshift  distribution, and may also affect galaxy clustering by changing  the growth factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' On the other hand, mild (∼ 10%) changes  to the width have a very strong impact on the amplitude of the  clustering auto-correlations, but leave the weak lensing kernel  almost unchanged (Ruiz-Zapatero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Including these free parameters amounts to burdening the  MCMC sampler with tens of extra parameters, which in-  evitably leads to a substantial slowdown of its convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Moreover, there is no guarantee that all the uncertainty in  the 𝑝(𝑧)s can be captured by these parameters, and several  alternative procedures have been developed to ensure this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Al-  though we will focus here on the shift-width parametrisation,  Hadzhiyska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (2020) and Ruiz-Zapatero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (2023) ex-  plore the most general case of treating the 𝑝(𝑧) as a step-wise  function, with the function values at each step treated as free  parameters, and show that the approximate marginalisation  described here leads to accurate results for both clustering  and shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' We will therefore focus here only on the simpler  shift-width parametrisation, to exemplify the performance of  the Laplace approximation in the case of calibratable nuisance  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  We stress that we can apply the same procedure to any other  parameter that appears to behave linearly in the theoretical  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  This is the case for any other nuisance parameter  with a sufficiently tight prior, and in fact this procedure is  routinely used for multiplicative bias parameters in cosmic  shear analyses (Hildebrandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The procedure is also  exact in the case of truly linear parameters, regardless of their  priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' A good example of this is the amplitude of shot noise or  stochastic contributions to galaxy clustering auto-correlations  (García-García et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Non-calibratable systematics: bias parameters  Most sources of astrophysical uncertainty cannot be well-  constrained from external data, and thus must be con-  strained at the same time as the cosmological parameters, and  marginalised over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In this case, the linearisation described in  the previous section is not appropriate, and we must resort to  numerical methods in order to obtain �𝑛∗ and the Laplace con-  tribution in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In the model introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='1,  these astrophysical uncertainties are described by the bias and  intrinsic alignment parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In this formalism, all tracers  can be expressed generically as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (39), where 𝑢𝛼  𝑀 and 𝑢𝛼  𝑘  are projected maps depending purely cosmological fields (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  depending only on the matter overdensity and the tidal field),  and all astrophysical uncertainties are incorporated in the 𝑏𝑢  𝛼,𝑘  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Although the bias/IA description used here covers a wide  range of state-of-the-art physical models used in current 3×2pt  analyses, it is mathematically exceptionally simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  From  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (39) we see that the cross-correlation between any two  such tracers (𝑢𝛼, 𝑤𝛽) is a simple quadratic function of the bias  parameters:  𝐶𝑢𝛼,𝑤 𝛽  ℓ  = 𝜖𝑢𝜖𝑤𝐶  𝑢𝛼  𝑀 ,𝑤 𝛽  𝑀  ℓ  +  ∑︁  𝑖  𝑏𝑢  𝛼,𝑖𝜖𝑤𝐶  𝑢𝛼  𝑖 ,𝑤 𝛽  𝑀  ℓ  +  ∑︁  𝑗  𝜖𝑢𝑏𝑤  𝛽, 𝑗𝐶  𝑢𝛼  𝑀 ,𝑤 𝛽  𝑗  ℓ  +  ∑︁  𝑖, 𝑗  𝑏𝑢  𝛼,𝑖𝑏𝑤  𝛽, 𝑗𝐶  𝑢𝛼  𝑖 ,𝑤 𝛽  𝑗  ℓ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  (46)  Here, 𝐶  𝑢𝛼  𝑀/𝑖,𝑤 𝛽  𝑀/𝑗  ℓ  are the power spectra between the cosmo-  logical projected fields 𝑢𝛼  𝑀/𝑖 and 𝑤𝛽  𝑀/𝑗, defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (39),  and the sums run over the associated bias terms as introduced  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Since these only involve radial projections of  purely cosmological quantities, they can be treated as tem-  plates that only depend on the cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Note  that, in principle, these templates also depend on the calibrat-  able nuisance parameters described in the previous section  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  through the modification in the radial kernels due to  𝑝(𝑧) uncertainties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' However, we assume that we have been  able to marginalise over these analytically as we described  above, and therefore they can be treated as fixed for all intents  and purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  The first derivative of the power spectrum with respect to  9  the bias parameters is thus a linear polynomial:  𝜕𝐶𝑢𝛼,𝑤 𝛽  ℓ  𝜕𝑏𝑟  𝛾,𝑘  = 𝛿𝐾  𝛼,𝛾𝛿𝐾  𝑢,𝑟  �  𝜖𝑤𝐶  𝑢𝛼  𝑘 ,𝑤 𝛽  𝑀  ℓ  +  ∑︁  𝑗  𝑏𝑤  𝛽, 𝑗𝐶  𝑢𝛼  𝑘 ,𝑤 𝛽  𝑗  ℓ  �  + 𝛿𝐾  𝛽,𝛾𝛿𝐾  𝑤,𝑟  �  𝜖𝑢𝐶  𝑢𝛼  𝑀 ,𝑤 𝛽  𝑘  ℓ  +  ∑︁  𝑖  𝑏𝑢  𝛼,𝑖𝐶  𝑢𝛼  𝑖 ,𝑤 𝛽  𝑘  ℓ  �  ,  (47)  where 𝛿𝐾 is the Kronecker delta, and 𝑏𝑟  𝛾,𝑘 denotes the 𝑘th  bias term of tracer type 𝑟 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=', galaxy overdensity or shear) in  redshift bin 𝛾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Finally, the Hessian is constant with respect to  the bias parameters  𝜕2𝐶𝑢𝛼,𝑤 𝛽  ℓ  𝜕𝑏𝑟  𝛾,𝑘𝜕𝑏𝑠𝜎,𝑚  =  (48)  �  𝛿𝐾  𝛾,𝛼𝛿𝐾  𝜎,𝛽𝛿𝐾  𝑟,𝑢𝛿𝐾  𝑠,𝑤 + 𝛿𝐾  𝛾,𝛽𝛿𝐾  𝜎,𝛼𝛿𝐾  𝑟,𝑤𝛿𝐾  𝑠,𝑢  �  𝐶  𝑟 𝛾  𝑘 ,𝑠𝜎  𝑚  ℓ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  These expressions are remarkably simple and fast to evalu-  ate, and thus computing the 𝜒2 and its derivatives (needed for  minimisation, and to calculate the Laplace contribution) can  be done extremely efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' As we will see, in practice we  find that finding the minimum of the 𝜒2 takes 𝑂(10 − 100)  Gauss-Newton iterations, each of which is orders of magnitude  faster than recomputing the power spectrum templates when  changing cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Computing the Laplace  approximation to the marginalised posterior at each sample  of the cosmological parameters is therefore virtually equiva-  lent to evaluating the joint posterior for new cosmological +  nuisance parameters if using brute-force marginalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Example Stage-III and Stage-IV datasets  To test the validity of the methodology described above in  practice, we will apply it to two different datasets:  Real data from the Dark Energy Survey Year-1 (DES-  Y1) data release (DES Collaboration 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  A simulated 3×2-point data vector mimicking the char-  acteristic of a Stage-IV survey such as LSST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  These datasets are representative of both the current and future  generation and thus span the plausible accuracy range for the  foreseable future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' We describe them next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The DES-Y1 data  We use the galaxy-galaxy, galaxy-shear, and shear-shear  power spectra and covariance matrix provided in García-  García et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (2021), constructed from the DES-Y1 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' DES  is a five-year photometric survey which has observed 5000  deg2 of the sky using five different filter bands (grizY) from  the 4m Blanco Telescope at the Cerro Tololo Inter-American  Observatory (CTIO), in Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' García-García et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (2021)  employed the publicly available key Y1KP catalogs4, covering  1786 deg2 before masking (DES Collaboration 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Drlica-  Wagner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  The galaxy clustering sample consists of luminous red  galaxies (LRGs) selected with the redMaGiC algorithm, di-  vided into 5 redshift bins, defined in Elvin-Poole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  4  https://des.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='ncsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='edu/releases/y1a1/  key-catalogs  In this analysis, we will make use of the fiducial redshift distri-  butions released by DES to model the angular power spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  We will also adopt the same galaxy weights to correct for sky  systematics (see Elvin-Poole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2018, for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  The shear sample is the official source sample used in the  DES-Y1 analysis (Zuntz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2018), including all cuts and  tomographic bin definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Galaxy shapes were determined  using the Metacalibration algorithm (Huff & Mandelbaum  2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Sheldon & Huff 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The sample is divided into four  tomographic bins, for which we use the official redshift distri-  butions provided with the Y1 release (Hoyle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' See  Nicola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (2021) for further details regarding the estimation  of the shear power spectra and covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Following the official DES-Y1 analysis, we use all cross-  correlations between different shear bins and between shear  and clustering bins, but only the auto-correlations between  clustering bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Synthetic Stage-IV data  We consider a futuristic, idealised data set that resembles the  characteristics of LSST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' It is important to test our method in the  low-noise regime, where the inferred posterior is even more  sensitive to redshift distribution uncertainties or, in general,  degeneracies between cosmological and nuisance parameters,  and where the final error budget is more dominated by these  effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  To define the clustering and shear samples we follow the  same procedure outlined in Nicola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The shear  sample is defined following the LSST Science Requirements  Document (The LSST Dark Energy Science Collaboration  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2018) (see Appendices D1 and D2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  We divide this  redshift distribution into 5 bins in photometric redshift space,  each containing the same number of sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' We assume a  Gaussian photometric redshift uncertainty with standard de-  viation 𝜎𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='05(1 + 𝑧), which thus defines the true-redshift  tails of the distribution in each tomographic bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The sample  has an overall angular number density of 27 gals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' arcmin−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  For galaxy clustering, we define a sample extending out to  𝑧 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='5 with a total density of 4 gals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' arcmin−2 (as would  be expected of an LRG-like sample for LSST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' This number  density and the associated redshift distribution were estimated  using measurements of the luminosity function for red galax-  ies as described in Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  The sample was  divided into 6 redshift bins equi-spaced in photometric red-  shift space, and assuming a photometric redshift uncertainty  of 𝜎𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='02(1 + 𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' To simplify the analysis, we assume a  constant linear galaxy bias 𝑏1 = 1, and set all higher-order bias  coefficients to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The results obtained in the next section  should be largely insensitive to this choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  For simplicity, we use a Gaussian covariance to describe the  uncertainties of the resulting data vector, calculated assuming  a sky fraction 𝑓sky = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The LSST data vector was generated  assuming a true cosmology with parameters  (Ω𝑚, Ω𝑏, ℎ, 𝑛𝑠, 𝜎8) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='96, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='8),  (49)  where Ω𝑚 and Ω𝑏 are the total matter and baryon fractions, ℎ is  the reduced Hubble parameter, 𝑛𝑠 is the scalar spectral index,  and 𝜎8 is the standard deviation of linear density perturbations  smoothed on spheres of radius 8 Mpc ℎ−1 at redshift 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Likelihood  To obtain constraints on cosmological parameters from the  real and synthetic data described in the previous section, we  10  Parameter priors  Parameter  Prior  Parameter  Prior  Cosmology  Redshift calibration  Ωm  𝑈 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='07, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='8)  Δ𝑧1s  N(Δ𝑧1𝑠,∗, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='016)  Ωb  𝑈 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='07)  Δ𝑧2s  N(Δ𝑧2𝑠,∗, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='013)  ℎ  𝑈 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='55, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='91)  Δ𝑧3s  N(Δ𝑧3𝑠,∗, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='011)  𝑛s  𝑈 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='87, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='07)  Δ𝑧4,5  s  N(Δ𝑧4,5  𝑠,∗, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='022)  𝜎8  𝑈 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='1)  Δ𝑧1g  N(Δ𝑧1𝑔,∗, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='007)  Bias parameters  Δ𝑧2g  N(Δ𝑧2𝑔,∗, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='007)  𝑏𝑖  1  N(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='5, 100)  Δ𝑧3g  N(Δ𝑧3𝑔,∗, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='006)  𝑏𝑖  2,𝑠,∇  N(0, 100)  Δ𝑧4,5,6  g  N(Δ𝑧4,5,6  𝑔,∗ , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='01)  𝐴IA,0  N(0, 100)  𝑤𝑖𝑧,g  N(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='00, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='08)  TABLE 1  Prior distributions for the nuisance parameters entering our  “3×2pt” analysis for each tracer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 𝑈 (𝑎, 𝑏) and N(𝜇, 𝜎) describe a  uniform distribution with boundaries (𝑎, 𝑏) and a Gaussian  distribution with mean 𝜇 and variance 𝜎, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The index 𝑖  in 𝑏𝑖g and 𝑚𝑖 runs over the different redshift bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Δ𝑧∗ denotes the  deviation from zero of the central/best-fit value of each redshift  uncertainty parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  will assume that the data vector (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' the clustering and shear  power spectra), follows a Gaussian likelihood as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (11),  with a parameter-independent covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The model will be  described by 5 cosmological parameters, listed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (49),  one or four linear parameters for each clustering redshift bins  (for linear and PT bias respectively), one intrinsic alignment  amplitude, one redshift shift parameter for each clustering and  cosmic shear bin, and one redshift distribution width parame-  ter for each clustering bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The priors used for all parameters  are provided in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The priors on cosmological param-  eters are roughly based on the choices made for the DES-Y1  analysis, except we sample over 𝜎8 instead of the scalar spec-  trum amplitude 𝐴𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The priors on the redshift shift parameters  are based on the calibration of the DES-Y1 data (Hoyle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  2018), and thus represent achievable calibration levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The  priors on the redshift width parameters are commensurate with  those used in the DES Year-3 analysis (DES Collaboration  2022) (the DES-Y1 analysis did not introduce width param-  eters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Finally, we place an uninformative Gaussian prior on  all the bias parameters, centered at zero and with a standard  deviation of 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The choice of using a very broad Gaussian  prior as opposed to simply a flat prior is intended to enforce  a smooth distribution as a function of these parameters, and  to potentially aid the Gauss-Newton iterator when minimising  the 𝜒2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  We employ the cobaya MCMC sampler (Torrado & Lewis  2019, 2021) with a convergence condition that the Gelman-  Rubin diagnostic, 𝑅, ought to satisfy 𝑅 − 1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' When  using the Laplace approximation, we minimise the 𝜒2 over  the nuisance parameters using a Gauss-Newton iterator, using  the analytical derivatives with respect to bias parameters as  described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='3, and modify the log-probability to  be sampled by cobaya to be that of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' We find that, in  order to reduce the number of steps taken by the Gauss-Newton  iterator, it is useful to determine a well educated global best-fit  for the full parameter space before taking any samples, and to  start the iterator from the corresponding best-fit value of the  nuisance parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Throughout, we made use of the fitting formula of Eisenstein  & Hu (1998) to calculate the linear matter power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' We  do this to speed up the calculations, and we have verified that  the results obtained on the DES-Y1 data are insensitive to this  choice compared to using a Boltzmann solver such as CLASS  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='80  S8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='4  Ωm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='4  Ωm  DES-Y1, p(z) marg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (x14 params.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=')  Brute-force  Analytical  Fixed p(z)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='— Contours comparing brute-force (silver) with analytic marginalisa-  tion over photo-𝑧 uncertainties (red;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Results are shown for  the DES-Y1 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' We find that the contours are virtually unchanged, demon-  strating the benefit of using an efficient analytic marginalisation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' We  also show for posterity the result of assuming negligible error on the photo-𝑧  distributions (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In reality, this assumption does not hold and potentially  leads to a bias in the inferred cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  (Blas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The non-linear matter power spectrum is  then computed using HALOFIT (Takahashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2012b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' RESULTS  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Calibratable nuisance parameters  We begin by focusing on calibratable systematics, for which  we will follow the procedure described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='2: we will  linearise the dependence of the theoretical prediction with re-  spect to these parameters, for which a relatively tight prior can  be obtained, and simply modify the covariance matrix as in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (28), with T evaluated at a fixed set of parameters (fix-  ing all cosmological parameters to the best-fit values found  by Planck (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2020) and all bias pa-  rameters to their best-fit values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' At this stage, we will thus  only consider the nuisance parameters describing the uncer-  tainty in the redshift distributions of the tracers under study,  described in the right column of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' All other parameters  (cosmological, bias, and intrinsic alignment parameters) will  for now be marginalised “brute-force” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' treating them as  free parameters in the MCMC chains).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' For simplicity, we will  consider only linear bias, using scales 𝑘 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='15 Mpc−1, as  described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  We first compare the performance of the method when ap-  plied to the 3×2pt analysis of DES-Y1 data (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2 and summarised in Table 2,  with the exact results shown as black contour lines, and the  results of the analytical marginalisation shown in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  We  find that using the analytic marginalisation technique not only  yields contours that are almost indistinguishable from those  obtained by the traditional approach, but also does so signif-  icantly faster (by a factor ∼ 10) with many fewer parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  The blue contours in the figure show the constraints found by  11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='795  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='805  S8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='7  h  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='98  ns  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='06  Ωb  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='31  Ωm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='31  Ωm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='06  Ωb  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='98  ns  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='7  h  LSST, p(z) marg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (x17 params.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=')  Brute-force  Analytical  Fixed p(z)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='— Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2, but results are shown for a futuristic data vector with minimal noise meant to mimic the expected precision of LSST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The redshift  distribution is less noisy compared with DES-Y1 and is split into 6 tomographic bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' We find that our analytic marginalisation recipe yields virtually equivalent  results to the standard method for marginalising over redshift uncertainties by sampling directly the shift and width redshift parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='79  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='81  S8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='28  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='32  Ωm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='33  Ωm  LSST, p(z) marg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (x17 params.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=', larger errors)  Brute-force  Analytical  Fixed p(z)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='— Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 3 except for the mean redshift distribution is the same  as that of DES-Y1, but the errors on the nuisance parameters are quadrupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  As before, we find that our method of analytic marginalisation works well  despite the larger errors on the 𝑝(𝑧) measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Thus, we assert that the  approximations in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='2 hold even in the conservative scenario of large  photometric uncertainties in futuristic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  fixing the nuisance parameters to their prior means, instead of  marginalising over them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In this case, we observe that the red-  shift distribution uncertainties cause only a mild broadening  of the marginalised contours, which is not very challenging  for the analytical approximate marginalisation to reproduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  In order to explore a more challenging scenario, we now  move on to the case of an LSST-like 3×2pt dataset, as defined  in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' We will assume the same prior uncertainties  used in the analysis of the DES-Y1 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' This will allow us to  quantify the validity of the analytical marginalisation approach  in a conservative scenario, in which, in spite of the much  higher sensitivity of Stage-IV data, the precision with which  we are able to calibrate redshift distributions has not improved  with respect to the performance achieved with current data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Furthermore, we will explore an additional case in which the  prior uncertainties are 4 times larger than the DES-Y1 ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  The reasoning behind this second test is two-fold: on one hand,  our marginalisation method is an approximation that works in  the regime where photometric uncertainties are linearisable  and testing when that assumption breaks is essential;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' on the  other hand, it is likely that, at the highest redshifts, and for  the faintest samples, the 𝑝(𝑧) uncertainties will be somewhat  larger for LSST than those of current surveys, especially at  high redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Hence, quadrupling the errors is an important  worst-case scenario to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  It is important to note that the results in this section are not  meant to be interpreted as forecasts on the constraining power  of LSST on cosmological parameters, but only as a quantifica-  tion of our ability to analytically marginalise over photometric  uncertainties when inferring the underlying cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  A  12  more thorough analysis would include a realistic treatment of  the LSST true redshift distribution and noise, and a more care-  ful treatment of galaxy bias and intrinsic alignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' As such,  the results presented here give us a conservative estimate of the  effect of analytic marginalisation on cosmology constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  We present the results of the first scenario (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=', 𝑝(𝑧) errors  matching DES) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The marginalised constraints on  all cosmological parameters are virtually unchanged when we  switch from the brute-force to the analytical marginalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  This latter method is therefore successful at recovering ac-  curate marginalised constraints on cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  The figure shows the constraints on all cosmological param-  eters, to highlight that the result extends even to the param-  eters that 3×2pt datasets are less sensitive to: Ω𝑏, 𝑛𝑠 and ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Also shown in blue are the constraints found assuming perfect  knowledge of the redshift distributions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' fixing all 𝑝(𝑧)  parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In this case, the uncertainties on the redshift dis-  tributions have a much larger effects than for the DES-Y1 data,  inflating the uncertainties on 𝑆8 by a factor ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Thus, even  though marginalising over the redshift distribution parameters  has an outsized effect on the final constraints, the analytical  approximation approach is able to capture it almost exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Similarly reassuring is our result of quadrupling the uncer-  tainty in the redshift nuisance parameters, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 4 in  the (Ω𝑚, 𝑆8) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In this case, the increased redshift distribu-  tion uncertainties broaden the final constraints on 𝑆8 by up to a  factor ∼10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In spite of this, we find that the analytic marginal-  isation method not only yields virtually the same constraints  on the cosmological parameters, but does so 3-10 times faster  than the traditional approach (see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' This implicitly  validates the approximation that a first-order expansion of the  theory data vector with respect to a change in redshift distribu-  tion is sufficient, even for prior uncertainties on 𝑝(𝑧) that are  substantially worse than those achieved by current datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Bias parameters: tight posteriors  Let us now focus on the bias parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In this case, the  absence of an informative prior prevents us from linearising  the dependence of the theory on these parameters, and we must  therefore calculate the profile and Laplace terms numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Moreover, since the dependence of the 𝜒2 on these param-  eters is not quadratic, marginalising over them may lead to  significant volume effects in the marginalised posterior for the  cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' We will however start by exploring  two situations in which the data is sensitive enough to measure  these bias parameters accurately, in which case, as we saw in  Section 2, the Laplace term and volume effects are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  In the first case, we make use of the DES-Y1 data, marginal-  ising only over a single linear galaxy bias parameter per clus-  tering bin, as well as an intrinsic alignment amplitude (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' a  total of 6 nuisance bias parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' This roughly coincides  with the analysis choices made for the official DES-Y1 anal-  ysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The results are shown in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The  exact marginalised constrains (solid black contours) are accu-  rately recovered by the Laplace approximation (red contours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  While the former are obtained by running an MCMC with 11  free parameters (5 cosmological, 6 nuisance 𝑝(𝑧) parameters  marginalised as described in the previous section), the latter in-  volve only a 5-dimensional parameter space, which is therefore  significantly simpler to explore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Concretely, the 5-parameter  chain converged 3 times faster than the 11-parameter chain  (see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The blue contours in the same figure shows  the constraints obtained after fixing the bias parameters to  the best-fit values found by DES (Elvin-Poole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Fixing the galaxy bias shifts the cosmological constraining  power from the cosmic shear data to the higher signal-to-noise  clustering data, thus significantly reducing the uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Note that, although the red contours show the result of the full  Laplace approximation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' profile + Laplace contributions),  the Laplace contribution is negligible, and the profile term is  enough to recover the marginalised posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  In order to explore the performance of the method with a  significantly larger number of nuisance parameters, while still  remaining in the regime where these parameters can be well  constrained by the data, we now move to the LSST-like syn-  thetic dataset, making use of the second-order perturbative  expansion of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (35) to describe galaxy bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In this case,  we include 4 free bias parameter in each of the 6 clustering  redshift bins, adding up to a total of 25 nuisance parameters  when combined with the intrinsic alignment amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The  results are shown in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 5, again as black  lines for the brute-force marginalisation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' considering all  30 free parameters, and as red contours for the 5-parameter  Laplace approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' As before, we find that the approxi-  mation is able to recover the marginalised constraints almost  exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  The impact of volume effects is also heavily sup-  pressed, shifting the marginalised contours by less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='3𝜎  (not shown in the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Note that, in both of these cases, besides the bias param-  eters, we have also marginalised over a number of redshift  distribution uncertainty parameters (14 for DES-Y1, 17 for  LSST), thus reducing the model dimensionality from 37 and  24 for LSST and DES-Y1, respectively, to only 5 cosmological  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Obtaining converged MCMC chains for these 5  cosmological parameters takes approximately 2 hours for the  LSST-like dataset, a factor ∼ 6 faster than the chains with only  the 𝑝(𝑧) parameters analytically marginalised over, and a fac-  tor ∼ 30 times faster than the full brute-force chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The mag-  nitude of these speed gains, however impressive, must be taken  with a pinch of salt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The performance of MCMC sampling  may depend significantly upon the design of the likelihood  code, and whether it allows the sampler to decompose the  space between “fast” and “slow” parameters, over-sampling  the former, and making use of “dragging” techniques (Lewis  2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Neal 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The fast-slow split allows one to effectively  marginalise over the fast subspace, and becomes particularly  powerful in the presence of a large number of nuisance pa-  rameters on which the likelihood has a computationally sim-  ple dependence (as is the case of the bias parameters in our  model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' To provide a fairer assessment of the computational  gains obtained using the Laplace approximation, we wrote  an optimised version of the 3×2-point likelihood that allows  cobaya to exploit the fast nature of the bias parameters as ef-  ficiently as possible (assuming all 𝑝(𝑧) parameters) are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  In this case, the speed-up factor for the cases explored here  ranged between ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='5 and ∼ 11, with the highest performance  improvement achieved for the LSST 2×2pt data discussed in  the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Ultimately the performance difference de-  pends on the design of the likelihood code, the complexity  in the parameter dependence of the likelihood, and the effi-  ciency of the 𝜒2 minimisation method used to calculate �𝑛∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  On this latter point, we find that the Gauss-Newton method  used here typically achieves convergence after only 10-20 it-  erations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Thus finding the best-fit bias parameters is signifi-  cantly faster than calculating the cosmology-dependent power  spectrum templates of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='80  S8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='4  Ωm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='4  Ωm  DES-Y1, linear bias marg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (x6 params.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=')  Brute-force  Analytical  Fixed bias  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='795  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='805  S8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
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+page_content='31  Ωm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='31  Ωm  LSST, PT bias marg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (x25 params.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=')  Brute-force  Analytical  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='— On the left, same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2, but here we marginalise over the bias parameters using the Laplace approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' This reduces the parameter space to  only the cosmological parameters without degrading the accuracy of the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' On the right, as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 3, but here we marginalise over the perturbation theory  bias parameters via the Laplace approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The agreement with the full numerical marginalisation (black) is almost perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='78  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='82  S8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='28  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
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+page_content='32  Ωm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='28  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
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+page_content='32  Ωm  LSST 2x2pt, PT bias marg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (x25 params.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=')  Brute-force  Brute-force (Jeffreys)  Analytical  Analytical (profile-only)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
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+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='4  Ωm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='4  Ωm  DES-Y1, PT bias marg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (x21 params.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=')  Brute-force  Brute-force (Jeffreys)  Analytical  Analytical (profile-only)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='— Demonstration of volume effects in the case of using less constraining data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In order to recover the best-fit parameter values (denoted with a star), when  applying brute-force marginalisation, one needs to make use of Jeffreys prior (compare solid with dashed black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Similarly, when analytically marginalising  over the PT biases, one needs to remove the Laplace term to avoid biasing the constraints (compare blue and red contours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The plot on the left shows the  constraints for a 2x2pt analysis with a LSST-like dataset, while on the right we show constraints using the DES-Y1 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In the latter case, the volume effects  become more pronounced, as the data has less constraining power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  14  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Bias parameters: loose posteriors and volume effects  In the cases explored in the previous sections, the data was  able to constrain the nuisance parameters sufficiently well, and  all volume effects associated with the choice of parametrisa-  tion were subdominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Let us now study the ability of the  Laplace approximation to capture these volume effects when  they become more prominent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  First, let us consider the case of a “2×2pt” combination  of the LSST data, consisting only of the galaxy-galaxy and  galaxy-shear power spectra, excluding the shear-shear auto-  correlation, and analysed under the perturbative bias expan-  sion of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The logic for focusing on this scenario is that,  in this case, the model relies completely on galaxy clustering  to constrain cosmological parameters, and is therefore more  sensitive to the complexity of the galaxy bias parametrisation  (which dominates the dimensionality of the nuisance parame-  ter space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The resulting constraints on Ω𝑚 and 𝑆8 are shown  in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The solid and dashed black contours  show the results using brute-force marginalisation, assuming  no prior and a Jeffreys prior respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The red contours  then show the same constraints using the Laplace approxima-  tion (including both profile and Laplace terms), while the blue  contour shows the result of including only the profile like-  lihood contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The position of the best-fit parameters  is indicated by the orange star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' We find that the profile-only  approximation is able to recover well the exact marginalised  constraints assuming a Jeffreys prior, and the full Laplace ap-  proximation recovers the constraints found in the absence of  this prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In this case we see a clear ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='8𝜎 shift in the cosmo-  logical constraints due to volume effects associated with the  large number of bias parameters, although the Laplace term is  able to capture this with high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  As a second example, we explore the possibility of using  the second-order perturbative bias expansion of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (35) to  analyse the DES-Y1 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The lower sensitivity of this data  compared to LSST should reduce its ability to constrain the  higher-order bias parameters, and increase the impact of vol-  ume effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The results are shown in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 6,  using the same color scheme as the last figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In this case we  can see that volume effects are a lot more significant, and can  shift the confidence contours of the cosmological parameters  by almost 3𝜎 with respect to their global best-fit value, marked  by the orange star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' We can also see that, in this more extreme  case, the Laplace approximation starts to fail, manifesting mild  shifts of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='3 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='5𝜎 with respect to the exact marginalised  constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' This provides us with a useful “rule of thumb”  to determine the reliability of the Laplace approximation: if  a significant (> 1𝜎) shift is found between the marginalised  contours obtained using the profile likelihood and those ob-  tained accounting also for the Laplace term, the approxima-  tion may start to fail, and a brute-force marginalisation over  the nuisance parameters using a Jeffreys prior is needed to  obtain accurate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' This test can be done without running  the MCMC twice: one can run MCMC chain using Laplace  approximation while also saving the profile likelihood values  at each MCMC step and employing importance sampling to  derive the profile likelihood posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Nevertheless, even in  these cases, we find that the Laplace approximation is able to  recover the region of parameter space preferred by the data  reasonably well, and can thus be used as a fast way to charac-  terise this region that can then be refined (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' via importance  sampling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  In an extreme case, such as the one we just discussed, it  is worth asking which of the constraints in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  6 are the  “correct” ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The presence of such large volume effects,  combined with the arbitrariness of using flat priors for the  nuisance parameters, would imply that the use of a Jeffreys  prior should produce the most correct results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' However, as  we noted in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='3, the Jeffreys prior is not guaranteed  to cancel out all volume effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' One might thus argue that  the profile likelihood, since it is by construction centered on  the best-fit parameters, could provide an equally reasonable  representation of the favoured confidence region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' One might,  however, view this choice critically due to the excessive gain  in precision over the marginal likelihood that is expected in  the case of many poorly constrained nuisance parameters (see  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Finally, we could adopt a completely frequentist  approach, determining the confidence intervals of all param-  eters by extracting the maximum-likelihood estimate for the  data and for a suite of simulated datasets compatible with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' CONCLUSIONS  Forecasts of the next decade in cosmology predict that mean-  ingful constraints on fundamental unknowns such as the mass  of neutrinos and the nature of dark matter and dark energy  will come from multi-scale, multi-tracer efforts, encompass-  ing a wide range of probes and redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' It is for this reason  that the efficient analysis of joint data sets combining low-  and high-redshift probes in an accurate manner is of great  importance to the field of large-scale structure analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' How-  ever, this comes at the cost of adding a colossal number of  nuisance parameters characterising the observational and the-  oretical systematic uncertainties of all probes involved, which  can noticeably slow down the sampling of the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  In galaxy clustering and weak lensing joint studies, the most  significant obstacles to overcome are the accurate modeling of  the redshift distribution, the galaxy bias relation, and intrin-  sic alignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Not doing so correctly can bias the inferred  cosmological parameters, but also accounting for these effects  via more elaborate models is a major inhibitor of efficient  sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  In this paper, we introduce a formal approach for speeding up  the sampling process in the presence of a large number of nui-  sance parameters, and investigate the accuracy of our method  when applying it to photometric survey data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In particular, we  study the current DES-Y1 data set as well as a synthetic data  vector from an LSST-like survey to validate whether the fast  analytic method proposed in this work is capable of reproduc-  ing the posterior contours and constraints one arrives at when  adopting the traditional method of diligently varying tens of  nuisance parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  In Section 2, we describe the general methodology behind  analytically marginalising over any nuisance parameter by ap-  proximating the marginal distribution, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (8), as consisting  of a “profile” term, centered on the best-fit nuisance param-  eters, and a “Laplace” term, associated with the quadratic  contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' We then consider the special case of a Gaussian  likelihood, showing that the Laplace term can be associated  with volume effects, which the profile term alone is free from,  being approximately equivalent to imposing a Jeffreys prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  This argument becomes exact when studying the case of nui-  sance parameters that contribute linearly to the theory vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  In Section 3, we introduce the cosmological analysis of pho-  tometric survey analysis and the specific problems associated  with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  In particular, we introduce the relevant summary  statistics utilised when performing “3×2pt” analysis and the  two dominant sets of nuisance parameters associated with that  15  analysis: redshift distribution (𝑝(𝑧)) calibration parameters  and bias/IA model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' We then provide details of  how our model enables a fast marginalisation over these and  how it can be applied to current and future data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  We summarise our results in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The first half of that  section deals with the 𝑝(𝑧) parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Since these parame-  ters have external priors, the dependence of the theory vector  on them can be linearised around the center of that prior, and  the Laplace term can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Their impact can then be  incorporated in a pre-sampling step by simply modifying the  covariance matrix (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' (28)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' We showed (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2 and  3), that this method is able to obtain constraints on cosmo-  logical parameters that are indistinguishable from those found  via brute-force marginalisation, while performing 5-10 times  better in terms of computational time, even when considering  calibration priors that are significantly worse than those that  can be achieved with current data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  We next focused on the marginalisation over bias parame-  ters, using the full Laplace approximation, finding the maxi-  mum of the conditional posterior distribution on the fly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' We  find that, when the data is able to place significant constraints  on all model parameters, the Laplace approximation is an ex-  cellent representation of the marginal distribution, and that the  Laplace term becomes subdominant with respect to the profile  likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In the presence of less constraining data, volume  effects associated with the choice of nuisance parametrisation  become relevant and, when mild, can be accurately captured  by the Laplace term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In these cases, we also find that the profile  likelihood is almost indistinguishable from the marginal distri-  bution when adopting a Jeffreys prior, cancelling these volume  effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' When volume effects become more relevant, signifi-  cantly shifting the parameter region favoured by the marginal  distribution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' by more than ∼ 1𝜎), the Laplace approxi-  mation begins to fail, although the position and extent of the  favoured region of parameter space are still qualitatively well  described by it (both with or without a Jeffreys prior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Thus,  even in this case, the method can be used to identify this region  and then refine it with a standard MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Through this paper we have assumed that the likelihood in  the nuisance parameters is unimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Obviously, for multi-  modal likelihoods with comparable posterior volumes in each  mode, the approximations we use will fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Since the likeli-  hood is in general quartic in bias parameters, the likelihood  can in principle have up to three local maxima in each bias  direction making multimodal likelihoods a possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' This  has been observed “in the wild” in fits to galaxy auto-power  spectrum using higher order bias parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=', Goldstein  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2022), but typically disappears with sufficiently aggres-  sive scale cuts and when employing the full 3×2pt data vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  These problems can be detected by initialising the Newton-  Raphson optimiser at different starting points and noting exis-  tence of multiple maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  The elimination of nuisance parameters is a nontrivial task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Caution must be taken when departing from our assumption of  Gaussian data with a model-independent covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  While in this specific application, we find the profile likeli-  hood to give unbiased parameter constraints and the marginal  distribution to be biased, this does not hold in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Com-  mon tasks in data analysis (such as the estimation of the vari-  ance from Gaussian data) have an opposite outcome, with the  marginal distribution peaking at the unbiased parameter esti-  mate, while in other cases neither method seems to perform  optimally (Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' It is therefore crucial to con-  sider the role of volume effects in every specific likelihood  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  The reduction in the dimensionality of the parameter space  afforded by the Laplace approximation is accompanied by a  boost in the speed with which convergence in the associated  MCMC chains is achieved, and we find improvements by a  factor ∼ 2 − 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Achieving this performance improvement is  greatly aided by the simplicity with which the theory predic-  tion depends on the bias and IA nuisance parameters, which  allows us to write down its derivatives analytically (see Sec-  tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='3), and to evaluate them quickly (in 𝑂(10−3 s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' There  may be more sophisticated astrophysical models that do not  conform to this simple structure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' physically motivated  models for the redshift dependence of bias terms, halo-based  models, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' ), and for which computing derivatives analyti-  cally is impractical or unfeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' A more general application  of this method in these situations would thus require the use of  automatic differentiation techniques (Margossian 2018), able  to efficiently calculate these gradients regardless of how the  model depends on the nuisance parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The Laplace ap-  proximation is by far not the only application that benefits  from having efficient access to derivatives of the likelihood  with respect to some parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  More general sampling  methods including HMC (MacKay 2002) or variational in-  ference (Blei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' 2016) often require the use of likelihood  gradients, as do efficient minimisation methods, or the calcu-  lation of Jeffreys priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Additionally, although here we have  focused on the volume effects associated with the marginali-  sation over nuisance parameters, the non-linear way in which  all cosmological parameters enter the likelihood implies that  degeneracies between them can also give rise to biasing vol-  ume effects within the subspace of cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  A Jeffreys prior (or any other form of correction for volume  effects) should therefore be used routinely in cosmological pa-  rameter inference, although this is normally prevented by the  need to estimate derivatives efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' The development of  fully automatically-differentiable cosmological theory predic-  tions is therefore of paramount importance in the context of  current and future experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  In conclusion, our method produces satisfactory results in  virtually all of the explored regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Thanks to the huge gain  in sampling efficiency this approach offers, it can be used to  enable the joint analysis of multiple galaxy surveys, which is  typically hindered by the gigantic number of nuisance param-  eters that need to be sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' Fast marginalisation methods  such as the one proposed in this work can help us address some  of the inconsistencies in the present analysis of cosmological  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' For example, current constraints find that both the ampli-  tude of clustering (𝑆8) and the energy density of matter (Ω𝑚)  inferred from the low-redshift analysis of galaxy and weak  lensing surveys are systemically lower than those obtained  from high-redshift probes such as the CMB (for a review, see  Perivolaropoulos & Skara 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' In the near term, we plan  to adopt the method proposed in this paper to speed up the  sampling process and analyze jointly the currently (publicly)  available photometric survey data in an effort to tackle some  of the obstacles standing in the way of gaining a fundamental  understanding of our Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  We would like to thank Deaglan Bartlett, Harry Desmond,  Simone Ferraro, Arrykrishna Mootoovaloo, and Martin White  for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' BH is supported by the Miller Institute  for Basic Research in Science and the Chamberlain Fellow-  16  ship and Lawrence Berkeley National Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' KW is funded by  a SISSA Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' SA is funded by a Kavli/IPMU  doctoral studentship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  DA is supported by the Science and  Technology Facilities Council through an Ernest Rutherford  Fellowship, grant reference ST/P004474.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' CGG is supported  by European Research Council Grant No: 693024 and the  Beecroft Trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' JRZ is supported by an STFC doctoral stu-  dentship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
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+page_content=' SUMMARY OF RESULTS  Table 2 lists the constraints on cosmological parameters,  and the time needed for the corresponding MCMC chains to  converge for the different cases explored in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='1, where  we marginalise over calibratable redshift distribution system-  atics using the linearisation technique described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
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+page_content='3, in which we make use of the full Laplace  approximation to marginalise over galaxy bias and intrinsic  alignment parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  This paper was built using the Open Journal of Astrophysics  LATEX template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
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+page_content='  17  Method  𝑆8  Ω𝑚  Ω𝑏  𝑛𝑠  ℎ  Converge time  DES 3×2pt: brute-force  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
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+page_content='017  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
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+page_content='042  –  –  –  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
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+page_content='780 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
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+page_content='029  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
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+page_content='7999 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='0021  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='3001 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='0057  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
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+page_content='3001 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='0057  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='0502 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
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+page_content='9597 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='0089  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
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+page_content='00 Hrs  LSST 3×2pt (4x): brute-force  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
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+page_content='0056  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
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+page_content='0501 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
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+page_content='9599 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='0097  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
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+page_content='19 Hrs  LSST 3×2pt (4x): analytic  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
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+page_content='0054  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
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+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='0503 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
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+page_content='9598 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='0099  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
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+page_content='035  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='36 Hrs  LSST 3×2pt: fixed 𝑧  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
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+page_content='0501 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='0036  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='9599 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='0068  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
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+page_content='020  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='023  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='39 Hrs  TABLE 2  Constraints on the cosmological parameters from our 3×2pt analysis applied to a current (DES-Y1) and future (LSST-like) data set, comparing  the performance of two marginalisation approaches: analytic and brute-force (described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' “Analytic” refers to the new method  proposed in this work, in which one accounts for photometric uncertainties prior to sampling the parameter space, whereas “brute-force”  refers to the standard method of introducing ∼10 “shift” and “width” redshift calibration parameters (see Table 1 and Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' “Fixed 𝑧”  considers the unrealistic scenario of completely ignoring photometric uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  Method  𝑆8  Ω𝑚  Ω𝑏  𝑛𝑠  ℎ  Converge time  DES-Y1 linear bias: brute-force  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='785 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='013  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
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+page_content='043  –  –  –  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='1 Hrs  DES-Y1 linear bias: analytical  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='785 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='012  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='302+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='030  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='041  –  –  –  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='5 Hrs  DES-Y1 linear bias: fixed 𝑏  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='7810 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='0090  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='2691 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='0094  –  –  –  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='1 Hrs  DES-Y1 PT bias: brute-force, no J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='780 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='012  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='347 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='031  –  –  –  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='5 Hrs  DES-Y1 PT bias: analytical, full Laplace  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='775 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='011  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='362 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='030  –  –  –  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='5 Hrs  DES-Y1 PT bias: brute-force, with J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
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+page_content='4 Hrs  TABLE 3  Constraints on the cosmological parameters from our 3×2pt analysis applied to a current (DES-Y1) and future (LSST-like) data set, comparing  the performance of two marginalisation approaches: analytic and brute-force (described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' “Analytic” refers to the new method  proposed in this work, in which one accounts for photometric uncertainties prior to sampling the parameter space, whereas “brute-force”  refers to the standard method of introducing ∼10 “shift” and “width” redshift calibration parameters (see Table 1 and Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
+page_content=' “Fixed 𝑧”  considers the unrealistic scenario of completely ignoring photometric uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFKT4oBgHgl3EQfvi5N/content/2301.11895v1.pdf'}
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+January 3, 2023 1:34
+ws-rv10x7-10x7
+Replica Symmetry Breaking & Far Beyond
+chapter1
+page 1
+Chapter 1
+Simulated annealing, optimization, searching for ground states
+Sergio Caracciolo†, Alexander K. Hartmann‡, Scott Kirkpatrick§, and Martin Weigel¶
+†Department of Physics, University of Milan and INFN Milan Section, Milan, Italy
+‡ Institut f¨ur Physik, Universit¨at Oldenburg, Oldenburg, Germany
+§School of Engineering and Computer Science, Hebrew University, Jerusalem, Israel
+¶Institut f¨ur Physik, Technische Universit¨at Chemnitz, Chemnitz, Germany
+The chapter starts with a historical summary of first attempts to optimize the spin
+glass Hamiltonian, comparing it to recent results on searching largest cliques in ran-
+dom graphs. Exact algorithms to find ground states in generic spin glass models are
+then explored in Section 1.2, while Section 1.3 is dedicated to the bidimensional case
+where polynomial algorithms exist and allow for the study of much larger systems.
+Finally Section 1.4 presents a summary of results for the assignment problem where
+the finite size corrections for the ground state can be studied in great detail.
+†sergio.caracciolo@mi.infn.it
+‡a.hartmann@uni-oldenburg.de
+§kirk@cs.huji.ac.il
+¶martin.weigel@physik.tu-chemnitz.de
+1
+arXiv:2301.00683v1  [cond-mat.dis-nn]  2 Jan 2023
+
+January 3, 2023 1:34
+ws-rv10x7-10x7
+Replica Symmetry Breaking & Far Beyond
+chapter1
+page 2
+2
+S. Caracciolo, A. K. Hartmann, S. Kirkpatrick, M. Weigel
+1.1. Introduction
+A theme which unites this chapter is the study by simulation of finite random systems in
+order to test the predictions and insights that arose as new kinds of order were defined
+and realized in spin glasses. The tools of this chapter have found use in many applied
+contexts. In physics typically phase spaces of physical systems have been considered.
+In applied mathematics, problems, either idealized or based on real-world data, are
+solved to optimize target functions or to satisfy constraints. Over the years is has been
+gradually accepted in mathematics and physics communities that these two realms are
+very similar to each other.
+Replicas, when introduced by Brout [1], were large patches of a perfectly ordered
+lattice. Edwards and Anderson (EA) [2] looked for order in the similarity of behavior
+across multiple identical copies of random spins and their interactions. In the Brout
+and EA work, the free energy was extracted as the linear term in an expansion of the
+partition function in powers of n, the number of replicas. EA defined order as a self-
+correlation of spin orientation in different replicas, since this could represent correlation
+over long times. Sherrington and Kirkpatrick(SK) [3], after simplifying the problem
+to an infinite ranged Ising system with i.i.d. random Gaussian interactions of strength
+1/
+√
+N in hopes of making it soluble, explicitly used the identity,
+ln Z = lim
+n→0
+Zn − 1
+n
+,
+(1.1)
+despite the obvious fact that the meaning of Zn anywhere except at integer values
+of n was unclear. A careful treatment of the extrapolation of the resulting equations
+for the free energy, F, showed an entropy at zero temperature which was negative,
+an impossibility in a discrete model, while other features such as a susceptibility cusp
+seemed plausible. The predicted ground state energy, E0 = (2/π)1/2 = −0.798..., could
+be tested in simulation.
+Fortunately, computers in 1975 had almost reached the point where they might be
+able to address questions about asymptotic values of quantities such as E0. Power-
+ful generally available computers such as IBM’s 370-168 and later 3033 could hold as
+much as 10 MB of data in random access memory, and process instructions at a few
+MIPS. (The first supercomputer, the Cray-1, in 1975 also had no more than about 10
+MB of RAM, but ran its instructions 4-6 times faster.)
+This amount of RAM per-
+mitted simulating a small number of samples with 500-1000 spins, so with a floating
+point coprocessor attached to their IBM mainframe to provide more Cray-like process-
+ing speeds, Kirkpatrick and Sherrington (KS) [4] in 1978 could report that E0 was in
+fact between −0.75 and −0.77, excluding the replica symmetric result. Improvements
+on that estimate have continued for the next 30+ years.
+The most recent estimate
+of the asymptotic result, using Parisi’s full RSB formulas and Pad´e approximants to
+extrapolate both upper and lower bounds to E0, yields the currently accepted value of
+−0.76321... [5]. The improvements in theory that have made this the accepted result
+required several decades, and are discussed elsewhere. Improving the experiments to
+test it also required some years for computing speeds to increase so that better statistics
+could emerge from simulations. Graphs with N up to 2000 sites have been studied [6],
+
+January 3, 2023 1:34
+ws-rv10x7-10x7
+Replica Symmetry Breaking & Far Beyond
+chapter1
+page 3
+Simulated annealing, optimization, searching for ground states
+3
+but understanding the size dependence of E0 remains a subject of research. Simulations
+agree with theory, and have grown more accurate as the bounds on the theoretical re-
+sult have also grown tighter. New methods of finding the ground state energies in the
+simulated model were required, intially and as the models got larger and the accuracy
+required grew narrower. These methods, such as simulated annealing, have taken on
+a life of their own, with many applications in complex systems outside of the simple
+models of statistical mechanics.
+Simulated annealing [7] is an obvious idea if you are statistical physicists, like KS
+when first studying their spin glass model, and also V. ˇCern`y [8] and K. Wilson [9].
+Kirkpatrick and ˇCern`y studied the Travelling Salesman as an easily described example.
+Wilson used annealing to optimize packing of parallel processor code generated in a
+compiler he wrote for his quantum chromodynamics simulations. All of us realized that
+allowing a Markov chain of states to evolve at a series of decreasing finite temperatures
+(annealing) would tend to reach larger and deeper minima of a complex function space
+with many local minima than simple gradient descent.
+At the time the preferred methods used gradient descent with many random restarts
+and tricks to jump to new starting points, applied only when a search gets stuck. The
+large number of possible local minima, mostly at higher energies, makes this ineffective.
+In addition, study of properties such as susceptibilities and specific heat (obtained from
+the fluctuations of a Boltzmann system) could guide the development of effective an-
+nealing schedules. Ranges of temperature at which large fluctuations were seen, possibly
+signalling a phase boundary, give a signal to slow the annealing process to allow large
+changes to occur.
+Kirkpatrick and IBM colleagues [7] were also at the time studying the problems
+presented by design automation tools used to place and route transistors and small VLSI
+circuits in IBM’s next-generation computers, and included simplified examples of both
+practical problems in their paper. These tools were successfully used and incorporated
+into IBM’s internal product development tool sets, and adopted in other industries.
+The IBM group learned important lessons from exposure to the engineering environ-
+ment. First, expressing complex constraints as energetic costs in a Hamiltonian frame-
+work provided valuable flexibility, and the annealing framework met the constraints
+simultaneously or in order of importance rather than one at a time. Second, they soon
+realized that finding the exact optimum in a problem like circuit design was not really
+the objective of an engineering team, who simply needed to find solutions that were
+good enough, soon enough, to ship as products. For that sort of objective, robustness
+of a methodology and the ability to incorporate many esoteric constraints was at least
+as important as its efficiency. Optimality of the solution obtained, if possible, was a
+bonus, but not the only objective.
+Simulated annealing has been accepted as another tool for at least approximately
+solving messy but useful and important problems.
+Probably its greater impact has
+been to stimulate the incorporation of statistical physics frameworks as an active part
+of applied mathematics. One recent review by a respected practitioner of constraint
+satisfaction [10] considers the 1990s to be the era of phase transitions, with the 2000s
+introducing more refined techniques such as survey propagation [11] and cavity con-
+
+January 3, 2023 1:34
+ws-rv10x7-10x7
+Replica Symmetry Breaking & Far Beyond
+chapter1
+page 4
+4
+S. Caracciolo, A. K. Hartmann, S. Kirkpatrick, M. Weigel
+ 10
+ 20
+ 30
+ 40
+ 50
+ 60
+ 100
+ 1000
+ 10000
+ 100000
+K
+N
+Kmax
+(N/e)0.5
+2 log2N
+log2N
+KHC SM1
+N0.5
+Fig. 1.1.
+Hidden clique sizes KHC of interest in G(N, p = 0.5) lie between the Kmax staircase and
+the proven or experimentally observed lower limits that can be found with spectral methods (dashed
+red line) or message-passing techniques (solid red line). Sizes of the smallest hidden cliques identified
+by a simple greedy search from all initial sites, stopping as soon as evidence for the hidden clique is
+found, are shown with blue dots and lie well below both limits.
+structions [12]. Phase diagrams with pictures of changes in the shapes of possible sets
+of solutions and different degrees of ergodicity in the phase space of a problem’s solu-
+tions [13, 14] now shape the discussions of the fundamental differences between different
+complexity classes of combinatorial and computational problems [15].
+The simplest problems in combinatorial optimization are proving explicitly some
+property of a random graph, or proving that the property doesn’t hold in that graph.
+An example would be testing if the maximum size of a clique, or completely connected
+cluster of sites, of size K exists in an Erd˝os-R´enyi graph G(N, p = 0.5) of size N, with
+half of the bonds, selected at random, the rest absent. This is a stylization of a common
+question data scientists might ask of the social networks in tracking interaction on
+today’s network, where such graphs may describe populations of billions, and the bonds
+might be demonstrated common interests or communication. We know what is possible
+by just calculating expectations. The largest cliques, of size Kmax, that form naturally
+in this model will not exceed 2 log2 N in size, with finite-size corrections making the
+actual limiting size of such a MaxClique much less. Simple linear algorithms, with costs
+proportional to the number of bonds, thus N 2, fail to find solutions with clique size K
+more than a few sites bigger than log2 N. A staircase of steps at which increasingly larger
+maximum cliques are found thus bounds a hard phase in which large numbers of cliques
+
+January 3, 2023 1:34
+ws-rv10x7-10x7
+Replica Symmetry Breaking & Far Beyond
+chapter1
+page 5
+Simulated annealing, optimization, searching for ground states
+5
+with sizes greater than log2 N must exist, but cannot be found without introducing
+some more expensive nonlinear search, such as backtracking.
+A popular extension of this problem is to add a hidden clique to the random graph
+which is larger than those which occur at random. Since this clique is unique, it should
+be easier to find, but the best methods proven to work with quasi-linear cost (but large
+prefactors) can only find hidden cliques of size greater than
+�
+N/e. So an easy region
+for this related problem exists when the hidden clique is bigger than this line. The
+resulting phase diagram is shown in Fig. 1.1 [16].
+Some recent work has returned to the original intuition of EA that correlations of
+activity between identical replicas of random, symmetry-less systems were an indica-
+tion of ordering that might serve in place of a traditional order parameter. Consider
+simulating the evolution of multiple copies of a random system at steadily lowered tem-
+peratures, or at a number of different temperatures, and passing information about
+improved solutions between the different replicas, a methodology called parallel tem-
+pering in physics [17]. This has proved capable of finding hidden cliques in very large
+graphs when the hidden clique barely exceeds the size of naturally occurring cliques [18],
+as well as to find maximum independent sets very efficiently [19]. Annealing multiple
+replicas in this way has the disadvantage that it cannot be proven to have only polyno-
+mial cost (in the size of the problem) asymptotically, although experience and empirical
+knowledge of the problem may allow tuning the method for affordable costs on large
+practical problems. It may also provide a path to overcoming the confusion that impedes
+the search for optimal configurations in combinatorial problems where many subopti-
+mal configurations overlap and confuse any local search, such as coloring or maximum
+clique [16].
+The extra search power that parallel tempering seems to provide suggests that there
+may be other ways to exploit the original insight of Edwards and Anderson, that or-
+dering in random systems might show up as a correlation between behavior observed
+in multiple copies or regions of a complex system. Replica symmetry breaking implies
+ordering which may take on a range of values, and study of the correlations seen in
+multiple replicas at low temperatures may add physical insight into such ordering. A
+further step could be to model systems which are basically similar but subject to weak
+local environmental corrections with a set of replicas, each slightly different. The ob-
+jective would be to identify from the differences between solutions in each replica a
+functional variation of the response of ordering to the different local influences. Such
+local variations are typical of much of the vast amounts of social data available for
+study in today’s highly instrumented world. This is just one more possible, but as yet
+unrealized, impact of spin glasses on applied mathematics and statistics.
+The original SK Ising model with Gaussian-distributed interactions is only one of
+the situations which has stimulated further exploration to tease out the lowest energy
+ground states. Different ways of combining different tools of optimization have proved
+valuable in this and different lessons emerge, as covered in the further sections of this
+chapter.
+
+January 3, 2023 1:34
+ws-rv10x7-10x7
+Replica Symmetry Breaking & Far Beyond
+chapter1
+page 6
+6
+S. Caracciolo, A. K. Hartmann, S. Kirkpatrick, M. Weigel
+1.2. Focusing on Ground States
+Consider the spin-glass (SG) model in zero field with Hamiltonian [2]
+H = −
+�
+⟨i,j⟩
+Jijsisj.
+(1.2)
+We focus on Ising spins si = ±1 placed on the vertices of a graph, but other symmetries
+of the order parameter have been considered. The Ising symmetry has the advantage
+that the related statistical physics problems are combinatorial. The sum ⟨i, j⟩ runs over
+all edges of the graph. For the Edwards-Anderson (EA) model the graph is a hyper-
+cubic lattice in d dimensions, but also other graph structures like the fully connected
+Sherrington-Kirkpatrick model corresponding to the d → ∞ limit of (1.2) have been
+studied.
+As discussed in Chapter 1, the physics of this mean-field problem is well
+understood, in many aspects since the 1980s. The situation in low dimensions is less
+clear and so much of the work in recent decades has focused on studying the problem
+in 2, 3 and 4 dimensions with numerical methods.
+Next to Monte Carlo simulations used to study the vicinity of the spin-glass tran-
+sition and the behavior in the ordered phase (see Chapter 5), much work has been
+invested in studying ground states and low-lying excitations above them in order to
+gauge the low-temperature behavior. Alternative theories for the nature of the spin-
+glass phase make contrasting predictions about the energetic and geometric properties
+of such excitations. Based on generalizations of Peierls’ argument for the stability of the
+ferromagnetic phase, researchers have argued that excitations with energies diverging
+with length will lead to spin-glass phases that are stable against thermal fluctuations,
+such that one of the prime goals of studying ground states for spin-glass systems is to
+understand the nature of the prevalent low-energy excitations and how their sizes and
+energies are related.
+The search for ground states for spin glasses with discrete symmetry amounts to
+a combinatorial optimization problem as there is a countable number of candidate so-
+lutions [20]. Such problems can hence be solved by a brute-force enumeration of all
+configurations, with an effort that grows exponentially with the number of spins. As
+discussed in section 1.3, a clever organization of this enumeration can lead to massive
+gains in efficiency of this process, but will not alter the overall exponential scaling of
+the process. This corresponds to the fact that the problem in d ≥ 3 is known to be NP
+hard [21]. In some special cases, however, the ground-state problem can be mapped onto
+auxiliary optimization problems that permit solutions in polynomial time. This will be
+explained in section 1.4. Beyond the ground states of spin glasses, many optimization
+problems have caught the attention of physicists [22, 23]. An example for a recently
+studied model is the assignment problem, which is discussed in section 1.5.
+1.3. Exact Algorithms for hard problems
+Here, exact and general algorithms for finding ground states (GSs) for Ising SGs are con-
+sidered, although the presented methods are very general. As an extension of Eq. (1.2),
+here also an interaction with local fields hi is included. For most studied systems no
+
+January 3, 2023 1:34
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+Replica Symmetry Breaking & Far Beyond
+chapter1
+page 7
+Simulated annealing, optimization, searching for ground states
+7
+local field is present, but for some of the algorithms presented below, such a term might
+arise for sub-problems to be solved. In this case, the energy of a SG is given by the
+Hamiltonian
+H(s) = −
+�
+i<j
+Jijsisj −
+�
+i
+hisi ,
+(1.3)
+where s = (s1, . . . , sN) is a configuration and the bonds Jij, corresponding to the edges
+of the graph, are quenched random variables. The sum runs here, in the most general
+case, over all pairs of spins. Thus, if all bonds are nonzero, the system is of mean-
+field type. For finite-dimensional lattices, most bonds are zero, suitably selected. The
+nonzero bonds are typically drawn from a Gaussian or a bimodal Jij = ±1 distribution.
+Here, three types of algorithms are presented, which have frequently been used to
+calculate exact GSs for three-dimensional SGs. First, the branch-and-bound method is
+explained, which is based on enumerating many states in a sophisticated way. Next, the
+linear programming approach is outlined, which is based on rewriting the Hamiltonian
+as a linear function, relaxing the condition si = ±1, plus adding additional constraints,
+called cutting planes or cuts. Finally, the combination of both approaches, the branch-
+and-cut algorithm is presented, which yields the currently fastest exact method to obtain
+spin-glass ground states.
+1.3.1. Branch-and-bound
+The basic idea of the branch-and-bound approach [24] to find the minima of Eq. (1.3) is
+to represent all spin configurations as a binary tree, where at each node the configuration
+space is, for a node-dependent selected spin i0, subdivided into the configurations with
+si0 = +1 and those with si0 = −1, respectively. The spin configurations are given by
+the 2N leafs of this tree. The simplest approach to the GS problem would be to obtain
+all configurations by enumeration, and simply pick those with the minimum energy,
+yielding an O(2N) running time.
+This running time can be improved, although still being exponential in the worst
+case, by omitting parts of this tree via considering bounds on the achievable energies
+in sub-trees [25]. Here we present the refined algorithm described in Ref. [26]. The
+branching is performed always on the last spin of the sub-problem, i.e., it starts with
+spin N. The energies for the sub problems with sN = +1 and sN = −1 can be written
+using Eq. (1.3) for s = (s1, . . . , sN−1) as follows:
+H+(s) = −
+�
+i<j
+′
+Jijsisj −
+�
+i
+′
+hisi −
+�
+i
+′
+JiNsi − hN ,
+(1.4)
+H−(s) = −
+�
+i<j
+′
+Jijsisj −
+�
+i
+′
+hisi +
+�
+i
+′
+JiNsi + hN ,
+(1.5)
+where the sums �′ run from 1 to N − 1. If one defines
+H∗
+N−1 =
+min
+s1,...,sN−1 −
+�
+i<j
+′
+Jijsisj ,
+(1.6)
+
+January 3, 2023 1:34
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+Replica Symmetry Breaking & Far Beyond
+chapter1
+page 8
+8
+S. Caracciolo, A. K. Hartmann, S. Kirkpatrick, M. Weigel
+one obtains, using the relation mins(f1(s)+f2(s)) ≥ mins f1(s)+mins f2(s), the bounds
+H+(s) ≥ H∗ +
+min
+s1,...,sN−1
+�
+i
+′
+(−hi − Jin)si − hn ,
+(1.7)
+H−(s) ≥ H∗ +
+min
+s1,...,sN−1 −
+�
+i
+′
+(−hi + Jin)si + hn .
+(1.8)
+These bounds are available because the minima H∗
+N−1, H∗
+N−2, . . . can be obtained recur-
+sively while the branching tree is built. Furthermore, the minimum of a linear function
+�
+s
+�
+i aisi, here ai = −hi − Jin or ai = −hi + Jin, respectively, is simply given by
+− �
+i |ai|. Thus, as a first way to restrict the size of the branching tree, if one or two
+of the branches exhibit bounds above a known threshold, the corresponding branches
+can be omitted. Such thresholds may come from low lying configurations found already
+in other branches or by heuristic algorithms, or simply given as part of the problem in
+case an enumeration of a specified range of low-lying configurations is sought.
+A second type of bound [26] works as follows. Using d(s) = −2�
+i
+′JiNsi − 2hN one
+can rewrite Eqs. (1.4) and (1.5) as H+(s) = H−(s) + 2d(s). Therefore, we obtain the
+implications
+max
+s1,...,sN−1 d(s) = 2
+�
+i
+|JiN| − 2hN ≤ 0 =⇒ H+(s) ≥ H−(s) ,
+min
+s1,...,sN−1 d(s) = −2
+�
+i
+|JiN| − 2hN ≥ 0 =⇒ H+(s) ≤ H−(s) .
+These bounds are easy to calculate and allow sometimes for omitting one of the two
+branches, before any branch has to be evaluated.
+Such branch-and-bound algorithms have been used, e.g., to analyze the low-
+temperature landscape [27–30] of SGs. More broadly, in the field of statistical mechanics
+of optimization problems [22, 23], the branch-and-bound approach has been applied to
+other problems like the satisfiability problem [31] or the vertex-cover problem [32]. For
+the latter one, a branch-and-bound approach was used, where the variable to branch
+on was not selected in a given order but determined by a local heuristics for further
+reduction of the branching tree. For simple variants of these algorithms, it has also been
+possible to calculate analytically the typical running time for ensembles of random prob-
+lems, which exhibit transitions between typically polynomial and typically exponential
+behavior [33, 34].
+1.3.2. Linear programming and cutting planes
+A different approach works by translating the quadratic Hamiltonian into a linear prob-
+lem. For convenience, here we consider the form of Eq. (1.2) Note that the field term
+present in Eq. (1.3) can be written as quadratic term by introducing a “ghost spin”
+s0 = 1 matching the given equation.
+We describe the system by a graph G = (V, E) where V denotes the set of sites
+where the spins are located and E the set of edges {i, j} between the interacting sites.
+For any set V ′ ⊂ V in G the cut δ(V ′) ⊂ E denotes the set of edges where one endpoint
+is in V ′ and the other is not, i.e., those connecting the two sets V ′ and V \ V ′. For
+
+January 3, 2023 1:34
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+Replica Symmetry Breaking & Far Beyond
+chapter1
+page 9
+Simulated annealing, optimization, searching for ground states
+9
+each configuration, the spins can be partitioned into two sets V + = {i|si = +1} of “up”
+spins and V + = {i|si = −1} = V \ V + of “down” spins. If two spins si and sj are
+oriented identically, they contribute the energy −Jij = +0 − Jij, while they contribute
+the energy Jij = 2Jij − Jij if they are oriented differently. Thus, using the cut, we can
+write
+H(s) = 2
+�
+{i,j}∈δ(V +)
+Jij −
+�
+ij
+Jij .
+(1.9)
+By introducing cij = −Jij, S = �
+ij Jij and variables xij = 1 if {i, j} ∈ δ(V +) and
+xij = 0 else, this reads
+H(x) = −2
+�
+i,j
+cijxij − S ,
+(1.10)
+where �
+i,j cijxij is called the weight of the cut represented by x. Therefore, since S
+is only a constant, finding the minimum energy of Eq. (1.2) corresponds to finding the
+maximum cut in a graph with edge weights cij. Note that Eq. (1.10) is a linear function
+with integer variables, which means we have transformed the quadratic optimization
+problem into a linear one, but with additional constraints since x must describe a cut.
+This problem is called integer linear program.
+To include the constraints describing the cut, we note that for any cycle in the graph
+G the cut must be crossed an even number of times. This can be conveniently described
+by linear inequalities [35] for the variables x as follows: For any cycle C ⊂ E, i.e., a
+path which starts and ends at the same vertex, and any odd cardinality subset Q ⊂ C,
+�
+{i,j}∈Q
+xij −
+�
+{i,j}∈C\Q
+xij ≤ |Q| − 1
+(1.11)
+must hold, which defines a cut comprehensively.
+The basic idea of the cutting-plane approach is now to relax the variables xij = 0, 1
+to 0 ≤ xij ≤ 1 and look for a maximum of Eq. (1.10) given the linear inequalities
+Eq. (1.11). The name cutting plane comes from the fact that all inequalities describe
+hyper-planes which cut off one part from the space of possible solutions. The resulting
+problem is called a linear program (LP). The good news is that LPs, i.e., with the relaxed
+variables, can be solved [36] in worst-case polynomial time in the problem size using the
+ellipsoid method. Still, in a practical context methods like the simplex approach [37] or
+the dual simplex approach, which exhibit no polynomial bound, perform much faster.
+Still, the bad news is that the number of inequalities is in principle exponentially large,
+which would yield an exponential running time right-on. Therefore, instead of adding all
+constraints immediately to the LP, one starts with no or few constraints and calculates
+a first solution. Since fewer constraints than necessary are contained in the relaxed
+problem, typically the obtained cut weight will be higher, i.e., an upper bound of the
+true maximum cut for the integer LP. Hence, the obtained solution will typically not
+correspond to a cut and not be pure integer-valued.
+Thus, there may be some of
+the exponentially many inequalities which are violated. The second basic idea of the
+cutting plane approach is to look specifically for violated inequalities, without searching
+all possible ones, and add the violated ones to a growing set of inequalities included in
+
+January 3, 2023 1:34
+ws-rv10x7-10x7
+Replica Symmetry Breaking & Far Beyond
+chapter1
+page 10
+10
+S. Caracciolo, A. K. Hartmann, S. Kirkpatrick, M. Weigel
+the LP. An exact polynomial algorithm to find violated inequalities is based on solving
+a series of shortest-path problems [35], which can be done in O(N 3). This is polynomial
+but rather slow. Fortunately, there exist, in particular for physical lattice structures,
+several heuristics which run in linear O(N) time [38] and are most of the time sufficient
+to generate additional inequalities for violated conditions. If at some point no further
+violated inequalities are found and the solution is fully integer-valued, a true optimum
+has been found, and the algorithm stops. But if on the contrary some variables are still
+non-integer, one can resort to branching as explained in the next subsection.
+Nevertheless, for some combinatorial problems it has been observed that a cutting
+plane approach alone may lead to a valid optimum solution of the integer problem.
+For example for vertex-cover problems on Erd˝os-R´enyi random graphs a different kind
+of cycle inequalities has been considered [39]. When varying the average number c of
+neighbor nodes in the graphs, a phase transition has been observed. In the thermody-
+namic limit N → ∞, for small values c < c∗ = e ≈ 2.71, typically all instances can be
+solved completely in a polynomial running time, while for larger values of c this is not
+possible. Interestingly, this easy-hard transition coincides with the critical connectivity
+where replica-symmetry breaking of the vertex-cover problem occurs [32]. At this point
+a complex structure of the solution space has been observed [40], where the so-called
+leaf-removal core [41] starts to percolate.
+Finally, note that there is a set of cutting planes, so called Gomory cuts [42], which
+can be constructed generally for all relaxed linear problems. They are based on iden-
+tifying violated inequalities directly from a given non-integer solution, actually in the
+so-called tableau [36] used by the simplex algorithm to solve the LP. It is proven that
+this, possibly exponentially large, set of inequalities is complete, i.e. any integer pro-
+gramming problem can be solved in principle by generating just these cutting planes.
+Still, in practice, any finite numerical accuracy leads to convergence problems, such
+that this and many other cutting-plane approaches turned out to be actually efficient
+in combination with branching, as explained next.
+1.3.3. Branch-and-cut
+If for a relaxed linear system describing maximum cuts the solution is still non-integer,
+one considers for some variable xij ̸= 0, 1 both possibilities xij = 0 and xij = 1 and
+solves the corresponding sub-problems recursively.
+This means one branches.
+The
+combination of these approaches is therefore called branch-and-cut [38, 43]. Note that
+here several other bounds, in addition to those mentioned above, can be used. This
+can be lower bounds obtained from the cut weight of any valid cut x∗ or upper bounds
+obtained from suitable relaxations like the LP.
+Currently, the branch-and-cut approach can be considered as the most powerful exact
+algorithm to obtain GSs for hard SG instances. A publicly accessible implementation
+of a branch-and-cut algorithm is the spin-glass server [44] hosted by the University of
+Bonn. The server was originally implemented at the University of Cologne by several
+members and collaborators of the research groups of F. Liers and M. J¨unger. At the
+server, you can submit SG instances as a file and, if the system is not too large, a
+corresponding GS will be returned.
+
+January 3, 2023 1:34
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+chapter1
+page 11
+Simulated annealing, optimization, searching for ground states
+11
+Jij
+Fig. 1.2.
+Mapping of the Ising ground-state problem to a matching problem on an auxiliary graph with
+Kasteleyn cities. Left: expansion of the Ising lattice, replacing each vertex by a complete graph K4.
+Edge weights in the K4 subgraphs are set to 0, the remaining weights are Jij. Middle: A minimum-
+weight perfect matching on the auxiliary graph. Right: Back-transformation from the decorated graph
+to the original lattice. The matching then results in a set of closed loops separating up from down
+spins.
+Branch-and-cut approaches have been applied to finite-dimensional SGs in several
+occasions. To our knowledge the largest three-dimensional instances were N = 123 as
+considered in a study of low-lying excitations [45].
+The approach has also been applied for a mean-field random-bond Ising model,
+which is a generalization of the standard SG with a variable fraction of negative bond
+as controlled by a non-zero mean of the bond values. Here, a phase transition of the
+typical branch-and-cut running time, as measured by the number of LPs solved, between
+an easy and a hard phase has been observed near the transition between ferromagnetic
+and SG behavior [46].
+1.4. Ground states in two dimensions
+We now turn to the case of ground-state problems permitting a polynomial-time solu-
+tion. For the EA model of Eq. (1.2) this is the case in dimensions d < 3. Since the 1d
+problem is rather trivial, the much more interesting case of this type is the EA model
+in two dimensions.
+A relevant mapping of the EA ground state to a minimum-weight perfect matching
+(MWPM) problem on an auxiliary graph was first proposed in Ref. [47]. It is based on
+the observation that frustrated plaquettes [48], i.e., elementary lattice faces including
+an odd number of antiferromagntic couplings, must have an odd number of broken
+bonds with Jijsisj < 0, while non-frustrated plaquettes have an even number of broken
+bonds. Hence a spin configuration can be depicted as a configuration of defect lines of
+broken bonds that start and end at frustrated plaquettes. A ground state configuration
+is then a perfect pairing (matching) of frustrated plaquettes through defect lines of
+minimum total weight. MWPM can be solved in polynomial time based on the blossom
+algorithm [49] and its variants [50].
+However, this approach is still not ideal as it
+operates on the complete graph of the F frustrated plaquettes with F(F −1) edges, and
+the edge weights need to be computed in a preparatory step using a suitable approach
+
+January 3, 2023 1:34
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+Replica Symmetry Breaking & Far Beyond
+chapter1
+page 12
+12
+S. Caracciolo, A. K. Hartmann, S. Kirkpatrick, M. Weigel
+such as Dijkstra’s algorithm [51] before attacking the matching problem. This approach
+allows one to study systems of a typical maximum linear size of L ≈ 500 [52].
+As was shown more recently [53, 54], an alternative mapping is significantly more
+efficient as it operates on a sparse graph with similar connectivity as the original lattice.
+It relies on an auxiliary graph that replaces each vertex of the lattice by a complete
+graph K4 of four nodes, also known as Kasteleyn city. Edge weights are set to Jij for
+the original edges and to zero for the internal K4 bonds. The solution of a matching
+problem on this graph then corresponds to a set of closed loops on the original lattice,
+separating domains of opposite spin orientations, cf. the illustration in Fig. 1.2. The
+configuration of minimum weight corresponds to a ground state of the spin-glass sample.
+Due to the sparsity of the auxiliary graph, significantly larger systems can be studied
+with this method as compared to the one proposed in [47], and calculations for square
+lattices up to L = 10 000 have been reported in Ref. [55].
+A widely applied method for studying the excitations out of the ground states that
+take a central role in the theory of the spin-glass phase, consists of systematic modifi-
+cations of boundary conditions (BCs). It is argued that the defect energy connected to
+a change from periodic to antiperiodic BCs in one direction,
+Edef = |EP − EAP|,
+(1.12)
+can act as a proxy for a typical low-energy excitation of the system [56]. Here, EP and
+EAP refers to the ground-state energy for periodic and antiperiodic BCs, respectively.
+One expects that Edef scales as [57, 58],
+Edef ∼ Lθ
+(1.13)
+with the spin-stiffness exponent θ, where θ > 0 should indicate the stability of the
+spin-glass phase at non-zero temperatures, while for θ < 0 the transition temperature
+TSG = 0 and θ = −1/ν governs the divergence of the spin-glass correlation length as
+T → TSG [59].
+For Gaussian exchange couplings Jij one finds a stiffness exponent
+θ ≈ −0.3 [52], with the most accurate estimate being [55],
+θ = −0.2793(3).
+(1.14)
+This is illustrated in Fig. 1.3(a) showing the scaling of defect energies over a wide range
+of system sizes. The change of boundary conditions induces a domain-wall defect that
+spans the system; a typical configuration of the overlap between the ground states for
+periodic and antiperiodic BCs is shown in the left panel of Fig. 1.4. The boundary of
+the flipped domain is a fractal curve, and the domain-wall length is hence expected to
+show fractal scaling of the form
+⟨ℓ⟩J = AℓLdf.
+(1.15)
+As is illustrated in Fig. 1.3(b), this is indeed borne out in the data to high accuracy,
+and the fractal dimension is estimated as df = 1.27319(9).
+For technical reasons the matching approach can only handle samples on planar
+graphs, i.e., lattices with periodic boundary conditions in at most one direction [47].
+More precisely, runs for systems with fully periodic boundaries yield the same result for
+
+January 3, 2023 1:34
+ws-rv10x7-10x7
+Replica Symmetry Breaking & Far Beyond
+chapter1
+page 13
+Simulated annealing, optimization, searching for ground states
+13
+10−1
+100
+101
+102
+103
+104
+(a)
+Gauss
+PFBC
+θ = −0.2793(3)
+⟨|∆E|⟩J
+L
+100
+101
+102
+103
+104
+105
+106
+107
+101
+102
+103
+104
+(b)
+Gauss
+PFBC
+df = 1.273 19(9)
+⟨ℓ⟩J
+L
+Fig. 1.3.
+(a) Scaling of defect energies Edef for the Gaussian EA model on the square lattice with
+periodic boundaries in x direction and free boundaries in y direction. The line shows a fit of the data to
+the functional form Edef = AθLθ +Cθ/L2 [55]. (b) Scaling of the domain-wall length between periodic
+and antiperiodic BCs. The line corresponds to a fit of the functional form (1.15) to the data.
+periodic and for antiperiodic BCs in each of the two directions, such that the configu-
+ration returned is a ground state for one out of four possible BCs. As pointed out in
+Ref. [53], the approach hence effectively optimizes over BCs as well as spin variables.
+To circumvent this problem and allow treatment of systems with fully periodic BCs
+with the resulting smaller scaling corrections, one may use a windowing technique as
+illustrated in the right panel of Fig. 1.4: since MWPM can be used to find exact ground
+states for planar graphs, in order to treat a periodic L × L system one applies it to a
+square subset of edge length L − 2 (“window”) while keeping the relative orientation of
+the outside spins fixed. Randomly displacing the window location over the (periodic)
+lattice, repeated applications of the window optimizations lead to a quick convergence of
+the result. This prescription results in a stochastic algorithm, whose success probability
+can be arbitrarily improved by using m independent runs,
+Ps({Jij}) = 1 − [1 − Pn({Jij})]m.
+(1.16)
+Numerically, one finds that the number of required repetitions for a given success prob-
+ability is independent of system size, such that the computational complexity of the
+algorithm remains the same as that of the MWPM approach for planar graphs, which
+scales as Lκ with κ ≈ 2.2 [55] for the Blossom V algorithm [50].
+The system with bimodal couplings can also be studied with matching techniques.
+Here, one finds no asymptotic decay of Edef, but a convergence to a positive limiting
+value as L → ∞ [52]. While this was initially taken as evidence for a lower-critical
+dimension dl = 2, it was later on realized that it is rather a signature of an additional
+zero-temperature renormalization-group fixed point that is not relevant for the physics
+at non-zero temperatures [60] and, instead, there is evidence for dl = 2.5 [61, 62]. A
+signature of this system is the extensive degeneracy of the ground state, leading to
+a finite ground-state entropy. This creates a difficulty for the matching approach as
+it does not pick the individual ground states with equal probabilities. In its simplest
+form, it is deterministic and will hence always return the same ground state. Some
+
+January 3, 2023 1:34
+ws-rv10x7-10x7
+Replica Symmetry Breaking & Far Beyond
+chapter1
+page 14
+14
+S. Caracciolo, A. K. Hartmann, S. Kirkpatrick, M. Weigel
+L
+L
+Fig. 1.4.
+Left: The domain wall separating regions of equal and opposite ground-state configurations
+for a specific disorder sample considered with periodic and with antiperiodic boundaries. Right: Setup
+used for the windowing technique to compute ground states for samples with fully periodic boundary
+conditions.
+simple randomizations through changing the order of considering bonds or adding some
+noise onto the couplings improve on this, but still lead to biased sampling methods [55].
+Unbiased sampling can be achieved via a suitably constructed Monte Carlo sampling
+technique in the ground-state manifold, thus allowing for estimates of the domain-wall
+fractal dimensions and related quantities [55].
+Matching techniques can also be used to sample other excitations than the domain-
+wall perturbations induced by a change of boundary conditions.
+Different types of
+droplet-shaped excitations can be studied by fixing the relative orientation of spins
+through “hard” bonds [63, 64]. Also, while matching is the most widely used technique
+for this problem, an alternative approach based on a calculation of the partition function
+through the use of Pfaffians allows to also study finite-temperature properties exactly
+and in polynomial time [65–67]. Recently, such methods have been extended to also
+enable the study of correlation functions and further related properties [68–70].
+1.5. The assignment problem
+In this section we will discuss some new results in the assignment problem. In order to
+define the question in its simplest version we have to consider a square matrix of size
+n, W := {wij}n
+i,j=1, with real entries. For each permutation π in the symmetric group
+Sn consider the matrix Π := {πij}n
+i,j=1 with entries zero or one, such that
+πij = δj,π(i) =
+�
+1
+if π(i) = j
+0
+elsewhere
+(1.17)
+
+LIJanuary 3, 2023 1:34
+ws-rv10x7-10x7
+Replica Symmetry Breaking & Far Beyond
+chapter1
+page 15
+Simulated annealing, optimization, searching for ground states
+15
+and ask for a permutation of the columns of W in order to get a minimum of the trace,
+that is of the Hamiltonian
+H(Π, W) := tr
+�
+W T · Π
+�
+=
+n
+�
+i,j=1
+wijπij =
+n
+�
+i,j=1
+wijδj,π(i) =
+n
+�
+i=1
+wi,π(i) .
+(1.18)
+Without loss of generality we can take the entries of W to be nonnegative: wij ≥ 0.
+Indeed, in combinatorial optimization they are usually referred to as costs. In different
+words, this is the classical matching problem on the bipartite complete graph Kn,n. The
+solution of the problem is a permutation π∗, with corresponding matrix Π∗(W) , that
+realizes the minimum cost and brings to the determination of
+H∗(W) := min
+π∈Sn H(Π, W) = H (Π∗(W), W)
+(1.19)
+the optimal value of the total cost. The optimal solution can be seen as the ground state
+of the Hamiltonian of a disordered system defined by the cost-matrix W, an analogy
+which can be made useful, for example, by simulated annealing [7].
+The introduction of a probability on the space of the possible costs allows the in-
+vestigation of the properties of the typical solutions and the artillery from statistical
+physics shows its whole power [22, 23, 71]. In their seminal works M´ezard and Parisi
+[72–74] (but see also [75]) could solve, by using the replica trick, the matching, the
+assignment and the Travelling Salesman Problem in the case in which the entries wij
+are independent random variables, equally distributed, with a probability distribution
+density of the form
+ρ(w) = wr
+∞
+�
+k=0
+ηkwk
+(1.20)
+with η0 ̸= 0. More precisely, in the asymptotic limit of an infinitely large size n, replica
+symmetry is not broken, and the optimal solution is determined by the application
+of the saddle-point method as the solution of an integral equation in which only the
+parameter r enters. In particular
+En := E [H∗(W)] ∼ n1−
+1
+r+1 ,
+(1.21)
+where the expectation value is taken on the possible weights and with ∼ we indicate
+that both sides of the relation scale with large n in the same way so that their ratio
+converges in the limit of an infinite number of points. For a recent work in which the
+first finite-size corrections are reconsidered after [76, 77] and the extension to non-integer
+value of the parameter r see [78].
+A new class of problems arises when the vertices of the graph Kn,n are identified
+with points in Ω ⊂ Rd seen as a subset of a Euclidean space. We thus have two sets of
+points of cardinality n: we denote them as the red points R, respectively blue points
+B, with positions xi ∈ Rd, respectively yi ∈ Rd, with i ∈ {1, . . . , n} = [n]. Then the
+cost of the assignment of the i-th red point with the j-th blue point is assumed to be
+a function of the Euclidean distance between the two points |xi − yj|. We shall restrict
+to the cases
+wij = f(|xi − yj|) = |xi − yj|p
+(1.22)
+
+January 3, 2023 1:34
+ws-rv10x7-10x7
+Replica Symmetry Breaking & Far Beyond
+chapter1
+page 16
+16
+S. Caracciolo, A. K. Hartmann, S. Kirkpatrick, M. Weigel
+parametrized by the real number p ∈ R. Let us introduce the empirical probability
+measures associated to the two sets of points
+ρR(x) = 1
+n
+n
+�
+i=1
+δ(x − xi) ;
+ρB(y) = 1
+n
+n
+�
+i=1
+δ(y − yj)
+(1.23)
+and look, for p ≥ 1, at the p-th Wasserstein (elsewhere associated to the names of Kan-
+torovich and Rubinstein) distance of two probability measures, that is at the variational
+problem
+Wp(ρ1, ρ2) :=
+�
+inf
+γ∈Γ(ρ1,ρ2)
+�
+dx dy γ(x, y) |x − y|p
+� 1
+p
+,
+(1.24)
+where Γ(ρ1, ρ2) is the set of measures on Ω×Ω with marginals ρ1 and ρ2. This is nothing
+but the optimal transport (or Monge-Kantorovich) problem of a unit mass distributed
+according to ρ1 to a distribution ρ2 [79, 80]. A function γ(x, y) defines a transportation
+plan, indeed, the marginality conditions
+�
+dx γ(x, y) = ρ2(y) ;
+�
+dy γ(x, y) = ρ1(x)
+(1.25)
+constraining the mass moved into the point y and from the point x are what is required.
+In the case of the two empirical probability measures the possible transportation plans
+reduce to the set of permutations and therefore their Wasserstein distance is simply
+related to the optimal cost [81]
+H∗(W) = n W p
+p (ρR, ρB).
+(1.26)
+A random matrix, whose elements depend on the Euclidean distance between points
+randomly distributed in space is called Euclidean. The spectra of Euclidean random
+matrices have been studied in [82].
+In order to fix the ideas, let us consider the case in which Ω = [0, 1]d, periodic
+boundary conditions are chosen, so that we are really on a torus of unit volume and the
+positions of red and blue points are taken at random with flat probability. A natural
+length-scale is therefore n− 1
+d and one can simply assume that for each red point there
+is at least a blue point to match in a ball of radius of order n− 1
+d , so that we can guess
+that
+En ∼ n1− p
+d .
+(1.27)
+But it is well known [83] that this can be true only in d ≥ 2. It was proven that in d = 2
+En ∼ n
+�log n
+n
+� p
+2
+(1.28)
+a logarithmic violation appears. A much more detailed analysis is possible in d = 1 [84]
+where an intriguing relation with Brownian processes is explored [85]. It is shown that,
+for a generic distribution probability ρ with cumulative Φ, the average total cost is
+En = n1− p
+2 2p
+√π Γ
+�p + 1
+2
+� � 1
+0
+dx[Φ(x)(1 − Φ(x))]
+p
+2
+ρp−1(x)
++ o(n− p
+2 )
+(1.29)
+
+January 3, 2023 1:34
+ws-rv10x7-10x7
+Replica Symmetry Breaking & Far Beyond
+chapter1
+page 17
+Simulated annealing, optimization, searching for ground states
+17
+at least when the integral is convergent. Otherwise an anomalous scaling emerges [86].
+In the case of open boundary conditions, with flat distribution, the average cost can be
+evaluated even for finite size by means of Selberg integrals [87]
+En = n Γ
+�
+1 + p
+2
+�
+p + 1
+Γ(n + 1)
+Γ
+�
+n + 1 + p
+2
+� .
+(1.30)
+Also the case p ⩽ 0 has been studied [88], where, instead, for flat distribution
+lim
+n→∞
+En
+n = 1
+2p .
+(1.31)
+When 0 < p < 1 the cost function becomes concave. In [89] it is argued that
+En ∼
+�
+�
+�
+�
+�
+�
+�
+n1−p
+for 0 < p < 1
+2 ,
+√n log n
+for p = 1
+2 ,
+√n
+for 1
+2 < p < 1 ,
+(1.32)
+so that a change in the phase diagram occurs at p = 1
+2.
+An amusing exact result has been obtained for the flat distribution in d = 2 for the
+particular value p = 2, in [90, 91], that is
+lim
+n→∞
+En
+log n = 1
+2π .
+(1.33)
+This has been rigorously proven in [92] (see also [93] for improvements). Let us follow
+the field theoretic approach introduced in [94] and let us introduce the vector transport
+field µ(x) which joins the red point in x to the blue point in y = x + µ(x) and consider
+the Lagrangian
+L[µ, φ] := 1
+2
+�
+µ2(x)ρR(dx) +
+�
+[φ(x + µ(x))ρR(dx) − φ(x)ρB(dx)]
+(1.34)
+to be minimized, where the scalar field φ(x) is a Lagrangian multiplier which implements
+a matching between blue and red points. In the limit of a large number of points n, when
+the red and blue points are extracted with the same distribution ρ so that δρ := ρR−ρB
+and the optimal µ goes to zero and we expect a good approximation by using the only
+the quadratic terms in the Lagrangian, that is
+L[µ, φ] :=
+� �1
+2µ2(x) + µ(x) · ∇φ(x)ρ(dx)
+�
++
+�
+φ(x)δρ(dx) ,
+(1.35)
+which has Euler-Lagrangian equations
+µ = −∇φ ,
+∇ · [ρ µ] = δρ ,
+(1.36)
+revealing a strict analogy with an electrostatic problem where µ plays the role of the
+electric field, φ is the scalar potential and indeed is the Lagrangian multiplier which
+implements the Gauss law, red and blue points have opposite unit charge, being null
+the total charge, while ρ is the dielectric function of a linear dielectric medium. As a
+consequence
+− ∇ · [ρ ∇φ] = ρ
+(1.37)
+
+January 3, 2023 1:34
+ws-rv10x7-10x7
+Replica Symmetry Breaking & Far Beyond
+chapter1
+page 18
+18
+S. Caracciolo, A. K. Hartmann, S. Kirkpatrick, M. Weigel
+is solved by means of the classical Green’s function Gρ(x, y) of the operator −∇·[ρ ∇•],
+so that an explicit approximate solution at fixed disorder is given by
+µ(x) =
+�
+∇xGρ(x, y) δρ(dy) .
+(1.38)
+After averaging over disorder, in the simple case of a flat measure (see [95] for non-
+constant case) we get
+lim
+n→∞ En(Ω) = −2 tr ∆−1
+Ω ,
+(1.39)
+where the Laplacian ∆Ω is defined on the domain Ω. This formula is correct in d = 1 but
+it simply provides a divergence on both sides for d > 1. It is an ultraviolet divergence
+which has been introduced by the linearization in the infinite number of modes, but the
+approximation cannot be true at very short distances. In the exact non-linear theory
+higher modes are cut off and we expect that
+En(Ω) = 2
+� ∞
+0+
+F
+� λ
+n
+�
+λ
+dNΩ(λ) ,
+(1.40)
+where NΩ(λ) is number of the eigenvalue less than λ for the Laplace operator and the
+unknown function F interpolates between 1 for λ ≲ n and 0 for λ ≳ n.
+By the Weyl law [96] on the asymptotics of the eigenvalue counting function for the
+Laplace-Beltrami operator we know that, for a 2-d manifold with unit volume (under
+Neumann boundary conditions which are appropriate for our problem)
+NΩ(λ) = 1
+4π
+�
+λ +
+√
+λ |∂Ω|
+�
++ o
+�√
+λ
+�
+.
+(1.41)
+In d = 2 it is easy now to evaluate the leading logarithmic singularity in the number of
+points and in agreement with [97] we expect that
+En(Ω) = 1
+2π log n + 2c∗(n) + 2cΩ + o(1) ,
+(1.42)
+where c∗(n) = O(log n) is a universal function not depending on Ω.
+As a further
+consequence,
+lim
+n→∞ [En(Ω) − En(Ω′)] = 2 lim
+n→∞
+� ∞
+0+
+F
+� λ
+n
+�
+λ
+[dNΩ(λ) − dNΩ′(λ)] ,
+(1.43)
+but the r.h.s. is convergent even in the absence of regularization thus
+lim
+n→∞ [En(Ω) − En(Ω′)] = 2
+� ∞
+0+
+dNΩ(λ) − dNΩ′(λ)
+λ
+,
+(1.44)
+an expression that has been tested in [98] on various 2d manifolds, by using both the
+Green’s function method as other classical tools as the Dedekind’s limit formulas, thus
+confirming the predictions of the field theoretic approach.
+For a similar method applied to a more general context see [99–101].
+Once more this relatively simple combinatorial optimization problem is revealing
+intriguing and fruitful connections with so many different research fields.
+
+January 3, 2023 1:34
+ws-rv10x7-10x7
+Replica Symmetry Breaking & Far Beyond
+chapter1
+page 19
+Simulated annealing, optimization, searching for ground states
+19
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+
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+page_content=' Hebrew University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Jerusalem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Israel  ¶Institut f¨ur Physik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Technische Universit¨at Chemnitz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Chemnitz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Germany  The chapter starts with a historical summary of first attempts to optimize the spin  glass Hamiltonian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
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+page_content='huji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='il  ¶martin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='weigel@physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='tu-chemnitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='de  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='00683v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='dis-nn]  2 Jan 2023  January 3, 2023 1:34  ws-rv10x7-10x7  Replica Symmetry Breaking & Far Beyond  chapter1  page 2  2  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Caracciolo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Hartmann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Kirkpatrick, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Weigel  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Introduction  A theme which unites this chapter is the study by simulation of finite random systems in  order to test the predictions and insights that arose as new kinds of order were defined  and realized in spin glasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' The tools of this chapter have found use in many applied  contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' In physics typically phase spaces of physical systems have been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  In applied mathematics, problems, either idealized or based on real-world data, are  solved to optimize target functions or to satisfy constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Over the years is has been  gradually accepted in mathematics and physics communities that these two realms are  very similar to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  Replicas, when introduced by Brout [1], were large patches of a perfectly ordered  lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Edwards and Anderson (EA) [2] looked for order in the similarity of behavior  across multiple identical copies of random spins and their interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' In the Brout  and EA work, the free energy was extracted as the linear term in an expansion of the  partition function in powers of n, the number of replicas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' EA defined order as a self-  correlation of spin orientation in different replicas, since this could represent correlation  over long times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Sherrington and Kirkpatrick(SK) [3], after simplifying the problem  to an infinite ranged Ising system with i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' random Gaussian interactions of strength  1/  √  N in hopes of making it soluble, explicitly used the identity,  ln Z = lim  n→0  Zn − 1  n  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='1)  despite the obvious fact that the meaning of Zn anywhere except at integer values  of n was unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' A careful treatment of the extrapolation of the resulting equations  for the free energy, F, showed an entropy at zero temperature which was negative,  an impossibility in a discrete model, while other features such as a susceptibility cusp  seemed plausible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' The predicted ground state energy, E0 = (2/π)1/2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='798.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=', could  be tested in simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  Fortunately, computers in 1975 had almost reached the point where they might be  able to address questions about asymptotic values of quantities such as E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Power-  ful generally available computers such as IBM’s 370-168 and later 3033 could hold as  much as 10 MB of data in random access memory, and process instructions at a few  MIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' (The first supercomputer, the Cray-1, in 1975 also had no more than about 10  MB of RAM, but ran its instructions 4-6 times faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=')  This amount of RAM per-  mitted simulating a small number of samples with 500-1000 spins, so with a floating  point coprocessor attached to their IBM mainframe to provide more Cray-like process-  ing speeds, Kirkpatrick and Sherrington (KS) [4] in 1978 could report that E0 was in  fact between −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='75 and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='77, excluding the replica symmetric result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Improvements  on that estimate have continued for the next 30+ years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  The most recent estimate  of the asymptotic result, using Parisi’s full RSB formulas and Pad´e approximants to  extrapolate both upper and lower bounds to E0, yields the currently accepted value of  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='76321.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' The improvements in theory that have made this the accepted result  required several decades, and are discussed elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Improving the experiments to  test it also required some years for computing speeds to increase so that better statistics  could emerge from simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Graphs with N up to 2000 sites have been studied [6],  January 3, 2023 1:34  ws-rv10x7-10x7  Replica Symmetry Breaking & Far Beyond  chapter1  page 3  Simulated annealing, optimization, searching for ground states  3  but understanding the size dependence of E0 remains a subject of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Simulations  agree with theory, and have grown more accurate as the bounds on the theoretical re-  sult have also grown tighter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' New methods of finding the ground state energies in the  simulated model were required, intially and as the models got larger and the accuracy  required grew narrower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' These methods, such as simulated annealing, have taken on  a life of their own, with many applications in complex systems outside of the simple  models of statistical mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  Simulated annealing [7] is an obvious idea if you are statistical physicists, like KS  when first studying their spin glass model, and also V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' ˇCern`y [8] and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Wilson [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  Kirkpatrick and ˇCern`y studied the Travelling Salesman as an easily described example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  Wilson used annealing to optimize packing of parallel processor code generated in a  compiler he wrote for his quantum chromodynamics simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' All of us realized that  allowing a Markov chain of states to evolve at a series of decreasing finite temperatures  (annealing) would tend to reach larger and deeper minima of a complex function space  with many local minima than simple gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  At the time the preferred methods used gradient descent with many random restarts  and tricks to jump to new starting points, applied only when a search gets stuck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' The  large number of possible local minima, mostly at higher energies, makes this ineffective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  In addition, study of properties such as susceptibilities and specific heat (obtained from  the fluctuations of a Boltzmann system) could guide the development of effective an-  nealing schedules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Ranges of temperature at which large fluctuations were seen, possibly  signalling a phase boundary, give a signal to slow the annealing process to allow large  changes to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  Kirkpatrick and IBM colleagues [7] were also at the time studying the problems  presented by design automation tools used to place and route transistors and small VLSI  circuits in IBM’s next-generation computers, and included simplified examples of both  practical problems in their paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' These tools were successfully used and incorporated  into IBM’s internal product development tool sets, and adopted in other industries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  The IBM group learned important lessons from exposure to the engineering environ-  ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' First, expressing complex constraints as energetic costs in a Hamiltonian frame-  work provided valuable flexibility, and the annealing framework met the constraints  simultaneously or in order of importance rather than one at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Second, they soon  realized that finding the exact optimum in a problem like circuit design was not really  the objective of an engineering team, who simply needed to find solutions that were  good enough, soon enough, to ship as products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' For that sort of objective, robustness  of a methodology and the ability to incorporate many esoteric constraints was at least  as important as its efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Optimality of the solution obtained, if possible, was a  bonus, but not the only objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  Simulated annealing has been accepted as another tool for at least approximately  solving messy but useful and important problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  Probably its greater impact has  been to stimulate the incorporation of statistical physics frameworks as an active part  of applied mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' One recent review by a respected practitioner of constraint  satisfaction [10] considers the 1990s to be the era of phase transitions, with the 2000s  introducing more refined techniques such as survey propagation [11] and cavity con-  January 3, 2023 1:34  ws-rv10x7-10x7  Replica Symmetry Breaking & Far Beyond  chapter1  page 4  4  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Caracciolo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Hartmann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Kirkpatrick, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Weigel  10  20  30  40  50  60  100  1000  10000  100000  K  N  Kmax  (N/e)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='5  2 log2N  log2N  KHC SM1  N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='5  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  Hidden clique sizes KHC of interest in G(N, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='5) lie between the Kmax staircase and  the proven or experimentally observed lower limits that can be found with spectral methods (dashed  red line) or message-passing techniques (solid red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Sizes of the smallest hidden cliques identified  by a simple greedy search from all initial sites, stopping as soon as evidence for the hidden clique is  found, are shown with blue dots and lie well below both limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  structions [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Phase diagrams with pictures of changes in the shapes of possible sets  of solutions and different degrees of ergodicity in the phase space of a problem’s solu-  tions [13, 14] now shape the discussions of the fundamental differences between different  complexity classes of combinatorial and computational problems [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  The simplest problems in combinatorial optimization are proving explicitly some  property of a random graph, or proving that the property doesn’t hold in that graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  An example would be testing if the maximum size of a clique, or completely connected  cluster of sites, of size K exists in an Erd˝os-R´enyi graph G(N, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='5) of size N, with  half of the bonds, selected at random, the rest absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' This is a stylization of a common  question data scientists might ask of the social networks in tracking interaction on  today’s network, where such graphs may describe populations of billions, and the bonds  might be demonstrated common interests or communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' We know what is possible  by just calculating expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' The largest cliques, of size Kmax, that form naturally  in this model will not exceed 2 log2 N in size, with finite-size corrections making the  actual limiting size of such a MaxClique much less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Simple linear algorithms, with costs  proportional to the number of bonds, thus N 2, fail to find solutions with clique size K  more than a few sites bigger than log2 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' A staircase of steps at which increasingly larger  maximum cliques are found thus bounds a hard phase in which large numbers of cliques  January 3, 2023 1:34  ws-rv10x7-10x7  Replica Symmetry Breaking & Far Beyond  chapter1  page 5  Simulated annealing, optimization, searching for ground states  5  with sizes greater than log2 N must exist, but cannot be found without introducing  some more expensive nonlinear search, such as backtracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  A popular extension of this problem is to add a hidden clique to the random graph  which is larger than those which occur at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Since this clique is unique, it should  be easier to find, but the best methods proven to work with quasi-linear cost (but large  prefactors) can only find hidden cliques of size greater than  �  N/e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' So an easy region  for this related problem exists when the hidden clique is bigger than this line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' The  resulting phase diagram is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='1 [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  Some recent work has returned to the original intuition of EA that correlations of  activity between identical replicas of random, symmetry-less systems were an indica-  tion of ordering that might serve in place of a traditional order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Consider  simulating the evolution of multiple copies of a random system at steadily lowered tem-  peratures, or at a number of different temperatures, and passing information about  improved solutions between the different replicas, a methodology called parallel tem-  pering in physics [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' This has proved capable of finding hidden cliques in very large  graphs when the hidden clique barely exceeds the size of naturally occurring cliques [18],  as well as to find maximum independent sets very efficiently [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Annealing multiple  replicas in this way has the disadvantage that it cannot be proven to have only polyno-  mial cost (in the size of the problem) asymptotically, although experience and empirical  knowledge of the problem may allow tuning the method for affordable costs on large  practical problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' It may also provide a path to overcoming the confusion that impedes  the search for optimal configurations in combinatorial problems where many subopti-  mal configurations overlap and confuse any local search, such as coloring or maximum  clique [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  The extra search power that parallel tempering seems to provide suggests that there  may be other ways to exploit the original insight of Edwards and Anderson, that or-  dering in random systems might show up as a correlation between behavior observed  in multiple copies or regions of a complex system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Replica symmetry breaking implies  ordering which may take on a range of values, and study of the correlations seen in  multiple replicas at low temperatures may add physical insight into such ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' A  further step could be to model systems which are basically similar but subject to weak  local environmental corrections with a set of replicas, each slightly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' The ob-  jective would be to identify from the differences between solutions in each replica a  functional variation of the response of ordering to the different local influences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Such  local variations are typical of much of the vast amounts of social data available for  study in today’s highly instrumented world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' This is just one more possible, but as yet  unrealized, impact of spin glasses on applied mathematics and statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  The original SK Ising model with Gaussian-distributed interactions is only one of  the situations which has stimulated further exploration to tease out the lowest energy  ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Different ways of combining different tools of optimization have proved  valuable in this and different lessons emerge, as covered in the further sections of this  chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  January 3, 2023 1:34  ws-rv10x7-10x7  Replica Symmetry Breaking & Far Beyond  chapter1  page 6  6  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Caracciolo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Hartmann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Kirkpatrick, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Weigel  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Focusing on Ground States  Consider the spin-glass (SG) model in zero field with Hamiltonian [2]  H = −  �  ⟨i,j⟩  Jijsisj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='2)  We focus on Ising spins si = ±1 placed on the vertices of a graph, but other symmetries  of the order parameter have been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' The Ising symmetry has the advantage  that the related statistical physics problems are combinatorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' The sum ⟨i, j⟩ runs over  all edges of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' For the Edwards-Anderson (EA) model the graph is a hyper-  cubic lattice in d dimensions, but also other graph structures like the fully connected  Sherrington-Kirkpatrick model corresponding to the d → ∞ limit of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='2) have been  studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  As discussed in Chapter 1, the physics of this mean-field problem is well  understood, in many aspects since the 1980s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' The situation in low dimensions is less  clear and so much of the work in recent decades has focused on studying the problem  in 2, 3 and 4 dimensions with numerical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  Next to Monte Carlo simulations used to study the vicinity of the spin-glass tran-  sition and the behavior in the ordered phase (see Chapter 5), much work has been  invested in studying ground states and low-lying excitations above them in order to  gauge the low-temperature behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Alternative theories for the nature of the spin-  glass phase make contrasting predictions about the energetic and geometric properties  of such excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Based on generalizations of Peierls’ argument for the stability of the  ferromagnetic phase, researchers have argued that excitations with energies diverging  with length will lead to spin-glass phases that are stable against thermal fluctuations,  such that one of the prime goals of studying ground states for spin-glass systems is to  understand the nature of the prevalent low-energy excitations and how their sizes and  energies are related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  The search for ground states for spin glasses with discrete symmetry amounts to  a combinatorial optimization problem as there is a countable number of candidate so-  lutions [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Such problems can hence be solved by a brute-force enumeration of all  configurations, with an effort that grows exponentially with the number of spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' As  discussed in section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='3, a clever organization of this enumeration can lead to massive  gains in efficiency of this process, but will not alter the overall exponential scaling of  the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' This corresponds to the fact that the problem in d ≥ 3 is known to be NP  hard [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' In some special cases, however, the ground-state problem can be mapped onto  auxiliary optimization problems that permit solutions in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' This will be  explained in section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Beyond the ground states of spin glasses, many optimization  problems have caught the attention of physicists [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' An example for a recently  studied model is the assignment problem, which is discussed in section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Exact Algorithms for hard problems  Here, exact and general algorithms for finding ground states (GSs) for Ising SGs are con-  sidered, although the presented methods are very general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' As an extension of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='2),  here also an interaction with local fields hi is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' For most studied systems no  January 3, 2023 1:34  ws-rv10x7-10x7  Replica Symmetry Breaking & Far Beyond  chapter1  page 7  Simulated annealing, optimization, searching for ground states  7  local field is present, but for some of the algorithms presented below, such a term might  arise for sub-problems to be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' In this case, the energy of a SG is given by the  Hamiltonian  H(s) = −  �  i<j  Jijsisj −  �  i  hisi ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='3)  where s = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' , sN) is a configuration and the bonds Jij, corresponding to the edges  of the graph, are quenched random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' The sum runs here, in the most general  case, over all pairs of spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Thus, if all bonds are nonzero, the system is of mean-  field type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' For finite-dimensional lattices, most bonds are zero, suitably selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' The  nonzero bonds are typically drawn from a Gaussian or a bimodal Jij = ±1 distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  Here, three types of algorithms are presented, which have frequently been used to  calculate exact GSs for three-dimensional SGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' First, the branch-and-bound method is  explained, which is based on enumerating many states in a sophisticated way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Next, the  linear programming approach is outlined, which is based on rewriting the Hamiltonian  as a linear function, relaxing the condition si = ±1, plus adding additional constraints,  called cutting planes or cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Finally, the combination of both approaches, the branch-  and-cut algorithm is presented, which yields the currently fastest exact method to obtain  spin-glass ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Branch-and-bound  The basic idea of the branch-and-bound approach [24] to find the minima of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='3) is  to represent all spin configurations as a binary tree, where at each node the configuration  space is, for a node-dependent selected spin i0, subdivided into the configurations with  si0 = +1 and those with si0 = −1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' The spin configurations are given by  the 2N leafs of this tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' The simplest approach to the GS problem would be to obtain  all configurations by enumeration, and simply pick those with the minimum energy,  yielding an O(2N) running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  This running time can be improved, although still being exponential in the worst  case, by omitting parts of this tree via considering bounds on the achievable energies  in sub-trees [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Here we present the refined algorithm described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' The  branching is performed always on the last spin of the sub-problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=', it starts with  spin N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' The energies for the sub problems with sN = +1 and sN = −1 can be written  using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='3) for s = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' , sN−1) as follows:  H+(s) = −  �  i<j  ′  Jijsisj −  �  i  ′  hisi −  �  i  ′  JiNsi − hN ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='4)  H−(s) = −  �  i<j  ′  Jijsisj −  �  i  ′  hisi +  �  i  ′  JiNsi + hN ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='5)  where the sums �′ run from 1 to N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' If one defines  H∗  N−1 =  min  s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=',sN−1 −  �  i<j  ′  Jijsisj ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='6)  January 3, 2023 1:34  ws-rv10x7-10x7  Replica Symmetry Breaking & Far Beyond  chapter1  page 8  8  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Caracciolo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Hartmann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Kirkpatrick, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Weigel  one obtains, using the relation mins(f1(s)+f2(s)) ≥ mins f1(s)+mins f2(s), the bounds  H+(s) ≥ H∗ +  min  s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=',sN−1  �  i  ′  (−hi − Jin)si − hn ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='7)  H−(s) ≥ H∗ +  min  s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=',sN−1 −  �  i  ′  (−hi + Jin)si + hn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='8)  These bounds are available because the minima H∗  N−1, H∗  N−2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' can be obtained recur-  sively while the branching tree is built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Furthermore, the minimum of a linear function  �  s  �  i aisi, here ai = −hi − Jin or ai = −hi + Jin, respectively, is simply given by  − �  i |ai|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Thus, as a first way to restrict the size of the branching tree, if one or two  of the branches exhibit bounds above a known threshold, the corresponding branches  can be omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Such thresholds may come from low lying configurations found already  in other branches or by heuristic algorithms, or simply given as part of the problem in  case an enumeration of a specified range of low-lying configurations is sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  A second type of bound [26] works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Using d(s) = −2�  i  ′JiNsi − 2hN one  can rewrite Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='4) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='5) as H+(s) = H−(s) + 2d(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Therefore, we obtain the  implications  max  s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=',sN−1 d(s) = 2  �  i  |JiN| − 2hN ≤ 0 =⇒ H+(s) ≥ H−(s) ,  min  s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=',sN−1 d(s) = −2  �  i  |JiN| − 2hN ≥ 0 =⇒ H+(s) ≤ H−(s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  These bounds are easy to calculate and allow sometimes for omitting one of the two  branches, before any branch has to be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  Such branch-and-bound algorithms have been used, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=', to analyze the low-  temperature landscape [27–30] of SGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' More broadly, in the field of statistical mechanics  of optimization problems [22, 23], the branch-and-bound approach has been applied to  other problems like the satisfiability problem [31] or the vertex-cover problem [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' For  the latter one, a branch-and-bound approach was used, where the variable to branch  on was not selected in a given order but determined by a local heuristics for further  reduction of the branching tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' For simple variants of these algorithms, it has also been  possible to calculate analytically the typical running time for ensembles of random prob-  lems, which exhibit transitions between typically polynomial and typically exponential  behavior [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Linear programming and cutting planes  A different approach works by translating the quadratic Hamiltonian into a linear prob-  lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' For convenience, here we consider the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='2) Note that the field term  present in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='3) can be written as quadratic term by introducing a “ghost spin”  s0 = 1 matching the given equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  We describe the system by a graph G = (V, E) where V denotes the set of sites  where the spins are located and E the set of edges {i, j} between the interacting sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  For any set V ′ ⊂ V in G the cut δ(V ′) ⊂ E denotes the set of edges where one endpoint  is in V ′ and the other is not, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=', those connecting the two sets V ′ and V \\ V ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' For  January 3, 2023 1:34  ws-rv10x7-10x7  Replica Symmetry Breaking & Far Beyond  chapter1  page 9  Simulated annealing, optimization, searching for ground states  9  each configuration, the spins can be partitioned into two sets V + = {i|si = +1} of “up”  spins and V + = {i|si = −1} = V \\ V + of “down” spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' If two spins si and sj are  oriented identically, they contribute the energy −Jij = +0 − Jij, while they contribute  the energy Jij = 2Jij − Jij if they are oriented differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Thus, using the cut, we can  write  H(s) = 2  �  {i,j}∈δ(V +)  Jij −  �  ij  Jij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='9)  By introducing cij = −Jij, S = �  ij Jij and variables xij = 1 if {i, j} ∈ δ(V +) and  xij = 0 else, this reads  H(x) = −2  �  i,j  cijxij − S ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='10)  where �  i,j cijxij is called the weight of the cut represented by x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Therefore, since S  is only a constant, finding the minimum energy of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='2) corresponds to finding the  maximum cut in a graph with edge weights cij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='10) is a linear function  with integer variables, which means we have transformed the quadratic optimization  problem into a linear one, but with additional constraints since x must describe a cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  This problem is called integer linear program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  To include the constraints describing the cut, we note that for any cycle in the graph  G the cut must be crossed an even number of times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' This can be conveniently described  by linear inequalities [35] for the variables x as follows: For any cycle C ⊂ E, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=', a  path which starts and ends at the same vertex, and any odd cardinality subset Q ⊂ C,  �  {i,j}∈Q  xij −  �  {i,j}∈C\\Q  xij ≤ |Q| − 1  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='11)  must hold, which defines a cut comprehensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  The basic idea of the cutting-plane approach is now to relax the variables xij = 0, 1  to 0 ≤ xij ≤ 1 and look for a maximum of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='10) given the linear inequalities  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' The name cutting plane comes from the fact that all inequalities describe  hyper-planes which cut off one part from the space of possible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' The resulting  problem is called a linear program (LP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' The good news is that LPs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=', with the relaxed  variables, can be solved [36] in worst-case polynomial time in the problem size using the  ellipsoid method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Still, in a practical context methods like the simplex approach [37] or  the dual simplex approach, which exhibit no polynomial bound, perform much faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  Still, the bad news is that the number of inequalities is in principle exponentially large,  which would yield an exponential running time right-on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Therefore, instead of adding all  constraints immediately to the LP, one starts with no or few constraints and calculates  a first solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Since fewer constraints than necessary are contained in the relaxed  problem, typically the obtained cut weight will be higher, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=', an upper bound of the  true maximum cut for the integer LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Hence, the obtained solution will typically not  correspond to a cut and not be pure integer-valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  Thus, there may be some of  the exponentially many inequalities which are violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' The second basic idea of the  cutting plane approach is to look specifically for violated inequalities, without searching  all possible ones, and add the violated ones to a growing set of inequalities included in  January 3, 2023 1:34  ws-rv10x7-10x7  Replica Symmetry Breaking & Far Beyond  chapter1  page 10  10  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Caracciolo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Hartmann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Kirkpatrick, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Weigel  the LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' An exact polynomial algorithm to find violated inequalities is based on solving  a series of shortest-path problems [35], which can be done in O(N 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' This is polynomial  but rather slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Fortunately, there exist, in particular for physical lattice structures,  several heuristics which run in linear O(N) time [38] and are most of the time sufficient  to generate additional inequalities for violated conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' If at some point no further  violated inequalities are found and the solution is fully integer-valued, a true optimum  has been found, and the algorithm stops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' But if on the contrary some variables are still  non-integer, one can resort to branching as explained in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  Nevertheless, for some combinatorial problems it has been observed that a cutting  plane approach alone may lead to a valid optimum solution of the integer problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  For example for vertex-cover problems on Erd˝os-R´enyi random graphs a different kind  of cycle inequalities has been considered [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' When varying the average number c of  neighbor nodes in the graphs, a phase transition has been observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' In the thermody-  namic limit N → ∞, for small values c < c∗ = e ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='71, typically all instances can be  solved completely in a polynomial running time, while for larger values of c this is not  possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Interestingly, this easy-hard transition coincides with the critical connectivity  where replica-symmetry breaking of the vertex-cover problem occurs [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' At this point  a complex structure of the solution space has been observed [40], where the so-called  leaf-removal core [41] starts to percolate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  Finally, note that there is a set of cutting planes, so called Gomory cuts [42], which  can be constructed generally for all relaxed linear problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' They are based on iden-  tifying violated inequalities directly from a given non-integer solution, actually in the  so-called tableau [36] used by the simplex algorithm to solve the LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' It is proven that  this, possibly exponentially large, set of inequalities is complete, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' any integer pro-  gramming problem can be solved in principle by generating just these cutting planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  Still, in practice, any finite numerical accuracy leads to convergence problems, such  that this and many other cutting-plane approaches turned out to be actually efficient  in combination with branching, as explained next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Branch-and-cut  If for a relaxed linear system describing maximum cuts the solution is still non-integer,  one considers for some variable xij ̸= 0, 1 both possibilities xij = 0 and xij = 1 and  solves the corresponding sub-problems recursively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  This means one branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  The  combination of these approaches is therefore called branch-and-cut [38, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Note that  here several other bounds, in addition to those mentioned above, can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' This  can be lower bounds obtained from the cut weight of any valid cut x∗ or upper bounds  obtained from suitable relaxations like the LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  Currently, the branch-and-cut approach can be considered as the most powerful exact  algorithm to obtain GSs for hard SG instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' A publicly accessible implementation  of a branch-and-cut algorithm is the spin-glass server [44] hosted by the University of  Bonn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' The server was originally implemented at the University of Cologne by several  members and collaborators of the research groups of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Liers and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' J¨unger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' At the  server, you can submit SG instances as a file and, if the system is not too large, a  corresponding GS will be returned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  January 3, 2023 1:34  ws-rv10x7-10x7  Replica Symmetry Breaking & Far Beyond  chapter1  page 11  Simulated annealing, optimization, searching for ground states  11  Jij  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  Mapping of the Ising ground-state problem to a matching problem on an auxiliary graph with  Kasteleyn cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Left: expansion of the Ising lattice, replacing each vertex by a complete graph K4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  Edge weights in the K4 subgraphs are set to 0, the remaining weights are Jij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Middle: A minimum-  weight perfect matching on the auxiliary graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Right: Back-transformation from the decorated graph  to the original lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' The matching then results in a set of closed loops separating up from down  spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  Branch-and-cut approaches have been applied to finite-dimensional SGs in several  occasions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' To our knowledge the largest three-dimensional instances were N = 123 as  considered in a study of low-lying excitations [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  The approach has also been applied for a mean-field random-bond Ising model,  which is a generalization of the standard SG with a variable fraction of negative bond  as controlled by a non-zero mean of the bond values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Here, a phase transition of the  typical branch-and-cut running time, as measured by the number of LPs solved, between  an easy and a hard phase has been observed near the transition between ferromagnetic  and SG behavior [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Ground states in two dimensions  We now turn to the case of ground-state problems permitting a polynomial-time solu-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' For the EA model of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='2) this is the case in dimensions d < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Since the 1d  problem is rather trivial, the much more interesting case of this type is the EA model  in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  A relevant mapping of the EA ground state to a minimum-weight perfect matching  (MWPM) problem on an auxiliary graph was first proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' It is based on  the observation that frustrated plaquettes [48], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=', elementary lattice faces including  an odd number of antiferromagntic couplings, must have an odd number of broken  bonds with Jijsisj < 0, while non-frustrated plaquettes have an even number of broken  bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Hence a spin configuration can be depicted as a configuration of defect lines of  broken bonds that start and end at frustrated plaquettes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' A ground state configuration  is then a perfect pairing (matching) of frustrated plaquettes through defect lines of  minimum total weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' MWPM can be solved in polynomial time based on the blossom  algorithm [49] and its variants [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  However, this approach is still not ideal as it  operates on the complete graph of the F frustrated plaquettes with F(F −1) edges, and  the edge weights need to be computed in a preparatory step using a suitable approach  January 3, 2023 1:34  ws-rv10x7-10x7  Replica Symmetry Breaking & Far Beyond  chapter1  page 12  12  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Caracciolo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Hartmann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Kirkpatrick, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Weigel  such as Dijkstra’s algorithm [51] before attacking the matching problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' This approach  allows one to study systems of a typical maximum linear size of L ≈ 500 [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  As was shown more recently [53, 54], an alternative mapping is significantly more  efficient as it operates on a sparse graph with similar connectivity as the original lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  It relies on an auxiliary graph that replaces each vertex of the lattice by a complete  graph K4 of four nodes, also known as Kasteleyn city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Edge weights are set to Jij for  the original edges and to zero for the internal K4 bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' The solution of a matching  problem on this graph then corresponds to a set of closed loops on the original lattice,  separating domains of opposite spin orientations, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' the illustration in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' The  configuration of minimum weight corresponds to a ground state of the spin-glass sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  Due to the sparsity of the auxiliary graph, significantly larger systems can be studied  with this method as compared to the one proposed in [47], and calculations for square  lattices up to L = 10 000 have been reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  A widely applied method for studying the excitations out of the ground states that  take a central role in the theory of the spin-glass phase, consists of systematic modifi-  cations of boundary conditions (BCs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' It is argued that the defect energy connected to  a change from periodic to antiperiodic BCs in one direction,  Edef = |EP − EAP|,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='12)  can act as a proxy for a typical low-energy excitation of the system [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Here, EP and  EAP refers to the ground-state energy for periodic and antiperiodic BCs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  One expects that Edef scales as [57, 58],  Edef ∼ Lθ  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='13)  with the spin-stiffness exponent θ, where θ > 0 should indicate the stability of the  spin-glass phase at non-zero temperatures, while for θ < 0 the transition temperature  TSG = 0 and θ = −1/ν governs the divergence of the spin-glass correlation length as  T → TSG [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  For Gaussian exchange couplings Jij one finds a stiffness exponent  θ ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='3 [52], with the most accurate estimate being [55],  θ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='2793(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='14)  This is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='3(a) showing the scaling of defect energies over a wide range  of system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' The change of boundary conditions induces a domain-wall defect that  spans the system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' a typical configuration of the overlap between the ground states for  periodic and antiperiodic BCs is shown in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' The boundary of  the flipped domain is a fractal curve, and the domain-wall length is hence expected to  show fractal scaling of the form  ⟨ℓ⟩J = AℓLdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='15)  As is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='3(b), this is indeed borne out in the data to high accuracy,  and the fractal dimension is estimated as df = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='27319(9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  For technical reasons the matching approach can only handle samples on planar  graphs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=', lattices with periodic boundary conditions in at most one direction [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  More precisely, runs for systems with fully periodic boundaries yield the same result for  January 3, 2023 1:34  ws-rv10x7-10x7  Replica Symmetry Breaking & Far Beyond  chapter1  page 13  Simulated annealing, optimization, searching for ground states  13  10−1  100  101  102  103  104  (a)  Gauss  PFBC  θ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='2793(3)  ⟨|∆E|⟩J  L  100  101  102  103  104  105  106  107  101  102  103  104  (b)  Gauss  PFBC  df = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='273 19(9)  ⟨ℓ⟩J  L  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  (a) Scaling of defect energies Edef for the Gaussian EA model on the square lattice with  periodic boundaries in x direction and free boundaries in y direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' The line shows a fit of the data to  the functional form Edef = AθLθ +Cθ/L2 [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' (b) Scaling of the domain-wall length between periodic  and antiperiodic BCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' The line corresponds to a fit of the functional form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='15) to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  periodic and for antiperiodic BCs in each of the two directions, such that the configu-  ration returned is a ground state for one out of four possible BCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' As pointed out in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' [53], the approach hence effectively optimizes over BCs as well as spin variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  To circumvent this problem and allow treatment of systems with fully periodic BCs  with the resulting smaller scaling corrections, one may use a windowing technique as  illustrated in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='4: since MWPM can be used to find exact ground  states for planar graphs, in order to treat a periodic L × L system one applies it to a  square subset of edge length L − 2 (“window”) while keeping the relative orientation of  the outside spins fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Randomly displacing the window location over the (periodic)  lattice, repeated applications of the window optimizations lead to a quick convergence of  the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' This prescription results in a stochastic algorithm, whose success probability  can be arbitrarily improved by using m independent runs,  Ps({Jij}) = 1 − [1 − Pn({Jij})]m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='16)  Numerically, one finds that the number of required repetitions for a given success prob-  ability is independent of system size, such that the computational complexity of the  algorithm remains the same as that of the MWPM approach for planar graphs, which  scales as Lκ with κ ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='2 [55] for the Blossom V algorithm [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  The system with bimodal couplings can also be studied with matching techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  Here, one finds no asymptotic decay of Edef, but a convergence to a positive limiting  value as L → ∞ [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' While this was initially taken as evidence for a lower-critical  dimension dl = 2, it was later on realized that it is rather a signature of an additional  zero-temperature renormalization-group fixed point that is not relevant for the physics  at non-zero temperatures [60] and, instead, there is evidence for dl = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='5 [61, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' A  signature of this system is the extensive degeneracy of the ground state, leading to  a finite ground-state entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' This creates a difficulty for the matching approach as  it does not pick the individual ground states with equal probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' In its simplest  form, it is deterministic and will hence always return the same ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Some  January 3, 2023 1:34  ws-rv10x7-10x7  Replica Symmetry Breaking & Far Beyond  chapter1  page 14  14  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Caracciolo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Hartmann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Kirkpatrick, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Weigel  L  L  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  Left: The domain wall separating regions of equal and opposite ground-state configurations  for a specific disorder sample considered with periodic and with antiperiodic boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Right: Setup  used for the windowing technique to compute ground states for samples with fully periodic boundary  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  simple randomizations through changing the order of considering bonds or adding some  noise onto the couplings improve on this, but still lead to biased sampling methods [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  Unbiased sampling can be achieved via a suitably constructed Monte Carlo sampling  technique in the ground-state manifold, thus allowing for estimates of the domain-wall  fractal dimensions and related quantities [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  Matching techniques can also be used to sample other excitations than the domain-  wall perturbations induced by a change of boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  Different types of  droplet-shaped excitations can be studied by fixing the relative orientation of spins  through “hard” bonds [63, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Also, while matching is the most widely used technique  for this problem, an alternative approach based on a calculation of the partition function  through the use of Pfaffians allows to also study finite-temperature properties exactly  and in polynomial time [65–67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Recently, such methods have been extended to also  enable the study of correlation functions and further related properties [68–70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' The assignment problem  In this section we will discuss some new results in the assignment problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' In order to  define the question in its simplest version we have to consider a square matrix of size  n, W := {wij}n  i,j=1, with real entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' For each permutation π in the symmetric group  Sn consider the matrix Π := {πij}n  i,j=1 with entries zero or one, such that  πij = δj,π(i) =  �  1  if π(i) = j  0  elsewhere  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='17)  LIJanuary 3, 2023 1:34  ws-rv10x7-10x7  Replica Symmetry Breaking & Far Beyond  chapter1  page 15  Simulated annealing, optimization, searching for ground states  15  and ask for a permutation of the columns of W in order to get a minimum of the trace,  that is of the Hamiltonian  H(Π, W) := tr  �  W T · Π  �  =  n  �  i,j=1  wijπij =  n  �  i,j=1  wijδj,π(i) =  n  �  i=1  wi,π(i) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='18)  Without loss of generality we can take the entries of W to be nonnegative: wij ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  Indeed, in combinatorial optimization they are usually referred to as costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' In different  words, this is the classical matching problem on the bipartite complete graph Kn,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' The  solution of the problem is a permutation π∗, with corresponding matrix Π∗(W) , that  realizes the minimum cost and brings to the determination of  H∗(W) := min  π∈Sn H(Π, W) = H (Π∗(W), W)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='19)  the optimal value of the total cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' The optimal solution can be seen as the ground state  of the Hamiltonian of a disordered system defined by the cost-matrix W, an analogy  which can be made useful, for example, by simulated annealing [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  The introduction of a probability on the space of the possible costs allows the in-  vestigation of the properties of the typical solutions and the artillery from statistical  physics shows its whole power [22, 23, 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' In their seminal works M´ezard and Parisi  [72–74] (but see also [75]) could solve, by using the replica trick, the matching, the  assignment and the Travelling Salesman Problem in the case in which the entries wij  are independent random variables, equally distributed, with a probability distribution  density of the form  ρ(w) = wr  ∞  �  k=0  ηkwk  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='20)  with η0 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' More precisely, in the asymptotic limit of an infinitely large size n, replica  symmetry is not broken, and the optimal solution is determined by the application  of the saddle-point method as the solution of an integral equation in which only the  parameter r enters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' In particular  En := E [H∗(W)] ∼ n1−  1  r+1 ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='21)  where the expectation value is taken on the possible weights and with ∼ we indicate  that both sides of the relation scale with large n in the same way so that their ratio  converges in the limit of an infinite number of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' For a recent work in which the  first finite-size corrections are reconsidered after [76, 77] and the extension to non-integer  value of the parameter r see [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  A new class of problems arises when the vertices of the graph Kn,n are identified  with points in Ω ⊂ Rd seen as a subset of a Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' We thus have two sets of  points of cardinality n: we denote them as the red points R, respectively blue points  B, with positions xi ∈ Rd, respectively yi ∈ Rd, with i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' , n} = [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Then the  cost of the assignment of the i-th red point with the j-th blue point is assumed to be  a function of the Euclidean distance between the two points |xi − yj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' We shall restrict  to the cases  wij = f(|xi − yj|) = |xi − yj|p  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='22)  January 3, 2023 1:34  ws-rv10x7-10x7  Replica Symmetry Breaking & Far Beyond  chapter1  page 16  16  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Caracciolo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Hartmann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Kirkpatrick, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Weigel  parametrized by the real number p ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Let us introduce the empirical probability  measures associated to the two sets of points  ρR(x) = 1  n  n  �  i=1  δ(x − xi) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  ρB(y) = 1  n  n  �  i=1  δ(y − yj)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='23)  and look, for p ≥ 1, at the p-th Wasserstein (elsewhere associated to the names of Kan-  torovich and Rubinstein) distance of two probability measures, that is at the variational  problem  Wp(ρ1, ρ2) :=  �  inf  γ∈Γ(ρ1,ρ2)  �  dx dy γ(x, y) |x − y|p  � 1  p  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='24)  where Γ(ρ1, ρ2) is the set of measures on Ω×Ω with marginals ρ1 and ρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' This is nothing  but the optimal transport (or Monge-Kantorovich) problem of a unit mass distributed  according to ρ1 to a distribution ρ2 [79, 80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' A function γ(x, y) defines a transportation  plan, indeed, the marginality conditions  �  dx γ(x, y) = ρ2(y) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  �  dy γ(x, y) = ρ1(x)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='25)  constraining the mass moved into the point y and from the point x are what is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  In the case of the two empirical probability measures the possible transportation plans  reduce to the set of permutations and therefore their Wasserstein distance is simply  related to the optimal cost [81]  H∗(W) = n W p  p (ρR, ρB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='26)  A random matrix, whose elements depend on the Euclidean distance between points  randomly distributed in space is called Euclidean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' The spectra of Euclidean random  matrices have been studied in [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  In order to fix the ideas, let us consider the case in which Ω = [0, 1]d, periodic  boundary conditions are chosen, so that we are really on a torus of unit volume and the  positions of red and blue points are taken at random with flat probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' A natural  length-scale is therefore n− 1  d and one can simply assume that for each red point there  is at least a blue point to match in a ball of radius of order n− 1  d , so that we can guess  that  En ∼ n1− p  d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='27)  But it is well known [83] that this can be true only in d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' It was proven that in d = 2  En ∼ n  �log n  n  � p  2  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='28)  a logarithmic violation appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' A much more detailed analysis is possible in d = 1 [84]  where an intriguing relation with Brownian processes is explored [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' It is shown that,  for a generic distribution probability ρ with cumulative Φ, the average total cost is  En = n1− p  2 2p  √π Γ  �p + 1  2  � � 1  0  dx[Φ(x)(1 − Φ(x))]  p  2  ρp−1(x)  + o(n− p  2 )  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='29)  January 3, 2023 1:34  ws-rv10x7-10x7  Replica Symmetry Breaking & Far Beyond  chapter1  page 17  Simulated annealing, optimization, searching for ground states  17  at least when the integral is convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Otherwise an anomalous scaling emerges [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  In the case of open boundary conditions, with flat distribution, the average cost can be  evaluated even for finite size by means of Selberg integrals [87]  En = n Γ  �  1 + p  2  �  p + 1  Γ(n + 1)  Γ  �  n + 1 + p  2  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='30)  Also the case p ⩽ 0 has been studied [88], where, instead, for flat distribution  lim  n→∞  En  n = 1  2p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='31)  When 0 < p < 1 the cost function becomes concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' In [89] it is argued that  En ∼  �  �  �  �  �  �  �  n1−p  for 0 < p < 1  2 ,  √n log n  for p = 1  2 ,  √n  for 1  2 < p < 1 ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='32)  so that a change in the phase diagram occurs at p = 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  An amusing exact result has been obtained for the flat distribution in d = 2 for the  particular value p = 2, in [90, 91], that is  lim  n→∞  En  log n = 1  2π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='33)  This has been rigorously proven in [92] (see also [93] for improvements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Let us follow  the field theoretic approach introduced in [94] and let us introduce the vector transport  field µ(x) which joins the red point in x to the blue point in y = x + µ(x) and consider  the Lagrangian  L[µ, φ] := 1  2  �  µ2(x)ρR(dx) +  �  [φ(x + µ(x))ρR(dx) − φ(x)ρB(dx)]  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='34)  to be minimized, where the scalar field φ(x) is a Lagrangian multiplier which implements  a matching between blue and red points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' In the limit of a large number of points n, when  the red and blue points are extracted with the same distribution ρ so that δρ := ρR−ρB  and the optimal µ goes to zero and we expect a good approximation by using the only  the quadratic terms in the Lagrangian, that is  L[µ, φ] :=  � �1  2µ2(x) + µ(x) · ∇φ(x)ρ(dx)  �  +  �  φ(x)δρ(dx) ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='35)  which has Euler-Lagrangian equations  µ = −∇φ ,  ∇ · [ρ µ] = δρ ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='36)  revealing a strict analogy with an electrostatic problem where µ plays the role of the  electric field, φ is the scalar potential and indeed is the Lagrangian multiplier which  implements the Gauss law, red and blue points have opposite unit charge, being null  the total charge, while ρ is the dielectric function of a linear dielectric medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' As a  consequence  − ∇ · [ρ ∇φ] = ρ  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='37)  January 3, 2023 1:34  ws-rv10x7-10x7  Replica Symmetry Breaking & Far Beyond  chapter1  page 18  18  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Caracciolo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Hartmann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Kirkpatrick, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Weigel  is solved by means of the classical Green’s function Gρ(x, y) of the operator −∇·[ρ ∇•],  so that an explicit approximate solution at fixed disorder is given by  µ(x) =  �  ∇xGρ(x, y) δρ(dy) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='38)  After averaging over disorder, in the simple case of a flat measure (see [95] for non-  constant case) we get  lim  n→∞ En(Ω) = −2 tr ∆−1  Ω ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='39)  where the Laplacian ∆Ω is defined on the domain Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' This formula is correct in d = 1 but  it simply provides a divergence on both sides for d > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' It is an ultraviolet divergence  which has been introduced by the linearization in the infinite number of modes, but the  approximation cannot be true at very short distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' In the exact non-linear theory  higher modes are cut off and we expect that  En(Ω) = 2  � ∞  0+  F  � λ  n  �  λ  dNΩ(λ) ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='40)  where NΩ(λ) is number of the eigenvalue less than λ for the Laplace operator and the  unknown function F interpolates between 1 for λ ≲ n and 0 for λ ≳ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  By the Weyl law [96] on the asymptotics of the eigenvalue counting function for the  Laplace-Beltrami operator we know that, for a 2-d manifold with unit volume (under  Neumann boundary conditions which are appropriate for our problem)  NΩ(λ) = 1  4π  �  λ +  √  λ |∂Ω|  �  + o  �√  λ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='41)  In d = 2 it is easy now to evaluate the leading logarithmic singularity in the number of  points and in agreement with [97] we expect that  En(Ω) = 1  2π log n + 2c∗(n) + 2cΩ + o(1) ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='42)  where c∗(n) = O(log n) is a universal function not depending on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  As a further  consequence,  lim  n→∞ [En(Ω) − En(Ω′)] = 2 lim  n→∞  � ∞  0+  F  � λ  n  �  λ  [dNΩ(λ) − dNΩ′(λ)] ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='43)  but the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' is convergent even in the absence of regularization thus  lim  n→∞ [En(Ω) − En(Ω′)] = 2  � ∞  0+  dNΩ(λ) − dNΩ′(λ)  λ  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='44)  an expression that has been tested in [98] on various 2d manifolds, by using both the  Green’s function method as other classical tools as the Dedekind’s limit formulas, thus  confirming the predictions of the field theoretic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  For a similar method applied to a more general context see [99–101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  Once more this relatively simple combinatorial optimization problem is revealing  intriguing and fruitful connections with so many different research fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
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+page_content='sciencemag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
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+page_content='long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
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+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
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+page_content=' Slides of lecture  CS6700 at Cornell University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
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+page_content=' Accessed:  2022-05-04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
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+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
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+page_content='  [88] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
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+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' D’Achille, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
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+page_content=' doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
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+page_content='96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='042102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  [89] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Caracciolo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' D’Achille, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
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+page_content=' Sportiello, The Dyck bound in the  concave 1-dimensional random assignment model, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' 53(6),  064001 (25pp), (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
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+page_content=' Caracciolo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Lucibello, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Parisi, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Sicuro, Scaling hypothesis for the Euclidean  bipartite matching problem, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
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+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='1103/PhysRevE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='012118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' URL https://journals-aps-org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='pros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='lib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='unimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
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+page_content='  1103/PhysRevE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='012118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
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+page_content=' Sicuro, Scaling hypothesis for the Euclidean bipartite matching  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Correlation functions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
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+page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='1103/PhysRevE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='062125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' URL https://journals-aps-org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
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+page_content='lib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='unimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  it/pre/abstract/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='1103/PhysRevE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
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+page_content=' Trevisan, A PDE approach to a 2-dimensional match-  ing problem, Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Theory Relat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' 173, 433–477, (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
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+page_content='  Ambrosio,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  Glaudo,  and  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content='  Trevisan,  On  the  optimal  map  in  the  2-  dimensional random matching problem,  Discrete Cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
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+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
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+page_content='  [100] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfyfkg/content/2301.00683v1.pdf'}
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+Local Inequities in the Relative Production of
+and Exposure to Vehicular Air Pollution in Los
+Angeles*
+Geof Boeing, Yougeng Lu, and Clemens Pilgram†
+University of Southern California
+Abstract
+Vehicular air pollution has created an ongoing air quality and public health
+crisis. Despite growing knowledge of racial injustice in exposure levels, less is
+known about the relationship between the production of and exposure to such
+pollution. This study assesses pollution burden by testing whether local popula-
+tions’ vehicular air pollution exposure is proportional to how much they drive.
+ThroughaLosAngeles, Californiacasestudyweexaminehowthisrelatestorace,
+ethnicity, and socioeconomic status—and how these relationships vary across
+the region. We fnd that, all else equal, tracts whose residents drive less are ex-
+posed to more air pollution, as are tracts with a less-White population. Com-
+muters from majority-White tracts disproportionately drive through non-White
+tracts, compared totheinverse. Decadesofracially-motivatedfreewayinfrastruc-
+ture planning and residential segregation shape today’s disparities in who pro-
+duces vehicular air pollution and who is exposed to it, but opportunities exist
+for urban planning and transport policy to mitigate this injustice.
+1. Introduction
+Twentieth century planners designed and constructed an enormous network of express-
+waystoopenupgrowingAmericanmetropolisestomotorists. Vastswathsofestablished
+*Citation info: Boeing, G., Y. Lu, and C. Pilgram. 2023. Local Inequities in the Relative Production
+of and Exposure to Vehicular Air Pollution in Los Angeles. Urban Studies, published online ahead of
+print. doi:10.1177/00420980221145403
+†Corresponding author’s email: boeing@usc.edu
+1
+arXiv:2301.00440v1  [stat.AP]  1 Jan 2023
+
+urban neighborhoods were bulldozed to clear new channels for suburban residents to
+drive to job centers. Yet some older neighborhoods survived relatively unscathed. For
+example, in Los Angeles, local residents organized to protest and eventually successfully
+cancel plans to extend State Route 2 through the afuent communities of Beverly Hills
+and Los Angeles’s westside (Perez, 2017). In contrast, similar grassroots eforts failed
+in Los Angeles’s eastside, where several major freeways carved up its less-afuent and
+less-White neighborhoods (Estrada, 2005). Race, wealth, and political power shaped
+the infrastructure planning that determines regional accessibility, travel behavior, and
+pollution exposure today.
+Exposure to air pollution from cars, especially particulate matter of 2.5 microns or
+smaller (PM2.5), poses a direct risk to human health (Habre et al., 2021; Thompson, 2018;
+Sarzynski, 2012). Much of the environmental justice literature around air pollution
+emphasizes community disparities in exposure, focusing on residential proximity to
+toxic release sites, refneries, road infrastructure, etc (Mikati et al., 2018; Schweitzer and
+Valenzuela, 2004; Reichmuth, 2019; Liévanos, 2019). These justice claims are straight-
+forward when identifying a stationary emissions source, like a refnery: the refnery
+exposes nearby residents to harmful pollutants through a spatial difusion process. Most
+empirical studies measure environmental risk through residential proximity to such
+stationary sources by assuming that proximity is a proxy for exposure (Pastor et al., 2001;
+Yuan, 2018).
+However, measuring exposure to emissions from mobile sources, such as cars, is
+less straightforward. Transport type and density along with local topography and mete-
+orological conditions infuence the degree, direction, and distance that air pollutants
+disperse (Lu et al., 2021; Houston et al., 2004). The political and planning decisions
+behind transport infrastructure placement impact environmental justice today through
+vehicular air pollution (Giles-Corti et al., 2022). Importantly, the polluting vehicles
+may be driven by people with diferent sociodemographic profles than the communi-
+ties they traverse and pollute. Despite our growing knowledge of pollution exposure
+injustice (Mikati et al., 2018; Liévanos, 2019; Schindler and Caruso, 2021), we know
+less about the relationship between who generates vehicular air pollution and who is
+exposed to that pollution. The individuals exposed to the most pollution may simply
+be producing the most themselves. A stronger measure of environmental justice would
+ask: how much vehicular air pollution are you exposed to in relation to how much you
+drive?
+This study asks this question in the context of race, class, and driving in Los Angeles
+County. Our data and methods are not unique to Los Angeles and can be applied else-
+where in the US. We focus on driving as roughly four in fve of the county’s commuters
+drive to work by themselves and another one in ten carpool (County of Los Angeles
+Open Data, 2019). We model local exposure to vehicular air pollution as a function of
+vehicle kilometers traveled (VKT), accounting for the roles of race, income, and other
+2
+
+covariates to better understand disparate infrastructure utilization and impacts. Using
+ordinary least squares (OLS) and geographically-weighted regression (GWR) we fnd
+that—all else equal—tracts that generate more vehicular travel tend to be exposed to
+less vehicular air pollution, and that non-White and poor communities are dispropor-
+tionately burdened with excess pollution. Our GWR models in particular allow us to
+explain substantially more of this variation than the OLS models that are standard in
+the environmental justice literature. To illustrate how vehicular air pollution disparities
+unfold spatially, we simulate car commutes and develop a novel inequity index that
+measures the extent to which commutes through a tract are disproportionately made by
+drivers of a certain race or ethnicity. We fnd that commuters from majority-White tracts
+disproportionately drive through non-White tracts, compared to the inverse. Moving
+beyond the original placement of infrastructure, we conclude by discussing possible
+interventions such as VKT taxes, tolls, emissions standards, and electrifcation.
+2. Background
+Urban motor vehicle transport generates air pollution that can harm human health.
+This relationship is mediated by the historical placement and current utilization of
+transport infrastructure by motorists.
+2.1. Driving, Emissions, and Health
+Manystudieshaveidentifedenvironmentalexternalitiesstemmingfromurbantransport—
+particularly motor vehicles—including greenhouse gas emissions and air, water, and
+noise pollution. Delucchi (2000) estimated motor vehicles’ environmental externalities
+and found that, compared to other environmental hazards, air pollution’s costs are the
+greatest—particularly for human health.
+Researchers have identifed cars and trucks as the most signifcant sources of urban
+air pollution (Karner et al., 2010; Rowangould, 2015; Pan et al., 2013) and several recent
+studies have quantifed the air pollution gradient near highways (Brugge et al., 2007;
+Seagram et al., 2019; Zhu et al., 2002). The US Environmental Protection Agency (EPA)
+notes that transport accounts for a large share of PM2.5 emissions (US EPA, 2021). For
+example, in Southern California, vehicular emissions alone generate roughly a third
+of the region’s PM2.5 (Habre et al., 2021; Hasheminassab et al., 2014), and transport
+produces 37% of California’s GHG emissions—more than any other sector in the state
+(California Air Resources Board, 2020).
+Air pollution correlates with an increased risk of lung cancer, asthma, bronchitis,
+and impaired cognitive development (Sunyer et al., 2015; Huang et al., 2019; Jagai et al.,
+2017). A large body of literature has documented the deleterious efects of PM2.5 expo-
+sure in particular (Feng et al., 2016; Thompson, 2018; Polichetti et al., 2009). Several
+3
+
+studies have demonstrated that children exposed to high-trafc roadways sufer from
+more respiratory ailments (Gauderman et al., 2005; Janssen et al., 2001; Oosterlee et al.,
+1996). Huang et al. (2019) found that higher daily PM2.5 concentrations correspond to
+higher pediatric respiratory rates. PM2.5 is associated with increased childhood asthma
+hospitalization (Lee et al., 2006) and exposure to vehicular emissions can both cause
+and exacerbate asthma (Künzli et al., 2003). Researchers have also found that children
+exposed to more vehicular air pollution demonstrate slower cognitive development
+(Sunyer et al., 2015). Further, air pollution exposure is associated with lung cancer
+incidence, especially among people who have never smoked (Jagai et al., 2017).
+2.2. Race and Class Disparities
+These health impacts pose an urgent problem that is unevenly distributed across popu-
+lations. Schweitzer and Valenzuela (2004) argue that transport justice fundamentally
+concerns the distribution of transport costs and benefts. Environmental hazards dis-
+proportionately impact non-White and low-income individuals relative to Whiter and
+more afuent ones (Bae et al., 2007; Liévanos, 2019; Houston et al., 2004; Tessum
+et al., 2021). In particular, non-White and low-income communities are disproportion-
+ately burdened by transport externalities while simultaneously sufering from worse
+accessibility (Schweitzer and Valenzuela, 2004; Rowangould et al., 2016; Yuan, 2018;
+Poorfakhraei et al., 2017). In tandem, this suggests that these communities do not
+beneft from transport infrastructure in proportion to the costs they bear.
+Recent studies have used many diferent methods to measure racial inequity in
+air pollution exposure, including descriptive analysis (Brunt et al., 2017; Milojevic
+et al., 2017), bivariate analysis (Rivas et al., 2017; Moreno-Jimenez et al., 2016), and
+multivariate analysis (Temam et al., 2017; Padilla et al., 2014; Kim and Kwan, 2021).
+Substantial heterogeneity exists across such studies in spatial scale (e.g., city, county,
+state), socioeconomic characteristics under consideration, and research design (e.g.,
+cross-sectional or panel data). Yet one consistent fnding in this literature has been
+that socially disadvantaged populations tend to be burdened with more air pollution
+exposure than other groups are.
+The confuence of transport, health, and racial injustice is exemplifed by Los Ange-
+les, a poster child for automobile dependence, air pollution, racial diversity, and wealth
+inequality. According to the US Census Bureau, in 2019, 74% of Los Angeles County
+residents were non-White. Latinos1 (49% of the total population) are the single largest
+group followed by Whites (26%), Asians (15%), and Blacks (9%). The region’s poverty
+rate is consistently higher than the national average. In Los Angeles, non-White and
+low-income populations are more likely to live in central neighborhoods for better
+accessibility and are exposed to twice the local trafc density as the rest of the region
+(Giuliano, 2003; Houston et al., 2004). Despite this exposure, they are less likely to own
+4
+
+anautomobileandmorelikelytocommutebypublictransit(PolicyLink,2017). Inother
+words, non-White and low-income populations live near more transport infrastructure
+yet beneft less from it.
+Decades of racist planning decisions in Los Angeles have contributed to today’s
+injustices in transport, health, and environmental quality. Twentieth century transport,
+housing, and land use planning accommodated White fight and industry at the expense
+of non-White communities (Estrada, 2005; Baum-Snow, 2007). Planners bulldozed
+central neighborhoods to construct an expansive freeway system and create regional
+access for new peripheral suburbs and their predominately White residents (Pulido,
+2000; Boeing, 2021)—often selecting freeway alignments through non-White neigh-
+borhoods for the sake of “slum clearance” or to protect business interests (Mohl, 2004;
+DiMento, 2009; Perez, 2017). Freeway construction fragmented and displaced local res-
+idents and subjected surrounding communities to decades of subsequent air pollution
+(Mohl, 2004). While some freeways were planned through Whiter and more afuent
+neighborhoods, they were often cancelled or rerouted due to these neighborhoods’
+political clout—such as with the aforementioned State Route 2 through Beverly Hills
+(Perez, 2017).
+3. Methods
+The literature suggests that some groups disproportionately beneft from automobility
+whileothergroupsdisproportionatelybearitsexternalcosts. Thepresentstudyadvances
+this literature through a Los Angeles case study investigating the relative production of
+and exposure to vehicular air pollution, using neighborhood-scale aggregations. Are
+diferent communities exposed to vehicular air pollution at a level proportional to how
+much they drive? If not, what is the relationship between race and this disparity, all
+else equal? Such knowledge can help guide equitable planning eforts, target restorative
+policy interventions, and set specifc environmental justice goals.
+3.1. Data
+This study focuses on Los Angeles County (Figure 1). Its sprawling structure, residential
+segregation, and transport infrastructure exemplify the spatial legacy of twentieth cen-
+tury American planning: today roughly 90% of its commuters drive or carpool (County
+of Los Angeles Open Data, 2019). We collect secondary data on air pollution, passenger
+vehicle travel, freight truck trafc, demographics, and street network design. We follow
+the literature by using census tract-level aggregations (e.g., Tayarani and Rowangould,
+2020; Lu, 2021; Nyhan et al., 2016), as fully disaggregate pollution exposure data exist
+only for sample populations and are generally unavailable to the public. Tracts ofer a
+5
+
+Figure 1. Los Angeles County’s vicinity and highway infrastructure.
+useful unit of analysis as they roughly represent neighborhoods and follow real-world
+physical and social boundaries. Table 1 describes our input data.
+We use PM2.5 concentrations from the Union of Concerned Scientists (UCS) as
+the primary air pollution exposure indicator. This data set combines emissions from
+on-road sources from the US EPA Emissions Inventory with the InMAP air pollution
+generation and transport model to estimate annual tract-level concentrations: Reich-
+muth (2019) and Tessum et al. (2017) detail this model and these data. This allows us
+to analyze local exposure to vehicular PM2.5. As a robustness check, we separately use
+CO2 emissions from on-road sources from the Database of Road Transport Emissions
+(DARTE), in place of PM2.5 concentrations (Gately et al., 2019). While the PM2.5 data
+measure concentration as the “stock” of air pollution, the CO2 data measure tailpipe
+emissions as the “fow” of vehicular CO2 into the air at a 1-kilometer resolution. Ex-
+amining this alternative “fow” indicator checks the robustness of our primary “stock”
+indicator’s results and provides insights into exposure to accumulated emissions by
+incorporating information on how pollutants disperse.
+We use tract-level resident VKT generation from the US Bureau of Transportation
+Statistics’ Local Area Transportation Characteristics for Households (LATCH) to
+proxy vehicular air pollution production. This data set combines National Household
+6
+
+Lancaster
+VENTURA
+SAN BERNARDINO
+COUNTY
+Santa Clarita.
+COUNTY
+San Fernando
+Valley
+Pasadena
+101)
+15
+2
+Hollywood Hills
+East Los
+SanGabriel
+1013
+405
+Malibu
+Valley
+10
+Angeles
+West Los
+Downtown
+Angeles
+605
+South Los
+Norwalk
+Angeles 105
+RIVERSIDE
+Legend
+COUNTY
+91
+Freeways
+ORANGE
+Other Major Roads
+COUNTY
+Long'Beach
+Urban Areas
+Palos Verdes
+405
+Peninsula
+ County BordersTable 1. Descriptions, units, and sources of tract-level variables. Source abbreviations
+are defined in the main text.
+Variable
+Description
+Units
+Source
+PM2.5
+annual average on-road PM2.5
+concentration (2014)
+µg/m3
+UCS
+CO2
+annual average road transport
+CO2 emissions (2014)
+metric tons
+DARTE
+VKT
+average daily vehicle travel gen-
+erated by median household
+(2017)
+kilometers
+LATCH
+Truck trafc volume
+daily distance traveled through
+tract by commercial trucks
+(2020)
+kilometers
+SCAG
+Distance to highway
+distance from tract centroid to
+nearest highway (2019)
+kilometers
+Census
+TIGER/Line
+Intersection density
+number of intersections per
+square kilometer (2020)
+intersections/km2
+OpenStreetMap
+Grade Mean
+average street segment incline
+(2020)
+n/a
+OpenStreetMap
+Proportion White
+proportion of White residents
+(2019)
+n/a
+ACS
+Median
+household
+income
+annual median household in-
+come (2019)
+10,000s of infation-
+adjusted 2018 USD
+ACS
+Proportion single-family
+proportion of housing units
+that are single-unit detached
+(2019)
+n/a
+ACS
+Median rooms per home
+median rooms per housing
+unit (2019)
+rooms
+ACS
+Mean household size
+average number of residents
+per housing unit (2019)
+residents
+ACS
+Population density
+residents per square kilometer
+(2019)
+1000s
+of
+residents/km2
+ACS
+Median home value
+median
+value
+of
+owner-
+occupied housing units (2019)
+10,000s of infation-
+adjusted 2018 USD
+ACS
+7
+
+Travel Survey responses with demographic data from the American Community Survey
+(ACS) to model household travel behavior and provide tract-level aggregations: the
+Bureau of Transportation Statistics (2018) provides full methodological and validation
+details. Some temporal mismatch exists between the UCS PM2.5 data’s 2014 vintage and
+the LATCH VKT data’s 2017 vintage. However, individual travel behavior and urban
+form change little over such a time span (Ramezani et al., 2021).
+We simulate commuting trips (as detailed in Section 3.3) to determine driving routes,
+by race, through diferent tracts using the US Census Bureau’s Longitudinal Employer-
+Household Dynamics (LEHD) Origin-Destination Employment Statistics (LODES).
+LODES is an administrative enumeration that includes private-sector employee home
+and work census blocks, ofering a rough approximation, with some bias, of commute
+origins and destinations (Boeing, 2018). We aggregate these origins and destinations to
+the tract-level to match the rest of the input data.
+We use modeled truck trafc volumes from the Southern California Association of
+Governments (SCAG) to control for freight truck trafc (SCAG, 2010). We measure
+tract-level intersection density and street grade using OpenStreetMap data and the
+OSMnx software to control for diferences in local street network structure (Boeing,
+2017). We use intersection density as a measure of street density and street grade as a
+measure of hilliness, which afects engine performance. We also collect tract-level race
+and income data from the 2018 5-year ACS, alongside a set of additional control variables
+summarized in Table 1. Some tracts lack observations across some of these variables.
+For consistent analysis across multiple specifcations, we retain only those tracts with
+observations across all variables in Table 1 (n=2238). Only 4% of the county’s tracts are
+thus excluded, but we note that they contain more multifamily housing, non-White
+residents, and lower income households than the retained tracts.
+3.2. Regression Analysis
+We estimate four regression models, two via OLS and two via GWR. Using OLS, we
+estimate the frst two models as specifed in Equation 1:
+log φ = β0 + β1 log τ + βX + ϵ
+(1)
+where log φ is the response (log PM2.5 concentration), β0 is the intercept, log τ is
+the log VKT generated by residents and β1 is its coefcient to be estimated, X is a matrix
+of n observations on k predictors and β is its vector of coefcients to be estimated, and
+ϵ is the error term.
+In Model 1, X includes a limited set of controls including the White proportion of
+the population, median household income, truck trafc volume, and distance to the
+nearesthighway. Wedefne“highway”asinterstatehighways, USroutes, andstateroutes,
+and measure the Euclidean distance between tract centroids and the nearest highway. In
+8
+
+Model 2, X includes a complete set of controls including all those from Model 1 as well
+as intersection density, mean street grade, proportion of single family homes, median
+number of rooms per home, mean household size, population density, and median
+home value. We log-transform predictors as needed for a linear relationship (as noted in
+Table 2) and estimate all models with robust standard errors due to heteroskedasticity.
+OLS estimates a global model with stationary parameters across the study region.
+However, such models cannot unpack spatially-varying statistical relationships. The
+literature suggests that air pollution exhibits spatial heterogeneity, but sophisticated
+models to investigate this remain rare in the literature. Given our interest in unpacking
+the potentially heterogeneous relationship between local PM2.5 concentration and VKT
+generated, we estimate GWR models of the general form specifed in Equation 2 to
+observe any such local variation in coefcients and goodness-of-ft:
+log φj = β0j + β1j log τj + βjXj + ϵj
+(2)
+where log φj is the response (log PM2.5 concentration in each tract j), β0j is the
+intercept, β1j is a local coefcient to be estimated, log τj is the log VKT generated
+by residents of j, βj is a vector of local coefcients to be estimated, Xj is a matrix of
+observations in the local neighborhood of j, and ϵj is the local error term. In Model
+3, Xj includes the limited set of controls from Model 1. In Model 4, Xj includes the
+complete set of controls from Model 2.
+GWR estimates separate models for the local neighborhood of each tract in the
+study region. For each such local regression, observations are weighted by a Gaussian
+distance-decay function centered on j. We use a spatially adaptive kernel that adjusts
+for the density of data at each regression location, due to the variability of census tract
+sizes across Los Angeles County. This determines a fxed number of nearest neighbors
+to adjust the bandwidth distance accordingly: tracts in dense areas have a narrower
+bandwidth and tracts in sparse areas have a wider one. We determine the optimal
+number of nearest neighbors according to the Akaike Information Criterion with
+small-sample correction (AICc) through iterative optimization. This results in 53 and
+73 nearest neighbors for each local estimation of Models 3 and 4, respectively.
+3.3. Commute Simulation
+To examine demographic diferences in driving patterns, we use block-level home and
+employment locations from LODES as a proxy to simulate commuters’ routes to work.
+Although this is a microsimulation of commutes, we are interested in its aggregate
+outcomes rather than any specifc individual commute. We solve routes by minimizing
+free-fow travel time, via Dijkstra’s algorithm and OSMnx (Boeing, 2017), from home
+blockcentroidsto work blockcentroids. We identify, atthe hometract level, all the other
+tracts through which its source trips pass. We assign each trip as White or non-White
+9
+
+probabilistically given the home tract’s White population proportion, and adjust the
+trip counts by the share of commuters who drive to work from each tract. Then we sum
+the total simulated kilometers driven through each tract, by White versus non-White
+commuters.
+Finally, to investigate trends in the racial disparity of commutes through each tract,
+we developed the inequity index defned in Equation 3:
+Ijg = Djg
+Dj
+− Cjg
+Cj
+(3)
+where the index Ijg quantifes the extent to which commuter vehicular distances
+traveled through tract j are disproportionately made by subgroup g (either White or
+non-White). Djg is the distance traveled through j by g, and Dj is the total distance
+traveled through j by all subgroups. Cjg is the number of g commuters living in j, and
+Cj is the total number of commuters living in j. Positive and negative values of Ijg
+indicate, respectively, disproportionately high and low distances traveled by subgroup
+g through tract j.
+4. Results
+4.1. OLS Results
+Table 2 reports the OLS results for Models 1 and 2, revealing a signifcant negative
+relationship between a tract’s vehicular PM2.5 concentration and its residents’ VKT
+generation. Controlling for race, income, truck trafc, and highway proximity in Model
+1, a 1% increase in VKT generation is associated with a 1.24% decrease in local PM2.5
+exposure. The full set of controls in Model 2 moderates the magnitude, but a 1%
+increase in VKT is still associated with a 0.62% decrease in local PM2.5 exposure. All
+else equal, tracts that generate more vehicular travel tend to be exposed to less vehicular
+air pollution—an important paradox we unpack in the discussion below.
+Both Models 1 and 2 reveal a signifcant negative relationship between a tract’s
+vehicular PM2.5 concentration and the White proportion of the population. Even
+when controlling for income, home value, truck trafc, and other covariates, Whiter
+tracts tend to be exposed to less vehicular air pollution. In other words, non-White
+communities, whether high-income or low-income, are exposed to more PM2.5 than
+otherwise-similar White communities. These models demonstrate that vehicular air
+pollution burdens distribute inequitably with regard to race across Los Angeles County
+as a whole.
+10
+
+Table 2. Regression model parameter estimates for Model 1 (basic OLS), Model 2 (OLS with full controls), Model 3 (basic GWR), and
+Model 4 (GWR with full controls). Standard error is shown in parentheses. Significance is denoted as * p < 0.05.
+(1)
+(2)
+(3)
+(4)
+estimate
+estimate
+mean
+min
+max
+t<-1.96
+t>1.96
+mean
+min
+max
+t<-1.96
+t>1.96
+Intercept
+6.13*
+1.31*
+2.02
+-1.58
+11.00
+2.32%
+70.60%
+1.37
+-1.98
+4.84
+2.32%
+57.42%
+(0.29)
+(0.38)
+VKT (log)
+-1.24*
+-0.62*
+-0.23
+-2.79
+0.68
+38.52%
+8.40%
+-0.18
+-1.46
+0.54
+33.91%
+6.52%
+(0.08)
+(0.10)
+Proportion White
+-0.61*
+-0.63*
+-0.14
+-1.36
+0.76
+32.17%
+20.38%
+-0.03
+-0.95
+1.22
+27.84%
+24.98%
+(0.06)
+(0.07)
+Median household income
+0.04*
+0.01
+-0.01
+-0.09
+0.25
+38.87%
+11.66%
+0.00
+-0.06
+0.10
+8.76%
+7.95%
+(0.01)
+(0.01)
+Truck trafc volume (log)
+-0.01*
+0.01
+0.00
+-0.05
+0.05
+6.88%
+13.94%
+0.01
+-0.05
+0.07
+7.86%
+21.94%
+(0.01)
+(0.00)
+Distance to highway
+-0.12*
+-0.09*
+-0.05
+-0.15
+0.02
+81.14%
+0.09%
+-0.04
+-0.12
+0.02
+76.68%
+0.09%
+(0.01)
+(0.01)
+Intersection density
+-0.00
+0.00
+-0.01
+0.00
+11.21%
+11.17%
+(0.00)
+Grade mean (log)
+-0.06*
+-0.04
+-0.39
+0.14
+37.85%
+19.71%
+(0.01)
+Prop single-family
+0.44*
+0.07
+-0.96
+0.83
+3.66%
+19.84%
+(0.07)
+Median rooms per home
+-0.22*
+-0.05
+-0.44
+0.12
+43.21%
+7.19%
+(0.02)
+Mean household size
+0.09*
+0.04
+-0.20
+0.33
+10.90%
+38.20%
+(0.02)
+Population density (log)
+0.07*
+0.02
+-0.10
+0.19
+1.30%
+19.57%
+(0.02)
+Median home value (log)
+0.66*
+0.05
+-0.41
+1.19
+15.19%
+23.82%
+(0.03)
+R2
+0.38
+0.54
+0.67
+0.00
+0.94
+0.74
+0.00
+0.97
+11
+
+Re-estimating these models using (log) on-road CO2 emissions as the response tells
+a similar story. A 1% increase in VKT generation is associated with a 0.52% decrease in
+local CO2 exposure, and 0.38% when including the full set of controls. A signifcant
+negative relationship is similarly found between the tracts’ CO2 exposure and their
+White population proportion.
+4.2. GWR Results
+Table 2 also reports the GWR results for Models 3 and 4 by summarizing the distribu-
+tions of their coefcients and goodness-of-ft measures. The GWR models perform
+much better than the OLS models due to spatial heterogeneity in the relationships
+studied. While Model 1 explains only 38% of the response’s variance, Model 3 (with
+the same predictors) explains 67% of its local variance on average. Similarly, Model 2
+explains 54% of the response’s variance, but Model 4 explains 74% locally on average.
+These GWR models unpack the spatial variation of individual predictors’ rela-
+tionships across the region. For example, in Model 1, the White proportion of the
+population is (uniformly) signifcantly and negatively associated with vehicular PM2.5
+exposure. However, Model 3 reveals a more nuanced and spatially varying relationship.
+Its coefcients for the White proportion of the population range from -1.36 to 0.76
+with a mean value of -0.14 (see Table 2). The relationship between these variables is not
+stationary across our study region, though it tends to be negative on average—similar to
+what Models 1 and 2 suggest globally. One potential explanation for this spatial variation
+could be sorting: people choose to live in diferent places for diferent reasons, including
+heterogeneous distaste for exposure to emissions and diferences in how they value
+accessibility to other locations. These variations in coefcient signifcance and direction
+highlight the importance of assessing these relationships locally as well as globally.
+Figure 2 depicts Model 4’s spatial distributions of local t-statistics and R2 values
+across the study region. The combined statistical relationship of the predictors with
+the response varies spatially and this fgure illustrates where the GWR models ofer a
+better ft than the OLS models could. Overall, the GWR R2 values improve on the
+OLS values in most tracts.
+The consistency of these model results across specifcations lends confdence in their
+estimates. In Model 3, 39% of tracts show a signifcant, negative relationship between
+resident VKT generation and vehicular PM2.5 exposure, while only 8% of tracts show
+a signifcant, positive relationship. With the full set of controls in Model 4, 34% of
+tracts show a signifcant, negative relationship between resident VKT generation and
+vehicularPM2.5 exposure, whileonly7%oftractsshowasignifcant, positiverelationship.
+Figure 2a illustrates where these relationships tend to be negative, including the low-
+income Latino eastside of Los Angeles and the high-income White communities along
+12
+
+Figure 2. Spatial distribution of Model 4’s GWR local t-statistics for a) log VKT, b)
+proportion White, c) median household income, d) log truck traffic volume, e) distance to
+highway, and f) local R2.
+the oceanfront. Such areas demonstrate the greatest injustice as those who drive more
+are exposed to less pollution, and vice versa.
+4.3. Commute Simulation
+The commute simulation results reveal unequal patterns in driving between White
+and non-White commuters which help explain the aforementioned exposure disparity.
+Figure 3 depicts the inequity index’s spatial distribution. Positive values indicate that
+commuters residing in a tract are less White than the commuters driving through
+the tract, while negative values indicate the opposite. It does not show where driving
+occurs: most driving is constrained to highway corridors (and thus non-White areas).
+Accordingly, more trips traverse the fgure’s red tracts than blue tracts.
+Countywide, the inequity index’s population-weighted mean is 0.0153, revealing
+that most tracts experienced disproportionately more travel by White commuters rela-
+tive to the local racial composition. This disparity is more than twice as large in tracts
+with highways (0.0307) than in those without (0.0147). Figure 3 illustrates the roles
+of residential segregation and highway placement in today’s driving patterns. Ma-
+jor highways stand out in particular. Tracts adjacent to Interstate 10—an east-west
+backbone—have a population-weighted mean of 0.0957, while those adjacent to Inter-
+states 110 and 105—the largest freeways through predominately Black and Latino South
+13
+
+a.
+a
+d
+e
+0.8
+0.6
+0.4
+0.2Figure 3. Tract-level inequity index: positive values indicate that the share of White
+commuters traversing the tract exceeds the share of White residents living in the tract.
+Negative values indicate the opposite.
+Los Angeles—have a value of 0.1288. Thus, commuters driving through these tracts
+are respectively 10 and 13 percentage points Whiter than the local resident population—
+particularly notable when considering that the countywide population is less than
+one-third White. Yet there are also examples of major highways on which commuters
+resemble or are less White than local residents. Tracts adjacent to Interstate 405 and
+US Route 101 have inequity index values of -0.0012 and -0.0999, respectively. Both
+highways traverse mountain passes through the majority-White, afuent Hollywood
+Hills and Santa Monica Mountains where alternative alignments were impossible or
+cost-prohibitive.
+14
+
+7
+0.4
+0.3
+Inequity Index to White Commuters
+0.2
+0.1
+0.0
+0.1
+-0.2
+-0.3
+-0.45. Discussion
+The research literature has explored race and class disparities in local vehicular air pol-
+lution exposure without accounting for local residents’ own contributions to their
+exposure. This study fnds that—all else equal—tracts whose residents drive less are
+exposed to more vehicular air pollution. Furthermore, tracts with a larger non-White
+population proportion—whether high- or low-income—experience more air pollution
+than do Whiter but otherwise similar tracts. Comparable trends hold in the GWR
+models on average as well. This reveals an injustice in pollution burden with a distinct
+racial dimension and these results are robust across multiple specifcations and both
+global and local estimations. Overall, our fndings reveal a systematic environmental
+injustice: road infrastructure benefts Whiter communities while exacting a cost on
+less-White communities.
+This study opens the door for future research. While the simulation analysis pro-
+vides evidence that unequal driving patterns could generate some of the observed dispar-
+ity, it does not rule out the infuence of other mechanisms such as tract-level diferences
+in wind patterns or fuel efciency. Future research should continue investigating these
+relationships. Nevertheless, our models consistently reveal the same broad story: tracts
+with a greater share of White residents and tracts with residents who drive more are
+exposed to less vehicular air pollution. One natural way to interpret this relationship
+is that people from majority-White neighborhoods do most of their “excess” driving
+through non-White neighborhoods and are thus exposed to less pollution at home
+despite producing more of it. However, there are other potential mechanisms that
+could explain the relationship. For example, majority-White neighborhoods could be
+located in areas where prevailing winds push pollution away and toward non-White
+neighborhoods, or majority-White neighborhoods’ residents could drive more fuel-
+efcient cars so that even though they drive more, they produce and are exposed to less
+pollution at home. However, the history of highway construction through non-White
+neighborhoods suggests that diferences in where travel occurs at least partially drive
+these results.
+Further, the commute simulation provides evidence that disparities in vehicular
+air pollution exposure at home result from where people drive. On average, White
+commuters traverse tracts that are far more non-White than the tracts where most White
+commuters live. This disparity does not exist in the opposite direction: on average,
+non-White commuters do not travel through tracts that are substantially Whiter than
+their home tracts. This also illustrates the role that highways play in mediating this
+disparity. When White commuters traverse non-White tracts, they do so predominantly
+through tracts that contain highways. In other words, White commuters receive the
+benefts of driving on a highway, but because those highways are predominantly in
+non-White neighborhoods, other racial groups bear external costs of that driving.
+15
+
+This disparity is less common in the opposite direction. On average, non-White
+commuters do not travel through tracts that are substantially Whiter than their home
+tracts—and even where they occasionally do, it is on freeway segments that follow
+alignments determined by topography. For example, by not building a freeway through
+Beverly Hills, planners caused a substantial share of commuters who would have taken
+this route to instead drive through majority non-White tracts on Interstate 10 instead.
+However, a similar rerouting of Interstate 405 or US Route 101 would not be possible
+without much greater and costlier detours. Transport planning has played a central—
+but not solo—role in these inequities: land use regulation, zoning, and housing policy
+interact with transport plans to generate these outcomes.
+Several policies can help mitigate this environmental injustice. First, policymakers
+could continue raising fuel efciency standards for new cars and encouraging vehicular
+electrifcation. Both would reduce on-road emissions without necessarily reducing the
+amount of driving. However, they would not eliminate rubber tires’ and brake dust’s
+substantial contributions to PM2.5 pollution. Second, policymakers could enact tolls
+or other forms of congestion taxes to reduce total driving or capture its externalities.
+Third, policymakers could discourage commuting altogether by incentivizing more
+people to work from home, such as through tax credits. The Covid-19 pandemic
+witnessed a surge in remote work. Continuing these work-from-home trends after
+the pandemic could reduce vehicular travel and air pollution, particularly by higher-
+income White collar workers with more fexible jobs. Finally, policymakers can address
+environmental injustice through the housing market. Permitting more residential
+construction in job-rich neighborhoods could reduce commute distances. Further,
+legalizing the construction of denser and more afordable housing in less-polluted,
+exclusive neighborhoods could reduce exposure disparities. Yet no single policy can
+eliminate longstanding systemic discrimination against low-income and non-White
+populations.
+6. Conclusion
+This paper extended quantitative research on transport-environmental justice through
+a spatial analysis of the production of and exposure to vehicular air pollution in Los
+Angeles. We assessed local exposure to vehicular air pollution adjusted by local produc-
+tion of VKT through four models and two estimation techniques. In particular, our
+GWR analysis allowed for local nuance to reveal spatial heterogeneity.
+We found that tracts that generate more vehicular travel tend to be exposed to less
+vehicular air pollution, all else equal. Race signifcantly predicts pollution exposure,
+even when controlling for a full set of related variables. Our commute simulation’s
+inequity index helps explain these fndings by illustrating the role of freeways and
+16
+
+residential sorting in commute patterns. On average, White commuters traverse tracts
+that are far more non-White than their home tracts, but non-White commuters do not
+travel through tracts that are substantially Whiter than their own. Dismantling decades
+of racially-motivated transport planning and segregation requires concerted efort by
+planners and policymakers to redress past harms and envision a more equitable future.
+Acknowledgments
+This research was supported by a grant from the US Department of Transportation
+and the Pacifc Southwest Region University Transportation Center. The authors wish
+to thank Peter Mannino, Nicholas Cerdera, and David Flores Moctezuma for research
+assistance.
+Notes
+1Throughout, we use “Latino” as shorthand for the Census Bureau’s designation of Hispanic or
+Latino of any race, “White” as shorthand for the Census Bureau’s designation of non-Hispanic or
+Latino White alone, and “Black” as shorthand for the Census Bureau’s designation of non-Hispanic
+or Latino Black or African-American alone.
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf,len=1100
+page_content='Local Inequities in the Relative Production of  and Exposure to Vehicular Air Pollution in Los  Angeles*  Geof Boeing, Yougeng Lu, and Clemens Pilgram†  University of Southern California  Abstract  Vehicular air pollution has created an ongoing air quality and public health  crisis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Despite growing knowledge of racial injustice in exposure levels, less is  known about the relationship between the production of and exposure to such  pollution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' This study assesses pollution burden by testing whether local popula-  tions’ vehicular air pollution exposure is proportional to how much they drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  ThroughaLosAngeles, Californiacasestudyweexaminehowthisrelatestorace,  ethnicity, and socioeconomic status—and how these relationships vary across  the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' We fnd that, all else equal, tracts whose residents drive less are ex-  posed to more air pollution, as are tracts with a less-White population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Com-  muters from majority-White tracts disproportionately drive through non-White  tracts, compared totheinverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Decadesofracially-motivatedfreewayinfrastruc-  ture planning and residential segregation shape today’s disparities in who pro-  duces vehicular air pollution and who is exposed to it, but opportunities exist  for urban planning and transport policy to mitigate this injustice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Introduction  Twentieth century planners designed and constructed an enormous network of express-  waystoopenupgrowingAmericanmetropolisestomotorists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Vastswathsofestablished  Citation info: Boeing, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Lu, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Pilgram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Local Inequities in the Relative Production  of and Exposure to Vehicular Air Pollution in Los Angeles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Urban Studies, published online ahead of  print.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='1177/00420980221145403  †Corresponding author’s email: boeing@usc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='edu  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='00440v1  [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='AP]  1 Jan 2023  urban neighborhoods were bulldozed to clear new channels for suburban residents to  drive to job centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Yet some older neighborhoods survived relatively unscathed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' For  example, in Los Angeles, local residents organized to protest and eventually successfully  cancel plans to extend State Route 2 through the afuent communities of Beverly Hills  and Los Angeles’s westside (Perez, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' In contrast, similar grassroots eforts failed  in Los Angeles’s eastside, where several major freeways carved up its less-afuent and  less-White neighborhoods (Estrada, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Race, wealth, and political power shaped  the infrastructure planning that determines regional accessibility, travel behavior, and  pollution exposure today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Exposure to air pollution from cars, especially particulate matter of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5 microns or  smaller (PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5), poses a direct risk to human health (Habre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Thompson, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Sarzynski, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Much of the environmental justice literature around air pollution  emphasizes community disparities in exposure, focusing on residential proximity to  toxic release sites, refneries, road infrastructure, etc (Mikati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Schweitzer and  Valenzuela, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Reichmuth, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Liévanos, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' These justice claims are straight-  forward when identifying a stationary emissions source, like a refnery: the refnery  exposes nearby residents to harmful pollutants through a spatial difusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Most  empirical studies measure environmental risk through residential proximity to such  stationary sources by assuming that proximity is a proxy for exposure (Pastor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Yuan, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  However, measuring exposure to emissions from mobile sources, such as cars, is  less straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Transport type and density along with local topography and mete-  orological conditions infuence the degree, direction, and distance that air pollutants  disperse (Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Houston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' The political and planning decisions  behind transport infrastructure placement impact environmental justice today through  vehicular air pollution (Giles-Corti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Importantly, the polluting vehicles  may be driven by people with diferent sociodemographic profles than the communi-  ties they traverse and pollute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Despite our growing knowledge of pollution exposure  injustice (Mikati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Liévanos, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Schindler and Caruso, 2021), we know  less about the relationship between who generates vehicular air pollution and who is  exposed to that pollution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' The individuals exposed to the most pollution may simply  be producing the most themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' A stronger measure of environmental justice would  ask: how much vehicular air pollution are you exposed to in relation to how much you  drive?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  This study asks this question in the context of race, class, and driving in Los Angeles  County.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Our data and methods are not unique to Los Angeles and can be applied else-  where in the US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' We focus on driving as roughly four in fve of the county’s commuters  drive to work by themselves and another one in ten carpool (County of Los Angeles  Open Data, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' We model local exposure to vehicular air pollution as a function of  vehicle kilometers traveled (VKT), accounting for the roles of race, income, and other  2  covariates to better understand disparate infrastructure utilization and impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Using  ordinary least squares (OLS) and geographically-weighted regression (GWR) we fnd  that—all else equal—tracts that generate more vehicular travel tend to be exposed to  less vehicular air pollution, and that non-White and poor communities are dispropor-  tionately burdened with excess pollution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Our GWR models in particular allow us to  explain substantially more of this variation than the OLS models that are standard in  the environmental justice literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' To illustrate how vehicular air pollution disparities  unfold spatially, we simulate car commutes and develop a novel inequity index that  measures the extent to which commutes through a tract are disproportionately made by  drivers of a certain race or ethnicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' We fnd that commuters from majority-White tracts  disproportionately drive through non-White tracts, compared to the inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Moving  beyond the original placement of infrastructure, we conclude by discussing possible  interventions such as VKT taxes, tolls, emissions standards, and electrifcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Background  Urban motor vehicle transport generates air pollution that can harm human health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  This relationship is mediated by the historical placement and current utilization of  transport infrastructure by motorists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Driving, Emissions, and Health  Manystudieshaveidentifedenvironmentalexternalitiesstemmingfromurbantransport—  particularly motor vehicles—including greenhouse gas emissions and air, water, and  noise pollution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Delucchi (2000) estimated motor vehicles’ environmental externalities  and found that, compared to other environmental hazards, air pollution’s costs are the  greatest—particularly for human health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Researchers have identifed cars and trucks as the most signifcant sources of urban  air pollution (Karner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Rowangould, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2013) and several recent  studies have quantifed the air pollution gradient near highways (Brugge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Seagram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' The US Environmental Protection Agency (EPA)  notes that transport accounts for a large share of PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5 emissions (US EPA, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' For  example, in Southern California, vehicular emissions alone generate roughly a third  of the region’s PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5 (Habre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Hasheminassab et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2014), and transport  produces 37% of California’s GHG emissions—more than any other sector in the state  (California Air Resources Board, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Air pollution correlates with an increased risk of lung cancer, asthma, bronchitis,  and impaired cognitive development (Sunyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Jagai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=',  2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' A large body of literature has documented the deleterious efects of PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5 expo-  sure in particular (Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Thompson, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Polichetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Several  3  studies have demonstrated that children exposed to high-trafc roadways sufer from  more respiratory ailments (Gauderman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Janssen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Oosterlee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=',  1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' (2019) found that higher daily PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5 concentrations correspond to  higher pediatric respiratory rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5 is associated with increased childhood asthma  hospitalization (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2006) and exposure to vehicular emissions can both cause  and exacerbate asthma (Künzli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Researchers have also found that children  exposed to more vehicular air pollution demonstrate slower cognitive development  (Sunyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Further, air pollution exposure is associated with lung cancer  incidence, especially among people who have never smoked (Jagai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Race and Class Disparities  These health impacts pose an urgent problem that is unevenly distributed across popu-  lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Schweitzer and Valenzuela (2004) argue that transport justice fundamentally  concerns the distribution of transport costs and benefts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Environmental hazards dis-  proportionately impact non-White and low-income individuals relative to Whiter and  more afuent ones (Bae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Liévanos, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Houston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Tessum  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' In particular, non-White and low-income communities are disproportion-  ately burdened by transport externalities while simultaneously sufering from worse  accessibility (Schweitzer and Valenzuela, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Rowangould et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Yuan, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Poorfakhraei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' In tandem, this suggests that these communities do not  beneft from transport infrastructure in proportion to the costs they bear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Recent studies have used many diferent methods to measure racial inequity in  air pollution exposure, including descriptive analysis (Brunt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Milojevic  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2017), bivariate analysis (Rivas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Moreno-Jimenez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2016), and  multivariate analysis (Temam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Padilla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Kim and Kwan, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Substantial heterogeneity exists across such studies in spatial scale (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', city, county,  state), socioeconomic characteristics under consideration, and research design (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=',  cross-sectional or panel data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Yet one consistent fnding in this literature has been  that socially disadvantaged populations tend to be burdened with more air pollution  exposure than other groups are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  The confuence of transport, health, and racial injustice is exemplifed by Los Ange-  les, a poster child for automobile dependence, air pollution, racial diversity, and wealth  inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' According to the US Census Bureau, in 2019, 74% of Los Angeles County  residents were non-White.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Latinos1 (49% of the total population) are the single largest  group followed by Whites (26%), Asians (15%), and Blacks (9%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' The region’s poverty  rate is consistently higher than the national average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' In Los Angeles, non-White and  low-income populations are more likely to live in central neighborhoods for better  accessibility and are exposed to twice the local trafc density as the rest of the region  (Giuliano, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Houston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Despite this exposure, they are less likely to own  4  anautomobileandmorelikelytocommutebypublictransit(PolicyLink,2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Inother  words, non-White and low-income populations live near more transport infrastructure  yet beneft less from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Decades of racist planning decisions in Los Angeles have contributed to today’s  injustices in transport, health, and environmental quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Twentieth century transport,  housing, and land use planning accommodated White fight and industry at the expense  of non-White communities (Estrada, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Baum-Snow, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Planners bulldozed  central neighborhoods to construct an expansive freeway system and create regional  access for new peripheral suburbs and their predominately White residents (Pulido,  2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Boeing, 2021)—often selecting freeway alignments through non-White neigh-  borhoods for the sake of “slum clearance” or to protect business interests (Mohl, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  DiMento, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Perez, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Freeway construction fragmented and displaced local res-  idents and subjected surrounding communities to decades of subsequent air pollution  (Mohl, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' While some freeways were planned through Whiter and more afuent  neighborhoods, they were often cancelled or rerouted due to these neighborhoods’  political clout—such as with the aforementioned State Route 2 through Beverly Hills  (Perez, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Methods  The literature suggests that some groups disproportionately beneft from automobility  whileothergroupsdisproportionatelybearitsexternalcosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Thepresentstudyadvances  this literature through a Los Angeles case study investigating the relative production of  and exposure to vehicular air pollution, using neighborhood-scale aggregations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Are  diferent communities exposed to vehicular air pollution at a level proportional to how  much they drive?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' If not, what is the relationship between race and this disparity, all  else equal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Such knowledge can help guide equitable planning eforts, target restorative  policy interventions, and set specifc environmental justice goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Data  This study focuses on Los Angeles County (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Its sprawling structure, residential  segregation, and transport infrastructure exemplify the spatial legacy of twentieth cen-  tury American planning: today roughly 90% of its commuters drive or carpool (County  of Los Angeles Open Data, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' We collect secondary data on air pollution, passenger  vehicle travel, freight truck trafc, demographics, and street network design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' We follow  the literature by using census tract-level aggregations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Tayarani and Rowangould,  2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Lu, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Nyhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2016), as fully disaggregate pollution exposure data exist  only for sample populations and are generally unavailable to the public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Tracts ofer a  5  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Los Angeles County’s vicinity and highway infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  useful unit of analysis as they roughly represent neighborhoods and follow real-world  physical and social boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Table 1 describes our input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  We use PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5 concentrations from the Union of Concerned Scientists (UCS) as  the primary air pollution exposure indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' This data set combines emissions from  on-road sources from the US EPA Emissions Inventory with the InMAP air pollution  generation and transport model to estimate annual tract-level concentrations: Reich-  muth (2019) and Tessum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' (2017) detail this model and these data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' This allows us  to analyze local exposure to vehicular PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' As a robustness check, we separately use  CO2 emissions from on-road sources from the Database of Road Transport Emissions  (DARTE), in place of PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5 concentrations (Gately et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' While the PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5 data  measure concentration as the “stock” of air pollution, the CO2 data measure tailpipe  emissions as the “fow” of vehicular CO2 into the air at a 1-kilometer resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Ex-  amining this alternative “fow” indicator checks the robustness of our primary “stock”  indicator’s results and provides insights into exposure to accumulated emissions by  incorporating information on how pollutants disperse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  We use tract-level resident VKT generation from the US Bureau of Transportation  Statistics’ Local Area Transportation Characteristics for Households (LATCH) to  proxy vehicular air pollution production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' This data set combines National Household  6  Lancaster  VENTURA  SAN BERNARDINO  COUNTY  Santa Clarita.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content="  COUNTY  San Fernando  Valley  Pasadena  101)  15  2  Hollywood Hills  East Los  SanGabriel  1013  405  Malibu  Valley  10  Angeles  West Los  Downtown  Angeles  605  South Los  Norwalk  Angeles 105  RIVERSIDE  Legend  COUNTY  91  Freeways  ORANGE  Other Major Roads  COUNTY  Long'Beach  Urban Areas  Palos Verdes  405  Peninsula  County BordersTable 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Descriptions, units, and sources of tract-level variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Source abbreviations  are defined in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Variable  Description  Units  Source  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5  annual average on-road PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='concentration (2014)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='µg/m3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='UCS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='CO2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='annual average road transport  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='CO2 emissions (2014)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='metric tons  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='DARTE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='VKT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='average daily vehicle travel gen-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='erated by median household  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='(2017)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='kilometers  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='LATCH  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='Truck trafc volume  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='daily distance traveled through  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='tract by commercial trucks  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='(2020)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='kilometers  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='SCAG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='Distance to highway  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='distance from tract centroid to  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='nearest highway (2019)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='kilometers  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='Census  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='TIGER/Line  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='Intersection density  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='number of intersections per  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='square kilometer (2020)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='intersections/km2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='OpenStreetMap  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='Grade Mean  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='average street segment incline  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='(2020)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='n/a  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='OpenStreetMap  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='Proportion White  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='proportion of White residents  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='(2019)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='n/a  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='ACS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='Median  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='household  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='income  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='annual median household in-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='come (2019)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='000s of infation-  adjusted 2018 USD  ACS  Proportion single-family  proportion of housing units  that are single-unit detached  (2019)  n/a  ACS  Median rooms per home  median rooms per housing  unit (2019)  rooms  ACS  Mean household size  average number of residents  per housing unit (2019)  residents  ACS  Population density  residents per square kilometer  (2019)  1000s  of  residents/km2  ACS  Median home value  median  value  of  owner-  occupied housing units (2019)  10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='000s of infation-  adjusted 2018 USD  ACS  7  Travel Survey responses with demographic data from the American Community Survey  (ACS) to model household travel behavior and provide tract-level aggregations: the  Bureau of Transportation Statistics (2018) provides full methodological and validation  details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Some temporal mismatch exists between the UCS PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5 data’s 2014 vintage and  the LATCH VKT data’s 2017 vintage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' However, individual travel behavior and urban  form change little over such a time span (Ramezani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  We simulate commuting trips (as detailed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='3) to determine driving routes,  by race, through diferent tracts using the US Census Bureau’s Longitudinal Employer-  Household Dynamics (LEHD) Origin-Destination Employment Statistics (LODES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  LODES is an administrative enumeration that includes private-sector employee home  and work census blocks, ofering a rough approximation, with some bias, of commute  origins and destinations (Boeing, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' We aggregate these origins and destinations to  the tract-level to match the rest of the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  We use modeled truck trafc volumes from the Southern California Association of  Governments (SCAG) to control for freight truck trafc (SCAG, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' We measure  tract-level intersection density and street grade using OpenStreetMap data and the  OSMnx software to control for diferences in local street network structure (Boeing,  2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' We use intersection density as a measure of street density and street grade as a  measure of hilliness, which afects engine performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' We also collect tract-level race  and income data from the 2018 5-year ACS, alongside a set of additional control variables  summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Some tracts lack observations across some of these variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  For consistent analysis across multiple specifcations, we retain only those tracts with  observations across all variables in Table 1 (n=2238).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Only 4% of the county’s tracts are  thus excluded, but we note that they contain more multifamily housing, non-White  residents, and lower income households than the retained tracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Regression Analysis  We estimate four regression models, two via OLS and two via GWR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Using OLS, we  estimate the frst two models as specifed in Equation 1:  log φ = β0 + β1 log τ + βX + ϵ  (1)  where log φ is the response (log PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5 concentration), β0 is the intercept, log τ is  the log VKT generated by residents and β1 is its coefcient to be estimated, X is a matrix  of n observations on k predictors and β is its vector of coefcients to be estimated, and  ϵ is the error term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  In Model 1, X includes a limited set of controls including the White proportion of  the population, median household income, truck trafc volume, and distance to the  nearesthighway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Wedefne“highway”asinterstatehighways, USroutes, andstateroutes,  and measure the Euclidean distance between tract centroids and the nearest highway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' In  8  Model 2, X includes a complete set of controls including all those from Model 1 as well  as intersection density, mean street grade, proportion of single family homes, median  number of rooms per home, mean household size, population density, and median  home value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' We log-transform predictors as needed for a linear relationship (as noted in  Table 2) and estimate all models with robust standard errors due to heteroskedasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  OLS estimates a global model with stationary parameters across the study region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  However, such models cannot unpack spatially-varying statistical relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' The  literature suggests that air pollution exhibits spatial heterogeneity, but sophisticated  models to investigate this remain rare in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Given our interest in unpacking  the potentially heterogeneous relationship between local PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5 concentration and VKT  generated, we estimate GWR models of the general form specifed in Equation 2 to  observe any such local variation in coefcients and goodness-of-ft:  log φj = β0j + β1j log τj + βjXj + ϵj  (2)  where log φj is the response (log PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5 concentration in each tract j), β0j is the  intercept, β1j is a local coefcient to be estimated, log τj is the log VKT generated  by residents of j, βj is a vector of local coefcients to be estimated, Xj is a matrix of  observations in the local neighborhood of j, and ϵj is the local error term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' In Model  3, Xj includes the limited set of controls from Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' In Model 4, Xj includes the  complete set of controls from Model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  GWR estimates separate models for the local neighborhood of each tract in the  study region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' For each such local regression, observations are weighted by a Gaussian  distance-decay function centered on j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' We use a spatially adaptive kernel that adjusts  for the density of data at each regression location, due to the variability of census tract  sizes across Los Angeles County.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' This determines a fxed number of nearest neighbors  to adjust the bandwidth distance accordingly: tracts in dense areas have a narrower  bandwidth and tracts in sparse areas have a wider one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' We determine the optimal  number of nearest neighbors according to the Akaike Information Criterion with  small-sample correction (AICc) through iterative optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' This results in 53 and  73 nearest neighbors for each local estimation of Models 3 and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Commute Simulation  To examine demographic diferences in driving patterns, we use block-level home and  employment locations from LODES as a proxy to simulate commuters’ routes to work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Although this is a microsimulation of commutes, we are interested in its aggregate  outcomes rather than any specifc individual commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' We solve routes by minimizing  free-fow travel time, via Dijkstra’s algorithm and OSMnx (Boeing, 2017), from home  blockcentroidsto work blockcentroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' We identify, atthe hometract level, all the other  tracts through which its source trips pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' We assign each trip as White or non-White  9  probabilistically given the home tract’s White population proportion, and adjust the  trip counts by the share of commuters who drive to work from each tract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Then we sum  the total simulated kilometers driven through each tract, by White versus non-White  commuters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Finally, to investigate trends in the racial disparity of commutes through each tract,  we developed the inequity index defned in Equation 3:  Ijg = Djg  Dj  − Cjg  Cj  (3)  where the index Ijg quantifes the extent to which commuter vehicular distances  traveled through tract j are disproportionately made by subgroup g (either White or  non-White).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Djg is the distance traveled through j by g, and Dj is the total distance  traveled through j by all subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Cjg is the number of g commuters living in j, and  Cj is the total number of commuters living in j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Positive and negative values of Ijg  indicate, respectively, disproportionately high and low distances traveled by subgroup  g through tract j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Results  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' OLS Results  Table 2 reports the OLS results for Models 1 and 2, revealing a signifcant negative  relationship between a tract’s vehicular PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5 concentration and its residents’ VKT  generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Controlling for race, income, truck trafc, and highway proximity in Model  1, a 1% increase in VKT generation is associated with a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='24% decrease in local PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5  exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' The full set of controls in Model 2 moderates the magnitude, but a 1%  increase in VKT is still associated with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='62% decrease in local PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5 exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' All  else equal, tracts that generate more vehicular travel tend to be exposed to less vehicular  air pollution—an important paradox we unpack in the discussion below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Both Models 1 and 2 reveal a signifcant negative relationship between a tract’s  vehicular PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5 concentration and the White proportion of the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Even  when controlling for income, home value, truck trafc, and other covariates, Whiter  tracts tend to be exposed to less vehicular air pollution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' In other words, non-White  communities, whether high-income or low-income, are exposed to more PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5 than  otherwise-similar White communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' These models demonstrate that vehicular air  pollution burdens distribute inequitably with regard to race across Los Angeles County  as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  10  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Regression model parameter estimates for Model 1 (basic OLS), Model 2 (OLS with full controls), Model 3 (basic GWR), and  Model 4 (GWR with full controls).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Standard error is shown in parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Significance is denoted as * p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  (1)  (2)  (3)  (4)  estimate  estimate  mean  min  max  t<-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='96  t>1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='96  mean  min  max  t<-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='96  t>1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='96  Intercept  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='13*  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='31*  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='02  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='58  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='32%  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='60%  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='37  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='98  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='84  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='32%  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='42%  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='29)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='38)  VKT (log)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='24*  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='62*  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='23  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='79  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='68  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='52%  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='40%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='18  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='46  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='54  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content='10)  Proportion White  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='61*  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content='76  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content='07)  Median household income  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='04*  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content='01)  Intersection density  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content='01)  Prop single-family  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='44*  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content='07)  Median rooms per home  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='22*  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content='02)  Mean household size  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='09*  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content='02)  Population density (log)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='07*  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content='02)  Median home value (log)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='66*  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content='19  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='19%  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content='03)  R2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='38  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content='74  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content='97  11  Re-estimating these models using (log) on-road CO2 emissions as the response tells  a similar story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' A 1% increase in VKT generation is associated with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='52% decrease in  local CO2 exposure, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='38% when including the full set of controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' A signifcant  negative relationship is similarly found between the tracts’ CO2 exposure and their  White population proportion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' GWR Results  Table 2 also reports the GWR results for Models 3 and 4 by summarizing the distribu-  tions of their coefcients and goodness-of-ft measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' The GWR models perform  much better than the OLS models due to spatial heterogeneity in the relationships  studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' While Model 1 explains only 38% of the response’s variance, Model 3 (with  the same predictors) explains 67% of its local variance on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Similarly, Model 2  explains 54% of the response’s variance, but Model 4 explains 74% locally on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  These GWR models unpack the spatial variation of individual predictors’ rela-  tionships across the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' For example, in Model 1, the White proportion of the  population is (uniformly) signifcantly and negatively associated with vehicular PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5  exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' However, Model 3 reveals a more nuanced and spatially varying relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Its coefcients for the White proportion of the population range from -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='36 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='76  with a mean value of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='14 (see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' The relationship between these variables is not  stationary across our study region, though it tends to be negative on average—similar to  what Models 1 and 2 suggest globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' One potential explanation for this spatial variation  could be sorting: people choose to live in diferent places for diferent reasons, including  heterogeneous distaste for exposure to emissions and diferences in how they value  accessibility to other locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' These variations in coefcient signifcance and direction  highlight the importance of assessing these relationships locally as well as globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Figure 2 depicts Model 4’s spatial distributions of local t-statistics and R2 values  across the study region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' The combined statistical relationship of the predictors with  the response varies spatially and this fgure illustrates where the GWR models ofer a  better ft than the OLS models could.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Overall, the GWR R2 values improve on the  OLS values in most tracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  The consistency of these model results across specifcations lends confdence in their  estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' In Model 3, 39% of tracts show a signifcant, negative relationship between  resident VKT generation and vehicular PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5 exposure, while only 8% of tracts show  a signifcant, positive relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' With the full set of controls in Model 4, 34% of  tracts show a signifcant, negative relationship between resident VKT generation and  vehicularPM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5 exposure, whileonly7%oftractsshowasignifcant, positiverelationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Figure 2a illustrates where these relationships tend to be negative, including the low-  income Latino eastside of Los Angeles and the high-income White communities along  12  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Spatial distribution of Model 4’s GWR local t-statistics for a) log VKT, b)  proportion White, c) median household income, d) log truck traffic volume, e) distance to  highway, and f) local R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  the oceanfront.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Such areas demonstrate the greatest injustice as those who drive more  are exposed to less pollution, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Commute Simulation  The commute simulation results reveal unequal patterns in driving between White  and non-White commuters which help explain the aforementioned exposure disparity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Figure 3 depicts the inequity index’s spatial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Positive values indicate that  commuters residing in a tract are less White than the commuters driving through  the tract, while negative values indicate the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' It does not show where driving  occurs: most driving is constrained to highway corridors (and thus non-White areas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Accordingly, more trips traverse the fgure’s red tracts than blue tracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Countywide, the inequity index’s population-weighted mean is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='0153, revealing  that most tracts experienced disproportionately more travel by White commuters rela-  tive to the local racial composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' This disparity is more than twice as large in tracts  with highways (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='0307) than in those without (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='0147).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Figure 3 illustrates the roles  of residential segregation and highway placement in today’s driving patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Ma-  jor highways stand out in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Tracts adjacent to Interstate 10—an east-west  backbone—have a population-weighted mean of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='0957, while those adjacent to Inter-  states 110 and 105—the largest freeways through predominately Black and Latino South  13  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  a  d  e  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='2Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Tract-level inequity index: positive values indicate that the share of White  commuters traversing the tract exceeds the share of White residents living in the tract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Negative values indicate the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Los Angeles—have a value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='1288.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Thus, commuters driving through these tracts  are respectively 10 and 13 percentage points Whiter than the local resident population—  particularly notable when considering that the countywide population is less than  one-third White.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Yet there are also examples of major highways on which commuters  resemble or are less White than local residents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Tracts adjacent to Interstate 405 and  US Route 101 have inequity index values of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='0012 and -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='0999, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Both  highways traverse mountain passes through the majority-White, afuent Hollywood  Hills and Santa Monica Mountains where alternative alignments were impossible or  cost-prohibitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  14  7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='3  Inequity Index to White Commuters  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Discussion  The research literature has explored race and class disparities in local vehicular air pol-  lution exposure without accounting for local residents’ own contributions to their  exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' This study fnds that—all else equal—tracts whose residents drive less are  exposed to more vehicular air pollution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Furthermore, tracts with a larger non-White  population proportion—whether high- or low-income—experience more air pollution  than do Whiter but otherwise similar tracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Comparable trends hold in the GWR  models on average as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' This reveals an injustice in pollution burden with a distinct  racial dimension and these results are robust across multiple specifcations and both  global and local estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Overall, our fndings reveal a systematic environmental  injustice: road infrastructure benefts Whiter communities while exacting a cost on  less-White communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  This study opens the door for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' While the simulation analysis pro-  vides evidence that unequal driving patterns could generate some of the observed dispar-  ity, it does not rule out the infuence of other mechanisms such as tract-level diferences  in wind patterns or fuel efciency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Future research should continue investigating these  relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Nevertheless, our models consistently reveal the same broad story: tracts  with a greater share of White residents and tracts with residents who drive more are  exposed to less vehicular air pollution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' One natural way to interpret this relationship  is that people from majority-White neighborhoods do most of their “excess” driving  through non-White neighborhoods and are thus exposed to less pollution at home  despite producing more of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' However, there are other potential mechanisms that  could explain the relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' For example, majority-White neighborhoods could be  located in areas where prevailing winds push pollution away and toward non-White  neighborhoods, or majority-White neighborhoods’ residents could drive more fuel-  efcient cars so that even though they drive more, they produce and are exposed to less  pollution at home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' However, the history of highway construction through non-White  neighborhoods suggests that diferences in where travel occurs at least partially drive  these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Further, the commute simulation provides evidence that disparities in vehicular  air pollution exposure at home result from where people drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' On average, White  commuters traverse tracts that are far more non-White than the tracts where most White  commuters live.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' This disparity does not exist in the opposite direction: on average,  non-White commuters do not travel through tracts that are substantially Whiter than  their home tracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' This also illustrates the role that highways play in mediating this  disparity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' When White commuters traverse non-White tracts, they do so predominantly  through tracts that contain highways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' In other words, White commuters receive the  benefts of driving on a highway, but because those highways are predominantly in  non-White neighborhoods, other racial groups bear external costs of that driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  15  This disparity is less common in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' On average, non-White  commuters do not travel through tracts that are substantially Whiter than their home  tracts—and even where they occasionally do, it is on freeway segments that follow  alignments determined by topography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' For example, by not building a freeway through  Beverly Hills, planners caused a substantial share of commuters who would have taken  this route to instead drive through majority non-White tracts on Interstate 10 instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  However, a similar rerouting of Interstate 405 or US Route 101 would not be possible  without much greater and costlier detours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Transport planning has played a central—  but not solo—role in these inequities: land use regulation, zoning, and housing policy  interact with transport plans to generate these outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Several policies can help mitigate this environmental injustice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' First, policymakers  could continue raising fuel efciency standards for new cars and encouraging vehicular  electrifcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Both would reduce on-road emissions without necessarily reducing the  amount of driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' However, they would not eliminate rubber tires’ and brake dust’s  substantial contributions to PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5 pollution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Second, policymakers could enact tolls  or other forms of congestion taxes to reduce total driving or capture its externalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Third, policymakers could discourage commuting altogether by incentivizing more  people to work from home, such as through tax credits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' The Covid-19 pandemic  witnessed a surge in remote work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Continuing these work-from-home trends after  the pandemic could reduce vehicular travel and air pollution, particularly by higher-  income White collar workers with more fexible jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Finally, policymakers can address  environmental injustice through the housing market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Permitting more residential  construction in job-rich neighborhoods could reduce commute distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Further,  legalizing the construction of denser and more afordable housing in less-polluted,  exclusive neighborhoods could reduce exposure disparities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Yet no single policy can  eliminate longstanding systemic discrimination against low-income and non-White  populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Conclusion  This paper extended quantitative research on transport-environmental justice through  a spatial analysis of the production of and exposure to vehicular air pollution in Los  Angeles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' We assessed local exposure to vehicular air pollution adjusted by local produc-  tion of VKT through four models and two estimation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' In particular, our  GWR analysis allowed for local nuance to reveal spatial heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  We found that tracts that generate more vehicular travel tend to be exposed to less  vehicular air pollution, all else equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Race signifcantly predicts pollution exposure,  even when controlling for a full set of related variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Our commute simulation’s  inequity index helps explain these fndings by illustrating the role of freeways and  16  residential sorting in commute patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' On average, White commuters traverse tracts  that are far more non-White than their home tracts, but non-White commuters do not  travel through tracts that are substantially Whiter than their own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Dismantling decades  of racially-motivated transport planning and segregation requires concerted efort by  planners and policymakers to redress past harms and envision a more equitable future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Acknowledgments  This research was supported by a grant from the US Department of Transportation  and the Pacifc Southwest Region University Transportation Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' The authors wish  to thank Peter Mannino, Nicholas Cerdera, and David Flores Moctezuma for research  assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Notes  1Throughout, we use “Latino” as shorthand for the Census Bureau’s designation of Hispanic or  Latino of any race, “White” as shorthand for the Census Bureau’s designation of non-Hispanic or  Latino White alone, and “Black” as shorthand for the Census Bureau’s designation of non-Hispanic  or Latino Black or African-American alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=', Scally, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', and Hayes, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Air  pollution, deprivation and health: understanding relationships to add value to local  air quality management policy and practice in Wales, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Journal of Public Health,  39(3):485–497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Bureau of Transportation Statistics (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' 2017 Local Area Transportation Character-  istics for Households Methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  California Air Resources Board (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Current California GHG Emission Inventory  Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  County of Los Angeles Open Data (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Commute Mode Share in LA County  (2005-2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Delucchi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Environmental externalities of motor-vehicle use in the US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Journal of Transport Economics and Policy, 34(2):135–168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  DiMento, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Stent (or dagger?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=') in the heart of town: Urban freeways in  Syracuse, 1944—1967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Journal of Planning History, 8(2):133–161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Estrada, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' If you build it, they will move: The Los Angeles freeway system  and the displacement of Mexican East Los Angeles, 1944-1972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Southern California  Quarterly, pages 287–315.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Feng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Gao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Liao, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Zhou, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', and Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' The health efects of  ambient PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5 and potential mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Ecotoxicology and Environmental Safety,  128:67–74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Gately, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Hutyra, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', and Wing, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' DARTE Annual On-road CO2 Emissions  on a 1-km Grid, Conterminous USA V2, 1980-2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Gauderman, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Avol, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Lurmann, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Kuenzli, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Gilliland, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Peters, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', and  McConnell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Childhood Asthma and Exposure to Trafc and Nitrogen  Dioxide:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Epidemiology, 16(6):737–743.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Giles-Corti, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Moudon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Lowe, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Cerin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Boeing, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Frumkin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Salvo,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Foster, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Kleeman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Bekessy, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', de Sá, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Nieuwenhuijsen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Higgs,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Hinckson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Adlakha, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Arundel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Oyeyemi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Nitvimol, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=',  and Sallis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' What next?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Expanding our view of city planning and global  health, and implementing and monitoring evidence-informed policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' The Lancet  Global Health, 10(6):e919–e926.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Giuliano, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Travel, location and race/ethnicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Transportation Research Part  A, 37:351–372.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Habre, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Girguis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Urman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Fruin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Lurmann, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Shafer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Gorski, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=',  Franklin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', McConnell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Avol, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', and Gilliland, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Contribution of  tailpipe and non-tailpipe trafc sources to quasi-ultrafne, fne and coarse particulate  matter in Southern California.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Journal of the Air & Waste Management Association,  71(2):209–230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Hasheminassab, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Daher, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Ostro, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', and Sioutas, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Long-term source  apportionment of ambient fne particulate matter (PM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5 ) in the Los Angeles Basin:  18  A focus on emissions reduction from vehicular sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Environmental Pollution,  193:54–64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Houston, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Ong, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', and Winer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Structural Disparities of Urban  Trafc in Southern California: Implications for Vehicle-Related Air Pollution Ex-  posure in Minority and High-Poverty Neighborhoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Journal of Urban Afairs,  26:565–592.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Huang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Ivey, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Hu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=' Source  apportionment of primary and secondary PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5: Associations with pediatric respira-  tory disease emergency department visits in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' State of Georgia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=' County-level cumulative environmental quality associated with cancer  incidence: Environment and Cancer Incidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Cancer, 123(15):2901–2908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content='  Assessment of exposure to trafc related air pollution of children attending schools  near motorways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Atmospheric Environment, 35(22):3875–3884.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=' Near-roadway air quality:  Synthesizing the fndings from real-world data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Environmental Science & Technology,  44(14):5334—5344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=' Assessment of sociodemographic disparities in environ-  mental exposure might be erroneous due to neighborhood efect averaging: Implica-  tions for environmental inequality research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Environmental Research, 195:110519.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=' Breathless in Los Angeles: The Exhausting Search  for Clean Air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' American Journal of Public Health, 93(9):1494–1499.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=' Clinical & Experimental  Allergy, 36(9):1138–1146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=' Racialized Structural Vulnerability: Neighborhood Racial  Composition, Concentrated Disadvantage, and Fine Particulate Matter in California.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  International Journal of Environmental Research and Public Health, 16(17):3196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=' Beyond air pollution at home: Assessment of personal exposure to  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5 using activity-based travel demand model and low-cost air sensor network  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=' Stop the Road: Freeway Revolts in American Cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content='  Moreno-Jimenez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Cañada-Torrecilla, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Vidal-Domínguez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=', andMartinez-Suarez, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='(2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Assessingenvironmentaljusticethroughpotential  exposure to air pollution: A socio-spatial analysis in Madrid and Barcelona, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Geoforum, 69:117–131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Nyhan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Grauwin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Britter, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Misstear, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', McNabola, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Laden, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Barrett,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' “Exposure Track”—The Impact of Mobile-Device-  Based Mobility Patterns on Quantifying Population Exposure to Air Pollution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Environmental Science & Technology, 50(17):9671–9681.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Oosterlee, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Drijver, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=' Chronic respira-  tory symptoms in children and adults living along streets with high trafc density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Occupational and Environmental Medicine, 53:241–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=', and Deguen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Air quality and social deprivation in four French  metropolitan areas: A localized spatio-temporal environmental inequality analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Environmental Research, 134:315–324.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=', Bartolome, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=', Edwards, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Investigation of roadside fne particulate matter concentration  surrounding major arterials in fve Southern Californian cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Journal of the Air &  Waste Management Association, 63(4):482–498.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Pastor, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Sadd, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=' (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Which Came First?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Toxic Facilities, Minority  Move-In, and Environmental Justice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Journal of Urban Afairs, 23(1):1–21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Perez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=' The Los AngelesFreeway and the Historyof Community Displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Toro Historical Review, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Polichetti, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Cocco, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Efects of  particulate matter (PM10, PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5 and PM1) on the cardiovascular system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Toxicology,  261(1-2):1–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  PolicyLink (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' An Equity Profle of the Los Angeles Region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Poorfakhraei, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=', and Rowangould, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=' Evaluating health out-  comes from vehicle emissions exposure in the long range regional transportation  planning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Journal of Transport & Health, 6:501–515.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Pulido, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Rethinking Environmental Racism: White Privilege and Urban  Development in Southern California.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Annals of the Association of American Geog-  raphers, page 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=' Transportation Research Part A: Policy and Practice, 147:28–48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Reichmuth, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=' Inequitable Exposure to Air Pollution from Vehicles in Califor-  nia Fact Sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Technical report, Union of Concerned Scientists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=' Exposure to air pollutants during  commuting in London: Are there inequalities among diferent socio-economic  groups?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Environment International, 101:143–157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=' Identifying environmental justice  communities for transportation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Transportation Research Part A, 88:151–162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Rowangould, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' A new approach for evaluating regional exposure to particulate  matter emissions from motor vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Transportation Research Part D, 34:307–317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Sarzynski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Bigger Is Not Always Better: A Comparative Analysis of Cities  and their Air Pollution Impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Urban Studies, 49(14):3121–3138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  SCAG (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Heavy Duty Truck Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Technical report, Southern California  Association of Governments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Schindler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' and Caruso, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Urban interventions to reduce pollution exposure  and improve spatial equity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Geographical Analysis, online before print.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Schweitzer, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' and Valenzuela, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Environmental Injustice and Transportation:  The Claims and the Evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Journal of Planning Literature, 18(4):383–398.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Seagram, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Brown, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=', Landsberg, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', and Eisinger, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  National Assessment of Near-Road Air Quality in 2016: Multi-Year Pollutant Trends  and Estimation of Near-Road PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5 Increment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Transportation Research Record,  2673(2):161–171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Sunyer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Esnaola, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Alvarez-Pedrerol, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Forns, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Rivas, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', López-Vicente, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=',  Suades-González, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Foraster, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Garcia-Esteban, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Basagaña, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', and et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Association between trafc-related air pollution in schools and cognitive develop-  ment in primary school children: A prospective cohort study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' PLOS Medicine,  12(3):e1001792.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Tayarani, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' and Rowangould, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Estimating exposure to fne particulate  matter emissions from vehicle trafc: Exposure misclassifcation and daily activity  patterns in a large, sprawling region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Environmental Research, 182:108999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Temam, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Burte, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Adam, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=', Bousquet, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Carsin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=',  Galobardes, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Keidel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Künzli, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=' Socioeconomic position and  outdoor nitrogen dioxide (NO2) exposure in Western Europe: A multi-city analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Environment International, 101:117–124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Tessum, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', and Marshall, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' InMAP: A model for air pollution  interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' PLOS ONE, 12(4):e0176131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  21  Tessum, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Paolella, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Chambliss, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Apte, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Hill, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', and Marshall,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='5 polluters disproportionately and systemically afect people of  color in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Science Advances, 7(18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Thompson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Airborne Particulate Matter: Human Exposure and Health  Efects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Journal of Occupational & Environmental Medicine, 60(5):392–423.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  US EPA (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Fast Facts: U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Transportation Sector Greenhouse Gas Emissions,  1990-2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Technical report, United States Environmental Protection Agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Yuan, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Mega freight generators in my backyard: A longitudinal study of  environmental justice in warehousing location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Land Use Policy, 76:130–143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  Zhu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Hinds, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=', and Sioutas, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Concentration and size dis-  tribution of ultrafne particles near a major highway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content=' Journal of the Air & Waste  Management Association, 52(9):1032—1042.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
+page_content='  22' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfkvi4/content/2301.00440v1.pdf'}
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+111
+s
+GaitSADA: Self-Aligned Domain Adaptation for mmWave
+Gait Recognition
+EKKASIT PINYOANUNTAPONG, AYMAN ALI, KALVIK JAKKALA, PU WANG, and MIN-
+WOO LEE, University of North Carolina at Charlotte, USA
+QUCHENG PENG and CHEN CHEN, University of Central Florida, USA
+ZHI SUN, Tsinghua University, China
+mmWave radar-based gait recognition is a novel user identification method that captures human gait biometrics
+from mmWave radar return signals. This technology offers privacy protection and is resilient to weather
+and lighting conditions. However, its generalization performance is yet unknown and limits its practical
+deployment. To address this problem, in this paper, a non-synthetic dataset is collected and analyzed to reveal
+the presence of spatial and temporal domain shifts in mmWave gait biometric data, which significantly impacts
+identification accuracy. To address this issue, a novel self-aligned domain adaptation method called GaitSADA
+is proposed. GaitSADA improves system generalization performance by using a two-stage semi-supervised
+model training approach. The first stage employs semi-supervised contrastive learning to learn a compact
+gait representation from both source and target domain data, aligning source-domain distributions implicitly.
+The second stage uses semi-supervised consistency training with centroid alignment to further enhance
+generalization performance in target domains by minimizing intra-class distance and maximizing inter-class
+distance using pseudo labeling and class centroid estimation. Experiments show that GaitSADA outperforms
+representative domain adaptation methods by an average of 15.41% in low data regimes.
+CCS Concepts: • Computer systems organization → Embedded systems; • Computing methodologies
+→ Artificial intelligence.
+Additional Key Words and Phrases: datasets, neural networks, gaze detection, text tagging
+ACM Reference Format:
+Ekkasit Pinyoanuntapong, Ayman Ali, Kalvik Jakkala, Pu Wang, Minwoo Lee, Qucheng Peng, Chen Chen,
+and Zhi Sun. 2018. GaitSADA: Self-Aligned Domain Adaptation for mmWave Gait Recognition. J. ACM 37, 4,
+Article 111 (August 2018), 27 pages. https://doi.org/XXXXXXX.XXXXXXX
+1
+INTRODUCTION
+User identification is an essential component for many Internet of Things (IoT) applications, e.g.
+personalized environmental control, security management, and access control for automatic doors.
+Short-distance biometrics such as fingerprints, face, iris, palm, and finger vein patterns typically
+demand active user participation within a certain proximity. In contrast, gait, i.e., human walking
+Authors’ addresses: Ekkasit Pinyoanuntapong, epinyoan@uncc.edu; Ayman Ali, aali26@uncc.edu; Kalvik Jakkala, kjakkala@
+uncc.edu; Pu Wang, pu.wang@uncc.edu; Minwoo Lee, minwoo.lee@uncc.edu, University of North Carolina at Charlotte, 9201,
+Charlotte, North Carolina, USA, 28223; Qucheng Peng, qucheng.peng@knights.ucf.edu; Chen Chen, chen.chen@crcv.ucf.edu,
+University of Central Florida, 4000, Orlando, Florida, USA, 32816 ; Zhi Sun, zhisun@tsinghua.edu.cn, Tsinghua University,
+Beijing, China , 100084 .
+Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee
+provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and
+the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored.
+Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires
+prior specific permission and/or a fee. Request permissions from permissions@acm.org.
+© 2018 Association for Computing Machinery.
+0004-5411/2018/8-ART111 $15.00
+https://doi.org/XXXXXXX.XXXXXXX
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+arXiv:2301.13384v1  [cs.CV]  31 Jan 2023
+
+111:2
+Pinyoanuntapong et al., Ekkasit Pinyoanuntapong, Ayman Ali, Kalvik Jakkala, Pu Wang, Minwoo Lee, Qucheng Peng,
+Chen Chen, and Zhi Sun
+style, can be exploited for user identification from a distance without the subject’s participation or
+interference. Moreover, it is shown in existing study [35] that human gait is very hard to spoof or
+mimic because our own gait works against us when we try to imitate someone elses gait.
+The majority of gait recognition techniques are vision-based, which directly perform identifica-
+tion tasks on video data. Appearance-based and model-based approaches are two general approaches
+for vision-based gait recognition. Appearance-based approaches [9][7][20] utilize silhouettes from
+a video sequence obtained from background subtraction. Although background subtraction is
+accurate in a lab context, it is difficult to use in a cluttered and dynamic real-world environment.
+Model-based approaches [25][31] try to tackle this issue by employing the existing pose estimators
+to improve robustness to environmental variation. However, all vision-based approaches require
+access to RGB visual images which are prone to performance loss from variations of ambient light
+settings and the social restriction from privacy concerns.
+Radar-based gait recognition is an emerging user identification solution that exploits radar
+return signals to capture human gait biometric [6, 18, 24, 26, 36, 41, 42]. Unlike camera surveillance
+systems, radar-based gait recognition systems have several unique advantages. First, they are
+privacy-preserving as they do not rely on images of human subjects. Next, they can operate in
+adverse weather and lighting conditions such as heavy fog, smoke, rain, and zero-light conditions.
+Finally, they can see through some opaque objects depending on its material and the frequency of the
+radar [1, 40]. Radar-based methods operate by emitting modulated electromagnetic signals, which
+are scattered and returned back by the subjects in the signal propagation path. The radar return
+signals capture the micro-Doppler (mD) signature, which can be represented as radar spectrogram.
+mD signature or radar spectrogram documents the time-dependent Doppler frequency shifts, which
+are caused by the small-scale vibrations or rotations of human body parts during walking. Each
+person has his/her unique walking style, i.e., gait, which leads to the unique mD signature that can
+be exploited for gait recognition.
+Existing radar-based gait recognition systems generally operate either in sub-6GHz WiFi bands
+[26, 36, 42] or the emerging mmWave bands (30 to 300 GHz) [6, 18, 24, 41]. Compared with sub-6GHz
+radar systems, mmWave radar can promise much fine-grained motion capture with significantly
+enhanced velocity and location resolutions. This feature is enabled by orderly shorter wavelength
+and wider bandwidth of mmWave radar signals, compared with the sub-6GHz counterparts. Besides
+promising motion capture capacities, mmWave radar system only needs a single radar device
+that integrates both radar transmitter and receiver, while sub-6GHz radar system is generally
+implemented by a pair of WiFi devices to act as distributed radar transmitter and receiver, which
+have to run modified WiFi firmware to extract mD signatures from CSI signals.
+The superior gait recognition performance of mmWave radar systems has been demonstrated in
+our previous research [18] and other related work [6, 24, 41]. However, the spatial and temporal
+generalization performance of this technology is still unknown. We found that deep neural networks
+(DNNs) trained on radar-based gait data suffer from substantial performance degradation with the
+progression of time and in new locations. Existing radar-based gait recognition systems proposed
+in the literature, test their performance on a dataset collected alongside the training data and fail
+to uncover the temporal degradation. Furthermore, they seldom address the spatial generalization
+issues from introducing new environments. The root cause of this generalization issue is domain
+shift [23]. Simply put, it occurs when the testing data distribution differs from that of the training
+data, i.e., the i.i.d. assumption is violated. Domain shift problem can be further exacerbated by
+the harmful environment reflections induced by mmWave signals. These environment reflections
+include static ones, which directly bounce from stationary obstructions e.g. walls, desks, and chairs,
+and dynamic ones, which indirectly bounce from other stationary obstructions and then bounce
+from human bodies. The harmful environment reflections carry delayed and distorted mD signature
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+GaitSADA: Self-Aligned Domain Adaptation for mmWave Gait Recognition
+111:3
+information, which cannot be completely removed. This leads to environment-induced temporal
+domain shift (TDS) and spatial domain shift (SDS). Moreover, human gait, although unique to each
+individual, could contain many variations, potentially as a consequence of a subject’s mood and
+clothing changes, further aggravate TDS and SDS.
+In this paper, we tackle the challenge of domain drift in mmWave radar-based gait recognition
+across multiple environments and dates. To address this issue, we propose a novel domain adaptation
+method, GaitSADA. GaitSADA is based on simple distribution alignment components and consists
+of a two-stage approach: pretraining and fine-tuning. In the pretraining stage, the model learns
+salient gait representations and noise invariance from both source and target domains using semi-
+supervised contrastive learning. In the fine-tuning stage, the model generalizes the learned gait
+representation to the target domain by leveraging pseudo-labels, semi-supervised consistency
+training, and centroid alignment. Experiments demonstrate that GaitSADA outperforms classic
+adversarial domain adaptation methods and achieves significant accuracy improvements in low
+data regimes and challenging domain drift conditions. Our contributions can be summarized as
+follows:
+• We curate a non-synthetic dataset consisting of mmWave radar-based gait biometric data. This
+dataset, for the first time, allows one to study and improve the spatio-temporal generalization
+performance of a radar-based biometric identification system.
+• We perform extensive research on the SDS and TDS from four different location environments
+on different days, experimenting with five state-of-the-art (SOTA) domain adaptation methods
+to mitigate these drifts and show the issue of accuracy drop in low data regimes of SOTA
+domain adaptation methods.
+• We propose a novel domain adaptation method for mmWave gait recognition system, self-
+aligned domain adaptation for mmWave gait recognition (GaitSADA), by jointly exploiting
+semi-supervised contrastive learning, semi-supervised consistency training, centroid align-
+ment and deliberately designed radar spectrogram augmentations. GaitSADA outperforms
+existing SOTA methods in 1-day, 2-day, and 3-day data by 15.41%, 5.83%, and 2.63% respec-
+tively on average of domain drift cases.
+2
+RELATED WORK
+Unsupervised domain adaptation (UDA), semi-supervised learning, and self-supervised learning
+are common strategies to reduce the dependency on expensive manually annotated data. UDA aims
+to mitigate the domain gap by transferring knowledge learned from label data in source domain
+and transfer to unlabeled data in target domain, while semi-supervised learning employs a few
+manually annotated samples and generalizes to large unlabeled samples without domain shift
+assumption. Self-supervised utilizes the unlabeled data to pre-train the model to learn the general
+semantics of the data.
+2.1
+Unsupervised Domain Adaptation (UDA)
+Supervised Learning’s primary objective is to learn an input-output mapping by minimizing the
+loss function E(𝑥,𝑦) ∼ 𝑃𝑋𝑌 (𝑓 (𝑥),𝑦) based on the training data distribution 𝑃𝑋𝑌 and evaluate on
+ˆ𝑃𝑋𝑌. This setting requires an independent and identically distributed (i.i.d.) assumption between
+𝑃𝑋𝑌 and ˆ𝑃𝑋𝑌 . However, the mmWave signal reflection is very sensitive to environments. The data
+distribution can be widely different from place to place depending on the surrounding environment.
+As shown in table 1, the distribution drifts occur both temporally and spatially which violates
+their i.i.d. assumption. The naive way to solve the problem is to train the model on source and
+target data, drawn from 𝑃𝑋𝑌 and ˆ𝑃𝑋𝑌, jointly using supervised learning with label data in both
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+111:4
+Pinyoanuntapong et al., Ekkasit Pinyoanuntapong, Ayman Ali, Kalvik Jakkala, Pu Wang, Minwoo Lee, Qucheng Peng,
+Chen Chen, and Zhi Sun
+domains. However, the privacy concern of biometric identification data provides another layer of
+complication to the identification data collecting process and causes the annotated data to be more
+expensive and time-consuming. As a result, it may not be easy to collect data with the subjects’
+identities (label data) in the target environment.
+Unsupervised domain adaptation can tackle the domain drift and the lack of labeled data problem
+by minimizing the discrepancy between source and target distributions without using expensive
+labeled data from the target domain. The simple methods learn the domain invariant of source
+and target domain by minimizing the distribution shift. DeepCORAL [29] directly matches the
+mean and covariance of the source and target distributions with a linear transformation. Many
+UDA techniques, inspired by Generative Adversarial Networks (GAN), apply an adversarial loss
+to confuse the domain discriminator network that tries to recognize the domain difference. [32]
+separates into two networks for source and target and forces the output representation to be in the
+same distribution. Domain-adversarial neural network (DANN) trains domain discriminator along
+with the feature extractor that confuses the domain discriminator by integrating a gradient reversal
+layer (GRL) [10]. Although the overall distribution is aligned, the class-wise discrepancy can still
+cause performance drop in the classification tasks. Conditional Adversarial Domain Adaptation
+(CDAN) [21] takes this into account by implementing multilinear conditioning of the feature
+representation and the class-wise output prediction. The other line of research in adversarial
+domain adaptation also incorporates pixel level in addition to feature level domain adaptation
+[4][17]. [39] aligns the centroid of the true source label and the target pseudo-label by minimizing
+their squared Euclidean distance.
+2.2
+Semi-supervised Learning
+Semi-supervised methods decrease the dependency on labeled data by combining normally smaller
+sets of labeled data with the vast amounts of unlabeled data that are available in many use cases.
+Since the classification loss is unknown for unlabeled data, the self-ensembling network trains itself
+via the pseudo label instead. The networks encourage consistent prediction despite the perturbation
+on data and networks. Virtual Adversarial Training (VAT) [22] adds only a small perturbation
+to the input data in the direction that produces maximum difference from the original input to
+promote local distributional smoothness. Mean Teacher [30] 𝜋-model and Temporal Ensembling
+Model [19], and Mutual Mean-Teacher [11] apply different augmentation and dropout for student
+and teacher networks. Also, a teacher network uses an exponential moving average (EMA) weights
+of the student model from the previous training epochs to train the noise invariant prediction to
+the slightly different network and data from a current student data. Noisy Student [38] takes a
+similar approach by pretraining the network with label data and using it as a teacher model to
+train the student model without the EMA weights. The trained student network is directly used
+as a teacher network to train an equal-or-larger student model and the iterative training process
+repeats. Recently, much research has shown that using only data augmentation with multiple
+identical networks works well without EMA weights or iterative training. UDA [37], FixMatch [28],
+MixMatch [3], and RemixMatch [2] utilize consistency training without the actual labeled data.
+Only high confidence pseudo labels, which represent the similarity of unseen data to the previously
+trained ones, are selected for training.
+Despite the different strategies of unsupervised domain adaptation (UDA) and semi-supervised
+learning, they share similar objectives and solutions. Specifically, if the target support covers source
+support, semi-supervised learning can be considered a special case of UDA problems. [43] shows
+that the state-of-the-art semi-supervised learning methods outperform the existing UDA methods
+on the challenging UDA task.
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+GaitSADA: Self-Aligned Domain Adaptation for mmWave Gait Recognition
+111:5
+2.3
+Self-supervised Contrastive Learning
+Similar to semi-supervised learning, self-supervised Learning utilizes large unlabeled data to
+improve performance from label data trained in a supervised learning fashion. However, self-
+supervised learning normally pre-trains unlabeled data to learn general data semantics before
+transferring pre-trained weights to further learn label data in a supervised learning fashion in the
+next stage. Many research [8], [13], [33], [14] enforce contrastive loss to learn the structure of the
+data without labels by imposing different augmentation to the same data and learning the similarity
+between the features of same data (positive samples). The challenge is that such similarity loss
+functions can trigger a dimensional collapse in which the model outputs the exact same embedding
+for all input. While MoCo [14] and BYOL [13] incorporate momentum encoder and stop gradient to
+prevent the collapse, SimCRL [8] and CPC [33] employ negative samples to discriminate the features
+of different data. SimCLR [8] proposed NT-Xent (the normalized temperature-scaled cross-entropy
+loss) which simply maximizes the similarity between the feature of the same data (positive samples)
+and minimizes the different ones (negative samples),
+𝐿𝑁𝑇−𝑋𝑒𝑛𝑡 = − 1
+𝐵
+𝐵
+∑︁
+𝑖=1
+𝑙𝑜𝑔
+𝑒 (𝑠𝑖𝑚(𝑓𝑖,^𝑓𝑖)/𝜏)
+�𝐾
+𝑗=1 1[𝑖≠𝑗 ]𝑒 (𝑠𝑖𝑚(𝑓𝑗, ^𝑓𝑗)/𝜏) ,
+(1)
+where 1[𝑖≠𝑗 ] ∈ {0, 1} denotes an indicator function to retain only negative samples, 𝑖 ≠ 𝑗, 𝜏 is a
+temperature parameter, and 𝑠𝑖𝑚(𝑓𝑖, ˆ𝑓𝑖) =
+𝑓 𝑇
+𝑖 ^𝑓𝑖
+∥𝑓𝑖 ∥
+���^𝑓𝑖
+��� represents cosine similarity. We modify this self-
+supervised learning method to pre-train target data in order to learn representations of unlabeled
+data as described in Section 4.2.
+3
+MMWAVE GAIT BIOMETRIC DATASET
+The dataset is created by exploiting the mmWave radar-based gait recognition system, STAR,
+developed in our previous work [18]. In this section, the preliminaries of mmWave radar sensing
+are first presented, the data collection and gait recognition procedure using STAR system is then
+introduced, and finally spatial-temporal domain drift issues are revealed.
+3.1
+Preliminaries of mmWave Radar Sensing
+Fig. 1. FMCW signal with linear ramp
+The transmission of a signal, which is reflected by objects along its route of propagation, is a key
+principle behind radar systems. One of the key advantages of a Frequency Modulated Continuous
+Wave (FMCW) radar system is its ability to simultaneously determine both the range and velocity
+of a moving target. As illustrated in Fig. 1, the frequency of the transmitted signal increases linearly
+over time during the transmission. When the chirp is reflected off an object, the frequency of
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+Transmit
+------ Receive111:6
+Pinyoanuntapong et al., Ekkasit Pinyoanuntapong, Ayman Ali, Kalvik Jakkala, Pu Wang, Minwoo Lee, Qucheng Peng,
+Chen Chen, and Zhi Sun
+the reflected signal will differ slightly from the transmitted signal due to the Doppler effect. This
+difference in frequency enables the radar to determine both the distance and velocity of the object.
+The rate of change of frequency, or the slope of the chirp, is defined as 𝑆 = 𝐵/𝑇𝑐, where 𝐵 is the
+bandwidth and 𝑇𝑐 is the duration. The beat signal is produced by combining the broadcast and
+received chirps in a frequency mixer. For a target at a distance 𝑑, the beat signal can be explained
+as follows:
+𝐴 exp(𝑗2𝜋(𝑓0𝑡 + 2𝑑/𝜆𝑐) = 𝐴 exp(𝑗2𝜋(2𝑑/𝜆𝑐)) exp(𝑗2𝜋𝑓0𝑡),
+(2)
+The wavelength of the carrier frequency is referred to as 𝜆𝑐 = 𝐶/𝑓𝑐 where 𝐶 is the speed of light. 𝐴
+denotes the signal attenuation gain. 𝑓0 = 2𝑑𝑆/𝐶 represents beat frequency. Therefore, the distance
+of the target can be simply obtained by
+𝑑 = 𝑓0𝐶
+2𝑆 .
+(3)
+The frequency of the beat signal will vary as the target is moving to a new location. The beat signal
+will change if the distance between the previous position and the new position is greater than the
+radar range resolution, which is directly related to the bandwidth 𝐵, i.e.,
+𝑑𝑟𝑒𝑠 = 𝐶
+2𝐵 .
+(4)
+A very broad bandwidth 𝐵 can be employed with mmWave FMCW radar due to the extremely high
+carrier frequency, which results in fine-grained range resolution.
+While the frequency of the beat signal is utilized to determine the target’s distance, its phase
+exp(𝑗2𝜋(2𝑑/𝜆𝑐)), as specified in Eq. (2), can also be utilized to estimate the velocity of the target,
+even in the presence of range resolution constraints that prevent detection of changes in the target’s
+position. In situations where the target moves a relatively small distance over a sequence of 𝑁 beat
+signals generated from transmitted and received chirps, all 𝑁 signals will exhibit varying phases
+but have the same beat frequency. Specifically, the phase of the beat signal 𝑛, where 𝑛 ∈ (1, 2, ..., 𝑁),
+is given by:
+exp(𝑗2𝜋(2𝑣/𝜆𝑐)𝑛𝑇𝑐),
+(5)
+where 2𝑣/𝜆𝑐 is Doppler frequency. For each distinguishable distance, the equation (5) reveals
+the phases of the beat signals, generating a new signal whose carrier frequency is the Doppler
+frequency. The various body parts of a walking human subject, such as arms, legs, feet, and torso,
+move at different velocities, leading to distinct Doppler frequencies. The radar is able to distinguish
+movements of two parts only if the velocity resolution surpasses the velocity difference between
+the two parts, which is determined by the observation duration, 𝑁𝑇𝑐, and the carrier wavelength,
+𝜆𝑐, as follows
+𝑣𝑟𝑒𝑠 =
+𝜆𝑐
+2𝑁𝑇𝑐
+.
+(6)
+The millimeter-scale wavelength of mmWave radar allows it to characterize the fine-grained
+motion dynamics of human gait in high resolution. For instance, 77GHz mmWave radar can reach
+over 12 times greater velocity resolution than sub-6GHz radar.
+3.2
+STAR System Overview
+STAR is the first one in the literature to exploit deep learning for mmWave radar-based gait
+recognition. As shown in Fig. 2, STAR consists of two components: radar spectrogram acquisition,
+spectrogram enhancement and CNN-based gait recognition. The spectrogram acquisition module
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+GaitSADA: Self-Aligned Domain Adaptation for mmWave Gait Recognition
+111:7
+Range 
+Estimation
+(1D-FFT)
+Static Cluster 
+Removal
+(Mean Subtraction) 
+Object 
+Detection
+(CA-CFAR)
+Doppler 
+Estimation
+(2D-FFT) 
+Sliding-
+Window based 
+Tracking
+Spectrogram 
+Generation
+(STFT)
+Normalization
+Mean Filtering
+2D Gaussian 
+Filtering 
+Feature Encoder 
+Gait Spectrogram
+Multiclass Classifier
+User ID
+Radar 
+Array
+Radar Spectrogram Acquisition 
+Spectrogram 
+Enhancement 
+Fig. 2. Overview of STAR system
+tracks and captures the radar signal returns that directly bounce off from human body, while
+filtering out undesired environment-induced radar returns. This is achieved by applying a sequence
+of radar widely-adopted radar signal processing algorithms including one-dimensional Fourier
+transform (ID-FFT) for range estimation, the mean subtraction for static reflection removal, CA-
+CFAR for robust target detection against a background of noise, clutter and interference, Doppler
+estimation via 2D-FFT to estimate target velocity, and sliding-window based tracking method to
+predict the target future state (e.g., location and velocity). The target tracking module acts a moving
+spatial passband filter so that only the radar returns from the target can be received. Performing
+short-time Fourier transform (STFT) on received radar returns generates the raw radar spectrogram,
+whose quality is further improved by exploiting the spectrogram enhancement module that consists
+of a sequence of frequency-domain noise removal processes [18]. The enhanced spectrograms are
+then encoded by a CNN-based feature encoder that maps the high-dimension spectrogram data
+into the compact low-dimension feature vector. Finally, the compact feature vectors are used to
+train a multiclass classifier that predicts the corresponding user identities.
+3.3
+Data Collection Setup
+We collected gait data from 10 volunteers between the ages of 18-35. Each subject’s data was
+collected in four different locations. The source location was a research space with cubicles, and
+the other three areas consisted of a server, conference, and an office room. By maintaining four
+distinct locations, we introduce SDS. In the source location, data was collected on 10 different days
+for each subject. 5 separate days of data was acquired for each of the three other locations, which
+are used as target domains. A participant can either walk towards the radar or walk away from
+the radar, each of which is counted as one walking instance and generates one spectrogram data
+sample. The data collection was limited to 100 data samples per person in the source location and
+50 data samples per person in each of the target locations on any given day. The gait spectrogram
+samples of a person are shown in Fig. 3.
+3.4
+Establishing the Presence of TDS and SDS
+We study the presence of TDS and SDS issues by training gait recognition model using the labelled
+data from source domain and test the model performance across different target domains. In
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+AA国口
+TIIWR1642.mmWaveSens0
+WalkingPath111:8
+Pinyoanuntapong et al., Ekkasit Pinyoanuntapong, Ayman Ali, Kalvik Jakkala, Pu Wang, Minwoo Lee, Qucheng Peng,
+Chen Chen, and Zhi Sun
+Source Day 1
+Source Day 2
+Source Day 3
+Source Day 4
+Source Day 5
+Server Day 1
+Server Day 2
+Server Day 3
+Server Day 4
+Server Day 5
+Office Day 1
+Office Day 2
+Office Day 3
+Office Day 4
+Office Day 5
+Conference Day 1
+Conference Day 2
+Conference Day 3
+Conference Day 4
+Conference Day 5
+Fig. 3. Gait spectrograms of the same person for 5 days at different locations, source (laboratory), conference
+room, server room, and office
+particular, the feature encoder is a modified version of ResNet-50 [15], as described in section 5.3,
+to extract features from mmWave biometric data samples. The loss function is a cross-entropy loss
+for multiclass classification. To establish the presence of SDS, we train the model on the data from 1
+to 3 days of source location and test the model on the data from other three unseen locations. SDS
+is evident from the performance degradation in all target domains as shown in table 1. To reveal
+the existence of TDS, we train the model on the data from 1 to 3 days of the source location and
+test the model on the data from the remaining days of the source location. TDS can be observed
+due to the degraded testing accuracy.
+4
+METHOD: GAITSADA
+4.1
+Method Overview
+Our novel method, self-aligned domain adaptation for mmWave gait recognition (GaitSADA),
+jointly exploits self- and semi-supervised learning techniques to realize the unsupervised domain
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+GaitSADA: Self-Aligned Domain Adaptation for mmWave Gait Recognition
+111:9
+Table 1. The result of supervised learning trained on source data and tested on different days of data (i.e.,
+temporal) and other different environments (i.e., server, conference, office). The accuracy drop reveals the
+domain discrepancy when the experiments are conducted on different days and environments. The results
+report in % of accuracy.
+# of training days (Source)
+Temporal
+Server
+Conference
+Office
+1-day
+76.05
+34.93
+47.00
+33.39
+2-days
+88.46
+53.05
+71.00
+49.92
+3-days
+95.25
+71.77
+85.20
+67.61
+adaptation by promoting intra-class compactness and inter-class distance while generalizing knowl-
+edge from label data to unlabeled data using pseudo-label. In particular, we leverage the unlaballed
+data from the target domains in two training stages: pretaining and fine-tuning as illustrated in
+Fig. 4.
+During pretraining stage, we propose a semi-supervised contrastive learning paradigm to learn
+the salient gait representation from a mixed dataset that consists of data from both source and and
+target domains. Our semi-supervised contrastive learning can explicitly avoid collapse problem
+without requiring the sophisticated techniques adopted by classic self-supervised (unsupervised)
+contrastive learning paradigms, e.g., negative samples, stop gradients, and moment encoders. This
+is achieved by maximizing the agreement between representations produced by encoders fed with
+different augmentations of the same spectrogam in the mixed dataset, while enforcing the encoder
+fed with labelled data in source domain to produce discriminative representations with correct
+class predictions.
+After the pretrained stage, the feature encoder implicitly aligns the data distributions of source
+domain and target domain in the feature space. As a result, the representations produced by such
+encoder are discriminative enough to provide relatively high-confidence pseudo labels for unlabelled
+data in the target domains, which is essential for the semi-supervised consistency training in the
+fine-tuning stage that can effectively regulate the features of the target domain data from the same
+class to be close to each other. However, classic semi-supervised consistency training only shows the
+best performance when the label and unlabelled data come from the same distribution. To address
+this issue, we propose the centroid alignment scheme that can be integrated with semi-supervised
+consistency training to mitigate source-target domain gap.
+4.2
+Pretraining via Semi-supervised contrastive Learning
+Supervised Learning with Additive Margin Softmax [34]. The main goal of this component
+is to promote intra-class compactness and inter-class separability. As the regular softmax loss is
+effective only in maximizing the inter-class difference, it is ineffective at minimizing the intra-class
+variation. Therefore, we adopt Additive Margin Softmax loss function to bring the classification
+boundary closer to the weight vector of each class by adding a margin 𝑚 to the boundary as
+𝐿𝑠𝑢𝑝𝑒𝑟𝑣𝑖𝑠𝑒𝑑 = − 1
+𝐵
+𝐵
+∑︁
+𝑖=1
+𝑙𝑜𝑔
+𝑒𝑠·(𝑐𝑜𝑠𝜃𝑦𝑖 )−𝑚
+𝑒𝑠·(𝑐𝑜𝑠𝜃𝑦𝑖 )−𝑚 + �𝐶
+𝑗=1,𝑗≠𝑦𝑖 𝑒𝑠·𝑐𝑜𝑠𝜃 𝑗 ,
+(7)
+where 𝑐𝑜𝑠𝜃𝑦𝑖 =
+𝑊 𝑇
+𝑦𝑖 𝑓𝑖
+∥𝑊𝑦𝑖 ∥ ∥𝑓𝑖 ∥ represents cosine similarity of the inner product between the normalized
+weight 𝑊𝑦𝑖 and the normalized feature 𝑓𝑖 of class 𝑖. Additional parameter 𝑠 is a scale parameter to
+control the temperature.
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+111:10
+Pinyoanuntapong et al., Ekkasit Pinyoanuntapong, Ayman Ali, Kalvik Jakkala, Pu Wang, Minwoo Lee, Qucheng Peng,
+Chen Chen, and Zhi Sun
+encoder
+encoder
+encoder
+Similarity loss
+Classifier
+AM loss
+Final loss
+Augmentation 
+Augmentation 
+Source 
+Domain
+(labelled) 
+Source + 
+Target 
+Domains
+(unlabeled) 
+encoder
+encoder
+encoder
+Classifier
+AM loss
+Final loss
+Augmentation 
+Source 
+Domain
+(labelled)  
+Target 
+Domain
+(unlabeled) 
+Classifier
+Classifier
+prediction
+prediction
+Pseudo label
+consistency 
+loss
+Centroid 
+Estimation 
+Centroid  
+loss
+w
+Augmentation 
+(a) Pretraining Stage: Semi-supervised Contrastive Learning
+(b) Fine-tuning Stage: Semi-supervised Consistency Training with Centroid Alignment
+Fig. 4. Overall Architecture of GaitSADA. (a) Stage 1. Pretraining via semi-supervised contrastive learning
+to learn a compact gait representation from both source and domain data, which also implicitly mitigates
+the source-domain distribution shift. (b) Stage 2. fine-tuning via semi-supervised consistency training with
+centroid alignment. The fine-tuning stage aims to further improve model generalization performance in
+target domains by minimizing intra-class distance and maximizing inter-class distance of the target domain
+samples assigned with pseudo labels, while explicitly minimizing the source-target domain gaps with centroid
+alignment. The encoders are sharing the weights.
+Positive Unlabelled contrastive Learning. The objective of the second component is to learn
+noise invariance by training the model to generate the same features regardless of the artificial noise
+that is added to the data without relying on label data. We carefully design multiple augmentations
+to simulate the radio wave noise such as frequency noise, temporal noise, different resolutions, and
+white noise as described in Section 5.2. The model learns noise invariance by minimizing cosine
+similarity between features from raw data 𝑓 and its augmented version ˆ𝑓 with the loss
+𝐿𝑠𝑒𝑙𝑓 = 1
+𝐵
+𝐵
+∑︁
+𝑖=1
+𝑓 𝑇
+𝑖 ˆ𝑓𝑖
+∥𝑓𝑖 ∥
+���ˆ𝑓𝑖
+���
+.
+(8)
+This loss is inspired by NT-Xent (eq. 1) from SimCLR [8] which is the contrastive self-supervised
+learning method that minimizes cosine similarity of positive samples (i.e., different augmentation
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+GaitSADA: Self-Aligned Domain Adaptation for mmWave Gait Recognition
+111:11
+of the same data) and maximizes the cosine similarity of negative samples (i.e., different data). The
+objective of negative samples in NT-Xent is to prevent dimensional collapse. However, since our Self
+Alignment loss is trained jointly with Supervised Learning loss, it can completely avoid the collapse
+issue. As a result, we are able to retain only the positive similarity and eliminate the negative
+marginal fraction. The final loss function in the first stage combines two loss functions to encourage
+the intra-class concentration and inter-class distance while being invariant to perturbation from
+noise by utilizing unlabeled data from target domain:
+𝐿𝑠𝑡𝑎𝑔𝑒1 = 𝐿𝑠𝑢𝑝𝑒𝑟𝑣𝑖𝑠𝑒𝑑 + 𝐿𝑠𝑒𝑙𝑓 .
+(9)
+4.3
+Fine-tuning via Consistency Training with Centroid Alignment
+In the first stage, the model has learned the well-aligned class-wise feature from label data in the
+source domain and unlabelled data from target domain. The objective of the second stage is to
+further regulate the feature of the target domain data of the same class to be close to each other.
+Consistency Training. Unlike the source domain, the labels in the target domain are unknown.
+Therefore, we cannot directly align target feature by class. To address this problem, we enforce
+consistency training via pseudo labels instead of the true labels of the target domain. Consistency
+training utilizes unlabeled data by relying on the assumption that when given perturbed variations
+of the same input, the model ought to provide similar predictions. The two identical models are
+used for consistency training. The first branch takes raw input data with no augmentation in order
+to provide accurate pseudo labels. We compute the model’s predicted class distribution of a given
+unlabeled spectrogram input: 𝑞𝑖 = 𝑝(𝑦𝑖|𝑢𝑖) as soft pseudo labels. The cross-entropy 𝐻 is applied
+against the pseudo labels distribution and the model’s prediction distribution of the augmented
+unlabeled data A(𝑢𝑖). The loss is defined as
+𝐿𝑐𝑜𝑛𝑠𝑖𝑠𝑡𝑒𝑛𝑐𝑦 = 1
+𝐵
+𝐵
+∑︁
+𝑖=1
+1(arg max
+𝑦 (𝑞𝑖) ≥ 𝜏)𝐻 (𝑞𝑖, ˆ𝑞𝑖),
+(10)
+where ˆ𝑞𝑖 = 𝑝(𝑦𝑖|A(𝑢𝑖)) represents the probability distribution of class 𝑦𝑖 given augmented input
+A(𝑢𝑖) and𝜏 is a scalar parameter denoting the threshold to filter only the high confidence prediction.
+To maintain the same standard of Additive Margin Softmax (Eq. (7)) in the first stage, we normalize
+weights and features of the classifier which can be referred to as cosine similarity 𝑐𝑜𝑠𝜃. The scale 𝑠
+remains the same, but we remove margin 𝑚 for accurate pseudo labels:
+𝑝(𝑦𝑖|·) =
+𝑒𝑠·(𝑐𝑜𝑠𝜃𝑦𝑖 )
+�𝑐
+𝑗=1 𝑒𝑠·𝑐𝑜𝑠𝜃 𝑗 ,
+(11)
+where 𝑐 is the number of classes. The predictions are likely to be inaccurate at the beginning of
+the training. For this, most semi-supervised learning algorithms [30][19][3][2] initialize training
+with a small weight 𝜆 for the consistency loss term and increase it over time. However, with the
+confidence threshold, FixMatch [28] suggests that the pseudo label produces a natural curriculum
+learning. As less pseudo-label data are available at first since the low confidence predictions are
+filtered out. When the model is more accurate, it then provides more predictions with a high degree
+of confidence. As a result, the threshold automatically increases the number of pseudo-label data
+without explicitly specifying incremental weight 𝜆.
+Centroid Alignment. While consistency training loss tends to move weights 𝑊 of the classifier
+toward target features, supervised learning loss still attempts to bring the weights back to source
+features. The weights can oscillate between these two groups as a result of the domain gap between
+the source and target features as shown in Fig. 5. To mitigate this issue, we impose centroid
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+111:12
+Pinyoanuntapong et al., Ekkasit Pinyoanuntapong, Ayman Ali, Kalvik Jakkala, Pu Wang, Minwoo Lee, Qucheng Peng,
+Chen Chen, and Zhi Sun
+Centroid
+Source Feature
+Target Feature
+Less Confidence Target Feature 
+Centroid of Source Feature
+Centroid of Target Feature
+Fig. 5. Centroid Alignment. The centroid of source feature tends to be closed to weight vector𝑊 following
+the supervised learning loss. On the other hand, the centroid of target features is likely to be further away
+due to the domain gap. Centroid Alignment encourages weight vector 𝑊 to move to a new centroid which is
+the center of both source and target features which can be a better representation as more target features are
+learned from consistency training.
+alignment loss to gradually bring the weights to the center of source and target features by applying
+Additive Margin Softmax to class-wise weights and class centroids as
+𝐿𝑐𝑒𝑛𝑡𝑟𝑜𝑖𝑑 = − 1
+𝐶
+𝐶
+∑︁
+𝑐=1
+𝑙𝑜𝑔
+𝑒𝑠·(𝑐𝑜𝑠𝜃𝑐)−𝑚
+𝑒𝑠·(𝑐𝑜𝑠𝜃𝑐)−𝑚 + �𝐶
+𝑗=1,𝑗≠𝑖 𝑒𝑠·𝑐𝑜𝑠𝜃 𝑗 ,
+(12)
+where 𝑐𝑜𝑠𝜃𝑐 =
+𝑊 𝑇
+𝑐 𝑍𝑐
+∥𝑊𝑐 ∥ ∥𝑍𝑐 ∥ is the inner product of normalized weights 𝑊𝑖 and centroids 𝑍𝑖 of each
+class. The class-wise centroid is from the Exponential Moving Average (EMA) of both source and
+target features with hyperparameter 𝛼 to control the rate of change as
+𝑍𝑐𝑖 = 𝛼 ˆ𝑍𝑐𝑖 + (1 − 𝛼)𝑍𝑐𝑖−1.
+(13)
+The current batch centroid ˆ𝑍𝑐𝑖 is the average features of source 𝑓𝑠𝑖 and target 𝑓𝑡𝑖. However, because
+the true class of the target domain is unknown, we cannot directly assign features to the class
+centroid. Instead, the hardmax (i.e., argmax) of the model’s prediction is used as a pseudo label of
+the class of each sample. Also, target features can be noisy at the beginning of the training, similar
+to consistency loss (Eq. (10)), only the high confidence labels are used for calculating the centroids
+to avoid false predictions of pseudo labels. Therefore,
+ˆ𝑍𝑐𝑖 =
+�𝐵
+𝑖=1 𝑓𝑠𝑖 + 1(arg max𝑦(𝑞𝑖) ≥ 𝜏)𝑓𝑡𝑖
+𝐵 + �𝐵
+𝑖=1 1(arg max𝑦(𝑞𝑖) ≥ 𝜏)
+.
+(14)
+Lastly, we impose Additive Margin Softmax for source label data as a supervised learning loss along
+with consistency training and Centroid Alignment loss to prevent dimensional collapse. So the
+final loss equation is shown as
+𝐿𝑠𝑡𝑎𝑔𝑒2 = 𝐿𝑠𝑢𝑝𝑒𝑟𝑣𝑖𝑠𝑒𝑑 + 𝐿𝑐𝑜𝑛𝑠𝑖𝑠𝑡𝑒𝑛𝑐𝑦 + 𝜆𝐿𝑐𝑒𝑛𝑡𝑟𝑜𝑖𝑑.
+(15)
+5
+EXPERIMENT SETUP
+Texas Instruments (TI) IWR1642EVM boost board interfaced with a DCA1000EVM board to collect
+the raw mmWave radar signal. The radar system consists of two transmitting antennas, four
+receiving antennas, and 120° view of the azimuth plane. The radar system supports up to a 4Ghz
+bandwidth operating on 77 GHz to 81 GHz. To configure FMCW wave parameters such as chirp
+width, repetition time, and chirp slope from our radar device, we use a Dell Latitude 7480 laptop
+with TI mmWave studio software as a control system.
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+GaitSADA: Self-Aligned Domain Adaptation for mmWave Gait Recognition
+111:13
+(a) Cutout
+Horizontal
+(b) Cutout
+Vertical
+(c) White Noise
+.
+(d) Zoom
+.
+(e) Shear
+.
+(f) Rotation
+.
+Fig. 6. Spectrogram of 6 augmentations.
+5.1
+Dataset and preprocessing
+The walking dataset is collected from four distinct locations such as laboratory, conference room,
+server room, and office. We recruit ten volunteers to walk in his/her natural ways. Each participant
+either walks toward or away from the radar. Data is preprocessed into 50 samples of five different
+days at each location, a total of 2500 samples (10 subjects × 50 samples × 5 days), except for the
+laboratory location, we collect additional data for another five more days (total of ten different days)
+in order to perform temporal domain shift experiments for a different time in the same location.
+We preprocess the data into 256×256×1 spectrogram and standardized values between -1 and 1.
+Laboratory location data is used as source data and the other location data (i.e. conference room,
+server room, and office) is referred to as target data. For the spatial domain drift experiment, day 1
+to day 3 samples are used as a training set for all source and target locations. The rest of the data is
+used for testing. For the temporal domain drift experiment, we employ 1 to 3 days of laboratory
+location as source data and the next consecutive days of laboratory location as target data. For
+example, considering the temporal 2-day case, the first and the second-day data of laboratory
+location is a source domain, the third and the fourth-day data is a target domain, and the fifth to
+the tenth-day data is utilized as a test set.
+5.2
+Spectrogram Augmentation
+Vertical noise stripe (Fig. 6b) simulates missing information in the random time interval of the
+spectrogram. Similarly, the horizontal noise stripe (Fig. 6b) represents the missing information in
+some frequency range. We implement horizontal and vertical noise together by randomly adding
+two of these noises per data with a probability of 1/3 to be horizontal and 1/3 to be vertical, the rest
+1/3 is for no cutout noise. The small side of the stripe is random between 2 to 8 pixels. The third
+augmentation is pixel-wised random numbers adjusted to the whole data uniformly simulating
+white noise (Fig. 6c) in a spectrogram. The uniformly random noise from -0.5 to 0.5 is added to
+the data with a probability of occurrence 2/3. Next is the zoom-in and out effect. Zoom-in will
+drop some information from the start and the end, also the lowest and the highest frequency of
+the spectrogram. Zoom effects (Fig. 6d) represent the data in different resolutions. The scale is
+applied to the data between 0.8 to 1.2 randomly. These four augmentations act as virtual noise
+to simulate the noisy real-world data thereby improving domain generalization. Not only these
+meaningful noises, but we also experiment with regular computer vision augmentations and we
+found that shearing (Fig. 6e) and rotation (Fig. 6f) help boost performance as well. Following
+Tensorflow ImageDataGenerator function, the shear angle in a counter-clockwise direction in
+degrees is random from 0 to 5. Also, 5-degree ranges are randomly applied for rotation. So we
+add these augmentations to our preprocess pipeline and then apply all augmentation to every
+spectrogram data with random magnitude. In addition to consistency training, augmentation can
+be used to generalize data in supervised learning during the first training stage.
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+111:14
+Pinyoanuntapong et al., Ekkasit Pinyoanuntapong, Ayman Ali, Kalvik Jakkala, Pu Wang, Minwoo Lee, Qucheng Peng,
+Chen Chen, and Zhi Sun
+5.3
+Model
+A feature extraction model is modified from Resnet-50 to support one channel instead of three
+channels and use SELU as an activation function. The other parts are similar to the original Resnet-
+50 [15] composed of five stages of convolution blocks and identical blocks including a stack of
+convolutional blocks, batch normalization, and activation with skip connections in each block and
+output 128 features. The classification model is simply a linear layer with 10 neurons to classify
+128 features from feature extraction into 10 classes.
+5.4
+Training Details
+We apply the same settings for all 1- to 3-day cases and all locations in both stages 1 and 2 to
+be consistent in all domain configurations. Every convolution layer applies 1e−4 of 𝑙2 kernel
+regularizer and trains for 10,000 epochs. The training optimizer is ADAM with an initial learning
+rate starting at 1e−3 and ending at 1e−5, applying the polynomial learning rate decay scheduler
+with learning rate restart. We use a batch size of 64. The confidence threshold of the consistency
+pseudo-label is 0.97. Additive Margin Softmax Loss applies 𝑙2-normalization to both input and
+weights to the fully connected layer head without bias, with 𝑠 = 10 and 𝑚 = 0.2. All losses in
+both stages 1 and 2 are applied with equal weights except centroid alignment loss with 𝛼 = 0.05 to
+gradually move centroids of the classes without breaking the current.
+5.5
+Evaluation
+We evaluate the model without any domain adaptation (supervised learning) as a baseline comparing
+with the five representative domain adaptation methods including GAN [16], GRL [10], ADDA [32],
+CDAN [21], and FixMatch [28] along with our method, GaitSADA, on the four different locations
+(laboratory, conference room, server room, and office) to explore the spatial domain adaptation
+using 1-3 days of data for training and the rest of data for testing.
+Generative Adversarial Networks (GAN). Generative Adversarial Network (GAN) [12] is a
+generative model which can construct new data instances in an unsupervised learning fashion by
+learning the distribution of the target data. The model separates into two sub-models. The first,
+generator 𝐺 produces the data by capturing the data distribution, and the second, discriminator
+𝐷 classifies if a sample comes from the true data or generated from the generator 𝐺. GAN can be
+repurposed for domain adaptation by considering the feature extraction model as a generative
+model 𝐺 generates features from source or target domain. While discriminator 𝐷 learns to classify
+a feature from source or target domain, the feature extractor 𝐺 tries to confuse the discriminator 𝐷
+by generating the features with the same distribution for both source and target, thereby closing
+the domain drift gap. So the value function 𝑉 (𝐺, 𝐷) is a two-player minimax game played by 𝐷
+and 𝐺, i.e., min𝐺 max𝐷 𝑉 (𝐷,𝐺) = E𝑥𝑠∼𝑋𝑠 [𝑙𝑜𝑔𝐷(𝐺(𝑥𝑠))] + E𝑥𝑡 ∼𝑋𝑡 [𝑙𝑜𝑔(1 − 𝐷(𝐺(𝑥𝑡)))], where 𝑥𝑠 is
+data drawn from source domain and similarly, 𝑥𝑡 is the data drawn from target domain.
+Gradient Reversal Layer (GRL). Similar to GAN, feature extractor 𝐺 (i.e., generator) and
+domain discriminator 𝐷 are trained to compete with each other. However, rather than minimizing
+domain discriminative loss and maximizing confusion loss separately, gradient reversal layer (GRL)
+[10] shows that this adaptation behavior can be implemented in a single step, practically with
+any feed-forward model by adding a GRL layer to train 𝐺 and 𝐷 at the same time in the opposite
+gradient direction. The GRL has no parameters associated with its component. For forward pass
+computation, the GRL is defined as an identity matrix. On the other hand, during backpropagation,
+GRL flips the gradient direction from subsequent level with the scale hyperparameter 𝜆 before
+passing the gradient to the preceding layer.
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+GaitSADA: Self-Aligned Domain Adaptation for mmWave Gait Recognition
+111:15
+Adversarial Discriminative Domain Adaptation (ADDA). Adversarial Discriminative Do-
+main Adaptation (ADDA) [32] improves GAN-based unsupervised domain adaptation by utilizing
+discriminative modeling, united weight sharing, and a GAN loss. Unlike GAN and GRL, feature
+extractor 𝐺, classier 𝐶, and domain discriminator 𝐷 are learned all together simultaneously. ADDA
+takes two separate pre-training stages. In the first stage, feature extractor 𝐺 and classifier 𝐶 are
+pre-trained with only source label data using a standard cross entropy loss. In the second stage,
+the weights of feature extractor 𝐺 are transferred from the first stage pre-trained. Similar to GAN,
+domain discriminative loss 𝐿𝑑𝑖𝑠 and confusion loss 𝐿𝑐𝑜𝑛𝑓 are trained in an adversarial manner.
+However, confusion loss 𝐿𝑐𝑜𝑛𝑓 in ADDA updates the feature extractor based on only target data. So
+the confusion loss 𝐿𝑐𝑜𝑛𝑓 reduces to min𝐺 𝐿𝑐𝑜𝑛𝑓 = − E𝑥𝑡 ∼𝑋𝑡 [𝑙𝑜𝑔𝐷(𝐺(𝑥𝑡))]. Finally, during test time,
+the pre-trained classifier 𝐶 in stage 1 and the pre-trained feature extractor in stage 2 are used for
+class prediction.
+Adversarial Domain Adaptation (CDAN). Conditional Adversarial Domain Adaptation (CDAN)
+[21] is another adversarial domain adaptation method utilizing multilinear conditioning to condi-
+tion discriminate domain by its class prediction. The aforementioned methods train the feature
+extractor 𝐺 and the domain discriminator 𝐷 directly from the probability distribution 𝑝(𝐺(𝑥))
+of output feature representation 𝐺(𝑥) without any condition. However, adapting the marginal
+probability distribution 𝑝(𝐺(𝑥)) may not be sufficient. CDAN utilizes a multilinear map, which can
+be defined as the outer product, to learn cross-covariance dependency E𝑥𝑦[𝐺(𝑥) ⊗ 𝐶(𝐺(𝑥))] of
+feature 𝐺(𝑥) and class prediction 𝐶(𝐺(𝑥)). Therefore, the domain discriminative loss 𝐿𝑑𝑖𝑠 can be
+written as: 𝐿𝑑𝑖𝑠 = − E𝑥𝑠∼𝑋𝑠 [𝑙𝑜𝑔𝐷(𝐺(𝑥𝑠) ⊗ 𝐶(𝐺(𝑥𝑠))] − E𝑥𝑡 ∼𝑋𝑡 [𝑙𝑜𝑔(1 − 𝐷(𝐺(𝑥𝑡) ⊗ 𝐶(𝐺(𝑥𝑡)))]. The
+confusion loss 𝐿𝑐𝑜𝑛𝑓 in CDAN is referred as the negative of −𝐿𝑑𝑖𝑠 and the classification loss 𝐿𝑐𝑙𝑠 is
+standard cross entropy loss. The minimax game of CDAN is: min𝐺 𝐿𝑐𝑙𝑠 − 𝛼𝐿𝑑𝑖𝑠 and min𝐷 𝐿𝑑𝑖𝑠.
+FixMatch. We repropose FixMatch [28], a semi-supervised learning method, for our unsuper-
+vised domain adaptation problem. FixMatch [28] learns target unlabeled data 𝑢𝑖 by generating
+pseudo-labels from models’ predictions on weakly-augmented data A(𝑢𝑖)) and filtering only high
+confidence data by selecting the prediction that has confidence more than the threshold. The cross-
+entropy 𝐻 is applied against the pseudo labels and the model’s prediction of the highly-augmented
+versions of the same unlabeled data ˆ
+A(𝑢𝑖), which can be referred to as consistency training to
+learn unlabeled data from its past knowledge: 𝐿𝐹𝑖𝑥𝑀𝑎𝑡𝑐ℎ = 1
+𝐵
+�𝐵
+𝑖=1 1(arg max𝑦(𝑞𝑖) ≥ 𝜏)𝐻 (𝑞𝑖, ˆ𝑞𝑖)
+where 𝑞𝑖 = 𝑝(𝑦𝑖|A(𝑢𝑖)) represents the prediction probability distribution of class 𝑦𝑖 from given
+weakly-augmented input A(𝑢𝑖) and similarly for prediction ˆ𝑞𝑖 = 𝑝(𝑦𝑖| ˆ
+A(𝑢𝑖)) based on highly-
+augmented data ˆ
+A(𝑢𝑖)). 𝜏 is a scalar hyperparameter representing the confidence threshold of
+retaining a pseudo-label
+6
+RESULTS
+The normal training settings such as optimizer, scheduler, and batch size, described in Section 5.4,
+are maintained the same across all experiments. We study naive supervised learning without
+domain adaptation to understand the domain discrepancy issue in low data regime, temporal case,
+and spatial case. As summarized in Table 2, our GaitSADA method outperforms the other five
+representative domain adaptation methods on all spatial domain adaptation (i.e., different locations)
+and temporal domain adaptation (i.e., different days on laboratory location).
+First, we show the problem of domain discrepancy of the mmWave data from different spatial
+and temporal domain shifts by training supervised learning of our modified version of Resnet-50 as
+a feature extraction only on label source data without any domain adaptation to the target domain.
+The loss function is a regular cross-entropy loss applied against ten subjects in source data.
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+111:16
+Pinyoanuntapong et al., Ekkasit Pinyoanuntapong, Ayman Ali, Kalvik Jakkala, Pu Wang, Minwoo Lee, Qucheng Peng,
+Chen Chen, and Zhi Sun
+Table 2. Result of training on 1 to 3 days on the data from laboratory location (source domain) while adapting
+to different 1 to 3 days data of same location (i.e., temporal target domain) and 1 to 3 days of different
+target locations, (i.e., server, conference, and office) with 5 representative methods, GRL [10], GAN 5.5, ADDA
+[32], CDAN [21], and FixMatch [28] along with "Supervised Learning" which train only on source label data
+without domain adaptation and lastly, our proposed method, GaitSADA. The results report in % of accuracy.
+# of training days
+Methods
+Temporal
+Server
+Conference
+Office
+Average
+1-day
+Supervised
+76.05
+34.93
+47.00
+33.39
+47.84
+GAN
+79.57
+38.14
+66.00
+32.93
+54.16
+GRL
+74.97
+40.14
+62.10
+36.14
+53.34
+ADDA
+80.96
+37.71
+65.21
+32.60
+54.12
+CDAN
+73.67
+45.15
+62.40
+43.24
+56.12
+FixMatch
+94.32
+44.24
+69.40
+49.05
+64.25
+GaitSADA (our)
+96.3
+75.98
+82.6
+63.76
+79.66
+2-day
+Supervised
+88.46
+53.05
+71.00
+49.92
+65.61
+GAN
+92.18
+59.96
+77.80
+59.76
+72.43
+GRL
+85.98
+69.07
+80.90
+59.96
+73.98
+ADDA
+88.55
+51.04
+75.63
+55.62
+67.71
+CDAN
+91.72
+63.36
+85.90
+63.16
+76.04
+FixMatch
+95.82
+75.38
+90.10
+82.58
+85.97
+GaitSADA (our)
+97.63
+79.88
+99.1
+90.59
+91.8
+3-day
+Supervised
+95.25
+71.77
+85.20
+67.61
+79.96
+GAN
+96.79
+78.18
+94.30
+85.89
+88.79
+GRL
+96.11
+84.58
+89.30
+75.58
+86.39
+ADDA
+97.38
+79.79
+94.79
+88.13
+90.02
+CDAN
+98.03
+91.39
+93.80
+86.19
+92.35
+FixMatch
+99.29
+93.29
+98.30
+88.89
+94.94
+GaitSADA (our)
+99.68
+93.69
+99.2
+97.7
+97.57
+The performance of 3-day cases is 79.96% on average for all locations and temporal case, then drop
+to 65.61% and 47.85% in 2-day and 1-day data which apparently shows that the amount of data helps
+mitigate the domain shift issues in both spatial and temporal cases. In the temporal case, the accuracy
+is higher than the spatial cases as the environment is the same, only the small features of subjects
+(e.g., clothes, body temperate) or subtle environment features (e.g. air moisture, temperature) change
+due to the different experiment times. The accuracy drops a lot in the conference room due to the
+different environments and drops dramatically in the server room and office as more obstruction
+and the smaller rooms cause the micro-Doppler to interfere with the reflection from the objects.
+GAN, GRL, and ADDA yield similar results. In 1-day scenario, their accuracies are relatively
+comparable, 54.16%, 53.34%, and 54.12% for GAN, GRL, and ADDA correspondingly. For 2-day
+instance, GAN and GRL are not significantly different, 72.43% and 73.98%. However, ADDA is a
+little bit lower at 67.71%. For 3-day scenario, GAN and GRL still perform near, 88.79% and 86.39%
+but ADDA accuracy spikes to 90.02%. GAN and GRL perform quite similarly owing to their simlar
+adaptation behavior as they alter the distribution by both source and target data. ADDA, on the
+other hand, updates feature extraction simply by target data toward source feature. Overall, these
+three techniques help alleviate the domain drift issue by boosting accuracy from the supervised
+method on average in all 3-day cases. However, we observe the accuracies drop compared to the
+supervised learning method in 1-day case for all these three methods, e.g., GAN and ADDA drop
+0.46% and 0.79% in office location, GRL drops 1.08% for the temporal case due to the intricate
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+GaitSADA: Self-Aligned Domain Adaptation for mmWave Gait Recognition
+111:17
+structure of minimax game between domain discriminator and feature extraction. The domain
+discriminator is unable to entirely minimize the domain shift while training on fewer data since it
+does not learn all feature distinctions of the source and target domains.
+CDAN has as better results comparing to GAN, GRL, and ADDA, 56.12%, 76.04%, and 92.35%
+on average of 1-day, 2-day, and 3-day respectively due to its ability to learn class-wise feature
+distribution cross-covariance dependency on class prediction and its feature, therefore, helping
+understanding the class-conditional domain drift. However, the accuracy drop happens in 1-day
+temporal case which CDAN accuracy is 73.67% while supervised learning method is 76.05%. The
+limited amount of training data is still a problem for CDAN as it requires more data to understand
+the distribution of source and target.
+FixMatch performs best comparing to the other representative methods even though its original
+purpose is for semi-supervised learning tasks, not unsupervised domain adaptation. Its average
+accuracies for 1-day, 2-day, and 3-day cases are 64.25%, 85.97%, and 94.94% respectively. Because
+FixMatch synthesizes data utilizing pseudo-label from target unlabeled data to train the model,
+it virtually trains on more data. We observe that the advantage of FixMatch presents more than
+the other domain adaptation methods in low data regime as in the temporal 1-day case, FixMatch
+gives 18.27% performance boost from supervised learning method while the other methods struggle
+with adapting for the small dataset. In 3-day case, this effect appears less, and the performance
+boost is down to 4.04%, as the larger dataset already mitigates the domain generalization problem.
+On the other hand, in the 1-day server location which is more challenging due to their different
+environments, FixMatch accuracy is less than CDAN method because the pseudo-label is filtered
+by the confidence threshold so that there is less synthetic data to train in the difficult environment.
+The average accuracies for 1-day, 2-day, and 3-day cases of GaitSADA are 80.41%, 91.80%, and
+97.57% respectively which increase accuracies of vanilla supervised learning method by a large
+margin. GaitSADA presents more advantages for fewer data. Specifically, the performance boosts
+by 32.56%, 26.19%, and 17.61% on average of 1-day, 2-day, and 3-day data respectively. Our results
+for the 1-day scenario clearly demonstrate a very strong accuracy improvement in the low data
+regime, especially in the difficult domain drift environments, due to the simplicity of the aligning
+components without a complex discriminating structure that requires more training data (e.g.,
+GAN, GRL, ADDA). The performance improves by 20.25%, 41.05%, 35.6, and 30.37% from temporal,
+server, conference, and office respectively.
+7
+VISUALIZATION
+7.1
+t-SNE
+We present t-SNE [5] plots to visualize the high-dimensional datasets in Fig. 7, Fig. 9 and Fig. 8. We
+convert high-dimensional features (128 dimensions) into a 2D space in such a way that similar data
+points are mapped into nearby points in the low-dimensional representation. This allows us to
+visualize the feature in a way that reveals the underlying structure and local relationships between
+the source and target in different classes to better understand the domain drift behavior in various
+locations and days. Source and target features are denoted by the symbol "o" and "x." Different
+colors are used to depict the various classes.
+Supervised learning. In the temporal case (Fig. 7a), most features are well-learned and sepa-
+rated by a considerable distance from other classes. In the temporal scenario, according to this
+visualization, each class has a gap between the others. However, some groups are still too close to
+one another and some target features ("x" symbols) are close to the incorrect class group. Conference
+(Fig. 7b) and Server (Fig. 7b), some of the target features ("x" symbol) are closer to the other color
+groups of source features ("o" symbol) more than the same color groups which clearly shows the
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+111:18
+Pinyoanuntapong et al., Ekkasit Pinyoanuntapong, Ayman Ali, Kalvik Jakkala, Pu Wang, Minwoo Lee, Qucheng Peng,
+Chen Chen, and Zhi Sun
+(a) Temporal
+(b) Server
+(c) Conference
+(d) Office
+Fig. 7. t-SNE [5] visualization for vanilla supervised learning training only on label source data for 3 days. 10
+colors represent 10 classes, while "o"s shows the source features and "x"s are target features.
+domain drift behavior. In office data (Fig. 7d), the visualization presents even stronger domain drift
+as the features are mixed together.
+Representative Methods. According to the visualization in A.1, t-SNE of GAN, GRL, and ADDA
+depict similar presentations. Each class in the temporal case is separated from the others better than
+supervised learning. However, some target features are still very near the incorrect class. In sever,
+conference, and office cases, features are better aligned even though the feature groups are not
+compact enough, which can cause misclassification. FixMatch exhibits the greatest compactness of
+each feature class. However, in the office case, each group is still close together.
+GaitSADA. Fig. 8 shows that the target features are well-adjusted to the source features. More-
+over, all classes are compact and strongly discriminated (i.e., wide spaces between each other
+classes). This clearly exhibits the benefit of our method that promotes intra-class compactness and
+inter-class separability while aligning source and target features to the class centers. Even our first
+stage (Additive Margin + Self Alignment) exhibits the compactness of the features (Fig. 9).
+7.2
+Grad-CAM
+Grad-CAM [27] visualizes the weighted feature maps to create a heatmap that highlights the most
+important regions to make a prediction. Fig. 10 illustrates diverse Grad-CAM saliency heatmaps for
+each algorithm. The other methods’ wrong predictions tend to focus only on a specific part of the
+spectrogram data, high-frequency or low-frequency part. The high-frequency part represents the
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+GaitSADA: Self-Aligned Domain Adaptation for mmWave Gait Recognition
+111:19
+(a) Temporal
+(b) Server
+(c) Conference
+(d) Office
+Fig. 8. t-SNE [5] visualization for the second stage of GaitSADA domain adaptation method training on label
+source data and unlabeled data for 3 days. 10 colors represent 10 classes, while "o"s shows the source features
+and "x"s are target features.
+fast-moving limbs i.e., arms and legs, indicating the importance of this region for gait recognition.
+However, Figures 10a, 10b, and 10f show the models tend to confuse high-frequency and the white
+noise in the background. Concentrating on this background noise area and ignoring the other
+parts lead to misprediction. Figures 10c, 10d, and 10k tend to focus only on the low-frequency
+part, the torso region, which may contain less unique information compared to other parts of
+the body. We visualize saliency of GaitSADA on the same data that the other methods predict
+incorrectly. In contrast to the other methods, GaitSADA looks broadly at multiple parts of the body
+without focusing too much on specific parts which allow our method to effectively use more unique
+information from different parts of the body for gait recognition and thus easily adapt to domain
+shift.
+7.3
+Ablation Study
+We experiment with 2-day data on laboratory location as source data while adapting to different
+days (i.e., temporal) and 3 different locations, (i.e., server, conference, and office) in the same setting
+as the experiments in Table 2 and examine the behavior of each component contributing to the final
+results in both jointly training and 2-stage training setting. We study six different combinations of
+components as shown in Table 3, i.e., supervised learning, additive margin softmax loss, contrastive
+learning, consistency training, and centroid alignment.
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+QX111:20
+Pinyoanuntapong et al., Ekkasit Pinyoanuntapong, Ayman Ali, Kalvik Jakkala, Pu Wang, Minwoo Lee, Qucheng Peng,
+Chen Chen, and Zhi Sun
+(a) Temporal
+(b) Server
+(c) Conference
+(d) Office
+Fig. 9. t-SNE [5] visualization for the first stage of GaitSADA domain adaptation method training on label
+source data and unlabeled data for 3 days. 10 colors represent 10 classes, while "o"s shows the source features
+and "x"s are target features.
+(a) Supervised
+(b) GAN
+(c) GRL
+(d) ADDA
+(e) CDAN
+(f) FixMatch
+(g) GaitSADA
+(h) GaitSADA
+(i) GaitSADA
+(j) GaitSADA
+(k) GaitSADA
+(l) GaitSADA
+Fig. 10. Grad-CAM [27] visualizations of the wrong prediction of other methods (a)-(f) comparing to our
+correct prediction of the same data (g)-(l). The red color area represents the more important weight contributing
+to the prediction.
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+(xx)
+XX++X
+C
+XtGaitSADA: Self-Aligned Domain Adaptation for mmWave Gait Recognition
+111:21
+Table 3. Ablation experiments of 2-day data on laboratory location as source data while adapting to different
+days (i.e., temporal) and 3 different locations, (i.e., server, conference, and office) using multiple combinations
+of Additive Margin, Contrastive learning, Consistency Training and Centroid Alignment. The results report in
+% of accuracy.
+Stage
+Components
+Temporal
+Server
+Conference
+Office
+Average
+1 stage
+(train jointly)
+1. supervised
+88.46
+53.05
+71
+49.92
+65.61
+2. supervised + AM
+91.8
+73.72
+82.8
+67.07
+78.85
+3. supervised + AM + contrastive learning
+94..21
+80.38
+89.37
+70.87
+83.71
+4. supervised + AM + consistency + centroid
+99.19
+52.15
+98.2
+46.55
+74.02
+2 stages
+5. (1st stage) supervised + AM + contrastive learning
+(2nd stage) supervised + AM + consistency
+94.68
+78.58
+90.3
+72.77
+84.08
+6. (1st stage) supervised + AM + contrastive learning
+(2nd stage) supervised + AM + consistency + centroid
+97.63
+79.88
+99.10
+90.59
+91.80
+(1) The first experiment is simply just supervised learning method to present the baseline of the
+ablation (supervised).
+(2) Additive margin softmax loss (AM) is added to the supervised learning in the second ex-
+periment. The AM component alone improves the performance by a large margin. In fact,
+this component already outperforms most of the other adversarial-based methods as AM
+loss increases inter-class separability and intra-class compactness without training multi-
+ple models (i.e. feature extractor and domain discriminator) to compete with each other as
+adversarial-based methods which require more data in order to learn the feature distribution.
+The performance boosts more in the large than the small domain drift as shown in the server
+case, the accuracy improves by 20.67% while in the temporal case improves by only 3.34%
+from supervised learning in the first experiment.
+(3) In the third experiment, we add a contrastive learning component along with AM which is
+exactly the same as in the first stage as of GaitSADA. Contrastive learning further improves
+by 4.86% on average as the model learns noise invariance on the target data.
+(4) The fourth experiment can be considered as the second stage of our GaitSADA without the
+first stage pre-train. In this experiment, consistency training and centroid alignment are
+trained jointly with supervised learning with additive margin softmax loss. This second stage
+ablation shows that without the first stage pre-training, in the less domain shift environment
+i.e., temporal and conference case, the accuracies are high. In temporal, the accuracy is even
+higher than one with the first stage pre-training experiment (the 6th experiment). However,
+in the large domain drift environments including server and office, the accuracies drop
+dramatically, even less than supervised learning only (the 1st experiment).
+(5) The fifth ablation study is GaitSADA without Centroid Alignment which shows that with-
+out Centroid Alignment the average performance decreases by 7.72% (drops from the 6th
+experiment).
+(6) The last one is our GaitSADA combines all components in 2 stages, which shows the overall
+best performance.
+8
+CONCLUSION
+In this paper, we designed a system for gait recognition using a commercial mmWave radar,
+conducted multiple times and environments to study the domain discrepancy in various cases,
+and proposed a novel domain adaptation method designed explicitly for mmWave signals. We
+experiment with five regular domain adaptation methods to close the domain gaps. The experiments
+show the limitations of adversarial-based domain adaptation methods which require more data in
+order to learn the data distribution due to the complicated minimax game of the feature extractor
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+111:22
+Pinyoanuntapong et al., Ekkasit Pinyoanuntapong, Ayman Ali, Kalvik Jakkala, Pu Wang, Minwoo Lee, Qucheng Peng,
+Chen Chen, and Zhi Sun
+and domain discriminator model. On the other hand, GaitSADA directly minimizes intra-class
+variation while maximizing inter-class deviation by utilizing self- and semi-supervised to learn
+target unlabelled data. Our proposed method outperforms standard domain adaptation methods
+by a large margin, particularly in the low data regime. Visualization and ablation studies further
+validate the effectiveness of GaitSADA.
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+GaitSADA: Self-Aligned Domain Adaptation for mmWave Gait Recognition
+111:23
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+
+GaitSADA: Self-Aligned Domain Adaptation for mmWave Gait Recognition
+111:25
+A
+SUPPLEMENTARY
+A.1
+t-SNE
+(a) Temporal
+(b) Server
+(c) Conference
+(d) Office
+Fig. 11. t-SNE visualization for 3-day GAN
+(a) Temporal
+(b) Server
+(c) Conference
+(d) Office
+Fig. 12. t-SNE visualization for 3-day GRL
+(a) Temporal
+(b) Server
+(c) Conference
+(d) Office
+Fig. 13. t-SNE visualization for 3-day ADDA
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+111:26
+Pinyoanuntapong et al., Ekkasit Pinyoanuntapong, Ayman Ali, Kalvik Jakkala, Pu Wang, Minwoo Lee, Qucheng Peng,
+Chen Chen, and Zhi Sun
+(a) Temporal
+(b) Server
+(c) Conference
+(d) Office
+Fig. 14. t-SNE visualization for 3-day CDAN
+(a) Temporal
+(b) Server
+(c) Conference
+(d) Office
+Fig. 15. t-SNE visualization for 3-day FixMatch
+A.2
+Grad-CAM
+Fig. 16. Grad-CAM of supervised learning method.
+Fig. 17. Grad-CAM of GAN.
+Fig. 18. Grad-CAM of GRL.
+Fig. 19. Grad-CAM of ADDA.
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+XGaitSADA: Self-Aligned Domain Adaptation for mmWave Gait Recognition
+111:27
+Fig. 20. Grad-CAM of GRL.
+Fig. 21. Grad-CAM of FixMatch.
+Fig. 22. Grad-CAM of GaitSADA.
+Received 20 February 2007; revised 12 March 2009; accepted 5 June 2009
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf,len=1294
+page_content='111  s  GaitSADA: Self-Aligned Domain Adaptation for mmWave  Gait Recognition  EKKASIT PINYOANUNTAPONG, AYMAN ALI, KALVIK JAKKALA, PU WANG, and MIN-  WOO LEE, University of North Carolina at Charlotte, USA  QUCHENG PENG and CHEN CHEN, University of Central Florida, USA  ZHI SUN, Tsinghua University, China  mmWave radar-based gait recognition is a novel user identification method that captures human gait biometrics  from mmWave radar return signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' This technology offers privacy protection and is resilient to weather  and lighting conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' However, its generalization performance is yet unknown and limits its practical  deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' To address this problem, in this paper, a non-synthetic dataset is collected and analyzed to reveal  the presence of spatial and temporal domain shifts in mmWave gait biometric data, which significantly impacts  identification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' To address this issue, a novel self-aligned domain adaptation method called GaitSADA  is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' GaitSADA improves system generalization performance by using a two-stage semi-supervised  model training approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The first stage employs semi-supervised contrastive learning to learn a compact  gait representation from both source and target domain data, aligning source-domain distributions implicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  The second stage uses semi-supervised consistency training with centroid alignment to further enhance  generalization performance in target domains by minimizing intra-class distance and maximizing inter-class  distance using pseudo labeling and class centroid estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Experiments show that GaitSADA outperforms  representative domain adaptation methods by an average of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='41% in low data regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  CCS Concepts: • Computer systems organization → Embedded systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' • Computing methodologies  → Artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  Additional Key Words and Phrases: datasets, neural networks, gaze detection, text tagging  ACM Reference Format:  Ekkasit Pinyoanuntapong, Ayman Ali, Kalvik Jakkala, Pu Wang, Minwoo Lee, Qucheng Peng, Chen Chen,  and Zhi Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' GaitSADA: Self-Aligned Domain Adaptation for mmWave Gait Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' ACM 37, 4,  Article 111 (August 2018), 27 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='org/XXXXXXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='XXXXXXX  1  INTRODUCTION  User identification is an essential component for many Internet of Things (IoT) applications, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  personalized environmental control, security management, and access control for automatic doors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  Short-distance biometrics such as fingerprints, face, iris, palm, and finger vein patterns typically  demand active user participation within a certain proximity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' In contrast, gait, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', human walking  Authors’ addresses: Ekkasit Pinyoanuntapong, epinyoan@uncc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
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+page_content=' Ayman Ali, aali26@uncc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
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+page_content=' Kalvik Jakkala, kjakkala@  uncc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
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+page_content=' Minwoo Lee, minwoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='lee@uncc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='edu, University of North Carolina at Charlotte, 9201,  Charlotte, North Carolina, USA, 28223;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Qucheng Peng, qucheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='peng@knights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
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+page_content=' Zhi Sun, zhisun@tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='cn, Tsinghua University,  Beijing, China , 100084 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee  provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and  the full citation on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Copyrights for components of this work owned by others than ACM must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  Abstracting with credit is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
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+page_content=' Request permissions from permissions@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
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+page_content='  © 2018 Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  0004-5411/2018/8-ART111 $15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='00  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='org/XXXXXXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='XXXXXXX  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='13384v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='CV]  31 Jan 2023  111:2  Pinyoanuntapong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', Ekkasit Pinyoanuntapong, Ayman Ali, Kalvik Jakkala, Pu Wang, Minwoo Lee, Qucheng Peng,  Chen Chen, and Zhi Sun  style, can be exploited for user identification from a distance without the subject’s participation or  interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Moreover, it is shown in existing study [35] that human gait is very hard to spoof or  mimic because our own gait works against us when we try to imitate someone elses gait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  The majority of gait recognition techniques are vision-based, which directly perform identifica-  tion tasks on video data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Appearance-based and model-based approaches are two general approaches  for vision-based gait recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Appearance-based approaches [9][7][20] utilize silhouettes from  a video sequence obtained from background subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Although background subtraction is  accurate in a lab context, it is difficult to use in a cluttered and dynamic real-world environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  Model-based approaches [25][31] try to tackle this issue by employing the existing pose estimators  to improve robustness to environmental variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' However, all vision-based approaches require  access to RGB visual images which are prone to performance loss from variations of ambient light  settings and the social restriction from privacy concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  Radar-based gait recognition is an emerging user identification solution that exploits radar  return signals to capture human gait biometric [6, 18, 24, 26, 36, 41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Unlike camera surveillance  systems, radar-based gait recognition systems have several unique advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' First, they are  privacy-preserving as they do not rely on images of human subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Next, they can operate in  adverse weather and lighting conditions such as heavy fog, smoke, rain, and zero-light conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  Finally, they can see through some opaque objects depending on its material and the frequency of the  radar [1, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Radar-based methods operate by emitting modulated electromagnetic signals, which  are scattered and returned back by the subjects in the signal propagation path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The radar return  signals capture the micro-Doppler (mD) signature, which can be represented as radar spectrogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  mD signature or radar spectrogram documents the time-dependent Doppler frequency shifts, which  are caused by the small-scale vibrations or rotations of human body parts during walking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Each  person has his/her unique walking style, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', gait, which leads to the unique mD signature that can  be exploited for gait recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  Existing radar-based gait recognition systems generally operate either in sub-6GHz WiFi bands  [26, 36, 42] or the emerging mmWave bands (30 to 300 GHz) [6, 18, 24, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Compared with sub-6GHz  radar systems, mmWave radar can promise much fine-grained motion capture with significantly  enhanced velocity and location resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' This feature is enabled by orderly shorter wavelength  and wider bandwidth of mmWave radar signals, compared with the sub-6GHz counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Besides  promising motion capture capacities, mmWave radar system only needs a single radar device  that integrates both radar transmitter and receiver, while sub-6GHz radar system is generally  implemented by a pair of WiFi devices to act as distributed radar transmitter and receiver, which  have to run modified WiFi firmware to extract mD signatures from CSI signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  The superior gait recognition performance of mmWave radar systems has been demonstrated in  our previous research [18] and other related work [6, 24, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' However, the spatial and temporal  generalization performance of this technology is still unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' We found that deep neural networks  (DNNs) trained on radar-based gait data suffer from substantial performance degradation with the  progression of time and in new locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Existing radar-based gait recognition systems proposed  in the literature, test their performance on a dataset collected alongside the training data and fail  to uncover the temporal degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Furthermore, they seldom address the spatial generalization  issues from introducing new environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The root cause of this generalization issue is domain  shift [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Simply put, it occurs when the testing data distribution differs from that of the training  data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' assumption is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Domain shift problem can be further exacerbated by  the harmful environment reflections induced by mmWave signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' These environment reflections  include static ones, which directly bounce from stationary obstructions e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' walls, desks, and chairs,  and dynamic ones, which indirectly bounce from other stationary obstructions and then bounce  from human bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The harmful environment reflections carry delayed and distorted mD signature  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  GaitSADA: Self-Aligned Domain Adaptation for mmWave Gait Recognition  111:3  information, which cannot be completely removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' This leads to environment-induced temporal  domain shift (TDS) and spatial domain shift (SDS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Moreover, human gait, although unique to each  individual, could contain many variations, potentially as a consequence of a subject’s mood and  clothing changes, further aggravate TDS and SDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  In this paper, we tackle the challenge of domain drift in mmWave radar-based gait recognition  across multiple environments and dates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' To address this issue, we propose a novel domain adaptation  method, GaitSADA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' GaitSADA is based on simple distribution alignment components and consists  of a two-stage approach: pretraining and fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' In the pretraining stage, the model learns  salient gait representations and noise invariance from both source and target domains using semi-  supervised contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' In the fine-tuning stage, the model generalizes the learned gait  representation to the target domain by leveraging pseudo-labels, semi-supervised consistency  training, and centroid alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Experiments demonstrate that GaitSADA outperforms classic  adversarial domain adaptation methods and achieves significant accuracy improvements in low  data regimes and challenging domain drift conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Our contributions can be summarized as  follows:  We curate a non-synthetic dataset consisting of mmWave radar-based gait biometric data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' This  dataset, for the first time, allows one to study and improve the spatio-temporal generalization  performance of a radar-based biometric identification system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  We perform extensive research on the SDS and TDS from four different location environments  on different days, experimenting with five state-of-the-art (SOTA) domain adaptation methods  to mitigate these drifts and show the issue of accuracy drop in low data regimes of SOTA  domain adaptation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  We propose a novel domain adaptation method for mmWave gait recognition system, self-  aligned domain adaptation for mmWave gait recognition (GaitSADA), by jointly exploiting  semi-supervised contrastive learning, semi-supervised consistency training, centroid align-  ment and deliberately designed radar spectrogram augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' GaitSADA outperforms  existing SOTA methods in 1-day, 2-day, and 3-day data by 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='41%, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='83%, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='63% respec-  tively on average of domain drift cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  2  RELATED WORK  Unsupervised domain adaptation (UDA), semi-supervised learning, and self-supervised learning  are common strategies to reduce the dependency on expensive manually annotated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' UDA aims  to mitigate the domain gap by transferring knowledge learned from label data in source domain  and transfer to unlabeled data in target domain, while semi-supervised learning employs a few  manually annotated samples and generalizes to large unlabeled samples without domain shift  assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Self-supervised utilizes the unlabeled data to pre-train the model to learn the general  semantics of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='1  Unsupervised Domain Adaptation (UDA)  Supervised Learning’s primary objective is to learn an input-output mapping by minimizing the  loss function E(𝑥,𝑦) ∼ 𝑃𝑋𝑌 (𝑓 (𝑥),𝑦) based on the training data distribution 𝑃𝑋𝑌 and evaluate on  ˆ𝑃𝑋𝑌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' This setting requires an independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=') assumption between  𝑃𝑋𝑌 and ˆ𝑃𝑋𝑌 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' However, the mmWave signal reflection is very sensitive to environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The data  distribution can be widely different from place to place depending on the surrounding environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  As shown in table 1, the distribution drifts occur both temporally and spatially which violates  their i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The naive way to solve the problem is to train the model on source and  target data, drawn from 𝑃𝑋𝑌 and ˆ𝑃𝑋𝑌, jointly using supervised learning with label data in both  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  111:4  Pinyoanuntapong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', Ekkasit Pinyoanuntapong, Ayman Ali, Kalvik Jakkala, Pu Wang, Minwoo Lee, Qucheng Peng,  Chen Chen, and Zhi Sun  domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' However, the privacy concern of biometric identification data provides another layer of  complication to the identification data collecting process and causes the annotated data to be more  expensive and time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' As a result, it may not be easy to collect data with the subjects’  identities (label data) in the target environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  Unsupervised domain adaptation can tackle the domain drift and the lack of labeled data problem  by minimizing the discrepancy between source and target distributions without using expensive  labeled data from the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The simple methods learn the domain invariant of source  and target domain by minimizing the distribution shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' DeepCORAL [29] directly matches the  mean and covariance of the source and target distributions with a linear transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Many  UDA techniques, inspired by Generative Adversarial Networks (GAN), apply an adversarial loss  to confuse the domain discriminator network that tries to recognize the domain difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' [32]  separates into two networks for source and target and forces the output representation to be in the  same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Domain-adversarial neural network (DANN) trains domain discriminator along  with the feature extractor that confuses the domain discriminator by integrating a gradient reversal  layer (GRL) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Although the overall distribution is aligned, the class-wise discrepancy can still  cause performance drop in the classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Conditional Adversarial Domain Adaptation  (CDAN) [21] takes this into account by implementing multilinear conditioning of the feature  representation and the class-wise output prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The other line of research in adversarial  domain adaptation also incorporates pixel level in addition to feature level domain adaptation  [4][17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' [39] aligns the centroid of the true source label and the target pseudo-label by minimizing  their squared Euclidean distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='2  Semi-supervised Learning  Semi-supervised methods decrease the dependency on labeled data by combining normally smaller  sets of labeled data with the vast amounts of unlabeled data that are available in many use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  Since the classification loss is unknown for unlabeled data, the self-ensembling network trains itself  via the pseudo label instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The networks encourage consistent prediction despite the perturbation  on data and networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Virtual Adversarial Training (VAT) [22] adds only a small perturbation  to the input data in the direction that produces maximum difference from the original input to  promote local distributional smoothness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Mean Teacher [30] 𝜋-model and Temporal Ensembling  Model [19], and Mutual Mean-Teacher [11] apply different augmentation and dropout for student  and teacher networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Also, a teacher network uses an exponential moving average (EMA) weights  of the student model from the previous training epochs to train the noise invariant prediction to  the slightly different network and data from a current student data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Noisy Student [38] takes a  similar approach by pretraining the network with label data and using it as a teacher model to  train the student model without the EMA weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The trained student network is directly used  as a teacher network to train an equal-or-larger student model and the iterative training process  repeats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Recently, much research has shown that using only data augmentation with multiple  identical networks works well without EMA weights or iterative training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' UDA [37], FixMatch [28],  MixMatch [3], and RemixMatch [2] utilize consistency training without the actual labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  Only high confidence pseudo labels, which represent the similarity of unseen data to the previously  trained ones, are selected for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  Despite the different strategies of unsupervised domain adaptation (UDA) and semi-supervised  learning, they share similar objectives and solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Specifically, if the target support covers source  support, semi-supervised learning can be considered a special case of UDA problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' [43] shows  that the state-of-the-art semi-supervised learning methods outperform the existing UDA methods  on the challenging UDA task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  GaitSADA: Self-Aligned Domain Adaptation for mmWave Gait Recognition  111:5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='3  Self-supervised Contrastive Learning  Similar to semi-supervised learning, self-supervised Learning utilizes large unlabeled data to  improve performance from label data trained in a supervised learning fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' However, self-  supervised learning normally pre-trains unlabeled data to learn general data semantics before  transferring pre-trained weights to further learn label data in a supervised learning fashion in the  next stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Many research [8], [13], [33], [14] enforce contrastive loss to learn the structure of the  data without labels by imposing different augmentation to the same data and learning the similarity  between the features of same data (positive samples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The challenge is that such similarity loss  functions can trigger a dimensional collapse in which the model outputs the exact same embedding  for all input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' While MoCo [14] and BYOL [13] incorporate momentum encoder and stop gradient to  prevent the collapse, SimCRL [8] and CPC [33] employ negative samples to discriminate the features  of different data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' SimCLR [8] proposed NT-Xent (the normalized temperature-scaled cross-entropy  loss) which simply maximizes the similarity between the feature of the same data (positive samples)  and minimizes the different ones (negative samples),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  𝐿𝑁𝑇−𝑋𝑒𝑛𝑡 = − 1  𝐵  𝐵  ∑︁  𝑖=1  𝑙𝑜𝑔  𝑒 (𝑠𝑖𝑚(𝑓𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='^𝑓𝑖)/𝜏)  �𝐾  𝑗=1 1[𝑖≠𝑗 ]𝑒 (𝑠𝑖𝑚(𝑓𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' ^𝑓𝑗)/𝜏) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  (1)  where 1[𝑖≠𝑗 ] ∈ {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 1} denotes an indicator function to retain only negative samples,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 𝑖 ≠ 𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 𝜏 is a  temperature parameter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' and 𝑠𝑖𝑚(𝑓𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' ˆ𝑓𝑖) =  𝑓 𝑇  𝑖 ^𝑓𝑖  ∥𝑓𝑖 ∥  ���^𝑓𝑖  ��� represents cosine similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' We modify this self-  supervised learning method to pre-train target data in order to learn representations of unlabeled  data as described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  3  MMWAVE GAIT BIOMETRIC DATASET  The dataset is created by exploiting the mmWave radar-based gait recognition system, STAR,  developed in our previous work [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' In this section, the preliminaries of mmWave radar sensing  are first presented, the data collection and gait recognition procedure using STAR system is then  introduced, and finally spatial-temporal domain drift issues are revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='1  Preliminaries of mmWave Radar Sensing  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' FMCW signal with linear ramp  The transmission of a signal, which is reflected by objects along its route of propagation, is a key  principle behind radar systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' One of the key advantages of a Frequency Modulated Continuous  Wave (FMCW) radar system is its ability to simultaneously determine both the range and velocity  of a moving target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 1, the frequency of the transmitted signal increases linearly  over time during the transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' When the chirp is reflected off an object, the frequency of  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  Transmit  ------ Receive111:6  Pinyoanuntapong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', Ekkasit Pinyoanuntapong, Ayman Ali, Kalvik Jakkala, Pu Wang, Minwoo Lee, Qucheng Peng,  Chen Chen, and Zhi Sun  the reflected signal will differ slightly from the transmitted signal due to the Doppler effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' This  difference in frequency enables the radar to determine both the distance and velocity of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  The rate of change of frequency, or the slope of the chirp, is defined as 𝑆 = 𝐵/𝑇𝑐, where 𝐵 is the  bandwidth and 𝑇𝑐 is the duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The beat signal is produced by combining the broadcast and  received chirps in a frequency mixer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' For a target at a distance 𝑑, the beat signal can be explained  as follows:  𝐴 exp(𝑗2𝜋(𝑓0𝑡 + 2𝑑/𝜆𝑐) = 𝐴 exp(𝑗2𝜋(2𝑑/𝜆𝑐)) exp(𝑗2𝜋𝑓0𝑡),  (2)  The wavelength of the carrier frequency is referred to as 𝜆𝑐 = 𝐶/𝑓𝑐 where 𝐶 is the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 𝐴  denotes the signal attenuation gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 𝑓0 = 2𝑑𝑆/𝐶 represents beat frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Therefore, the distance  of the target can be simply obtained by  𝑑 = 𝑓0𝐶  2𝑆 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  (3)  The frequency of the beat signal will vary as the target is moving to a new location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The beat signal  will change if the distance between the previous position and the new position is greater than the  radar range resolution, which is directly related to the bandwidth 𝐵, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=',  𝑑𝑟𝑒𝑠 = 𝐶  2𝐵 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  (4)  A very broad bandwidth 𝐵 can be employed with mmWave FMCW radar due to the extremely high  carrier frequency, which results in fine-grained range resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  While the frequency of the beat signal is utilized to determine the target’s distance, its phase  exp(𝑗2𝜋(2𝑑/𝜆𝑐)), as specified in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' (2), can also be utilized to estimate the velocity of the target,  even in the presence of range resolution constraints that prevent detection of changes in the target’s  position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' In situations where the target moves a relatively small distance over a sequence of 𝑁 beat  signals generated from transmitted and received chirps, all 𝑁 signals will exhibit varying phases  but have the same beat frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Specifically, the phase of the beat signal 𝑛, where 𝑛 ∈ (1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', 𝑁),  is given by:  exp(𝑗2𝜋(2𝑣/𝜆𝑐)𝑛𝑇𝑐),  (5)  where 2𝑣/𝜆𝑐 is Doppler frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' For each distinguishable distance, the equation (5) reveals  the phases of the beat signals, generating a new signal whose carrier frequency is the Doppler  frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The various body parts of a walking human subject, such as arms, legs, feet, and torso,  move at different velocities, leading to distinct Doppler frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The radar is able to distinguish  movements of two parts only if the velocity resolution surpasses the velocity difference between  the two parts, which is determined by the observation duration, 𝑁𝑇𝑐, and the carrier wavelength,  𝜆𝑐, as follows  𝑣𝑟𝑒𝑠 =  𝜆𝑐  2𝑁𝑇𝑐  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  (6)  The millimeter-scale wavelength of mmWave radar allows it to characterize the fine-grained  motion dynamics of human gait in high resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' For instance, 77GHz mmWave radar can reach  over 12 times greater velocity resolution than sub-6GHz radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='2  STAR System Overview  STAR is the first one in the literature to exploit deep learning for mmWave radar-based gait  recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 2, STAR consists of two components: radar spectrogram acquisition,  spectrogram enhancement and CNN-based gait recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The spectrogram acquisition module  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  GaitSADA: Self-Aligned Domain Adaptation for mmWave Gait Recognition  111:7  Range  Estimation  (1D-FFT)  Static Cluster  Removal  (Mean Subtraction)  Object  Detection  (CA-CFAR)  Doppler  Estimation  (2D-FFT)  Sliding-  Window based  Tracking  Spectrogram  Generation  (STFT)  Normalization  Mean Filtering  2D Gaussian  Filtering  Feature Encoder  Gait Spectrogram  Multiclass Classifier  User ID  Radar  Array  Radar Spectrogram Acquisition  Spectrogram  Enhancement  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Overview of STAR system  tracks and captures the radar signal returns that directly bounce off from human body, while  filtering out undesired environment-induced radar returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' This is achieved by applying a sequence  of radar widely-adopted radar signal processing algorithms including one-dimensional Fourier  transform (ID-FFT) for range estimation, the mean subtraction for static reflection removal, CA-  CFAR for robust target detection against a background of noise, clutter and interference, Doppler  estimation via 2D-FFT to estimate target velocity, and sliding-window based tracking method to  predict the target future state (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', location and velocity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The target tracking module acts a moving  spatial passband filter so that only the radar returns from the target can be received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Performing  short-time Fourier transform (STFT) on received radar returns generates the raw radar spectrogram,  whose quality is further improved by exploiting the spectrogram enhancement module that consists  of a sequence of frequency-domain noise removal processes [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The enhanced spectrograms are  then encoded by a CNN-based feature encoder that maps the high-dimension spectrogram data  into the compact low-dimension feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Finally, the compact feature vectors are used to  train a multiclass classifier that predicts the corresponding user identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='3  Data Collection Setup  We collected gait data from 10 volunteers between the ages of 18-35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Each subject’s data was  collected in four different locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The source location was a research space with cubicles, and  the other three areas consisted of a server, conference, and an office room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' By maintaining four  distinct locations, we introduce SDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' In the source location, data was collected on 10 different days  for each subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 5 separate days of data was acquired for each of the three other locations, which  are used as target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' A participant can either walk towards the radar or walk away from  the radar, each of which is counted as one walking instance and generates one spectrogram data  sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The data collection was limited to 100 data samples per person in the source location and  50 data samples per person in each of the target locations on any given day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The gait spectrogram  samples of a person are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='4  Establishing the Presence of TDS and SDS  We study the presence of TDS and SDS issues by training gait recognition model using the labelled  data from source domain and test the model performance across different target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' In  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  AA国口  TIIWR1642.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='mmWaveSens0  WalkingPath111:8  Pinyoanuntapong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', Ekkasit Pinyoanuntapong, Ayman Ali, Kalvik Jakkala, Pu Wang, Minwoo Lee, Qucheng Peng,  Chen Chen, and Zhi Sun  Source Day 1  Source Day 2  Source Day 3  Source Day 4  Source Day 5  Server Day 1  Server Day 2  Server Day 3  Server Day 4  Server Day 5  Office Day 1  Office Day 2  Office Day 3  Office Day 4  Office Day 5  Conference Day 1  Conference Day 2  Conference Day 3  Conference Day 4  Conference Day 5  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Gait spectrograms of the same person for 5 days at different locations, source (laboratory), conference  room, server room, and office  particular, the feature encoder is a modified version of ResNet-50 [15], as described in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='3,  to extract features from mmWave biometric data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The loss function is a cross-entropy loss  for multiclass classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' To establish the presence of SDS, we train the model on the data from 1  to 3 days of source location and test the model on the data from other three unseen locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' SDS  is evident from the performance degradation in all target domains as shown in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' To reveal  the existence of TDS, we train the model on the data from 1 to 3 days of the source location and  test the model on the data from the remaining days of the source location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' TDS can be observed  due to the degraded testing accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  4  METHOD: GAITSADA  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='1  Method Overview  Our novel method, self-aligned domain adaptation for mmWave gait recognition (GaitSADA),  jointly exploits self- and semi-supervised learning techniques to realize the unsupervised domain  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  GaitSADA: Self-Aligned Domain Adaptation for mmWave Gait Recognition  111:9  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The result of supervised learning trained on source data and tested on different days of data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=',  temporal) and other different environments (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', server, conference, office).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The accuracy drop reveals the  domain discrepancy when the experiments are conducted on different days and environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The results  report in % of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  # of training days (Source)  Temporal  Server  Conference  Office  1-day  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='05  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='93  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='00  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='39  2-days  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='46  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='05  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='00  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='92  3-days  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='25  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='77  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='20  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='61  adaptation by promoting intra-class compactness and inter-class distance while generalizing knowl-  edge from label data to unlabeled data using pseudo-label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' In particular, we leverage the unlaballed  data from the target domains in two training stages: pretaining and fine-tuning as illustrated in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  During pretraining stage, we propose a semi-supervised contrastive learning paradigm to learn  the salient gait representation from a mixed dataset that consists of data from both source and and  target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Our semi-supervised contrastive learning can explicitly avoid collapse problem  without requiring the sophisticated techniques adopted by classic self-supervised (unsupervised)  contrastive learning paradigms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', negative samples, stop gradients, and moment encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' This  is achieved by maximizing the agreement between representations produced by encoders fed with  different augmentations of the same spectrogam in the mixed dataset, while enforcing the encoder  fed with labelled data in source domain to produce discriminative representations with correct  class predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  After the pretrained stage, the feature encoder implicitly aligns the data distributions of source  domain and target domain in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' As a result, the representations produced by such  encoder are discriminative enough to provide relatively high-confidence pseudo labels for unlabelled  data in the target domains, which is essential for the semi-supervised consistency training in the  fine-tuning stage that can effectively regulate the features of the target domain data from the same  class to be close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' However, classic semi-supervised consistency training only shows the  best performance when the label and unlabelled data come from the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' To address  this issue, we propose the centroid alignment scheme that can be integrated with semi-supervised  consistency training to mitigate source-target domain gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='2  Pretraining via Semi-supervised contrastive Learning  Supervised Learning with Additive Margin Softmax [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The main goal of this component  is to promote intra-class compactness and inter-class separability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' As the regular softmax loss is  effective only in maximizing the inter-class difference, it is ineffective at minimizing the intra-class  variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Therefore, we adopt Additive Margin Softmax loss function to bring the classification  boundary closer to the weight vector of each class by adding a margin 𝑚 to the boundary as  𝐿𝑠𝑢𝑝𝑒𝑟𝑣𝑖𝑠𝑒𝑑 = − 1  𝐵  𝐵  ∑︁  𝑖=1  𝑙𝑜𝑔  𝑒𝑠·(𝑐𝑜𝑠𝜃𝑦𝑖 )−𝑚  𝑒𝑠·(𝑐𝑜𝑠𝜃𝑦𝑖 )−𝑚 + �𝐶  𝑗=1,𝑗≠𝑦𝑖 𝑒𝑠·𝑐𝑜𝑠𝜃 𝑗 ,  (7)  where 𝑐𝑜𝑠𝜃𝑦𝑖 =  𝑊 𝑇  𝑦𝑖 𝑓𝑖  ∥𝑊𝑦𝑖 ∥ ∥𝑓𝑖 ∥ represents cosine similarity of the inner product between the normalized  weight 𝑊𝑦𝑖 and the normalized feature 𝑓𝑖 of class 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Additional parameter 𝑠 is a scale parameter to  control the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  111:10  Pinyoanuntapong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Ekkasit Pinyoanuntapong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Ayman Ali,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Kalvik Jakkala,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Pu Wang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Minwoo Lee,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Qucheng Peng,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  Chen Chen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' and Zhi Sun  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='encoder  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='encoder  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='encoder  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='Similarity loss  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='Classifier  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='AM loss  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='Final loss  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='Augmentation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='Augmentation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='Source  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='Domain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='(labelled)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='Source +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='Target  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='Domains  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='(unlabeled)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='encoder  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='encoder  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='encoder  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='Classifier  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='AM loss  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='Final loss  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='Augmentation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='Source  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='Domain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='(labelled)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='Target  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='Domain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='(unlabeled)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='Classifier  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='Classifier  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='prediction  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='prediction  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='Pseudo label  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='consistency  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='loss  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='Centroid  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='Estimation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='Centroid  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='loss  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='w  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='Augmentation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='(a) Pretraining Stage: Semi-supervised Contrastive Learning  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='(b) Fine-tuning Stage: Semi-supervised Consistency Training with Centroid Alignment  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Overall Architecture of GaitSADA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' (a) Stage 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Pretraining via semi-supervised contrastive learning  to learn a compact gait representation from both source and domain data, which also implicitly mitigates  the source-domain distribution shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' (b) Stage 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' fine-tuning via semi-supervised consistency training with  centroid alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The fine-tuning stage aims to further improve model generalization performance in  target domains by minimizing intra-class distance and maximizing inter-class distance of the target domain  samples assigned with pseudo labels, while explicitly minimizing the source-target domain gaps with centroid  alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The encoders are sharing the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  Positive Unlabelled contrastive Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The objective of the second component is to learn  noise invariance by training the model to generate the same features regardless of the artificial noise  that is added to the data without relying on label data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' We carefully design multiple augmentations  to simulate the radio wave noise such as frequency noise, temporal noise, different resolutions, and  white noise as described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The model learns noise invariance by minimizing cosine  similarity between features from raw data 𝑓 and its augmented version ˆ𝑓 with the loss  𝐿𝑠𝑒𝑙𝑓 = 1  𝐵  𝐵  ∑︁  𝑖=1  𝑓 𝑇  𝑖 ˆ𝑓𝑖  ∥𝑓𝑖 ∥  ���ˆ𝑓𝑖  ���  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  (8)  This loss is inspired by NT-Xent (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 1) from SimCLR [8] which is the contrastive self-supervised  learning method that minimizes cosine similarity of positive samples (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', different augmentation  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  GaitSADA: Self-Aligned Domain Adaptation for mmWave Gait Recognition  111:11  of the same data) and maximizes the cosine similarity of negative samples (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', different data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The  objective of negative samples in NT-Xent is to prevent dimensional collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' However, since our Self  Alignment loss is trained jointly with Supervised Learning loss, it can completely avoid the collapse  issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' As a result, we are able to retain only the positive similarity and eliminate the negative  marginal fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The final loss function in the first stage combines two loss functions to encourage  the intra-class concentration and inter-class distance while being invariant to perturbation from  noise by utilizing unlabeled data from target domain:  𝐿𝑠𝑡𝑎𝑔𝑒1 = 𝐿𝑠𝑢𝑝𝑒𝑟𝑣𝑖𝑠𝑒𝑑 + 𝐿𝑠𝑒𝑙𝑓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  (9)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='3  Fine-tuning via Consistency Training with Centroid Alignment  In the first stage, the model has learned the well-aligned class-wise feature from label data in the  source domain and unlabelled data from target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The objective of the second stage is to  further regulate the feature of the target domain data of the same class to be close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  Consistency Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Unlike the source domain, the labels in the target domain are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  Therefore, we cannot directly align target feature by class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' To address this problem, we enforce  consistency training via pseudo labels instead of the true labels of the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Consistency  training utilizes unlabeled data by relying on the assumption that when given perturbed variations  of the same input, the model ought to provide similar predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The two identical models are  used for consistency training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The first branch takes raw input data with no augmentation in order  to provide accurate pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' We compute the model’s predicted class distribution of a given  unlabeled spectrogram input: 𝑞𝑖 = 𝑝(𝑦𝑖|𝑢𝑖) as soft pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The cross-entropy 𝐻 is applied  against the pseudo labels distribution and the model’s prediction distribution of the augmented  unlabeled data A(𝑢𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The loss is defined as  𝐿𝑐𝑜𝑛𝑠𝑖𝑠𝑡𝑒𝑛𝑐𝑦 = 1  𝐵  𝐵  ∑︁  𝑖=1  1(arg max  𝑦 (𝑞𝑖) ≥ 𝜏)𝐻 (𝑞𝑖, ˆ𝑞𝑖),  (10)  where ˆ𝑞𝑖 = 𝑝(𝑦𝑖|A(𝑢𝑖)) represents the probability distribution of class 𝑦𝑖 given augmented input  A(𝑢𝑖) and𝜏 is a scalar parameter denoting the threshold to filter only the high confidence prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  To maintain the same standard of Additive Margin Softmax (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' (7)) in the first stage, we normalize  weights and features of the classifier which can be referred to as cosine similarity 𝑐𝑜𝑠𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The scale 𝑠  remains the same, but we remove margin 𝑚 for accurate pseudo labels:  𝑝(𝑦𝑖|·) =  𝑒𝑠·(𝑐𝑜𝑠𝜃𝑦𝑖 )  �𝑐  𝑗=1 𝑒𝑠·𝑐𝑜𝑠𝜃 𝑗 ,  (11)  where 𝑐 is the number of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The predictions are likely to be inaccurate at the beginning of  the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' For this, most semi-supervised learning algorithms [30][19][3][2] initialize training  with a small weight 𝜆 for the consistency loss term and increase it over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' However, with the  confidence threshold, FixMatch [28] suggests that the pseudo label produces a natural curriculum  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' As less pseudo-label data are available at first since the low confidence predictions are  filtered out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' When the model is more accurate, it then provides more predictions with a high degree  of confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' As a result, the threshold automatically increases the number of pseudo-label data  without explicitly specifying incremental weight 𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  Centroid Alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' While consistency training loss tends to move weights 𝑊 of the classifier  toward target features, supervised learning loss still attempts to bring the weights back to source  features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The weights can oscillate between these two groups as a result of the domain gap between  the source and target features as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' To mitigate this issue, we impose centroid  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  111:12  Pinyoanuntapong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', Ekkasit Pinyoanuntapong, Ayman Ali, Kalvik Jakkala, Pu Wang, Minwoo Lee, Qucheng Peng,  Chen Chen, and Zhi Sun  Centroid  Source Feature  Target Feature  Less Confidence Target Feature  Centroid of Source Feature  Centroid of Target Feature  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Centroid Alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The centroid of source feature tends to be closed to weight vector𝑊 following  the supervised learning loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' On the other hand, the centroid of target features is likely to be further away  due to the domain gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Centroid Alignment encourages weight vector 𝑊 to move to a new centroid which is  the center of both source and target features which can be a better representation as more target features are  learned from consistency training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  alignment loss to gradually bring the weights to the center of source and target features by applying  Additive Margin Softmax to class-wise weights and class centroids as  𝐿𝑐𝑒𝑛𝑡𝑟𝑜𝑖𝑑 = − 1  𝐶  𝐶  ∑︁  𝑐=1  𝑙𝑜𝑔  𝑒𝑠·(𝑐𝑜𝑠𝜃𝑐)−𝑚  𝑒𝑠·(𝑐𝑜𝑠𝜃𝑐)−𝑚 + �𝐶  𝑗=1,𝑗≠𝑖 𝑒𝑠·𝑐𝑜𝑠𝜃 𝑗 ,  (12)  where 𝑐𝑜𝑠𝜃𝑐 =  𝑊 𝑇  𝑐 𝑍𝑐  ∥𝑊𝑐 ∥ ∥𝑍𝑐 ∥ is the inner product of normalized weights 𝑊𝑖 and centroids 𝑍𝑖 of each  class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The class-wise centroid is from the Exponential Moving Average (EMA) of both source and  target features with hyperparameter 𝛼 to control the rate of change as  𝑍𝑐𝑖 = 𝛼 ˆ𝑍𝑐𝑖 + (1 − 𝛼)𝑍𝑐𝑖−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  (13)  The current batch centroid ˆ𝑍𝑐𝑖 is the average features of source 𝑓𝑠𝑖 and target 𝑓𝑡𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' However, because  the true class of the target domain is unknown, we cannot directly assign features to the class  centroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Instead, the hardmax (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', argmax) of the model’s prediction is used as a pseudo label of  the class of each sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Also, target features can be noisy at the beginning of the training, similar  to consistency loss (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' (10)), only the high confidence labels are used for calculating the centroids  to avoid false predictions of pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Therefore,  ˆ𝑍𝑐𝑖 =  �𝐵  𝑖=1 𝑓𝑠𝑖 + 1(arg max𝑦(𝑞𝑖) ≥ 𝜏)𝑓𝑡𝑖  𝐵 + �𝐵  𝑖=1 1(arg max𝑦(𝑞𝑖) ≥ 𝜏)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  (14)  Lastly, we impose Additive Margin Softmax for source label data as a supervised learning loss along  with consistency training and Centroid Alignment loss to prevent dimensional collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' So the  final loss equation is shown as  𝐿𝑠𝑡𝑎𝑔𝑒2 = 𝐿𝑠𝑢𝑝𝑒𝑟𝑣𝑖𝑠𝑒𝑑 + 𝐿𝑐𝑜𝑛𝑠𝑖𝑠𝑡𝑒𝑛𝑐𝑦 + 𝜆𝐿𝑐𝑒𝑛𝑡𝑟𝑜𝑖𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  (15)  5  EXPERIMENT SETUP  Texas Instruments (TI) IWR1642EVM boost board interfaced with a DCA1000EVM board to collect  the raw mmWave radar signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The radar system consists of two transmitting antennas, four  receiving antennas, and 120° view of the azimuth plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The radar system supports up to a 4Ghz  bandwidth operating on 77 GHz to 81 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' To configure FMCW wave parameters such as chirp  width, repetition time, and chirp slope from our radar device, we use a Dell Latitude 7480 laptop  with TI mmWave studio software as a control system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  GaitSADA: Self-Aligned Domain Adaptation for mmWave Gait Recognition  111:13  (a) Cutout  Horizontal  (b) Cutout  Vertical  (c) White Noise  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  (d) Zoom  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  (e) Shear  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  (f) Rotation  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Spectrogram of 6 augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='1  Dataset and preprocessing  The walking dataset is collected from four distinct locations such as laboratory, conference room,  server room, and office.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' We recruit ten volunteers to walk in his/her natural ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Each participant  either walks toward or away from the radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Data is preprocessed into 50 samples of five different  days at each location, a total of 2500 samples (10 subjects × 50 samples × 5 days), except for the  laboratory location, we collect additional data for another five more days (total of ten different days)  in order to perform temporal domain shift experiments for a different time in the same location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  We preprocess the data into 256×256×1 spectrogram and standardized values between -1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  Laboratory location data is used as source data and the other location data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' conference room,  server room, and office) is referred to as target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' For the spatial domain drift experiment, day 1  to day 3 samples are used as a training set for all source and target locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The rest of the data is  used for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' For the temporal domain drift experiment, we employ 1 to 3 days of laboratory  location as source data and the next consecutive days of laboratory location as target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' For  example, considering the temporal 2-day case, the first and the second-day data of laboratory  location is a source domain, the third and the fourth-day data is a target domain, and the fifth to  the tenth-day data is utilized as a test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='2  Spectrogram Augmentation  Vertical noise stripe (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 6b) simulates missing information in the random time interval of the  spectrogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Similarly, the horizontal noise stripe (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 6b) represents the missing information in  some frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' We implement horizontal and vertical noise together by randomly adding  two of these noises per data with a probability of 1/3 to be horizontal and 1/3 to be vertical, the rest  1/3 is for no cutout noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The small side of the stripe is random between 2 to 8 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The third  augmentation is pixel-wised random numbers adjusted to the whole data uniformly simulating  white noise (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 6c) in a spectrogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The uniformly random noise from -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='5 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='5 is added to  the data with a probability of occurrence 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Next is the zoom-in and out effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Zoom-in will  drop some information from the start and the end, also the lowest and the highest frequency of  the spectrogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Zoom effects (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 6d) represent the data in different resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The scale is  applied to the data between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='8 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='2 randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' These four augmentations act as virtual noise  to simulate the noisy real-world data thereby improving domain generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Not only these  meaningful noises, but we also experiment with regular computer vision augmentations and we  found that shearing (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 6e) and rotation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 6f) help boost performance as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Following  Tensorflow ImageDataGenerator function, the shear angle in a counter-clockwise direction in  degrees is random from 0 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Also, 5-degree ranges are randomly applied for rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' So we  add these augmentations to our preprocess pipeline and then apply all augmentation to every  spectrogram data with random magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' In addition to consistency training, augmentation can  be used to generalize data in supervised learning during the first training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  111:14  Pinyoanuntapong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', Ekkasit Pinyoanuntapong, Ayman Ali, Kalvik Jakkala, Pu Wang, Minwoo Lee, Qucheng Peng,  Chen Chen, and Zhi Sun  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='3  Model  A feature extraction model is modified from Resnet-50 to support one channel instead of three  channels and use SELU as an activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The other parts are similar to the original Resnet-  50 [15] composed of five stages of convolution blocks and identical blocks including a stack of  convolutional blocks, batch normalization, and activation with skip connections in each block and  output 128 features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The classification model is simply a linear layer with 10 neurons to classify  128 features from feature extraction into 10 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='4  Training Details  We apply the same settings for all 1- to 3-day cases and all locations in both stages 1 and 2 to  be consistent in all domain configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Every convolution layer applies 1e−4 of 𝑙2 kernel  regularizer and trains for 10,000 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The training optimizer is ADAM with an initial learning  rate starting at 1e−3 and ending at 1e−5, applying the polynomial learning rate decay scheduler  with learning rate restart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' We use a batch size of 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The confidence threshold of the consistency  pseudo-label is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Additive Margin Softmax Loss applies 𝑙2-normalization to both input and  weights to the fully connected layer head without bias, with 𝑠 = 10 and 𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' All losses in  both stages 1 and 2 are applied with equal weights except centroid alignment loss with 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='05 to  gradually move centroids of the classes without breaking the current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='5  Evaluation  We evaluate the model without any domain adaptation (supervised learning) as a baseline comparing  with the five representative domain adaptation methods including GAN [16], GRL [10], ADDA [32],  CDAN [21], and FixMatch [28] along with our method, GaitSADA, on the four different locations  (laboratory, conference room, server room, and office) to explore the spatial domain adaptation  using 1-3 days of data for training and the rest of data for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  Generative Adversarial Networks (GAN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Generative Adversarial Network (GAN) [12] is a  generative model which can construct new data instances in an unsupervised learning fashion by  learning the distribution of the target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The model separates into two sub-models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The first,  generator 𝐺 produces the data by capturing the data distribution, and the second, discriminator  𝐷 classifies if a sample comes from the true data or generated from the generator 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' GAN can be  repurposed for domain adaptation by considering the feature extraction model as a generative  model 𝐺 generates features from source or target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' While discriminator 𝐷 learns to classify  a feature from source or target domain, the feature extractor 𝐺 tries to confuse the discriminator 𝐷  by generating the features with the same distribution for both source and target, thereby closing  the domain drift gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' So the value function 𝑉 (𝐺, 𝐷) is a two-player minimax game played by 𝐷  and 𝐺, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', min𝐺 max𝐷 𝑉 (𝐷,𝐺) = E𝑥𝑠∼𝑋𝑠 [𝑙𝑜𝑔𝐷(𝐺(𝑥𝑠))] + E𝑥𝑡 ∼𝑋𝑡 [𝑙𝑜𝑔(1 − 𝐷(𝐺(𝑥𝑡)))], where 𝑥𝑠 is  data drawn from source domain and similarly, 𝑥𝑡 is the data drawn from target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  Gradient Reversal Layer (GRL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Similar to GAN, feature extractor 𝐺 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', generator) and  domain discriminator 𝐷 are trained to compete with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' However, rather than minimizing  domain discriminative loss and maximizing confusion loss separately, gradient reversal layer (GRL)  [10] shows that this adaptation behavior can be implemented in a single step, practically with  any feed-forward model by adding a GRL layer to train 𝐺 and 𝐷 at the same time in the opposite  gradient direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The GRL has no parameters associated with its component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' For forward pass  computation, the GRL is defined as an identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' On the other hand, during backpropagation,  GRL flips the gradient direction from subsequent level with the scale hyperparameter 𝜆 before  passing the gradient to the preceding layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  GaitSADA: Self-Aligned Domain Adaptation for mmWave Gait Recognition  111:15  Adversarial Discriminative Domain Adaptation (ADDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Adversarial Discriminative Do-  main Adaptation (ADDA) [32] improves GAN-based unsupervised domain adaptation by utilizing  discriminative modeling, united weight sharing, and a GAN loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Unlike GAN and GRL, feature  extractor 𝐺, classier 𝐶, and domain discriminator 𝐷 are learned all together simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' ADDA  takes two separate pre-training stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' In the first stage, feature extractor 𝐺 and classifier 𝐶 are  pre-trained with only source label data using a standard cross entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' In the second stage,  the weights of feature extractor 𝐺 are transferred from the first stage pre-trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Similar to GAN,  domain discriminative loss 𝐿𝑑𝑖𝑠 and confusion loss 𝐿𝑐𝑜𝑛𝑓 are trained in an adversarial manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  However, confusion loss 𝐿𝑐𝑜𝑛𝑓 in ADDA updates the feature extractor based on only target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' So  the confusion loss 𝐿𝑐𝑜𝑛𝑓 reduces to min𝐺 𝐿𝑐𝑜𝑛𝑓 = − E𝑥𝑡 ∼𝑋𝑡 [𝑙𝑜𝑔𝐷(𝐺(𝑥𝑡))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Finally, during test time,  the pre-trained classifier 𝐶 in stage 1 and the pre-trained feature extractor in stage 2 are used for  class prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  Adversarial Domain Adaptation (CDAN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Conditional Adversarial Domain Adaptation (CDAN)  [21] is another adversarial domain adaptation method utilizing multilinear conditioning to condi-  tion discriminate domain by its class prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The aforementioned methods train the feature  extractor 𝐺 and the domain discriminator 𝐷 directly from the probability distribution 𝑝(𝐺(𝑥))  of output feature representation 𝐺(𝑥) without any condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' However, adapting the marginal  probability distribution 𝑝(𝐺(𝑥)) may not be sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' CDAN utilizes a multilinear map, which can  be defined as the outer product, to learn cross-covariance dependency E𝑥𝑦[𝐺(𝑥) ⊗ 𝐶(𝐺(𝑥))] of  feature 𝐺(𝑥) and class prediction 𝐶(𝐺(𝑥)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Therefore, the domain discriminative loss 𝐿𝑑𝑖𝑠 can be  written as: 𝐿𝑑𝑖𝑠 = − E𝑥𝑠∼𝑋𝑠 [𝑙𝑜𝑔𝐷(𝐺(𝑥𝑠) ⊗ 𝐶(𝐺(𝑥𝑠))] − E𝑥𝑡 ∼𝑋𝑡 [𝑙𝑜𝑔(1 − 𝐷(𝐺(𝑥𝑡) ⊗ 𝐶(𝐺(𝑥𝑡)))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The  confusion loss 𝐿𝑐𝑜𝑛𝑓 in CDAN is referred as the negative of −𝐿𝑑𝑖𝑠 and the classification loss 𝐿𝑐𝑙𝑠 is  standard cross entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The minimax game of CDAN is: min𝐺 𝐿𝑐𝑙𝑠 − 𝛼𝐿𝑑𝑖𝑠 and min𝐷 𝐿𝑑𝑖𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  FixMatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' We repropose FixMatch [28], a semi-supervised learning method, for our unsuper-  vised domain adaptation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' FixMatch [28] learns target unlabeled data 𝑢𝑖 by generating  pseudo-labels from models’ predictions on weakly-augmented data A(𝑢𝑖)) and filtering only high  confidence data by selecting the prediction that has confidence more than the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The cross-  entropy 𝐻 is applied against the pseudo labels and the model’s prediction of the highly-augmented  versions of the same unlabeled data ˆ  A(𝑢𝑖),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' which can be referred to as consistency training to  learn unlabeled data from its past knowledge: 𝐿𝐹𝑖𝑥𝑀𝑎𝑡𝑐ℎ = 1  𝐵  �𝐵  𝑖=1 1(arg max𝑦(𝑞𝑖) ≥ 𝜏)𝐻 (𝑞𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' ˆ𝑞𝑖)  where 𝑞𝑖 = 𝑝(𝑦𝑖|A(𝑢𝑖)) represents the prediction probability distribution of class 𝑦𝑖 from given  weakly-augmented input A(𝑢𝑖) and similarly for prediction ˆ𝑞𝑖 = 𝑝(𝑦𝑖| ˆ  A(𝑢𝑖)) based on highly-  augmented data ˆ  A(𝑢𝑖)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 𝜏 is a scalar hyperparameter representing the confidence threshold of  retaining a pseudo-label  6  RESULTS  The normal training settings such as optimizer, scheduler, and batch size, described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='4,  are maintained the same across all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' We study naive supervised learning without  domain adaptation to understand the domain discrepancy issue in low data regime, temporal case,  and spatial case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' As summarized in Table 2, our GaitSADA method outperforms the other five  representative domain adaptation methods on all spatial domain adaptation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', different locations)  and temporal domain adaptation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', different days on laboratory location).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  First, we show the problem of domain discrepancy of the mmWave data from different spatial  and temporal domain shifts by training supervised learning of our modified version of Resnet-50 as  a feature extraction only on label source data without any domain adaptation to the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  The loss function is a regular cross-entropy loss applied against ten subjects in source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  111:16  Pinyoanuntapong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', Ekkasit Pinyoanuntapong, Ayman Ali, Kalvik Jakkala, Pu Wang, Minwoo Lee, Qucheng Peng,  Chen Chen, and Zhi Sun  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Result of training on 1 to 3 days on the data from laboratory location (source domain) while adapting  to different 1 to 3 days data of same location (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', temporal target domain) and 1 to 3 days of different  target locations, (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', server, conference, and office) with 5 representative methods, GRL [10], GAN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='5, ADDA  [32], CDAN [21], and FixMatch [28] along with "Supervised Learning" which train only on source label data  without domain adaptation and lastly, our proposed method, GaitSADA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The results report in % of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  # of training days  Methods  Temporal  Server  Conference  Office  Average  1-day  Supervised  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='05  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='93  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='00  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='39  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='84  GAN  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='57  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='14  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='00  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='93  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='16  GRL  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='97  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='14  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='10  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='14  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='34  ADDA  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='96  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
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+page_content='67  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
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+page_content='32  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
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+page_content='05  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='25  GaitSADA (our)  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='3  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
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+page_content='66  2-day  Supervised  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='46  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='05  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='00  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
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+page_content='18  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='96  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='80  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='76  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='43  GRL  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='98  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='07  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='90  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='96  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='98  ADDA  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='55  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='04  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='63  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='62  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='71  CDAN  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='72  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='36  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='90  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='16  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='04  FixMatch  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='82  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='38  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='10  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='58  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='97  GaitSADA (our)  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='63  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='88  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='1  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='59  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='8  3-day  Supervised  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='25  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='77  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='20  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='61  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='96  GAN  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='79  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='18  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='30  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='89  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='79  GRL  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='11  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='58  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='30  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='58  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='39  ADDA  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='38  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='79  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='79  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='13  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='02  CDAN  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='03  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='39  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='80  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='19  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='35  FixMatch  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='29  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='29  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='30  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='89  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='94  GaitSADA (our)  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='68  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='69  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='2  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='7  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='57  The performance of 3-day cases is 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='96% on average for all locations and temporal case, then drop  to 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='61% and 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='85% in 2-day and 1-day data which apparently shows that the amount of data helps  mitigate the domain shift issues in both spatial and temporal cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' In the temporal case, the accuracy  is higher than the spatial cases as the environment is the same, only the small features of subjects  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', clothes, body temperate) or subtle environment features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' air moisture, temperature) change  due to the different experiment times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The accuracy drops a lot in the conference room due to the  different environments and drops dramatically in the server room and office as more obstruction  and the smaller rooms cause the micro-Doppler to interfere with the reflection from the objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  GAN, GRL, and ADDA yield similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' In 1-day scenario, their accuracies are relatively  comparable, 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='16%, 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='34%, and 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='12% for GAN, GRL, and ADDA correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' For 2-day  instance, GAN and GRL are not significantly different, 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='43% and 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='98%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' However, ADDA is a  little bit lower at 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='71%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' For 3-day scenario, GAN and GRL still perform near, 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='79% and 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='39%  but ADDA accuracy spikes to 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='02%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' GAN and GRL perform quite similarly owing to their simlar  adaptation behavior as they alter the distribution by both source and target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' ADDA, on the  other hand, updates feature extraction simply by target data toward source feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Overall, these  three techniques help alleviate the domain drift issue by boosting accuracy from the supervised  method on average in all 3-day cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' However, we observe the accuracies drop compared to the  supervised learning method in 1-day case for all these three methods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', GAN and ADDA drop  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='46% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='79% in office location, GRL drops 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='08% for the temporal case due to the intricate  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  GaitSADA: Self-Aligned Domain Adaptation for mmWave Gait Recognition  111:17  structure of minimax game between domain discriminator and feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The domain  discriminator is unable to entirely minimize the domain shift while training on fewer data since it  does not learn all feature distinctions of the source and target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  CDAN has as better results comparing to GAN, GRL, and ADDA, 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='12%, 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='04%, and 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='35%  on average of 1-day, 2-day, and 3-day respectively due to its ability to learn class-wise feature  distribution cross-covariance dependency on class prediction and its feature, therefore, helping  understanding the class-conditional domain drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' However, the accuracy drop happens in 1-day  temporal case which CDAN accuracy is 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='67% while supervised learning method is 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='05%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The  limited amount of training data is still a problem for CDAN as it requires more data to understand  the distribution of source and target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  FixMatch performs best comparing to the other representative methods even though its original  purpose is for semi-supervised learning tasks, not unsupervised domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Its average  accuracies for 1-day, 2-day, and 3-day cases are 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='25%, 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='97%, and 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='94% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Because  FixMatch synthesizes data utilizing pseudo-label from target unlabeled data to train the model,  it virtually trains on more data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' We observe that the advantage of FixMatch presents more than  the other domain adaptation methods in low data regime as in the temporal 1-day case, FixMatch  gives 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='27% performance boost from supervised learning method while the other methods struggle  with adapting for the small dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' In 3-day case, this effect appears less, and the performance  boost is down to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='04%, as the larger dataset already mitigates the domain generalization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  On the other hand, in the 1-day server location which is more challenging due to their different  environments, FixMatch accuracy is less than CDAN method because the pseudo-label is filtered  by the confidence threshold so that there is less synthetic data to train in the difficult environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  The average accuracies for 1-day, 2-day, and 3-day cases of GaitSADA are 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='41%, 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='80%, and  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='57% respectively which increase accuracies of vanilla supervised learning method by a large  margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' GaitSADA presents more advantages for fewer data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Specifically, the performance boosts  by 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='56%, 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='19%, and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='61% on average of 1-day, 2-day, and 3-day data respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Our results  for the 1-day scenario clearly demonstrate a very strong accuracy improvement in the low data  regime, especially in the difficult domain drift environments, due to the simplicity of the aligning  components without a complex discriminating structure that requires more training data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=',  GAN, GRL, ADDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The performance improves by 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='25%, 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='05%, 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='6, and 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='37% from temporal,  server, conference, and office respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  7  VISUALIZATION  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='1  t-SNE  We present t-SNE [5] plots to visualize the high-dimensional datasets in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 7, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 9 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' We  convert high-dimensional features (128 dimensions) into a 2D space in such a way that similar data  points are mapped into nearby points in the low-dimensional representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' This allows us to  visualize the feature in a way that reveals the underlying structure and local relationships between  the source and target in different classes to better understand the domain drift behavior in various  locations and days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Source and target features are denoted by the symbol "o" and "x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='" Different  colors are used to depict the various classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  Supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' In the temporal case (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 7a), most features are well-learned and sepa-  rated by a considerable distance from other classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' In the temporal scenario, according to this  visualization, each class has a gap between the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' However, some groups are still too close to  one another and some target features ("x" symbols) are close to the incorrect class group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Conference  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 7b) and Server (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 7b), some of the target features ("x" symbol) are closer to the other color  groups of source features ("o" symbol) more than the same color groups which clearly shows the  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  111:18  Pinyoanuntapong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', Ekkasit Pinyoanuntapong, Ayman Ali, Kalvik Jakkala, Pu Wang, Minwoo Lee, Qucheng Peng,  Chen Chen, and Zhi Sun  (a) Temporal  (b) Server  (c) Conference  (d) Office  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' t-SNE [5] visualization for vanilla supervised learning training only on label source data for 3 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 10  colors represent 10 classes, while "o"s shows the source features and "x"s are target features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  domain drift behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' In office data (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 7d), the visualization presents even stronger domain drift  as the features are mixed together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  Representative Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' According to the visualization in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='1, t-SNE of GAN, GRL, and ADDA  depict similar presentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Each class in the temporal case is separated from the others better than  supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' However, some target features are still very near the incorrect class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' In sever,  conference, and office cases, features are better aligned even though the feature groups are not  compact enough, which can cause misclassification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' FixMatch exhibits the greatest compactness of  each feature class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' However, in the office case, each group is still close together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  GaitSADA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 8 shows that the target features are well-adjusted to the source features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' More-  over, all classes are compact and strongly discriminated (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', wide spaces between each other  classes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' This clearly exhibits the benefit of our method that promotes intra-class compactness and  inter-class separability while aligning source and target features to the class centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Even our first  stage (Additive Margin + Self Alignment) exhibits the compactness of the features (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='2  Grad-CAM  Grad-CAM [27] visualizes the weighted feature maps to create a heatmap that highlights the most  important regions to make a prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 10 illustrates diverse Grad-CAM saliency heatmaps for  each algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The other methods’ wrong predictions tend to focus only on a specific part of the  spectrogram data, high-frequency or low-frequency part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The high-frequency part represents the  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  GaitSADA: Self-Aligned Domain Adaptation for mmWave Gait Recognition  111:19  (a) Temporal  (b) Server  (c) Conference  (d) Office  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' t-SNE [5] visualization for the second stage of GaitSADA domain adaptation method training on label  source data and unlabeled data for 3 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 10 colors represent 10 classes, while "o"s shows the source features  and "x"s are target features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  fast-moving limbs i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', arms and legs, indicating the importance of this region for gait recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  However, Figures 10a, 10b, and 10f show the models tend to confuse high-frequency and the white  noise in the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Concentrating on this background noise area and ignoring the other  parts lead to misprediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Figures 10c, 10d, and 10k tend to focus only on the low-frequency  part, the torso region, which may contain less unique information compared to other parts of  the body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' We visualize saliency of GaitSADA on the same data that the other methods predict  incorrectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' In contrast to the other methods, GaitSADA looks broadly at multiple parts of the body  without focusing too much on specific parts which allow our method to effectively use more unique  information from different parts of the body for gait recognition and thus easily adapt to domain  shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='3  Ablation Study  We experiment with 2-day data on laboratory location as source data while adapting to different  days (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', temporal) and 3 different locations, (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', server, conference, and office) in the same setting  as the experiments in Table 2 and examine the behavior of each component contributing to the final  results in both jointly training and 2-stage training setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' We study six different combinations of  components as shown in Table 3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', supervised learning, additive margin softmax loss, contrastive  learning, consistency training, and centroid alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  QX111:20  Pinyoanuntapong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', Ekkasit Pinyoanuntapong, Ayman Ali, Kalvik Jakkala, Pu Wang, Minwoo Lee, Qucheng Peng,  Chen Chen, and Zhi Sun  (a) Temporal  (b) Server  (c) Conference  (d) Office  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' t-SNE [5] visualization for the first stage of GaitSADA domain adaptation method training on label  source data and unlabeled data for 3 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 10 colors represent 10 classes, while "o"s shows the source features  and "x"s are target features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  (a) Supervised  (b) GAN  (c) GRL  (d) ADDA  (e) CDAN  (f) FixMatch  (g) GaitSADA  (h) GaitSADA  (i) GaitSADA  (j) GaitSADA  (k) GaitSADA  (l) GaitSADA  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Grad-CAM [27] visualizations of the wrong prediction of other methods (a)-(f) comparing to our  correct prediction of the same data (g)-(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The red color area represents the more important weight contributing  to the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  (xx)  XX++X  C  XtGaitSADA: Self-Aligned Domain Adaptation for mmWave Gait Recognition  111:21  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Ablation experiments of 2-day data on laboratory location as source data while adapting to different  days (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', temporal) and 3 different locations, (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', server, conference, and office) using multiple combinations  of Additive Margin, Contrastive learning, Consistency Training and Centroid Alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The results report in  % of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  Stage  Components  Temporal  Server  Conference  Office  Average  1 stage  (train jointly)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' supervised  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='46  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='05  71  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='92  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='61  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' supervised + AM  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='8  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='72  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='8  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='07  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='85  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' supervised + AM + contrastive learning  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='.21  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='38  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='37  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='87  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='71  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' supervised + AM + consistency + centroid  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='19  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='15  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='2  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='55  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='02  2 stages  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' (1st stage) supervised + AM + contrastive learning  (2nd stage) supervised + AM + consistency  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='68  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='58  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='3  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='77  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='08  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' (1st stage) supervised + AM + contrastive learning  (2nd stage) supervised + AM + consistency + centroid  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='63  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='88  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='10  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='59  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='80  (1) The first experiment is simply just supervised learning method to present the baseline of the  ablation (supervised).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  (2) Additive margin softmax loss (AM) is added to the supervised learning in the second ex-  periment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The AM component alone improves the performance by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' In fact,  this component already outperforms most of the other adversarial-based methods as AM  loss increases inter-class separability and intra-class compactness without training multi-  ple models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' feature extractor and domain discriminator) to compete with each other as  adversarial-based methods which require more data in order to learn the feature distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  The performance boosts more in the large than the small domain drift as shown in the server  case, the accuracy improves by 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='67% while in the temporal case improves by only 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='34%  from supervised learning in the first experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  (3) In the third experiment, we add a contrastive learning component along with AM which is  exactly the same as in the first stage as of GaitSADA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Contrastive learning further improves  by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='86% on average as the model learns noise invariance on the target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  (4) The fourth experiment can be considered as the second stage of our GaitSADA without the  first stage pre-train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' In this experiment, consistency training and centroid alignment are  trained jointly with supervised learning with additive margin softmax loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' This second stage  ablation shows that without the first stage pre-training, in the less domain shift environment  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', temporal and conference case, the accuracies are high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' In temporal, the accuracy is even  higher than one with the first stage pre-training experiment (the 6th experiment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' However,  in the large domain drift environments including server and office, the accuracies drop  dramatically, even less than supervised learning only (the 1st experiment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  (5) The fifth ablation study is GaitSADA without Centroid Alignment which shows that with-  out Centroid Alignment the average performance decreases by 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='72% (drops from the 6th  experiment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  (6) The last one is our GaitSADA combines all components in 2 stages, which shows the overall  best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  8  CONCLUSION  In this paper, we designed a system for gait recognition using a commercial mmWave radar,  conducted multiple times and environments to study the domain discrepancy in various cases,  and proposed a novel domain adaptation method designed explicitly for mmWave signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' We  experiment with five regular domain adaptation methods to close the domain gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' The experiments  show the limitations of adversarial-based domain adaptation methods which require more data in  order to learn the data distribution due to the complicated minimax game of the feature extractor  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  111:22  Pinyoanuntapong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', Ekkasit Pinyoanuntapong, Ayman Ali, Kalvik Jakkala, Pu Wang, Minwoo Lee, Qucheng Peng,  Chen Chen, and Zhi Sun  and domain discriminator model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' On the other hand, GaitSADA directly minimizes intra-class  variation while maximizing inter-class deviation by utilizing self- and semi-supervised to learn  target unlabelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Our proposed method outperforms standard domain adaptation methods  by a large margin, particularly in the low data regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Visualization and ablation studies further  validate the effectiveness of GaitSADA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
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+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  GaitSADA: Self-Aligned Domain Adaptation for mmWave Gait Recognition  111:23  REFERENCES  [1] Fadel Adib, Chen-Yu Hsu, Hongzi Mao, Dina Katabi, and Fredo Durand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
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+page_content=' Hovy, Minh-Thang Luong, and Quoc V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Le.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
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+page_content=' Self-Training With Noisy Student Improves  ImageNet Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) (2019), 10684–  10695.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
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+page_content=' In Proceedings of the 35th International Conference on Machine Learning (Proceedings of Machine  Learning Research, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
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+page_content=' IEEE  Transactions on Information Forensics and Security 12, 5 (2017), 1141–1155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
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+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' MU-ID: Multi-user Identification Through  Gaits Using Millimeter Wave Radios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' In IEEE INFOCOM 2020 - IEEE Conference on Computer Communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
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+page_content=' Pathak, and Prasant Mohapatra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' WiWho: WiFi-Based Person Identification in Smart  Spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' In 2016 15th ACM/IEEE International Conference on Information Processing in Sensor Networks (IPSN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='1109/IPSN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
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+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Semi-supervised Models are Strong  Unsupervised Domain Adaptation Learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' ArXiv abs/2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='00417 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  GaitSADA: Self-Aligned Domain Adaptation for mmWave Gait Recognition  111:25  A  SUPPLEMENTARY  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='1  t-SNE  (a) Temporal  (b) Server  (c) Conference  (d) Office  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' t-SNE visualization for 3-day GAN  (a) Temporal  (b) Server  (c) Conference  (d) Office  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' t-SNE visualization for 3-day GRL  (a) Temporal  (b) Server  (c) Conference  (d) Office  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' t-SNE visualization for 3-day ADDA  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  111:26  Pinyoanuntapong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=', Ekkasit Pinyoanuntapong, Ayman Ali, Kalvik Jakkala, Pu Wang, Minwoo Lee, Qucheng Peng,  Chen Chen, and Zhi Sun  (a) Temporal  (b) Server  (c) Conference  (d) Office  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' t-SNE visualization for 3-day CDAN  (a) Temporal  (b) Server  (c) Conference  (d) Office  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' t-SNE visualization for 3-day FixMatch  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='2  Grad-CAM  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Grad-CAM of supervised learning method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Grad-CAM of GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Grad-CAM of GRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Grad-CAM of ADDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  XGaitSADA: Self-Aligned Domain Adaptation for mmWave Gait Recognition  111:27  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Grad-CAM of GRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Grad-CAM of FixMatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Grad-CAM of GaitSADA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content='  Received 20 February 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' revised 12 March 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' accepted 5 June 2009  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf'}
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+arXiv:2301.11909v1  [eess.SY]  27 Jan 2023
+Quantized Deep Path-following Control on a Microcontroller
+Pablo Zometa1 and Timm Faulwasser2
+Abstract— Model predictive Path-Following Control (MPFC)
+is a viable option for motion systems in many application
+domains. However, despite considerable progress on tailored
+numerical methods for predictive control, the real-time im-
+plementation of predictive control and MPFC on small-scale
+autonomous platforms with low-cost embedded hardware re-
+mains challenging. While usual stabilizing MPC formulations
+lead to static feedback laws, the MPFC feedback turns out to
+be dynamic as the path parameter acts as an internal controller
+variable. In this paper, we leverage deep learning to implement
+predictive path-following control on microcontrollers. We show
+that deep neural networks can approximate the dynamic
+MPFC feedback law accurately. Moreover, we illustrate and
+tackle the challenges that arise if the target platform employs
+limited precision arithmetic. Specifically, we draw upon a post-
+stabilization with an additional feedback law to attenuate
+undesired quantization effects. Simulation examples underpin
+the efficacy of the proposed approach.
+I. INTRODUCTION
+Nonlinear Model Predictive Control (NMPC) is a control
+method that can handle nonlinear system dynamics as well
+as input and state constraints. In its base variant NMPC for
+setpoint stabilization yields a static feedback law. Another
+variant is Model predictive Path-Following Control (MPFC),
+which has been successfully applied to motion control of
+robots to precisely follow a geometric reference path [1],
+[2], [3]. In MPFC the considered reference is a geometric
+path and timing along the path is computed at the run-time
+of the controller. Hence and in contrast to NMPC for setpoint
+stabilization, the MPFC is a dynamic feedback strategy as
+the reference position is an internal controller memory [4].
+An often cited disadvantage of NMPC is its high com-
+putational cost, which significantly limits its use in low-
+cost computing hardware like MicroController Units (MCU).
+The Optimization Engine (OpEn) [5] and acados [6], two
+popular state-of-the-art NMPC solvers, can efficiently run
+on embedded hardware like a Raspberry Pi (a single-board
+computer). However, at the time of this writing, none of them
+can run out of the box on 32-bit MCUs.
+To overcome the high computational demands of NMPC,
+the use of deep neural networks as a way to quickly find
+an approximate solution to the NMPC problem has been
+proposed [7], [8], [9]. In particular, [8] explores a robust
+multi-stage NMPC on an MCU using a Deep Neural Net-
+work (DNN) using single-precision floating-point arithmetic
+during network inference.
+1PZ is with the Faculty of Engineering, German International University
+in Berlin, Germany. pablo.zometa@giu-berlin.de
+2TF
+is
+with
+the
+Institute
+of
+Energy
+Systems,
+Energy
+Efficiency
+and
+Energy
+Economics,
+TU
+Dortmund,
+Germany.
+timm.faulwasser@ieee.org
+X
+Y
+L
+R
+ℓ
+ℓ
+s
+ˆX
+ˆY
+ϕ
+qy
+qx
+q
+−2 −1
+0
+1
+2
+−2
+0
+2
+X
+Y
+Fig. 1: Left: differential drive robot and its coordinate
+systems. Right: the path at scale, an ellipse. The robot’s left
+and right wheels are marked L and R, respectively.
+Moreover, to further increase the efficiency of DNNs, the
+use of quantization—i.e., storing the network parameters
+using fixed-point representation instead of floating point—
+has been explored [10]. Compared to a regular DNN, a
+quantized DNN executes much faster, requires less memory,
+and is more energy efficient—there is the downside of some
+loss of numerical accuracy [10].
+The present paper investigates the use of quantized deep
+neural networks for model predictive path-following control
+of mobile robots. Our main contribution is two-fold: first,
+we propose a way to generate the training set that takes
+into account the path to be followed, and second, we extend
+the DNN with a simple controller to make up for errors
+introduced by the quantized DNN approximation.
+Using the proposed approach with hardware-in-the-loop
+simulations running on an MCU, we show that a quantized
+deep neural network requiring less than 5 kB of storage
+memory achieves a good path following performance while
+being several orders of magnitude faster than OpEn.
+The remainder of the paper is organized as follows:
+Section II recalls MPFC applied to a mobile robot. Section
+III discusses quantized DNNs. Section IV introduces an
+approach to efficiently approximate the MPFC problem using
+quantized DNN, followed by the results (Section V) and
+conclusions (Section VI).
+II. PATH FOLLOWING CONTROL OF A MOBILE ROBOT
+This section summarizes the main idea of MPFC according
+to [11], and its application to differential drive robots [3].
+A. System Description
+Fig. 1 shows a schematic of a differential drive robot.
+The global (inertial) frame is defined by the axes XY ,
+
+whereas the local frame attached to the robot is defined by
+the axes ˆX ˆY . The position of the robot in the global frame
+is represented by the Cartesian coordinates of point q (the
+origin of the local frame). The robot’s pose ξ in the inertial
+frame is represented by its Cartesian position q = [qx qy]T
+and orientation ϕ, that is ξ = [qx qy ϕ]T. We represent the
+robot dynamics as the rate of change of the pose in terms of
+the robot’s forward speed s, and its angular velocity ω:
+˙ξ = f(ξ, u) =
+
+
+s cos(ϕ)
+s sin(ϕ)
+ω
+
+ ,
+ξ(0) = ξ0,
+(1)
+with ξ ∈ X ⊆ R3, and u = [s ω]T ∈ PC(U) ⊂ R2. We use
+PC(U) to denote that the inputs are piece-wise continuous
+and take values from a compact set U.
+B. The State-Space Path-Following Problem
+We recall the path-following problem in the state space of
+the robot model (1) as introduced by [11]. The path-following
+problem aims at making the system (1) follow a geometric
+reference without explicit timing requirements, i.e., when to
+be where on the path is not specified. The reference is given
+by
+P = {ξ ∈ R3 | ∃ θ ∈ R �→ ξ = p(θ)}.
+The variable θ(t) ∈ R is the path parameter, and p(θ(t)) ∈
+R3 is a parameterization of P. Note that although θ is
+dependent on time, its time evolution t �→ θ is not specified.
+Thus, the control inputs u ∈ PC(U) and the timing θ : R+
+0 →
+R+
+0 are chosen such that they follow the path as closely as
+possible.
+Problem 1. (State-space path following with speed assign-
+ment)
+1) Convergence to the path: the robot’s state ξ converges
+to the path P such that
+lim
+t→∞ ∥ξ(t) − p(θ)∥ = 0.
+2) Constraint satisfaction: the constraints on the states ξ ∈
+X and inputs u ∈ U are satisfied at all times.
+3) Velocity convergence: the path velocity ˙θ converges to
+a predefined profile such that
+lim
+t→∞ ∥ ˙θ(t) − vr(t)∥ = 0.
+Here we consider path parametrizations of the form
+p(θ) = [px(θ) py(θ) pϕ(θ)]
+T,
+(2)
+pϕ(θ) = arctan
+�p′
+y
+p′x
+�
+, p′
+x = ∂px
+∂θ , p′
+y = ∂py
+∂θ ,
+where px(θ) and py(θ) are at least twice continuously
+differentiable (see [3]). We denote pxy = [px py]T as the
+vector of Cartesian coordinates of the path.
+The path parameter θ is considered a virtual state, which
+is controlled by the virtual input v. Here the dynamics of θ
+are chosen as a single integrator:
+˙θ = v,
+θ(0) = θ0,
+where v ∈ PC(V), V .= [0, ¯v], and ¯v ∈ R.
+The path following problem is formulated using the aug-
+mented system
+˙z = f(z, w) =
+
+
+˙qx
+˙qy
+˙ϕ
+˙θ
+
+ =
+
+
+s cos(ϕ)
+s sin(ϕ)
+ω
+v
+
+ ,
+with the augmented state vector z = [ξT θ]T = [qx qy ϕ θ]T ∈
+Z = X ×R+
+0 and the augmented input vector w = [uT v]T =
+[s ω v]T ∈ PC(U × V) ⊂ R3.
+System (1) is differentially flat, and [qx qy]T is one of its
+flat outputs [12]. Therefore there is an input ur = [sr ωr]
+which guarantees that path (2) is followed by the system.
+The vector ur is used as a reference for the input vectors
+and can be built by observing that the first two equations of
+system (1) satisfy s2 = ˙q2
+x + ˙q2
+y, and thus:
+sr(θ, v) =
+��dpx(θ(t))
+dt
+�2
++
+�dpy(θ(t))
+dt
+�2
+= v
+�
+(p′x)2 +
+�
+p′y
+�2.
+(3)
+Furthermore, from the last equation of system (1) we have
+ω = ˙ϕ, which yields
+ωr(θ, v) = dpϕ(θ(t))
+dt
+= v
+�
+(p′
+x)2 +
+�
+p′
+y
+�2�−1 �
+p′
+xp′′
+y − p′
+yp′′
+x
+�
+,
+with p′′
+x = ∂2px
+∂θ2 ,
+and p′′
+y = ∂2py
+∂θ2 .
+(4)
+Further details on the derivation can be found in [13], [3].
+C. Model Predictive Path Following Control (MPFC)
+This section is based on the state-space MPFC scheme
+proposed in [11]. For paths defined in output spaces, we
+refer to [4], [2].
+The sampling period is δ > 0, and the prediction horizon
+is T = Nδ, with N ∈ N. The extended state at the current
+sampling time tk = kδ is denoted zk =
+�
+ξ(tk)
+θ(tk)
+�
+and
+the extended control input is w =
+�u
+v�
+. We consider the
+stage cost
+ℓ(z, w) =
+����
+ξ − p(θ)
+θ
+����
+2
+Q
++
+����
+u − ur(θ, v)
+v − vr
+����
+2
+R
+,
+with Q = QT ⪰ 0 and R = RT ≻ 0, i.e., symmetric positive
+(semi)definite diagonal matrices. The Optimal Control Prob-
+lem (OCP) to be solved repeatedly at each sampling instant
+tk and using zk as parametric data reads
+w∗ = arg min
+w∈PC(W)
+� T
+0
+ℓ(z(τ), w(τ))dτ
+subject to
+˙z(τ) = f(z(τ), w(τ)),
+z(0) = zk,
+z(τ) ∈ Z, w(τ) ∈ W.
+(5)
+Although this OCP is formulated in continuous time, our
+MPFC implementation is done in discrete time with w∗ =
+
+{w∗
+0, . . . , w∗
+N−1} ∈ WN a sequence of N input vectors.
+Typically, in MPC we only apply to the controlled system the
+first vector w = w∗
+0 in the sequence w∗. The MPFC feedback
+controller based on (5) can be expressed as the function
+w =
+�
+u
+v
+�
+= M(z).
+(6)
+Observe that w entails the robot command u and the virtual
+control v, which controls the evolution of the path parameter
+θ, cf. (II-B). Hence only u is applied to the robot.
+III. FEEDFORWARD NEURAL NETWORKS
+Next, we recall the basics of how a function can be ap-
+proximated by feedforward neural networks, the advantages
+of using deep architectures, and how to quantize them.
+A. Deep Neural Networks
+The use of feedforward Neural Networks (NN) is moti-
+vated by their universal function approximation properties
+[14]. In particular, we are interested in approximating the
+MPFC feedback (6). Our goal is to train an NN that ap-
+proximates M(z) by defining the mapping wD = D(z; Θ),
+where Θ represents a set of NΘ unknown parameters, which
+are learned during training. Once we have a trained network,
+we can use the D(z; Θ) to infer the values of wD ≈ M(z).
+To train our network, we rely on a training data set
+T = {ν1, ν2, . . . , νNT }, with νj =
+�
+zj
+M(zj)
+�
+∈ R7,
+j ∈ J = {1, . . ., NT }, and NT is large enough. The training
+algorithm aims to find the values of Θ that make D(zj; Θ) ≈
+M(zj), ∀ j ∈ J using some statistical measure like the Mean
+Squared Error (MSE). It is common to use a gradient-based
+optimization algorithm during training to minimize the MSE.
+The trained network is said to generalize well if D(z; Θ) is
+still a good approximation of M(z) for values of z not seen
+during training, in particular those relevant to the application.
+In general, an NN consists of H+2 layers: one input layer,
+one output layer, and H ≥ 1 hidden layers. Each layer k
+consists of nk units called neurons. Commonly, if there are
+only one or two hidden layers, the network is referred to
+as shallow, otherwise, it is called a Deep Neural Network
+(DNN). The advantage of a DNN, compared to a shallow
+network, is that it can approximate a function like (6) with
+similar accuracy but with fewer parameters NΘ as fewer
+neurons (and hence parameters) are considered per layer. We
+refer to [15] for details.
+Starting with the input z = h0 as the first layer, the output
+of layer k = 1, 2, . . ., H + 1 is
+hk = β(bk + W khk−1),
+(7)
+with bk ∈ Rnk a vector called bias and W k ∈ Rnk×nk−1 a
+matrix called weights, and the function β(·) is a saturating
+activation function. The last layer is the output layer wD =
+hH+1. Note that Θ = {b1, W 1, . . . , bH+1, W H+1}, and the
+number of parameters of the network is given by:
+NΘ =
+H+1
+�
+k=1
+nk(1 + nk−1).
+For example, a network with 1 hidden layer would be
+described as wD = D(z; Θ) = β(b2 + W 2β(b1 + W 1z)).
+A frequently used activation function is the Rectifying
+Linear Unit (ReLU) ([15, p. 171]), defined as β(h) =
+max(0, h), where max(·) is computed element-wise. Other
+common activation functions include the tangent hyperbolic
+and the sigmoid function.
+B. Network Training
+In practice, to find the set of parameters Θ that make D(·)
+approximate M(·) sufficiently well the higher-level set of so-
+called hyper-parameters needs to be determined. Common
+hyper-parameters include the network architecture (H, nk,
+β), and the gradient-based optimization algorithm parameters
+(e.g., the step size, also called the learning rate) to name
+just a few. A suitable combination of hyper-parameters is
+typically determined experimentally [16].
+It is helpful to normalize the training set to improve the
+numerical properties of the network. Here, we represent
+the training set as a matrix T ∈ R7×NT for simplicity in
+notation. For each column j, and row i of T we have:
+¯νj
+i = N(νj
+i ; µi, σi) = νj
+i − µi
+σi
+,
+where µi is the mean and σi is the standard deviation of row
+i. Note that νj represents column j of T . After applying this
+transformation, we obtain a normalized data set TN that has
+each row i with ¯µi = 0 and ¯σi = 1. To recover the original
+set T , we apply the inverse transformation:
+νj
+i = N −1(¯νj
+i ; µi, σi) = ¯νj
+i σi + µi.
+These operations must be applied to the extended robot
+state z =
+�
+ξ
+θ
+�
+∈ R4 and the extended input vector wD =
+�u
+v�
+∈ R3 during inference. That is ¯zi = N(zi; µi, σi), for
+i = 1, 2, 3, 4, and wD
+i = N −1( ¯wi; µi+4, σi+4), for i = 1, 2, 3
+(refer to Fig. 4a).
+C. Quantized DNN (QDNN)
+Quantization refers to storing the parameters of the net-
+work (weights and biases) as integer values. The main
+advantages are reduced memory required to store the NΘ
+parameters, faster execution, and higher energy efficiency
+during inference. The main disadvantage is the loss of
+accuracy in the inference [10].
+It is common to use an 8-bit integer representation (i8)
+to store the parameters set Θ. The network is trained first
+using floating point numbers often with single precision
+(32 bits). After the training is completed, the parameters Θ
+are quantized to an i8 approximation. There are different
+quantization methods [10]. Here we have used a uniform
+asymmetric quantization. That means that during inference,
+
+the normalized inputs ¯z in the network must be transformed
+from a floating point number to an integer using
+ˆz = Q(z; ˜a,ˆb) = i8(˜a¯z) + ˆb,
+(8)
+where ˜a is a floating point scaling, ˆb is an integer offset,
+and i8 refers to a mapping from floating point to 8-bit
+integer representation. Similarly, the output of the network ˆw
+must be transformed from an 8-bit integer to a floating-point
+normalized output ¯w, i.e., it must be dequantized using
+¯w = Q−1( ˆw; ˜c, ˆd) = f32( ˆw − ˆd)˜c,
+(9)
+where ˜c is a floating point scaling, ˆd is an integer offset, and
+f32 refers to a mapping from an 8-bit integer to a single-
+precision floating-point representation. The scaling and offset
+parameters are determined during the quantization of Θ. Fig.
+4a depicts how the robot state z (input to the network) and
+input vector wD (output of the network) are numerically
+transformed.
+IV. QDNN-BASED MPFC
+We now turn to a practical approach to approximate the
+MPFC problem presented in Section II using QDNNs as
+described in Section III. We denote this approach as MPFC-
+QDNN. This section also discusses how to augment the
+MPFC-QDNN with an online feedback controller to improve
+the accuracy of the path-following control. We denote this
+approach as MPFC-QDNN+P.
+A. Generating a Training Set for MPFC
+Although it is possible to find a network that approximates
+M(z) ∀ z ∈ Z, this typically would require a network and
+set T larger than necessary for the path-following problem.
+Under normal circumstances, a mobile robot following a path
+will mostly take poses ξ that are close to the reference path
+p(θ). Based on this, a smaller set ZC ⊂ Z can be used to
+significantly reduce the size of the network and the training
+set, without affecting the performance of the MPFC near the
+path. However, if the robot is driven far away from the path
+(e.g., due to large disturbances), the MPFC-QDNN may not
+be able to bring the robot back to following the path.
+To generate a set T appropriate for MPFC, we propose to
+use a corridor centered around the path (see Fig. 2). To build
+the set T , we select specific values of the path parameter θi,
+i = 0, 1, ..., Np, and compute the path vector p(θi). At each
+θi, we build a corridor C(θi) ∈ R3×Nc using a set of Nc
+points in the vicinity of p(θi).
+We propose a corridor in the form of a cuboid centered
+around p(θi) along the orthonormal vectors ⃗t,⃗n,⃗o (see Fig.
+2), with width 2cW , length 2cL, and height 2cH. The points
+Cj(θ) = [pj
+t pj
+n pj
+o]T are equidistant along each axis, with
+
+
+−cW⃗t
+−cL⃗n
+−cH⃗o
+
+ ≤
+
+
+pj
+t
+pj
+n
+pj
+o
+
+ ≤
+
+
+cW⃗t
+cL⃗n
+cH⃗o
+
+ .
+The set ZC has NpNc elements. The corridor can be
+defined in many different ways (e.g., using randomly selected
+points inside an ellipsoid). Here we have presented one way
+0.11
+0.1
+0.09
+0.075
+0.05
+−0.2
+0
+0.2
+1
+⃗n(p(θi))
+⃗t(p(θi))
+X [m]
+Y [m]
+p(θ)
+C(θi)
+p(θi)
+Fig. 2: Simplified 2-dimensional visualization of data point
+used to build the training set T . The vectors ⃗n, ⃗t, ⃗o (coming
+out of the page) are orthonormal. Only px and py are shown
+(the orientation pϕ is not depicted). The figure shows the
+corridor for three values of θi, and at each point p(θi) (cross)
+a corridor C(θi) of 9 points (dots) is constructed. The width
+(0.02 m in the figure), and the length (0.4 m) of the corridor
+are measured normal (along ⃗n) and tangential (along ⃗t) to
+the path at p(θi), respectively.
+that has worked well in our experiments (a cuboid grid with
+equidistant points). Determining the optimal way to construct
+the set ZC is beyond the scope of this work.
+The size of the corridor plays an important role in how
+well the MPFC-QDNN can follow the path in practice. If the
+corridor is too narrow, the network is not able to follow the
+path at all, due to inevitable errors inherent in any feedback
+control system. A broad corridor is thus preferred. However,
+that may require more data points in the set, and perhaps a
+larger network, to make the approximation D(·) useful.
+B. Path-Following Error
+Although MPFC can follow the reference path P very
+accurately, at any time t there might be an error e in the
+robot’s Cartesian position q(t) with respect to the reference
+point in the path p(θ(t)). In the XY coordinates, the error
+is given by:
+eXY (q(t), pxy(θ(t))) = q(t) − pxy(θ(t)).
+This error vector can be expressed in the basis formed by
+the orthonormal vectors ⃗t(pϕ(θ)) and ⃗n(pϕ(θ)), which are
+tangential and normal to the path at pxy(θ), respectively
+(refer to Fig. 3). That is:
+e(q(t), p(θ(t))) = en(q, p)⃗n(pϕ) + et(q, p)⃗t(pϕ),
+(10)
+where the scalars en and et are the projection of eXY onto
+each orthonormal vector, computed by the dot product
+en(q, p) = (q − pxy)
+T ⃗n(pϕ),
+et(q, p) = (q − pxy)
+T ⃗t(pϕ).
+
+0
+5
+·10−2
+1
+2
+⃗n(pϕ)
+⃗t(pϕ)
+e(q, p)
+et(q, p)
+en(q, p)
+X [m]
+Y [m]
+p(θ(t))
+pxy(θk)
+q(tk)
+Fig. 3: At any given time tk, the robot’s position in the
+global frame XY is given by q(tk). The unit vectors ⃗t(θk)
+(not shown at scale), and ⃗n(θk) are tangential and normal to
+the path at point pxy(θk), respectively, and θk = θ(tk). The
+vector e(q, p) = q − pxy = et(q, p)⃗t(pϕ) + en(p, q)⃗n(pϕ) is
+the current robot’s position with respect to the path.
+In the case of an ellipse, the tangential and normal vectors
+are given by:
+⃗t(pϕ) =
+�cos(pϕ)
+sin(pϕ)
+�
+,
+⃗n(pϕ) =
+�− sin(pϕ)
+cos(pϕ)
+�
+.
+C. Augmented Control Scheme
+Due to the MPFC-QDNN being an approximation of the
+MPFC, the path-following error e resulting from D(z; Θ) is
+significantly larger than the error observed under the original
+MPFC controller M(z) (see Section V). To compensate
+this error we extend the MPFC-QDNN controller with an
+additional linear feedback which acts on the tangential com-
+ponent et through the forward speed of the robot s, and on
+the normal component en through the robot’s angular speed
+ω. Put differently, the compensation term wP is added to the
+control vector, i.e., w = wD + wP . Here wP = [sP ωP 0]T,
+with sP = Ptet, and ωP = Pnen, where Pt, and Pn are
+the proportional gains (see Fig. 4). We denote this approach
+MPFC-QDNN-P. We selected static feedback mainly due to
+its simplicity and effectiveness as shown in Section V.
+D. Implementation
+We consider an ellipse as the path (see Fig. 1), which is
+defined by the parametrization
+p(θ) =
+�
+0.1 cos(θ)
+2 sin(θ)
+arctan
+�
+−0.1 sin(θ)
+2 cos(θ)
+��T
+,
+which yields the input references (3) and (4) as
+sr(θ, v) = 2v
+�
+1 − 0.9975 sin2(θ),
+ωr(θ, v) = 2v
+�
+40 − 39.9 sin2(θ)
+�−1 .
+To generate the training set T , we use a corridor consisting
+of a cuboid of width 2cW = 0.02, length 2cL = 0.2, and
+height 2cH =
+2π
+3 . Each axis is split into 5, 5, and 40
+equidistant points (Nc = 1000), respectively. We split the
+path in Np = 4000 equidistant segments between 0 ≤ θ ≤
+2π, which corresponds to a full turn around the path. The
+subset of states in the corridor ZC consists of NpNc = 4E6
+points.
+To solve the MPFC problem (5), and consequently find
+w = M(z), ∀ z ∈ ZC according to (6), we use the
+Optimization Engine (OpEn) [5], a fast solver for optimal
+control problems. The training set T consists of NpNc pair
+of vectors z, M(z) for all z ∈ ZC. We use a discretization
+time δ = 0.01 s, and a horizon length T = 0.6 s in (5).
+We use a random search approach to find the hyper-
+parameters of a network that is a good approximation to
+M under the constraint that the number of parameters NΘ
+should remain small. i.e., to reduce the size of the network
+in the MCU’s ROM. Random search typically delivers better
+results than manual or grid search for the same amount
+of computation during training [16]. The selected hyper-
+parameters were the number of hidden layers H, the number
+of units in each hidden layer nk, k = 1, . . . , H, and the learn-
+ing rate of the optimization algorithm (see the Appendix).
+To find the hyper-parameters we use KerasTuner [17]. To
+perform the training and the quantization of the network
+we use the deep learning framework Keras/TensorFlow [18],
+[19].
+We implement a Hardware-In-the-Loop (HIL) simulation
+where the MPFC-QDNN+P is deployed on an STM32F407
+MCU, which is based on a Cortex-M4 processor core running
+at 168 MHz, which includes a single-precision floating-point
+unit and 1 MB flash ROM. The robot dynamics are simulated
+on a PC, see the Appendix for details.
+The QDNN consists of 9 hidden layers with roughly 4700
+parameters, using 8-bit integers to store the parameters (i.e. 1
+parameter requires 1 byte of ROM). The quantized parameter
+set Θ requires less than 5 kB of the MCU’s flash memory.
+V. RESULTS
+As our reference implementation (denoted MPFC-OpEn),
+we use OpEn (the same solver used for training) to solve the
+MPFC problem (5). The advantages of MPFC are illustrated
+in Fig. 5. The input w computed by OpEn to steer the robot
+along the path in Fig. 6 shows that when the path curvature is
+tight, i.e. top (θ = π
+2 ) and bottom (θ = 3π
+2 ) of the ellipse, the
+path speed v is reduced, and consequently the robot’s forward
+speed s is also reduced. This allows the robot to follow the
+tight curve. Similarly, v is reduced when the constraints on
+s are active (e.g. θ = π) because the robot cannot otherwise
+closely follow the path.
+The main advantage of using a DNN on an MCU is that
+it is relatively easy to implement quantization (8), inference
+(7), and dequantization (9) sequentially for all layers in the
+network. Furthermore, for a small network like the one used
+here (4700 parameters), the inference is executed much faster
+than solving the OCP (5).
+Table I shows the execution time for different implemen-
+tations of MPFC. Our experiments ran on a PC with Ubuntu
+
+MPFC-QDNN
+z
+N
+Q
+QDNN
+Q−1
+N −1
+wD
+¯z
+ˆz
+ˆw
+¯w
+(a) The quantized deep neural network inference chain used to
+approximate the MPFC (denoted MPFC-QDNN on the right figure).
+Robot
+�
+MPFC-
+QDNN
++
+E
+P
+w
+s
+ω
+v
+ξ
+θ
+z
+e
+wP
+wD
+MPFC-QDNN+P
+(b) The proposed controller approach (MPFC-QDNN+P)
+Fig. 4: Block diagram of the proposed combined controller. Inside the dashed block (MPFC-QDNN) is the MPFC controller
+approximated by a QDNN. The extended state z is fed into the MPFC-QDNN. The state is first normalized (¯z) then
+quantized (ˆz), and finally fed into the QDNN block for inference of the approximate control input ˆw. Dequantization and
+denormalization are applied to get the approximated augmented control input wD. The block E computes the robot’s path
+error vector e (10) used by the P controllers. The vector wP is added to wD to compute the input w. The forward velocity
+s, and angular velocity ω are applied to the robot, whereas the path velocity v is integrated to compute the path variable θ.
+0
+π
+2
+π
+3π
+2
+2π
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+θ
+s [m/s]
+ω [rad/s]
+v [s−1]
+Fig. 5: Control inputs vs. path parameter. When the path
+curvature is tight (near π
+2 , and 3π
+2 ) the path speed v, and
+the robot’s forward speed s are reduced. The path speed v is
+also reduced when the input constraints are active (near π).
+Linux 22.04-LTS, and a x86-64 processor with a 2.4 GHz
+clock. Compared to MPFC-OpEn, the average execution time
+of the MPFC-QDNN+P implementation is about three orders
+of magnitude faster on the PC.
+The QDNN implementation using 8-bit integers requires
+on average 230 microseconds to execute on the MCU.
+Currently, running OpEn on an MCU is not supported.
+Fig. 6 shows a comparison of the path in the Cartesian
+XY plane followed by the simulated robot using different
+implementations. The absolute Cartesian position error is
+shown in Fig. 7, with a summary presented in Table II.
+All implementations can follow the path, with OpEn being
+the most accurate. When the worst-case error is considered,
+using a regular (non-quantized) DNN is two orders of
+magnitude worse than the OpEn implementation. The QDNN
+Implementation
+Mean [s]
+Std. [s]
+Worst [s]
+OpEn (PC)
+1.2E-3
+5.9E-4
+7.8E-3
+QDNN+P (PC)
+7.3E-6
+2.9E-6
+3.2E-5
+OpEn (MCU)
+-
+-
+-
+QDNN+P (MCU)
+2.3E-4
+2.1E-6
+2.4E-4
+TABLE I: Mean, standard deviation (Std.), worst-case ex-
+ecution (Worst) time in seconds for 10000 steps close to
+the path. The table is split into MPFC implementations on
+a personal computer (PC, top part) and a microcontroller
+(MCU, bottom part). Note that the execution of QDNN+P
+on the MCU is temporally deterministic. In the PC case, the
+QDNN+P has more variability due to the operating system.
+Mean
+Max.
+OpEn
+1.9E-4
+3.3E-4
+DNN
+7.5E-3
+2.1E-2
+QDNN
+1.6E-2
+4.8E-2
+QDNN+P
+6.1E-4
+4.9E-3
+TABLE II: Mean and maximum Cartesian error in the path.
+implementation has worse overall performance than the non-
+quantized network. Finally, the proposed addition of two P
+controllers to the QDNN reduces its worst-case error by an
+order of magnitude and outperforms the DNN.
+VI. CONCLUSIONS
+The paper presented a model predictive path following
+implementation using quantized deep neural networks aug-
+mented with a controller for quantization error compensation.
+We showed a practical way to select the training set, and
+how to design the error compensation controller. Compared
+to a traditional MPFC using online optimization, our pro-
+posed approach requires only a fraction of the memory and
+runs several orders of magnitude faster on PC simulations.
+Although the path-following accuracy is slightly degraded,
+we believe the performance may still be good for low-cost
+applications. With a hardware-in-the-loop implementation
+
+−0.1
+−0.05
+0
+0.05
+0.1
+−2
+−1
+0
+1
+2
+θ = 0
+θ = π
+2
+θ = π
+θ = 3π
+2
+X [m]
+Y [m]
+OpEn
+DNN
+QDNN
+QDNN+P
+Fig. 6: The path followed by the robot using different
+MPFC implementations: OpEn (visually indistinguishable
+from the reference path p(θ)), the approximation using a
+deep neural network (DNN, 32-bit float), a quantized deep
+neural network (QDNN, 8-bit integer), and the proposed
+QDNN plus P control approach (QDNN+P).
+using a microcontroller, we showed the effectiveness of this
+approach for low-cost embedded devices. Future work will
+discuss how to handle different path geometries with one
+trained QDNN and how to give performance guarantees.
+APPENDIX
+The hyperparameters of the QDNN network are the learn-
+ing rate = 4.5E − 4, the activation function (ReLU), the
+number of hidden layers H = 9, and the units on each layer:
+input layer 4 units, followed by the hidden layers with 48,
+16, 24, 16, 16, 40, 24, 16, and 24 units, and output layer 3
+units.
+The parameters of OCP (5) are the matrices Q
+=
+diag(2E5, 2E5, 1E5, 0), R = diag(1E1, 5E3, 1E5), the
+box sets Z = {z ∈ R4 | z ≤ z ≤ z}, with z =
+[−5, −15, − π
+2 , −5], z = [5, 15, π
+2 , −5], and W = {w ∈
+R3 | w ≤ w ≤ w}, with w = [−0.26, −0.455, 0], w =
+[0.26, 0.455, 0.15], the discretization time δ = 0.01s, and the
+horizon length steps N = 60.
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+
diff --git a/ptFKT4oBgHgl3EQfyy7U/content/tmp_files/load_file.txt b/ptFKT4oBgHgl3EQfyy7U/content/tmp_files/load_file.txt
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@@ -0,0 +1,525 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf,len=524
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='11909v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='SY]  27 Jan 2023  Quantized Deep Path-following Control on a Microcontroller  Pablo Zometa1 and Timm Faulwasser2  Abstract— Model predictive Path-Following Control (MPFC)  is a viable option for motion systems in many application  domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' However, despite considerable progress on tailored  numerical methods for predictive control, the real-time im-  plementation of predictive control and MPFC on small-scale  autonomous platforms with low-cost embedded hardware re-  mains challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' While usual stabilizing MPC formulations  lead to static feedback laws, the MPFC feedback turns out to  be dynamic as the path parameter acts as an internal controller  variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' In this paper, we leverage deep learning to implement  predictive path-following control on microcontrollers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' We show  that deep neural networks can approximate the dynamic  MPFC feedback law accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Moreover, we illustrate and  tackle the challenges that arise if the target platform employs  limited precision arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Specifically, we draw upon a post-  stabilization with an additional feedback law to attenuate  undesired quantization effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Simulation examples underpin  the efficacy of the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' INTRODUCTION  Nonlinear Model Predictive Control (NMPC) is a control  method that can handle nonlinear system dynamics as well  as input and state constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' In its base variant NMPC for  setpoint stabilization yields a static feedback law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Another  variant is Model predictive Path-Following Control (MPFC),  which has been successfully applied to motion control of  robots to precisely follow a geometric reference path [1],  [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' In MPFC the considered reference is a geometric  path and timing along the path is computed at the run-time  of the controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Hence and in contrast to NMPC for setpoint  stabilization, the MPFC is a dynamic feedback strategy as  the reference position is an internal controller memory [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  An often cited disadvantage of NMPC is its high com-  putational cost, which significantly limits its use in low-  cost computing hardware like MicroController Units (MCU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  The Optimization Engine (OpEn) [5] and acados [6], two  popular state-of-the-art NMPC solvers, can efficiently run  on embedded hardware like a Raspberry Pi (a single-board  computer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' However, at the time of this writing, none of them  can run out of the box on 32-bit MCUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  To overcome the high computational demands of NMPC,  the use of deep neural networks as a way to quickly find  an approximate solution to the NMPC problem has been  proposed [7], [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' In particular, [8] explores a robust  multi-stage NMPC on an MCU using a Deep Neural Net-  work (DNN) using single-precision floating-point arithmetic  during network inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  1PZ is with the Faculty of Engineering, German International University  in Berlin, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' pablo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='zometa@giu-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='de  2TF  is  with  the  Institute  of  Energy  Systems,  Energy  Efficiency  and  Energy  Economics,  TU  Dortmund,  Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  timm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='faulwasser@ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='org  X  Y  L  R  ℓ  ℓ  s  ˆX  ˆY  ϕ  qy  qx  q  −2 −1  0  1  2  −2  0  2  X  Y  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' 1: Left: differential drive robot and its coordinate  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Right: the path at scale, an ellipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The robot’s left  and right wheels are marked L and R, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  Moreover, to further increase the efficiency of DNNs, the  use of quantization—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=', storing the network parameters  using fixed-point representation instead of floating point—  has been explored [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Compared to a regular DNN, a  quantized DNN executes much faster, requires less memory,  and is more energy efficient—there is the downside of some  loss of numerical accuracy [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  The present paper investigates the use of quantized deep  neural networks for model predictive path-following control  of mobile robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Our main contribution is two-fold: first,  we propose a way to generate the training set that takes  into account the path to be followed, and second, we extend  the DNN with a simple controller to make up for errors  introduced by the quantized DNN approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  Using the proposed approach with hardware-in-the-loop  simulations running on an MCU, we show that a quantized  deep neural network requiring less than 5 kB of storage  memory achieves a good path following performance while  being several orders of magnitude faster than OpEn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  The remainder of the paper is organized as follows:  Section II recalls MPFC applied to a mobile robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Section  III discusses quantized DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Section IV introduces an  approach to efficiently approximate the MPFC problem using  quantized DNN, followed by the results (Section V) and  conclusions (Section VI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' PATH FOLLOWING CONTROL OF A MOBILE ROBOT  This section summarizes the main idea of MPFC according  to [11], and its application to differential drive robots [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' System Description  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' 1 shows a schematic of a differential drive robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  The global (inertial) frame is defined by the axes XY ,  whereas the local frame attached to the robot is defined by  the axes ˆX ˆY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The position of the robot in the global frame  is represented by the Cartesian coordinates of point q (the  origin of the local frame).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The robot’s pose ξ in the inertial  frame is represented by its Cartesian position q = [qx qy]T  and orientation ϕ, that is ξ = [qx qy ϕ]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' We represent the  robot dynamics as the rate of change of the pose in terms of  the robot’s forward speed s, and its angular velocity ω:  ˙ξ = f(ξ, u) =  \uf8ee  \uf8f0  s cos(ϕ)  s sin(ϕ)  ω  \uf8f9  \uf8fb ,  ξ(0) = ξ0,  (1)  with ξ ∈ X ⊆ R3, and u = [s ω]T ∈ PC(U) ⊂ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' We use  PC(U) to denote that the inputs are piece-wise continuous  and take values from a compact set U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The State-Space Path-Following Problem  We recall the path-following problem in the state space of  the robot model (1) as introduced by [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The path-following  problem aims at making the system (1) follow a geometric  reference without explicit timing requirements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=', when to  be where on the path is not specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The reference is given  by  P = {ξ ∈ R3 | ∃ θ ∈ R �→ ξ = p(θ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  The variable θ(t) ∈ R is the path parameter, and p(θ(t)) ∈  R3 is a parameterization of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Note that although θ is  dependent on time, its time evolution t �→ θ is not specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  Thus, the control inputs u ∈ PC(U) and the timing θ : R+  0 →  R+  0 are chosen such that they follow the path as closely as  possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' (State-space path following with speed assign-  ment)  1) Convergence to the path: the robot’s state ξ converges  to the path P such that  lim  t→∞ ∥ξ(t) − p(θ)∥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  2) Constraint satisfaction: the constraints on the states ξ ∈  X and inputs u ∈ U are satisfied at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  3) Velocity convergence: the path velocity ˙θ converges to  a predefined profile such that  lim  t→∞ ∥ ˙θ(t) − vr(t)∥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  Here we consider path parametrizations of the form  p(θ) = [px(θ) py(θ) pϕ(θ)]  T,  (2)  pϕ(θ) = arctan  �p′  y  p′x  �  , p′  x = ∂px  ∂θ , p′  y = ∂py  ∂θ ,  where px(θ) and py(θ) are at least twice continuously  differentiable (see [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' We denote pxy = [px py]T as the  vector of Cartesian coordinates of the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  The path parameter θ is considered a virtual state, which  is controlled by the virtual input v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Here the dynamics of θ  are chosen as a single integrator:  ˙θ = v,  θ(0) = θ0,  where v ∈ PC(V), V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='= [0, ¯v], and ¯v ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  The path following problem is formulated using the aug-  mented system  ˙z = f(z, w) =  \uf8ee  \uf8ef\uf8ef\uf8f0  ˙qx  ˙qy  ˙ϕ  ˙θ  \uf8f9  \uf8fa\uf8fa\uf8fb =  \uf8ee  \uf8ef\uf8ef\uf8f0  s cos(ϕ)  s sin(ϕ)  ω  v  \uf8f9  \uf8fa\uf8fa\uf8fb ,  with the augmented state vector z = [ξT θ]T = [qx qy ϕ θ]T ∈  Z = X ×R+  0 and the augmented input vector w = [uT v]T =  [s ω v]T ∈ PC(U × V) ⊂ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  System (1) is differentially flat, and [qx qy]T is one of its  flat outputs [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Therefore there is an input ur = [sr ωr]  which guarantees that path (2) is followed by the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  The vector ur is used as a reference for the input vectors  and can be built by observing that the first two equations of  system (1) satisfy s2 = ˙q2  x + ˙q2  y, and thus:  sr(θ, v) =  ��dpx(θ(t))  dt  �2  +  �dpy(θ(t))  dt  �2  = v  �  (p′x)2 +  �  p′y  �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  (3)  Furthermore, from the last equation of system (1) we have  ω = ˙ϕ, which yields  ωr(θ, v) = dpϕ(θ(t))  dt  = v  �  (p′  x)2 +  �  p′  y  �2�−1 �  p′  xp′′  y − p′  yp′′  x  �  ,  with p′′  x = ∂2px  ∂θ2 ,  and p′′  y = ∂2py  ∂θ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  (4)  Further details on the derivation can be found in [13], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Model Predictive Path Following Control (MPFC)  This section is based on the state-space MPFC scheme  proposed in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' For paths defined in output spaces, we  refer to [4], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  The sampling period is δ > 0, and the prediction horizon  is T = Nδ, with N ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The extended state at the current  sampling time tk = kδ is denoted zk =  �  ξ(tk)  θ(tk)  �  and  the extended control input is w =  �u  v�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' We consider the  stage cost  ℓ(z, w) =  ����  ξ − p(θ)  θ  ����  2  Q  +  ����  u − ur(θ, v)  v − vr  ����  2  R  ,  with Q = QT ⪰ 0 and R = RT ≻ 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=', symmetric positive  (semi)definite diagonal matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The Optimal Control Prob-  lem (OCP) to be solved repeatedly at each sampling instant  tk and using zk as parametric data reads  w∗ = arg min  w∈PC(W)  � T  0  ℓ(z(τ), w(τ))dτ  subject to  ˙z(τ) = f(z(τ), w(τ)),  z(0) = zk,  z(τ) ∈ Z, w(τ) ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  (5)  Although this OCP is formulated in continuous time, our  MPFC implementation is done in discrete time with w∗ =  {w∗  0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' , w∗  N−1} ∈ WN a sequence of N input vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  Typically, in MPC we only apply to the controlled system the  first vector w = w∗  0 in the sequence w∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The MPFC feedback  controller based on (5) can be expressed as the function  w =  �  u  v  �  = M(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  (6)  Observe that w entails the robot command u and the virtual  control v, which controls the evolution of the path parameter  θ, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' (II-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Hence only u is applied to the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' FEEDFORWARD NEURAL NETWORKS  Next, we recall the basics of how a function can be ap-  proximated by feedforward neural networks, the advantages  of using deep architectures, and how to quantize them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Deep Neural Networks  The use of feedforward Neural Networks (NN) is moti-  vated by their universal function approximation properties  [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' In particular, we are interested in approximating the  MPFC feedback (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Our goal is to train an NN that ap-  proximates M(z) by defining the mapping wD = D(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Θ),  where Θ represents a set of NΘ unknown parameters, which  are learned during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Once we have a trained network,  we can use the D(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Θ) to infer the values of wD ≈ M(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  To train our network, we rely on a training data set  T = {ν1, ν2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' , νNT }, with νj =  �  zj  M(zj)  �  ∈ R7,  j ∈ J = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=', NT }, and NT is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The training  algorithm aims to find the values of Θ that make D(zj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Θ) ≈  M(zj), ∀ j ∈ J using some statistical measure like the Mean  Squared Error (MSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' It is common to use a gradient-based  optimization algorithm during training to minimize the MSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  The trained network is said to generalize well if D(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Θ) is  still a good approximation of M(z) for values of z not seen  during training, in particular those relevant to the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  In general, an NN consists of H+2 layers: one input layer,  one output layer, and H ≥ 1 hidden layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Each layer k  consists of nk units called neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Commonly, if there are  only one or two hidden layers, the network is referred to  as shallow, otherwise, it is called a Deep Neural Network  (DNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The advantage of a DNN, compared to a shallow  network, is that it can approximate a function like (6) with  similar accuracy but with fewer parameters NΘ as fewer  neurons (and hence parameters) are considered per layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' We  refer to [15] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  Starting with the input z = h0 as the first layer, the output  of layer k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=', H + 1 is  hk = β(bk + W khk−1),  (7)  with bk ∈ Rnk a vector called bias and W k ∈ Rnk×nk−1 a  matrix called weights, and the function β(·) is a saturating  activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The last layer is the output layer wD =  hH+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Note that Θ = {b1, W 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' , bH+1, W H+1}, and the  number of parameters of the network is given by:  NΘ =  H+1  �  k=1  nk(1 + nk−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  For example, a network with 1 hidden layer would be  described as wD = D(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Θ) = β(b2 + W 2β(b1 + W 1z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  A frequently used activation function is the Rectifying  Linear Unit (ReLU) ([15, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' 171]), defined as β(h) =  max(0, h), where max(·) is computed element-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Other  common activation functions include the tangent hyperbolic  and the sigmoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Network Training  In practice, to find the set of parameters Θ that make D(·)  approximate M(·) sufficiently well the higher-level set of so-  called hyper-parameters needs to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Common  hyper-parameters include the network architecture (H, nk,  β), and the gradient-based optimization algorithm parameters  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=', the step size, also called the learning rate) to name  just a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' A suitable combination of hyper-parameters is  typically determined experimentally [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  It is helpful to normalize the training set to improve the  numerical properties of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Here, we represent  the training set as a matrix T ∈ R7×NT for simplicity in  notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' For each column j, and row i of T we have:  ¯νj  i = N(νj  i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' µi, σi) = νj  i − µi  σi  ,  where µi is the mean and σi is the standard deviation of row  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Note that νj represents column j of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' After applying this  transformation, we obtain a normalized data set TN that has  each row i with ¯µi = 0 and ¯σi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' To recover the original  set T , we apply the inverse transformation:  νj  i = N −1(¯νj  i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' µi, σi) = ¯νj  i σi + µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  These operations must be applied to the extended robot  state z =  �  ξ  θ  �  ∈ R4 and the extended input vector wD =  �u  v�  ∈ R3 during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' That is ¯zi = N(zi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' µi, σi), for  i = 1, 2, 3, 4, and wD  i = N −1( ¯wi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' µi+4, σi+4), for i = 1, 2, 3  (refer to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Quantized DNN (QDNN)  Quantization refers to storing the parameters of the net-  work (weights and biases) as integer values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The main  advantages are reduced memory required to store the NΘ  parameters, faster execution, and higher energy efficiency  during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The main disadvantage is the loss of  accuracy in the inference [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  It is common to use an 8-bit integer representation (i8)  to store the parameters set Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The network is trained first  using floating point numbers often with single precision  (32 bits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' After the training is completed, the parameters Θ  are quantized to an i8 approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' There are different  quantization methods [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Here we have used a uniform  asymmetric quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' That means that during inference,  the normalized inputs ¯z in the network must be transformed  from a floating point number to an integer using  ˆz = Q(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' ˜a,ˆb) = i8(˜a¯z) + ˆb,  (8)  where ˜a is a floating point scaling, ˆb is an integer offset,  and i8 refers to a mapping from floating point to 8-bit  integer representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Similarly, the output of the network ˆw  must be transformed from an 8-bit integer to a floating-point  normalized output ¯w, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=', it must be dequantized using  ¯w = Q−1( ˆw;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' ˜c, ˆd) = f32( ˆw − ˆd)˜c,  (9)  where ˜c is a floating point scaling, ˆd is an integer offset, and  f32 refers to a mapping from an 8-bit integer to a single-  precision floating-point representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The scaling and offset  parameters are determined during the quantization of Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  4a depicts how the robot state z (input to the network) and  input vector wD (output of the network) are numerically  transformed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' QDNN-BASED MPFC  We now turn to a practical approach to approximate the  MPFC problem presented in Section II using QDNNs as  described in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' We denote this approach as MPFC-  QDNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' This section also discusses how to augment the  MPFC-QDNN with an online feedback controller to improve  the accuracy of the path-following control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' We denote this  approach as MPFC-QDNN+P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Generating a Training Set for MPFC  Although it is possible to find a network that approximates  M(z) ∀ z ∈ Z, this typically would require a network and  set T larger than necessary for the path-following problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  Under normal circumstances, a mobile robot following a path  will mostly take poses ξ that are close to the reference path  p(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Based on this, a smaller set ZC ⊂ Z can be used to  significantly reduce the size of the network and the training  set, without affecting the performance of the MPFC near the  path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' However, if the robot is driven far away from the path  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=', due to large disturbances), the MPFC-QDNN may not  be able to bring the robot back to following the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  To generate a set T appropriate for MPFC, we propose to  use a corridor centered around the path (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' To build  the set T , we select specific values of the path parameter θi,  i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=', Np, and compute the path vector p(θi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' At each  θi, we build a corridor C(θi) ∈ R3×Nc using a set of Nc  points in the vicinity of p(θi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  We propose a corridor in the form of a cuboid centered  around p(θi) along the orthonormal vectors ⃗t,⃗n,⃗o (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  2), with width 2cW , length 2cL, and height 2cH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The points  Cj(θ) = [pj  t pj  n pj  o]T are equidistant along each axis, with  \uf8ee  \uf8f0  −cW⃗t  −cL⃗n  −cH⃗o  \uf8f9  \uf8fb ≤  \uf8ee  \uf8f0  pj  t  pj  n  pj  o  \uf8f9  \uf8fb ≤  \uf8ee  \uf8f0  cW⃗t  cL⃗n  cH⃗o  \uf8f9  \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  The set ZC has NpNc elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The corridor can be  defined in many different ways (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=', using randomly selected  points inside an ellipsoid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Here we have presented one way  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='075  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='05  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='2  1  ⃗n(p(θi))  ⃗t(p(θi))  X [m]  Y [m]  p(θ)  C(θi)  p(θi)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' 2: Simplified 2-dimensional visualization of data point  used to build the training set T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The vectors ⃗n, ⃗t, ⃗o (coming  out of the page) are orthonormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Only px and py are shown  (the orientation pϕ is not depicted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The figure shows the  corridor for three values of θi, and at each point p(θi) (cross)  a corridor C(θi) of 9 points (dots) is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The width  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='02 m in the figure), and the length (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='4 m) of the corridor  are measured normal (along ⃗n) and tangential (along ⃗t) to  the path at p(θi), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  that has worked well in our experiments (a cuboid grid with  equidistant points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Determining the optimal way to construct  the set ZC is beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  The size of the corridor plays an important role in how  well the MPFC-QDNN can follow the path in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' If the  corridor is too narrow, the network is not able to follow the  path at all, due to inevitable errors inherent in any feedback  control system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' A broad corridor is thus preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' However,  that may require more data points in the set, and perhaps a  larger network, to make the approximation D(·) useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Path-Following Error  Although MPFC can follow the reference path P very  accurately, at any time t there might be an error e in the  robot’s Cartesian position q(t) with respect to the reference  point in the path p(θ(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' In the XY coordinates, the error  is given by:  eXY (q(t), pxy(θ(t))) = q(t) − pxy(θ(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  This error vector can be expressed in the basis formed by  the orthonormal vectors ⃗t(pϕ(θ)) and ⃗n(pϕ(θ)), which are  tangential and normal to the path at pxy(θ), respectively  (refer to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' That is:  e(q(t), p(θ(t))) = en(q, p)⃗n(pϕ) + et(q, p)⃗t(pϕ),  (10)  where the scalars en and et are the projection of eXY onto  each orthonormal vector, computed by the dot product  en(q, p) = (q − pxy)  T ⃗n(pϕ),  et(q, p) = (q − pxy)  T ⃗t(pϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  0  5  10−2  1  2  ⃗n(pϕ)  ⃗t(pϕ)  e(q, p)  et(q, p)  en(q, p)  X [m]  Y [m]  p(θ(t))  pxy(θk)  q(tk)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' 3: At any given time tk, the robot’s position in the  global frame XY is given by q(tk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The unit vectors ⃗t(θk)  (not shown at scale), and ⃗n(θk) are tangential and normal to  the path at point pxy(θk), respectively, and θk = θ(tk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The  vector e(q, p) = q − pxy = et(q, p)⃗t(pϕ) + en(p, q)⃗n(pϕ) is  the current robot’s position with respect to the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  In the case of an ellipse, the tangential and normal vectors  are given by:  ⃗t(pϕ) =  �cos(pϕ)  sin(pϕ)  �  ,  ⃗n(pϕ) =  �− sin(pϕ)  cos(pϕ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Augmented Control Scheme  Due to the MPFC-QDNN being an approximation of the  MPFC, the path-following error e resulting from D(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Θ) is  significantly larger than the error observed under the original  MPFC controller M(z) (see Section V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' To compensate  this error we extend the MPFC-QDNN controller with an  additional linear feedback which acts on the tangential com-  ponent et through the forward speed of the robot s, and on  the normal component en through the robot’s angular speed  ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Put differently, the compensation term wP is added to the  control vector, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=', w = wD + wP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Here wP = [sP ωP 0]T,  with sP = Ptet, and ωP = Pnen, where Pt, and Pn are  the proportional gains (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' We denote this approach  MPFC-QDNN-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' We selected static feedback mainly due to  its simplicity and effectiveness as shown in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Implementation  We consider an ellipse as the path (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' 1), which is  defined by the parametrization  p(θ) =  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='1 cos(θ)  2 sin(θ)  arctan  �  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='1 sin(θ)  2 cos(θ)  ��T  ,  which yields the input references (3) and (4) as  sr(θ, v) = 2v  �  1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='9975 sin2(θ),  ωr(θ, v) = 2v  �  40 − 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='9 sin2(θ)  �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  To generate the training set T , we use a corridor consisting  of a cuboid of width 2cW = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='02, length 2cL = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='2, and  height 2cH =  2π  3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Each axis is split into 5, 5, and 40  equidistant points (Nc = 1000), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' We split the  path in Np = 4000 equidistant segments between 0 ≤ θ ≤  2π, which corresponds to a full turn around the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The  subset of states in the corridor ZC consists of NpNc = 4E6  points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  To solve the MPFC problem (5), and consequently find  w = M(z), ∀ z ∈ ZC according to (6), we use the  Optimization Engine (OpEn) [5], a fast solver for optimal  control problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The training set T consists of NpNc pair  of vectors z, M(z) for all z ∈ ZC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' We use a discretization  time δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='01 s, and a horizon length T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='6 s in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  We use a random search approach to find the hyper-  parameters of a network that is a good approximation to  M under the constraint that the number of parameters NΘ  should remain small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=', to reduce the size of the network  in the MCU’s ROM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Random search typically delivers better  results than manual or grid search for the same amount  of computation during training [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The selected hyper-  parameters were the number of hidden layers H, the number  of units in each hidden layer nk, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' , H, and the learn-  ing rate of the optimization algorithm (see the Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  To find the hyper-parameters we use KerasTuner [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' To  perform the training and the quantization of the network  we use the deep learning framework Keras/TensorFlow [18],  [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  We implement a Hardware-In-the-Loop (HIL) simulation  where the MPFC-QDNN+P is deployed on an STM32F407  MCU, which is based on a Cortex-M4 processor core running  at 168 MHz, which includes a single-precision floating-point  unit and 1 MB flash ROM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The robot dynamics are simulated  on a PC, see the Appendix for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  The QDNN consists of 9 hidden layers with roughly 4700  parameters, using 8-bit integers to store the parameters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' 1  parameter requires 1 byte of ROM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The quantized parameter  set Θ requires less than 5 kB of the MCU’s flash memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' RESULTS  As our reference implementation (denoted MPFC-OpEn),  we use OpEn (the same solver used for training) to solve the  MPFC problem (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The advantages of MPFC are illustrated  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The input w computed by OpEn to steer the robot  along the path in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' 6 shows that when the path curvature is  tight, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' top (θ = π  2 ) and bottom (θ = 3π  2 ) of the ellipse, the  path speed v is reduced, and consequently the robot’s forward  speed s is also reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' This allows the robot to follow the  tight curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Similarly, v is reduced when the constraints on  s are active (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' θ = π) because the robot cannot otherwise  closely follow the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  The main advantage of using a DNN on an MCU is that  it is relatively easy to implement quantization (8), inference  (7), and dequantization (9) sequentially for all layers in the  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Furthermore, for a small network like the one used  here (4700 parameters), the inference is executed much faster  than solving the OCP (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  Table I shows the execution time for different implemen-  tations of MPFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Our experiments ran on a PC with Ubuntu  MPFC-QDNN  z  N  Q  QDNN  Q−1  N −1  wD  ¯z  ˆz  ˆw  ¯w  (a) The quantized deep neural network inference chain used to  approximate the MPFC (denoted MPFC-QDNN on the right figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  Robot  �  MPFC-  QDNN  +  E  P  w  s  ω  v  ξ  θ  z  e  wP  wD  MPFC-QDNN+P  (b) The proposed controller approach (MPFC-QDNN+P)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' 4: Block diagram of the proposed combined controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Inside the dashed block (MPFC-QDNN) is the MPFC controller  approximated by a QDNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The extended state z is fed into the MPFC-QDNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The state is first normalized (¯z) then  quantized (ˆz), and finally fed into the QDNN block for inference of the approximate control input ˆw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Dequantization and  denormalization are applied to get the approximated augmented control input wD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The block E computes the robot’s path  error vector e (10) used by the P controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The vector wP is added to wD to compute the input w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The forward velocity  s, and angular velocity ω are applied to the robot, whereas the path velocity v is integrated to compute the path variable θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  0  π  2  π  3π  2  2π  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='5  θ  s [m/s]  ω [rad/s]  v [s−1]  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' 5: Control inputs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' path parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' When the path  curvature is tight (near π  2 , and 3π  2 ) the path speed v, and  the robot’s forward speed s are reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The path speed v is  also reduced when the input constraints are active (near π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  Linux 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='04-LTS, and a x86-64 processor with a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='4 GHz  clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Compared to MPFC-OpEn, the average execution time  of the MPFC-QDNN+P implementation is about three orders  of magnitude faster on the PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  The QDNN implementation using 8-bit integers requires  on average 230 microseconds to execute on the MCU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  Currently, running OpEn on an MCU is not supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' 6 shows a comparison of the path in the Cartesian  XY plane followed by the simulated robot using different  implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The absolute Cartesian position error is  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' 7, with a summary presented in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  All implementations can follow the path, with OpEn being  the most accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' When the worst-case error is considered,  using a regular (non-quantized) DNN is two orders of  magnitude worse than the OpEn implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The QDNN  Implementation  Mean [s]  Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' [s]  Worst [s]  OpEn (PC)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='2E-3  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='9E-4  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='8E-3  QDNN+P (PC)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='3E-6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='9E-6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='2E-5  OpEn (MCU) QDNN+P (MCU)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='3E-4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='1E-6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='4E-4  TABLE I: Mean, standard deviation (Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' ), worst-case ex-  ecution (Worst) time in seconds for 10000 steps close to  the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The table is split into MPFC implementations on  a personal computer (PC, top part) and a microcontroller  (MCU, bottom part).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Note that the execution of QDNN+P  on the MCU is temporally deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' In the PC case, the  QDNN+P has more variability due to the operating system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  Mean  Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  OpEn  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='9E-4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='3E-4  DNN  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='5E-3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='1E-2  QDNN  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='6E-2  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='8E-2  QDNN+P  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='1E-4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='9E-3  TABLE II: Mean and maximum Cartesian error in the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  implementation has worse overall performance than the non-  quantized network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Finally, the proposed addition of two P  controllers to the QDNN reduces its worst-case error by an  order of magnitude and outperforms the DNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' CONCLUSIONS  The paper presented a model predictive path following  implementation using quantized deep neural networks aug-  mented with a controller for quantization error compensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  We showed a practical way to select the training set, and  how to design the error compensation controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Compared  to a traditional MPFC using online optimization, our pro-  posed approach requires only a fraction of the memory and  runs several orders of magnitude faster on PC simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  Although the path-following accuracy is slightly degraded,  we believe the performance may still be good for low-cost  applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' With a hardware-in-the-loop implementation  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='05  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='1  −2  −1  0  1  2  θ = 0  θ = π  2  θ = π  θ = 3π  2  X [m]  Y [m]  OpEn  DNN  QDNN  QDNN+P  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' 6: The path followed by the robot using different  MPFC implementations: OpEn (visually indistinguishable  from the reference path p(θ)), the approximation using a  deep neural network (DNN, 32-bit float), a quantized deep  neural network (QDNN, 8-bit integer), and the proposed  QDNN plus P control approach (QDNN+P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  using a microcontroller, we showed the effectiveness of this  approach for low-cost embedded devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Future work will  discuss how to handle different path geometries with one  trained QDNN and how to give performance guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  APPENDIX  The hyperparameters of the QDNN network are the learn-  ing rate = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='5E − 4, the activation function (ReLU), the  number of hidden layers H = 9, and the units on each layer:  input layer 4 units, followed by the hidden layers with 48,  16, 24, 16, 16, 40, 24, 16, and 24 units, and output layer 3  units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  The parameters of OCP (5) are the matrices Q  =  diag(2E5, 2E5, 1E5, 0), R = diag(1E1, 5E3, 1E5), the  box sets Z = {z ∈ R4 | z ≤ z ≤ z}, with z =  [−5, −15, − π  2 , −5], z = [5, 15, π  2 , −5], and W = {w ∈  R3 | w ≤ w ≤ w}, with w = [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='26, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='455, 0], w =  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='26, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='455, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='15], the discretization time δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='01s, and the  horizon length steps N = 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
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+page_content=' Faulwasser and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Findeisen, “Nonlinear model predictive control  for constrained output path following,” IEEE Transactions on Auto-  matic Control, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' 61, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' 1026–1039, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  0  π  2  π  3π  2  2π  0  1  2  3  10−4  OpEn  0  π  2  π  3π  2  2π  0  1  2  10−2  DNN  0  π  2  π  3π  2  2π  0  2  4  10−2  QDNN  0  π  2  π  3π  2  2π  0  2  4  10−3  θ  QDNN+P  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' 7: Cartesian position error with respect to the reference  path for different implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The MPFC solution as  computed by OpEn is the most accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' The next most  accurate is the proposed approach (QDNN+P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  [5] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Sopasakis, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Fresk, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Patrinos, “OpEn: Code generation for  embedded nonconvex optimization,” in IFAC World Congress, Berlin,  Germany, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  [6] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Verschueren, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Frison, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Kouzoupis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Frey, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' van Duijkeren,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Zanelli, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Novoselnik, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Albin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Quirynen, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Diehl,  “acados – a modular open-source framework for fast embedded  optimal control,” Mathematical Programming Computation, Oct 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='  [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content=' Available: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFKT4oBgHgl3EQfyy7U/content/2301.11909v1.pdf'}
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diff --git a/rNFAT4oBgHgl3EQffh2A/content/tmp_files/2301.08582v1.pdf.txt b/rNFAT4oBgHgl3EQffh2A/content/tmp_files/2301.08582v1.pdf.txt
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+Quantum Hall effect and Landau levels without spatial long-range correlations
+Isac Sahlberg, Moein N. Ivaki, Kim P¨oyh¨onen and Teemu Ojanen
+Computational Physics Laboratory, Physics Unit, Faculty of Engineering and Natural Sciences,
+Tampere University, P.O. Box 692, FI-33014 Tampere, Finland and
+Helsinki Institute of Physics P.O. Box 64, FI-00014, Helsinki, Finland
+The spectrum of charged particles in translation-invariant systems in a magnetic field is charac-
+terized by the Landau levels, which play a fundamental role in the thermodynamic and transport
+properties of solids. The topological nature and the approximate degeneracy of the Landau lev-
+els are known to also survive on crystalline lattices with discrete translation symmetry when the
+magnetic flux through a primitive cell is small compared to the flux quantum. Here we show that
+the notion of Landau levels and the quantum Hall effect can be generalized to 2d non-crystalline
+lattices without spatial long-range order.
+Remarkably, even when the spatial correlations decay
+over microscopic distances, 2d systems can exhibit a number of well-resolved Landau-like bands.
+The existence of these bands imply that non-crystalline systems in magnetic fields can support the
+hallmark quantum effects which have been typically associated with crystalline solids.
+Introduction– There are few notions more fundamental
+and important to condensed-matter physics than Landau
+levels [1]. A spectrum of charged particles in a magnetic
+field is squeezed into a sequence of flat bands En that
+can support extensive degeneracy, giving rise to a rich
+variety of phenomena in solids [2, 3]. The classical ex-
+amples include quantum oscillations in thermodynamic
+and transport quantities, such as the de Haas-van Alpen
+and the Shubnikov-de Haas effects, while the discovery
+of the quantum Hall effect ushered in the modern age of
+topological matter [4–6]. In gapless systems, the Landau
+levels are characterized by their linear dependence on the
+magnetic field En ∝ B, and their behavior as a function
+of the level index n which depends on the specific dis-
+persion of the particles. The two most studied cases are
+the parabolic dispersion and the Dirac-type dispersion
+which correspond to En ∝ n and En ∝ √n. The dif-
+ferent n-dependence makes it explicit that the Landau
+level spectrum depends on the underlying band struc-
+ture. This naturally raises the question of what happens
+to the Landau levels and the related phenomena in non-
+crystalline systems, where the notion of a band structure
+does not apply. More generally, how are the magnetic-
+field-related phenomena modified in non-crystalline sys-
+tems? Although the effects of disorder on Landau levels
+have been studied in great detail [7–10], these studies do
+not apply to systems where a band structure cannot be
+assumed as a starting point for the analysis.
+In this work we study the fate of the Landau levels and
+quantum Hall effect on non-crystalline 2d lattices with-
+out long-range spatial order. On crystalline lattices, the
+Landau levels emerge in the low-field regime where the
+magnetic flux through a primitive cell is small compared
+to a flux quantum φ0 = h/e [11]. These states, as their
+continuum counterparts, disperse linearly as a function
+of the magnetic field, are (nearly) flat and carry a unit
+Chern number [6]. We introduce a family of random lat-
+tices (depicted in the insets in Figs. 1 a-c which can be
+tuned continuously from a crystalline lattice, through a
+FIG. 1. a-c: Averaged density distribution function g(r) =
+�
+i δ(r−ri) of lattice sites along a horizontal cut in the studied
+lattices for three different values of the structural disorder
+parameter ε. The insets show an example of a small lattice
+configuration, and the red points indicate the direction for
+which the distribution was obtained. In the crystalline case
+ε = 0, the distances are all integer multiples of the lattice
+constant, and the distribution remains uniform for all r. For
+non-zero ε, the long-range correlations decay after a length
+scale ξ, indicated by the black arrow for the ε = 0.2 case.
+d-f: Density of states ρ(E) for magnetic flux φ = 1/10 per
+particle and L = 300, for the same ε as in the left panel.
+arXiv:2301.08582v1  [cond-mat.mes-hall]  20 Jan 2023
+
+d
+a
+m=0
+b
+e
+=0.1
+6
+= 0.2
+b
+10
+15
+¥20　2530　35
+-3
+-2
+0
+5
+0
+E
+r2
+FIG. 2.
+a:
+Generation of lattices with tunable short- to
+medium-range spatial correlations. b: Example of a lattice
+with ε = 0.2 and linear size L = 100.
+The circles on the
+boundary layer represent the local DOS for flux φ = 1/10
+and energy E = −1.9, revealing the topological edge modes
+for the Chern number C = 2 state. Size and colors of circles
+represent the magnitude of local DOS at each site.
+paracrystalline state [12, 13], to an amorphous phase [14].
+While these lattices do not exhibit spatial long-range or-
+der, the density distribution function of a paracrystal ex-
+hibits residual periodic medium-range correlations [15–
+17].
+We establish that even when the scale of spatial
+correlations approaches the order of the average lattice
+constant i) the spectrum can still support well-resolved
+Landau-like bands shown in Figs. 1 d-f; ii) these bands
+disperse linearly in the magnetic field at low flux and
+support unit Chern numbers; iii) the system supports
+a quantized conductivity and topological edge modes.
+These facts suggest that also other phenomena arising
+from the Landau levels, typically associated with band
+structures, can survive even on random lattices without
+long-range spatial order.
+Model– The studied non-crystalline lattice geometries
+are characterized by short range square-lattice correla-
+tions, which decay over a characteristic correlation length
+ξ. The correlation length is controlled by the parameter
+ε, which characterizes a family of random lattices with
+the same statistical properties. In Figs. 1 a-c we have
+plotted representatives of random lattices with different
+ε. The case ε = 0 corresponds to a pristine square lattice
+with infinite correlation length, while ε = 0.1 and ε = 0.2
+lead to random lattices with ξ/a0 ∼ 30 and ξ/a0 ∼ 10,
+respectively, in the horizontal direction as indicated in
+Fig. 1. Setting the lattice constant a0 to 1, the generation
+of the lattices and the role of ε is illustrated in Fig. 2 a.
+Starting from the origin point A, the adjacent site B is
+obtained by first translating A by ˆex, drawing an ε-circle
+(blue) about that point, and randomly drawing a point
+B inside that circle, from a uniform distribution. The
+upper neighbour C is obtained analogously but translat-
+ing A by vector ˆey instead. The diagonal neighbour D
+is obtained by translating C by ˆex and B by ˆey, forming
+an ε-circle (purple) centered at the midpoint of the two
+translated points, and randomly placing D inside the cir-
+cle. Repeating this process, one can generate arbitrarily
+large random lattices, one realization of which is illus-
+trated in Fig. 2 b. The lattices generated by this prescrip-
+tion have the property that the local environment around
+any chosen interior point appears a perturbed square lat-
+tice. However, the lattice points more than distance ξ
+away no longer appear at their square lattice positions
+with respect to a reference point, making it impossible
+to uniquely identify them with the square lattice sites.
+Thus, despite the short-range correlations, all geometries
+with ε > 0 have no long-range correlation and cannot
+be regarded as perturbed crystalline lattices.
+Lattices
+with finite-range crystalline density correlations that are
+washed away for long distances, as seen in Fig. 1, are re-
+ferred to as paracrystals [12, 13]. The studied lattices are
+homogeneous with statistically indistinguishable interior
+points; however, they are not statistically isotropic, as
+indicated by the large-scale diagonal patterns, visible in
+Fig. 2 b, which breaks the four-fold rotation symmetry.
+We now define a tight-binding model which describes
+hopping in a perpendicular magnetic field, generalizing
+the much-studied Hofstadter model [11] to the random
+lattices discussed above.
+We would like the coordina-
+tion number to remain fixed for all studied geometries.
+Indeed, for sufficiently small ε, the four closest points
+of any interior point are the four nearest square lattice
+neighbours with perturbed distances. Thus, we can de-
+fine the Hamiltonian as
+H = t
+�
+<ij>
+e−iΦij ˆc†
+i ˆcj,
+(1)
+where t is the hopping parameter, ˆc†
+i (ˆci) is the cre-
+ation (annihilation) operator at site i, and the hop-
+pings connect each site to its four closest points, as the
+sum is over all neighboring pairs.
+The Peierls phases
+Φij = e/ℏ
+� j
+i A·dl accommodate the magnetic field which
+arises from the vector potential A. The magnetic field
+is assumed to be spatially homogeneous, so the vector
+potential can be written as A(r) =
+B
+2 (xˆey − yˆex) in
+the symmetric gauge and A(r) = Bxˆey in the Landau
+gauge.
+We have employed different gauges to confirm
+that the results for various observables presented below
+are gauge-invariant.
+For the studied random lattices,
+the average distance between the closest lattice points
+is fixed by a square lattice constant a0. Also, the av-
+erage magnetic flux per site remains fixed to its square
+lattice value φ = Ba2
+0/φ0 for lattices corresponding to
+different ε, which makes it an appropriate quantity to
+parametrize the properties of the system. One could also
+introduce a variable hopping strength; however, due to
+small variations in bond lengths, we elect to keep the
+number of free parameters as low as possible and focus
+on the crucial magnetic-field-dependent modifications of
+complex phases.
+
+b3
+FIG. 3. a-c: Phase diagram for a 70 × 70 system as a function of the structural disorder parameter ε, obtained from the
+averaged local Chern marker. For b and c, C(r) is evaluated at every parameter pair for 200 different lattice configurations, and
+100 randomly chosen points in the bulk of each lattice configuration. Colors are assigned for quantized C(r), meaning values
+within 3% of the indicated integers; higher Chern numbers are omitted, as they are barely discernable. d-f: Density of states
+ρ(E) for L = 500 along the vertical cut φ = 1/20 indicated in white in the panel above.
+Landau levels and Quantum Hall effect– We are now
+ready to explore the physical properties of the model.
+We concentrate on the low flux regime φ ≪ 1, which
+is relevant for real solids and makes connection to the
+Landau levels in the continuum.
+We start by evaluating the topological phase diagram,
+determined by the Chern numbers, and the density of
+states (DOS) defined as
+ρ(E) = L−2 �
+n
+δ(E − En).
+(2)
+Additionally, an efficient method to extract the phase
+diagram in random systems is to evaluate the local Chern
+marker [18–20] and average it over many lattice sites, and
+over different random configurations to obtain the Chern
+number. The local Chern marker is defined by
+C(r) = −2πi
+�
+dr′ [X(r, r′)Y(r′, r) − Y(r, r′)X(r′, r)] ,
+(3)
+where X(r, r′) =
+�
+dr′′P(r, r′′)x′′P(r′′, r′) is the pro-
+jected x coordinate, and Y(r, r′) is similarly defined pro-
+jected y coordinate.
+Here P(r, r′) denotes the matrix
+elements of the projection operator to the filled energy
+bands. In practice, to avoid diagonalization of the full
+system, we calculate the DOS and the Chern marker by
+employing the Kernel Polynomial Method (KPM) [21].
+While the calculation of the DOS using the KPM is well-
+documented [21], the derivation of the kernel polynomial
+formula for the Chern marker and the details of the con-
+figuration averaging are presented in the SI [22].
+In Figs. 3 a-c we have illustrated how the topological
+phase diagram evolves as the spatial correlation length is
+decreased. The corresponding DOS for each case is pre-
+sented in Figs. 3 d-f. The DOS and the phase diagram
+are symmetric with respect to E = 0, so we plot them
+only for negative energies. In the crystalline limit ε = 0,
+we recover the almost flat Landau bands, which disperse
+linearly in the low flux regime as expected. These states
+also carry a unit Chern number, as indicated by the unit
+jumps in the Chern marker at energies corresponding to
+the Landau bands.
+For finite ε, the DOS correspond-
+ing to individual Landau bands becomes significantly
+broadened, however, the main features of the phase di-
+agram survive even in the presence of strong random-
+ness ε = 0.2. The relevant length scales for the problem
+are the average lattice constant a0, the magnetic length
+LB = 1/√2πφ (the characteristic length of the Landau
+orbits), and the spatial correlation length ξ of the lat-
+tice. For the parameters corresponding to Figs. 3 c and f,
+they satisfy a0 ≲ LB ∼ 0.1ξ, indicating that the Landau
+level picture and the quantum Hall effect persists even
+when the correlation length approaches the microscopic
+length scales of the problem. The broadening caused by
+the random lattice geometry on the Landau bands ap-
+pears qualitatively similar to that caused by disorder on
+crystalline lattices and in continuum [7]. However, this
+
+b
+a
+c
+2
+6
+5
+4
+F-3
+3
+2
+.1
+=0
+ε=0.1
+=0.2
+-4
+0.05
+0.1
+0.05
+0.1
+0.05
+0.1
+f
+p
+e
+0
+-3
+-2
+-2
+-1
+0
+-1
+0
+-1
+E4
+FIG. 4. Averaged local Chern marker C(r) as function of energy for different ε for fluxes φ = 1/10 (a) and φ = 1/20 (b). In a
+we plot the Hall conductivity σxy evaluated at selected energies and averaged over random lattice positions deep in the bulk for
+the ε = 0.2 case. The inset in b illustrates the absolute deviation in the averaged local Chern marker values |∆C(r)| between
+ε = 0.2 and the crystalline case. Here the linear size of the system is L = 200, and the local Chern values are averaged over
+n = 20 random configurations and 150 random lattice positions for each configuration. Error bars are not visible in this scale.
+observation is far from self-evident, since disorder can
+be regarded as a mere perturbation to the underlying
+structure which always retains an approximate transla-
+tion invariance. In contrast, the finite correlation length
+in non-crystalline lattices cannot be regarded as a pertur-
+bative effect and there exists no systematic approach in
+which a perfect crystal would serve as a starting point for
+the analysis. Furthermore, the studied model is funda-
+mentally different from those obtained by perturbing the
+sites of a crystalline system independently around their
+regular positions. Such lattices support arbitrary long-
+range spatial order with ξ = ∞, and the density distri-
+bution function associated with them resembles Fig. 1 a,
+except with broadened delta peaks.
+A non-zero Chern number gives rise to the quantum
+Hall effect, indicating the existence of a quantized Hall
+conductivity in the system. As shown in Fig. 4 a, this
+is confirmed by a direct evaluation of the Hall conduc-
+tivity using a method developed in Ref. [23] based on a
+real-space Kubo-Bastin formalism [24–26]. The evolution
+of the quantized plateaus as a function of ε, illustrated
+in Fig. 4 b, indicates that the overlapping higher Landau
+bands will be the first affected by randomness. When the
+Landau level broadening increases, the plateaus shrink,
+and when the broadening becomes comparable to the
+level separation they eventually disappear [27]. In ad-
+dition to the quantized Hall conductance, the non-trivial
+Chern numbers also give rise to topologically protected
+edge modes at energies that fall in the mobility gap be-
+tween the Landau levels [28]. These edge modes are re-
+vealed by the local DOS as seen in Fig. 2 b. The in-
+tensity of the local DOS varies randomly along the edge,
+reflecting the irregular geometry of the lattice, vanishing
+rapidly beyond the surface layer.
+Discussion– While the existence of amorphous topo-
+logical states has been established in various systems
+recently [29–41], the present work is the first study of
+the Landau levels as a function of spatial correlations.
+It should be emphasized that our findings of the non-
+crystalline Landau bands have significant implications
+beyond the quantum Hall effect. Since the birth of mod-
+ern solid state physics, it has been known that many
+thermodynamic and transport quantities are determined
+by the DOS at the Fermi level, resulting in quantum oscil-
+lations when the Landau levels are pushed through the
+Fermi level by varying the magnetic field [2, 3]. Qual-
+itatively similar oscillations persist in the studied sys-
+tems, as indicated by the Landau peaks in the DOS that
+also disperse as a function of the magnetic field.
+The
+Landau level formation and the resulting quantum os-
+cillations are typically considered as a consequence of a
+band structure, so it is remarkable that the quantum os-
+cillations may survive in systems where the notion of a
+band structure does not apply. Approximately flat bands
+which carry finite Chern numbers are also a prerequisite
+for more complex interaction-induced phases, suggesting
+that the lack of a long-range spatial order does not pose
+a fundamental obstacle for formation of fractional quan-
+tum Hall states [1, 2].
+While in this work the focus was on 2d systems, the
+phenomena resulting from the existence of Landau levels
+have a wide variety of consequences in 3d systems [3, 42].
+Our results indicate that, at least qualitatively, many
+hallmark features of 3d electrons in magnetic fields can be
+expected to survive in non-crystalline 3d systems without
+long-range order. However, a systematic study of these
+features in 3d quickly becomes computationally challeng-
+ing.
+Summary– In this work we studied the evolution of
+Landau levels and topology as a function of randomness
+in non-crystalline 2d lattices without long-range order.
+We discovered that, even when the spatial correlations
+
+Φ= 0.1
+b
+Φ = 0.05
+a
+4
+0=3
+ ε= 0.1
+3
+5
++-- = 0.2
+可ary(r),8 = 0.2
+E
+C
+3
+1
+0
+-2
+-2
+-3
+-3
+-1
+E
+E5
+decay over microscopic length scales, the Landau levels
+persist as broadened but well-defined features. Moreover,
+we showed that the magnetic field dependence and the
+topological nature of the Landau bands remain intact,
+giving rise to the quantum Hall effect with quantized con-
+ductance and topological edge modes. The fact that the
+lowest Landau bands can be resolved, despite the lack of
+long range order, imply that the fundamental quantum
+effects that have been traditionally considered as prop-
+erties of crystalline solids in the magnetic field, can also
+be observed in non-crystalline systems.
+Acknowledgement– This work was supported by the
+Academy of Finland through the project number 331094.
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+
+7
+SUPPLEMENTAL INFORMATION to “Quantum Hall effect and Landau levels
+without spatial long-range correlations”
+LOCAL CHERN MARKER FROM KERNEL POLYNOMIAL EXPANSION
+The local Chern marker can be defined by the equation
+C(r) = −2πi
+�
+dr′ [X(r, r′)Y(r′, r) − Y(r, r′)X(r′, r)] ≡ Cxy(r) − Cyx(r),
+(4)
+where
+X(r, r′) =
+�
+dr′′P(r, r′′)x′′P(r′′, r′),
+(5)
+and analogously for Y(r, r′), in terms of the projection operator to occupied bands
+P(r, r′) =
+� Emax
+Emin
+dEθ(EF − E)
+�
+n
+δ(E − En) ⟨r⟩ n ⟨n⟩ r′,
+(6)
+with n labeling the eigenstates of the Hamiltonian. In order to implement KPM we rescale energies to lie in the
+interval (−1, 1):
+H = (H − b)/a
+(7)
+a = (Emax − Emin)/(2 − η)
+(8)
+b = 1
+2(Emax + Emin).
+(9)
+Together with the substitution ϵ = (E − b)/a, we get
+P(r, r′) =
+� 1
+−1
+dϵθ(ϵF − ϵ)
+�
+n
+δ(ϵ − ϵn) ⟨r⟩ n ⟨n⟩ r′,
+(10)
+and hence, after performing the radial integrals,
+Cxy =
+�
+{−1,1}3 d3ϵ
+�
+nmk
+⟨r⟩ n ⟨n| ˆx |m⟩ ⟨m| ˆy |k⟩ ⟨k⟩ rδ(ϵ1 − ϵn)δ(ϵ2 − ϵm)δ(ϵ3 − ϵk)θ(ϵF − ϵ1)θ(ϵF − ϵ2)θ(ϵF − ϵ3). (11)
+Note that the integration here is over the numbered epsilons, while the ones labelled by letters are instead summed
+over. We now approximate this result by replacing the quantities
+f(ϵ1) =
+�
+n
+|n⟩ ⟨n| δ(ϵ1 − ϵn),
+(12)
+with their truncated Chebyshev expansions, i.e.
+f(ϵ1) ≈ fKPM(ϵ1) =
+N−1
+�
+n=0
+2µnhngnTn(ϵ1)
+π(1 + δn,0)
+�
+1 − ϵ2
+1
+.
+(13)
+Here µn are the moments
+µn =
+� 1
+−1
+dϵf(ϵ1)Tn(ϵ1) = Tn(H),
+(14)
+and gn represent the kernel terms improving the accuracy of the truncated sum. Inserting this into the expression for
+Cxy, we get
+Cxy =
+�
+{−1,1}3 d3ϵ
+�
+nmk
+⟨r| fKPM(ϵ1)ˆxfKPM(ϵ2)ˆyfKPM(ϵ3) |r⟩ θ(ϵF − ϵ1)θ(ϵF − ϵ2)θ(ϵF − ϵ3)
+= ⟨r| T (H)ˆxT (H)ˆyT (H) |r⟩ ,
+(15)
+
+8
+where
+T (H) =
+�
+n
+2gnTn(H)
+(1 + δn,0)
+� 1
+−1
+dϵTn(ϵ)θ(ϵF − ϵ)
+π
+√
+1 − ϵ2
+.
+(16)
+Approximating the integral and subsequently making use of the discrete Fourier transform (see Ref. 21), we obtain
+T (H) ≈
+�
+n
+2gnTn(H)
+(1 + δn,0)
+1
+M
+M−1
+�
+s=0
+cos
+�nπ(s + 1
+2)
+M
+�
+θ
+�
+ϵF − cos
+�π(s + 1
+2)
+M
+��
+=
+�
+n
+2gnTn(H)
+M(1 + δn,0)Tn(H)S(n),
+(17)
+where we assume M > N.
+We note that, as a function of s, the argument inside the step function increases monotonically; as |ϵF | < 1,
+the overall effect of the step function is hence to exclude terms from the beginning of the sum.
+Let us denote
+Γ ≡ ⌈ M
+π arccos ϵF − 1
+2⌉. Then,
+S(n) =
+M−1
+�
+s=Γ
+cos
+�nπ(s + 1
+2)
+M
+�
+= Re
+�M−1
+�
+s=Γ
+e
+nπ(s+ 1
+2 )
+M
+�
+= Re
+�
+�e
+inπ
+2M e
+inπΓ
+M
+M−Γ−1
+�
+j=0
+e
+nπj
+M
+�
+� .
+(18)
+This yields
+�
+S(n) = M − Γ,
+n = 0
+S(n) = − 1
+2
+sin nπΓ
+M
+sin nπ
+2M ,
+n > 0.
+(19)
+Absorbing a minus sign into T , we finally get
+C(r) = 4πIm ⟨r| T (H)ˆxT (H)ˆyT (H) |r⟩
+(20)
+where, using T0(x) = 1 and g0 = 1,
+T (H) ≈ −1 + Γ
+M +
+N−1
+�
+n=1
+gn
+M
+sin nπΓ
+M
+sin nπ
+2M
+Tn(H).
+(21)
+CALCULATION DETAILS
+For quantities such as the Chern marker and conductivity tensor which are evaluated locally, only a finite number
+of lattice points in the bulk are sufficient to fully describe topological features of the system. The local markers
+vary significantly throughout the lattices with open boundaries, but can form small regions where the changes in the
+values are almost continuous. This means that choosing to average many points next to each other may be a biased
+sample of the distribution, depending on the configuration. Therefore, rather than e.g. choosing a large region near
+the center, we randomly choose points within the bulk. This approach has the benefit of sampling the entirety of the
+lattice, excluding only a finite border close to the edges. We typically choose N ∼ 100 points randomly from the bulk
+where we evaluate the local marker, yielding an appropriate average for the quantity at hand. In other words, it is an
+unbiased sampling of the full distribution of, say, local Chern marker values C(r), and thus we can obtain a proper
+average value with less computational cost, especially when we scale up the system size. We repeat this sampling
+for a number of configurations, and the distributions converge together remarkably quickly, as shown in Fig. 5. This
+observation provides a powerful and efficient tool to study larger systems. The number of random samples needed
+to describe the correct distributions and the corresponding mean values remains roughly constant, provided N is not
+too small.
+Moreover, for an accurate description within KPM, in our calculations we set the number of moments Nm used in
+the expansion to Nm ≈ L2/3 for systems which host L2 electronic states. Additionally, in studying DOS, the accuracy
+in calculating stochastic traces depends on the product Nr Nc, where Nr is the number of random phase vectors and
+Nc is the number of independent random lattice configurations. In our numerical simulations NrNc ∼ 103. We also
+studied the averaged Hall conductivity σxy(r, E) of the bulk, similarly to the calculation of Chern markers by random
+sampling of lattice points away from the boundaries, and found that it matches well with the predicted Chern number.
+This hints toward the equivalence of the local bulk conductivity, which is a physical observable, and the local Chern
+marker.
+
+9
+FIG. 5. a: Sampling of the local Chern marker for all bulk points in a given configuration. b: Local Chern marker shown for
+a subset of N = 100 points. The parameters used are φ = 0.1 and E = −3 with the structural disorder parameter ε = 0.2. c:
+The normalized full distribution of local marker values, shown in white, as well as the partial distribution of random points in
+light red, for the single configuration used in a and b. d: Same as c, but for a set of 10 randomly generated configurations.
+The distributions match very well and yield the same mean value for the Chern number with a high accuracy.
+
+b
+a
+40 
+40.
+1.1
+1.1
+30
+30
+1.0
+r
+1.0
+r
+20 
+20
+C
+C
+10 
+10
+0.9
+0.9
+0
+0
+0
+10
+20
+30
+40
+0
+10
+20
+30
+40
+p
+C
+Cbulk,i
+= Cbulk,i(r)
+= 0.994
+= 0.974
+= 0.993
+Cpoints,i = Cpoints,i(r) = 0.979
+Cave,points
+0.8
+0.9
+1.0
+1.1
+1.2
+0.8
+0.9
+1.0
+1.1
+1.2
\ No newline at end of file
diff --git a/rNFAT4oBgHgl3EQffh2A/content/tmp_files/load_file.txt b/rNFAT4oBgHgl3EQffh2A/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..98ea8ec9cf2363a7fdc488cddff356a6af9450b5
--- /dev/null
+++ b/rNFAT4oBgHgl3EQffh2A/content/tmp_files/load_file.txt
@@ -0,0 +1,399 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf,len=398
+page_content='Quantum Hall effect and Landau levels without spatial long-range correlations  Isac Sahlberg, Moein N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' Ivaki, Kim P¨oyh¨onen and Teemu Ojanen  Computational Physics Laboratory, Physics Unit, Faculty of Engineering and Natural Sciences,  Tampere University, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' Box 692, FI-33014 Tampere, Finland and  Helsinki Institute of Physics P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' Box 64, FI-00014, Helsinki, Finland  The spectrum of charged particles in translation-invariant systems in a magnetic field is charac-  terized by the Landau levels, which play a fundamental role in the thermodynamic and transport  properties of solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' The topological nature and the approximate degeneracy of the Landau lev-  els are known to also survive on crystalline lattices with discrete translation symmetry when the  magnetic flux through a primitive cell is small compared to the flux quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' Here we show that  the notion of Landau levels and the quantum Hall effect can be generalized to 2d non-crystalline  lattices without spatial long-range order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  Remarkably, even when the spatial correlations decay  over microscopic distances, 2d systems can exhibit a number of well-resolved Landau-like bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  The existence of these bands imply that non-crystalline systems in magnetic fields can support the  hallmark quantum effects which have been typically associated with crystalline solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  Introduction– There are few notions more fundamental  and important to condensed-matter physics than Landau  levels [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' A spectrum of charged particles in a magnetic  field is squeezed into a sequence of flat bands En that  can support extensive degeneracy, giving rise to a rich  variety of phenomena in solids [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' The classical ex-  amples include quantum oscillations in thermodynamic  and transport quantities, such as the de Haas-van Alpen  and the Shubnikov-de Haas effects, while the discovery  of the quantum Hall effect ushered in the modern age of  topological matter [4–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' In gapless systems, the Landau  levels are characterized by their linear dependence on the  magnetic field En ∝ B, and their behavior as a function  of the level index n which depends on the specific dis-  persion of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' The two most studied cases are  the parabolic dispersion and the Dirac-type dispersion  which correspond to En ∝ n and En ∝ √n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' The dif-  ferent n-dependence makes it explicit that the Landau  level spectrum depends on the underlying band struc-  ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' This naturally raises the question of what happens  to the Landau levels and the related phenomena in non-  crystalline systems, where the notion of a band structure  does not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' More generally, how are the magnetic-  field-related phenomena modified in non-crystalline sys-  tems?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' Although the effects of disorder on Landau levels  have been studied in great detail [7–10], these studies do  not apply to systems where a band structure cannot be  assumed as a starting point for the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  In this work we study the fate of the Landau levels and  quantum Hall effect on non-crystalline 2d lattices with-  out long-range spatial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' On crystalline lattices, the  Landau levels emerge in the low-field regime where the  magnetic flux through a primitive cell is small compared  to a flux quantum φ0 = h/e [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' These states, as their  continuum counterparts, disperse linearly as a function  of the magnetic field, are (nearly) flat and carry a unit  Chern number [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' We introduce a family of random lat-  tices (depicted in the insets in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' 1 a-c which can be  tuned continuously from a crystalline lattice, through a  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' a-c: Averaged density distribution function g(r) =  �  i δ(r−ri) of lattice sites along a horizontal cut in the studied  lattices for three different values of the structural disorder  parameter ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' The insets show an example of a small lattice  configuration, and the red points indicate the direction for  which the distribution was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' In the crystalline case  ε = 0, the distances are all integer multiples of the lattice  constant, and the distribution remains uniform for all r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' For  non-zero ε, the long-range correlations decay after a length  scale ξ, indicated by the black arrow for the ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='2 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  d-f: Density of states ρ(E) for magnetic flux φ = 1/10 per  particle and L = 300, for the same ε as in the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='08582v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='mes-hall]  20 Jan 2023  d  a  m=0  b  e  =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='1  6  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='2  b  10  15  ¥20 2530 35  3  2  0  5  0  E  r2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  a:  Generation of lattices with tunable short- to  medium-range spatial correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' b: Example of a lattice  with ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='2 and linear size L = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  The circles on the  boundary layer represent the local DOS for flux φ = 1/10  and energy E = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='9, revealing the topological edge modes  for the Chern number C = 2 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' Size and colors of circles  represent the magnitude of local DOS at each site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  paracrystalline state [12, 13], to an amorphous phase [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  While these lattices do not exhibit spatial long-range or-  der, the density distribution function of a paracrystal ex-  hibits residual periodic medium-range correlations [15–  17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  We establish that even when the scale of spatial  correlations approaches the order of the average lattice  constant i) the spectrum can still support well-resolved  Landau-like bands shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' 1 d-f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' ii) these bands  disperse linearly in the magnetic field at low flux and  support unit Chern numbers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' iii) the system supports  a quantized conductivity and topological edge modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  These facts suggest that also other phenomena arising  from the Landau levels, typically associated with band  structures, can survive even on random lattices without  long-range spatial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  Model– The studied non-crystalline lattice geometries  are characterized by short range square-lattice correla-  tions, which decay over a characteristic correlation length  ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' The correlation length is controlled by the parameter  ε, which characterizes a family of random lattices with  the same statistical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' 1 a-c we have  plotted representatives of random lattices with different  ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' The case ε = 0 corresponds to a pristine square lattice  with infinite correlation length, while ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='1 and ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='2  lead to random lattices with ξ/a0 ∼ 30 and ξ/a0 ∼ 10,  respectively, in the horizontal direction as indicated in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' Setting the lattice constant a0 to 1, the generation  of the lattices and the role of ε is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' 2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  Starting from the origin point A, the adjacent site B is  obtained by first translating A by ˆex, drawing an ε-circle  (blue) about that point, and randomly drawing a point  B inside that circle, from a uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' The  upper neighbour C is obtained analogously but translat-  ing A by vector ˆey instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' The diagonal neighbour D  is obtained by translating C by ˆex and B by ˆey, forming  an ε-circle (purple) centered at the midpoint of the two  translated points, and randomly placing D inside the cir-  cle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' Repeating this process, one can generate arbitrarily  large random lattices, one realization of which is illus-  trated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' 2 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' The lattices generated by this prescrip-  tion have the property that the local environment around  any chosen interior point appears a perturbed square lat-  tice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' However, the lattice points more than distance ξ  away no longer appear at their square lattice positions  with respect to a reference point, making it impossible  to uniquely identify them with the square lattice sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  Thus, despite the short-range correlations, all geometries  with ε > 0 have no long-range correlation and cannot  be regarded as perturbed crystalline lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  Lattices  with finite-range crystalline density correlations that are  washed away for long distances, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' 1, are re-  ferred to as paracrystals [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' The studied lattices are  homogeneous with statistically indistinguishable interior  points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' however, they are not statistically isotropic, as  indicated by the large-scale diagonal patterns, visible in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' 2 b, which breaks the four-fold rotation symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  We now define a tight-binding model which describes  hopping in a perpendicular magnetic field, generalizing  the much-studied Hofstadter model [11] to the random  lattices discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  We would like the coordina-  tion number to remain fixed for all studied geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  Indeed, for sufficiently small ε, the four closest points  of any interior point are the four nearest square lattice  neighbours with perturbed distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' Thus, we can de-  fine the Hamiltonian as  H = t  �  <ij>  e−iΦij ˆc†  i ˆcj,  (1)  where t is the hopping parameter, ˆc†  i (ˆci) is the cre-  ation (annihilation) operator at site i, and the hop-  pings connect each site to its four closest points, as the  sum is over all neighboring pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  The Peierls phases  Φij = e/ℏ  � j  i A·dl accommodate the magnetic field which  arises from the vector potential A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' The magnetic field  is assumed to be spatially homogeneous, so the vector  potential can be written as A(r) =  B  2 (xˆey − yˆex) in  the symmetric gauge and A(r) = Bxˆey in the Landau  gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  We have employed different gauges to confirm  that the results for various observables presented below  are gauge-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  For the studied random lattices,  the average distance between the closest lattice points  is fixed by a square lattice constant a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' Also, the av-  erage magnetic flux per site remains fixed to its square  lattice value φ = Ba2  0/φ0 for lattices corresponding to  different ε, which makes it an appropriate quantity to  parametrize the properties of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' One could also  introduce a variable hopping strength;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' however, due to  small variations in bond lengths, we elect to keep the  number of free parameters as low as possible and focus  on the crucial magnetic-field-dependent modifications of  complex phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  b3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' a-c: Phase diagram for a 70 × 70 system as a function of the structural disorder parameter ε, obtained from the  averaged local Chern marker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' For b and c, C(r) is evaluated at every parameter pair for 200 different lattice configurations, and  100 randomly chosen points in the bulk of each lattice configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' Colors are assigned for quantized C(r), meaning values  within 3% of the indicated integers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' higher Chern numbers are omitted, as they are barely discernable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' d-f: Density of states  ρ(E) for L = 500 along the vertical cut φ = 1/20 indicated in white in the panel above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  Landau levels and Quantum Hall effect– We are now  ready to explore the physical properties of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  We concentrate on the low flux regime φ ≪ 1, which  is relevant for real solids and makes connection to the  Landau levels in the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  We start by evaluating the topological phase diagram,  determined by the Chern numbers, and the density of  states (DOS) defined as  ρ(E) = L−2 �  n  δ(E − En).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  (2)  Additionally, an efficient method to extract the phase  diagram in random systems is to evaluate the local Chern  marker [18–20] and average it over many lattice sites, and  over different random configurations to obtain the Chern  number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' The local Chern marker is defined by  C(r) = −2πi  �  dr′ [X(r, r′)Y(r′, r) − Y(r, r′)X(r′, r)] ,  (3)  where X(r, r′) =  �  dr′′P(r, r′′)x′′P(r′′, r′) is the pro-  jected x coordinate, and Y(r, r′) is similarly defined pro-  jected y coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  Here P(r, r′) denotes the matrix  elements of the projection operator to the filled energy  bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' In practice, to avoid diagonalization of the full  system, we calculate the DOS and the Chern marker by  employing the Kernel Polynomial Method (KPM) [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  While the calculation of the DOS using the KPM is well-  documented [21], the derivation of the kernel polynomial  formula for the Chern marker and the details of the con-  figuration averaging are presented in the SI [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' 3 a-c we have illustrated how the topological  phase diagram evolves as the spatial correlation length is  decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' The corresponding DOS for each case is pre-  sented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' 3 d-f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' The DOS and the phase diagram  are symmetric with respect to E = 0, so we plot them  only for negative energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' In the crystalline limit ε = 0,  we recover the almost flat Landau bands, which disperse  linearly in the low flux regime as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' These states  also carry a unit Chern number, as indicated by the unit  jumps in the Chern marker at energies corresponding to  the Landau bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  For finite ε, the DOS correspond-  ing to individual Landau bands becomes significantly  broadened, however, the main features of the phase di-  agram survive even in the presence of strong random-  ness ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' The relevant length scales for the problem  are the average lattice constant a0, the magnetic length  LB = 1/√2πφ (the characteristic length of the Landau  orbits), and the spatial correlation length ξ of the lat-  tice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' For the parameters corresponding to Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' 3 c and f,  they satisfy a0 ≲ LB ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='1ξ, indicating that the Landau  level picture and the quantum Hall effect persists even  when the correlation length approaches the microscopic  length scales of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' The broadening caused by  the random lattice geometry on the Landau bands ap-  pears qualitatively similar to that caused by disorder on  crystalline lattices and in continuum [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' However, this  b  a  c  2  6  5  4  F-3  3  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='1  =0  ε=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='1  =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='2  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='1  f  p  e  0  3  2  2  1  0  1  0  1  E4  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' Averaged local Chern marker C(r) as function of energy for different ε for fluxes φ = 1/10 (a) and φ = 1/20 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' In a  we plot the Hall conductivity σxy evaluated at selected energies and averaged over random lattice positions deep in the bulk for  the ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='2 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' The inset in b illustrates the absolute deviation in the averaged local Chern marker values |∆C(r)| between  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='2 and the crystalline case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' Here the linear size of the system is L = 200, and the local Chern values are averaged over  n = 20 random configurations and 150 random lattice positions for each configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' Error bars are not visible in this scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  observation is far from self-evident, since disorder can  be regarded as a mere perturbation to the underlying  structure which always retains an approximate transla-  tion invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' In contrast, the finite correlation length  in non-crystalline lattices cannot be regarded as a pertur-  bative effect and there exists no systematic approach in  which a perfect crystal would serve as a starting point for  the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' Furthermore, the studied model is funda-  mentally different from those obtained by perturbing the  sites of a crystalline system independently around their  regular positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' Such lattices support arbitrary long-  range spatial order with ξ = ∞, and the density distri-  bution function associated with them resembles Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' 1 a,  except with broadened delta peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  A non-zero Chern number gives rise to the quantum  Hall effect, indicating the existence of a quantized Hall  conductivity in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' 4 a, this  is confirmed by a direct evaluation of the Hall conduc-  tivity using a method developed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' [23] based on a  real-space Kubo-Bastin formalism [24–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' The evolution  of the quantized plateaus as a function of ε, illustrated  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' 4 b, indicates that the overlapping higher Landau  bands will be the first affected by randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' When the  Landau level broadening increases, the plateaus shrink,  and when the broadening becomes comparable to the  level separation they eventually disappear [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' In ad-  dition to the quantized Hall conductance, the non-trivial  Chern numbers also give rise to topologically protected  edge modes at energies that fall in the mobility gap be-  tween the Landau levels [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' These edge modes are re-  vealed by the local DOS as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' 2 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' The in-  tensity of the local DOS varies randomly along the edge,  reflecting the irregular geometry of the lattice, vanishing  rapidly beyond the surface layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  Discussion– While the existence of amorphous topo-  logical states has been established in various systems  recently [29–41], the present work is the first study of  the Landau levels as a function of spatial correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  It should be emphasized that our findings of the non-  crystalline Landau bands have significant implications  beyond the quantum Hall effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' Since the birth of mod-  ern solid state physics, it has been known that many  thermodynamic and transport quantities are determined  by the DOS at the Fermi level, resulting in quantum oscil-  lations when the Landau levels are pushed through the  Fermi level by varying the magnetic field [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' Qual-  itatively similar oscillations persist in the studied sys-  tems, as indicated by the Landau peaks in the DOS that  also disperse as a function of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  The  Landau level formation and the resulting quantum os-  cillations are typically considered as a consequence of a  band structure, so it is remarkable that the quantum os-  cillations may survive in systems where the notion of a  band structure does not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' Approximately flat bands  which carry finite Chern numbers are also a prerequisite  for more complex interaction-induced phases, suggesting  that the lack of a long-range spatial order does not pose  a fundamental obstacle for formation of fractional quan-  tum Hall states [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  While in this work the focus was on 2d systems, the  phenomena resulting from the existence of Landau levels  have a wide variety of consequences in 3d systems [3, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  Our results indicate that, at least qualitatively, many  hallmark features of 3d electrons in magnetic fields can be  expected to survive in non-crystalline 3d systems without  long-range order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' However, a systematic study of these  features in 3d quickly becomes computationally challeng-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  Summary– In this work we studied the evolution of  Landau levels and topology as a function of randomness  in non-crystalline 2d lattices without long-range order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  We discovered that, even when the spatial correlations  Φ= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='1  b  Φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='05  a  4  0=3  ε= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='1  3  5  +-- = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='2  可ary(r),8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='2  E  C  3  1  0  2  2  3  3  1  E  E5  decay over microscopic length scales, the Landau levels  persist as broadened but well-defined features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' Moreover,  we showed that the magnetic field dependence and the  topological nature of the Landau bands remain intact,  giving rise to the quantum Hall effect with quantized con-  ductance and topological edge modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' The fact that the  lowest Landau bands can be resolved, despite the lack of  long range order, imply that the fundamental quantum  effects that have been traditionally considered as prop-  erties of crystalline solids in the magnetic field, can also  be observed in non-crystalline systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  Acknowledgement– This work was supported by the  Academy of Finland through the project number 331094.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  [1] Zyun Francis Ezawa, Quantum Hall effects: Recent the-  oretical and experimental developments (World Scientific  Publishing Company, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  [2] Steven M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' Girvin and Kun Yang, Modern Condensed-  Matter Physics (Cambridge University Press, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
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+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' 130, 026202 (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  [42] Johannes Gooth, Stanislaw Galeski,  and Tobias Meng,  “Quantum-hall physics and three dimensions,” arXiv  preprint arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='06248 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  7  SUPPLEMENTAL INFORMATION to “Quantum Hall effect and Landau levels  without spatial long-range correlations”  LOCAL CHERN MARKER FROM KERNEL POLYNOMIAL EXPANSION  The local Chern marker can be defined by the equation  C(r) = −2πi  �  dr′ [X(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' r′)Y(r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' r) − Y(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' r′)X(r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' r)] ≡ Cxy(r) − Cyx(r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  (4)  where  X(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' r′) =  �  dr′′P(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' r′′)x′′P(r′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' r′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  (5)  and analogously for Y(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' r′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' in terms of the projection operator to occupied bands  P(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' r′) =  � Emax  Emin  dEθ(EF − E)  �  n  δ(E − En) ⟨r⟩ n ⟨n⟩ r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  (6)  with n labeling the eigenstates of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' In order to implement KPM we rescale energies to lie in the  interval (−1, 1):  H = (H − b)/a  (7)  a = (Emax − Emin)/(2 − η)  (8)  b = 1  2(Emax + Emin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  (9)  Together with the substitution ϵ = (E − b)/a, we get  P(r, r′) =  � 1  −1  dϵθ(ϵF − ϵ)  �  n  δ(ϵ − ϵn) ⟨r⟩ n ⟨n⟩ r′,  (10)  and hence, after performing the radial integrals,  Cxy =  �  {−1,1}3 d3ϵ  �  nmk  ⟨r⟩ n ⟨n| ˆx |m⟩ ⟨m| ˆy |k⟩ ⟨k⟩ rδ(ϵ1 − ϵn)δ(ϵ2 − ϵm)δ(ϵ3 − ϵk)θ(ϵF − ϵ1)θ(ϵF − ϵ2)θ(ϵF − ϵ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' (11)  Note that the integration here is over the numbered epsilons, while the ones labelled by letters are instead summed  over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' We now approximate this result by replacing the quantities  f(ϵ1) =  �  n  |n⟩ ⟨n| δ(ϵ1 − ϵn),  (12)  with their truncated Chebyshev expansions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  f(ϵ1) ≈ fKPM(ϵ1) =  N−1  �  n=0  2µnhngnTn(ϵ1)  π(1 + δn,0)  �  1 − ϵ2  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  (13)  Here µn are the moments  µn =  � 1  −1  dϵf(ϵ1)Tn(ϵ1) = Tn(H),  (14)  and gn represent the kernel terms improving the accuracy of the truncated sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' Inserting this into the expression for  Cxy, we get  Cxy =  �  {−1,1}3 d3ϵ  �  nmk  ⟨r| fKPM(ϵ1)ˆxfKPM(ϵ2)ˆyfKPM(ϵ3) |r⟩ θ(ϵF − ϵ1)θ(ϵF − ϵ2)θ(ϵF − ϵ3)  = ⟨r| T (H)ˆxT (H)ˆyT (H) |r⟩ ,  (15)  8  where  T (H) =  �  n  2gnTn(H)  (1 + δn,0)  � 1  −1  dϵTn(ϵ)θ(ϵF − ϵ)  π  √  1 − ϵ2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  (16)  Approximating the integral and subsequently making use of the discrete Fourier transform (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' 21), we obtain  T (H) ≈  �  n  2gnTn(H)  (1 + δn,0)  1  M  M−1  �  s=0  cos  �nπ(s + 1  2)  M  �  θ  �  ϵF − cos  �π(s + 1  2)  M  ��  =  �  n  2gnTn(H)  M(1 + δn,0)Tn(H)S(n),  (17)  where we assume M > N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  We note that, as a function of s, the argument inside the step function increases monotonically;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' as |ϵF | < 1,  the overall effect of the step function is hence to exclude terms from the beginning of the sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  Let us denote  Γ ≡ ⌈ M  π arccos ϵF − 1  2⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' Then,  S(n) =  M−1  �  s=Γ  cos  �nπ(s + 1  2)  M  �  = Re  �M−1  �  s=Γ  e  nπ(s+ 1  2 )  M  �  = Re  �  �e  inπ  2M e  inπΓ  M  M−Γ−1  �  j=0  e  nπj  M  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  (18)  This yields  �  S(n) = M − Γ,  n = 0  S(n) = − 1  2  sin nπΓ  M  sin nπ  2M ,  n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  (19)  Absorbing a minus sign into T , we finally get  C(r) = 4πIm ⟨r| T (H)ˆxT (H)ˆyT (H) |r⟩  (20)  where, using T0(x) = 1 and g0 = 1,  T (H) ≈ −1 + Γ  M +  N−1  �  n=1  gn  M  sin nπΓ  M  sin nπ  2M  Tn(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  (21)  CALCULATION DETAILS  For quantities such as the Chern marker and conductivity tensor which are evaluated locally, only a finite number  of lattice points in the bulk are sufficient to fully describe topological features of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' The local markers  vary significantly throughout the lattices with open boundaries, but can form small regions where the changes in the  values are almost continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' This means that choosing to average many points next to each other may be a biased  sample of the distribution, depending on the configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' Therefore, rather than e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' choosing a large region near  the center, we randomly choose points within the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' This approach has the benefit of sampling the entirety of the  lattice, excluding only a finite border close to the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' We typically choose N ∼ 100 points randomly from the bulk  where we evaluate the local marker, yielding an appropriate average for the quantity at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' In other words, it is an  unbiased sampling of the full distribution of, say, local Chern marker values C(r), and thus we can obtain a proper  average value with less computational cost, especially when we scale up the system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' We repeat this sampling  for a number of configurations, and the distributions converge together remarkably quickly, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' This  observation provides a powerful and efficient tool to study larger systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' The number of random samples needed  to describe the correct distributions and the corresponding mean values remains roughly constant, provided N is not  too small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  Moreover, for an accurate description within KPM, in our calculations we set the number of moments Nm used in  the expansion to Nm ≈ L2/3 for systems which host L2 electronic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' Additionally, in studying DOS, the accuracy  in calculating stochastic traces depends on the product Nr Nc, where Nr is the number of random phase vectors and  Nc is the number of independent random lattice configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' In our numerical simulations NrNc ∼ 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' We also  studied the averaged Hall conductivity σxy(r, E) of the bulk, similarly to the calculation of Chern markers by random  sampling of lattice points away from the boundaries, and found that it matches well with the predicted Chern number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  This hints toward the equivalence of the local bulk conductivity, which is a physical observable, and the local Chern  marker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  9  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' a: Sampling of the local Chern marker for all bulk points in a given configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' b: Local Chern marker shown for  a subset of N = 100 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' The parameters used are φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='1 and E = −3 with the structural disorder parameter ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' c:  The normalized full distribution of local marker values, shown in white, as well as the partial distribution of random points in  light red, for the single configuration used in a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content=' d: Same as c, but for a set of 10 randomly generated configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  The distributions match very well and yield the same mean value for the Chern number with a high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  b  a  40  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='1  30  30  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='0  r  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='0  r  20  20  C  C  10  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='9  0  0  0  10  20  30  40  0  10  20  30  40  p  C  Cbulk,i  = Cbulk,i(r)  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='994  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='974  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='993  Cpoints,i = Cpoints,i(r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='979  Cave,points  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
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+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
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+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFAT4oBgHgl3EQffh2A/content/2301.08582v1.pdf'}
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diff --git a/rdE4T4oBgHgl3EQfVAyt/content/tmp_files/2301.05021v1.pdf.txt b/rdE4T4oBgHgl3EQfVAyt/content/tmp_files/2301.05021v1.pdf.txt
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+0 
+ 
+SYNTHESES AND EVALUATION OF Fe2+, Co2+ AND Ni2+ COMPLEXES WITH 
+DERIVATIVES OF VANILLIN – TRYPTOPHAN SCHIFF BASE LIGAND 
+SALISU ABUBAKAR 1, Prof. G. A. Shallangwa⁎ 2 and Dr. Abdulkadir Ibrahim2 
+1Department of Chemistry Federal College of Education, P.M.B. 1041, Zaria 
+2 Department of Chemistry, Ahmadu Bello University, Zaria, Nigeria. 
+ 
+*Corresponding Author’s abubakarsalisu@abu.edu.ng +2348136135156   
+Abstract  
+This research focused on synthesis, characterization and determination of biological   activities of 
+Fe2+ Co2+ and Ni2+ complexes of a novel L- Tryptophan-Vanillin schiff base ligand. They were 
+synthesized and characterized by determination of their Purity, Solubility, Elemental analyses 
+using phase match XRD data, FT-IR spectra, Molar conductivities, Magnetic susceptibilities, 
+PXRD and Biological activities. Results from the analyses revealed that, mpt of schiff base ligand 
+(SL) = 84 – 85 oC, Fe2+-SL = 245 - 246 oC, Co2+-SL= 271 - 272 oC while Ni2+ - SL is >350 oC. 
+Molar conductivities were found to be 10300, 5000, 17300 and 52900 Sm2 mol-1 for the schiff base 
+and the complexes respectively. It indicates that, the complexes are non-electrolytic in nature. 
+Magnetic susceptibilities of the complexes found to be higher than the spin only values due to 
+unpaired electrons (Fe2+ = 5.73 B.M, Co2+= 4.52 B.M while for Ni2+ = 3.46 B.M, suggesting an 
+octahedral geometries. Electronic absorption (λmax) for Fe2+ = 480, Co2+ = 520nm while Ni2+ shows 
+two bands at 480 and 570 which signifies, n-π* and π-π* transition respectively. Their crystalinity 
+index was also assessed using pxrd technique, it shows that complexes are 75, 80 and 85 % 
+crystalline with average crystallite size of 21.63 nm. Antimicrobial test results revealed that, Co2+ 
+and Fe2+ complexes have excellent activities against gram negative bacteria (E. coli, F. Shigella . 
+and S. Typi). They all shows efficient activities against some gram positive bacteria such as: 
+MRSA, Klebsiella P. and S. Pnemoniae and some fungi spp A. niger and C. albicans. Therefore 
+based on the findings from the study it was concluded that, schiff base is bidentate ligand and 
+octahedral geometries, paramagnetic susceptibilities were suggested for the complexes. 
+Keywords: Schiff base, Complexes, FT-IR Spectra, Characterization, Magnetic susceptibility, 
+Molar conductivity, diffraction, Absorption, Biological activities, Vanillin and Mole ratio. 
+ 
+ 
+ 
+
+1 
+ 
+ 
+1.0 Introduction  
+Schiff bases from amino acids and flavones such as vanillin and quercetin and their 
+complexes have numerous biological activities (Akhter et al., 2017b; Sani and Hussain, 2017). 
+The high affinity for the chelation of the schiff bases towards the transition metals is utilized in 
+preparing their solid complexes and due to its polyhydroxylated chemical structure, vanillin easily 
+forms complexes with species having free orbitals which may be occupied (Barbosa et al., 2019).  
+The potentials of vanillin schiff base ligands in inorganic chemistry can be understood very 
+well from the properties exhibited by its complexes such as: magnetics; luminescence, chirality, 
+catalysis, cytotoxicity and ferro electricity among others (Al-jeboori and Al-shimiesawi, 2021; 
+Neacşu et al., 2018 and Barbosa et - al., 2019). Vanillin derivatives are found to be  active against 
+both gram-positive and gram-negative bacterial strains and has been shown to be effective against 
+some fungal spp ( Sheyin et al., 2018). 
+L-Tryptophan (Trp) is an essential amino acid which is required for the biosynthesis of 
+proteins in the body. It is very significant for nitrogen balance, maintenance of muscle mass and 
+body weight in humans (Yan et al., 2010).  
+1.1 The Chemistry of Schiff base Formation 
+In reactions that yield schiff base ligands, the electrophilic carbon atom of carbonyl 
+compounds will experience a nucleophilic attack by amines to give a compound with C=O being 
+replaced by C=N. The reaction is reversible and occurs as shown in eqn (1) or (2) depending 
+whether the carbonyl is an aldehyde or a ketone ( Usman et al., ) 
+
+2 
+ 
+ 
+ 
+Condensation products of vanillin with amines confers biological activities as well as 
+having good complexation ability with metal ions (Saranraj and Devi, 2018). This study focused 
+on synthesis, characterization and test for antimicrobial activities of a schiff base ligand derived 
+from condensation of o-vanillin with L– tryptophan and complexes of their ligands with Fe2+, Co 
+2+ and Ni2+ ions. Transition metal complexes with bidentate schiff base ligands that contains 
+nitrogen, oxygen, or sulphur donor atoms contribute immensely in biological systems (Malik et 
+al., 2018; Maalik et al., 2014). 
+Numerous researches carried out on transition metal complexes derived from schiff base 
+ligands have pointed out their significance due to their properties and wider biological applications 
+(Nagesh and Mruthyunjayaswamy, 2015; Agbese et al, 2018). Fugu et al. (2013) reported that 
+schiff base complexes derived from 4-hydroxysalicylaldehyde and amines have strong anticancer 
+activity. Schiff bases of ketones, their derivatives, and their metal complexes are active anticancer 
+and antioxidant agents (Dhanaraj and Johnson, 2014). 
+Transition metal ions are amongst the important species needed to keep human body 
+healthy and active because several important biological functions depend upon their presence and    
+their absence may lead to deficiency diseases (Arvind  and Andreas  2018). Mustafa and Alsharif 
+
+3 
+ 
+(2018) opined that, most notable metals that exist in human body are in the form of ions such as:  
+Fe2+, Co2+, Ni2+, Ca2+, Cu2+, Zn2+, and Cr2+. Transition metal complexes are observed to be 
+generally more biologically active than the uncomplexed ligands, they are formed due to the  
+presence of the metal moieties (Mohamed and Omar, 2006; Abu Bakar et al., 2010a).  
+2.0 Experimental  
+All chemicals used for this research are of analytical grade (AR) obtained from Sigma 
+Aldrich and other reliable chemical vendors. The chemicals were used as purchased. 
+2.1 Materials  
+The following materials were used for the study: Mettler Analytical weighing balance (MT 
+200B 0.0001-200G), spectrophotometer (JENWAY-7315), magnetic stirrer (Stuart), fume 
+cupboard (Biobase), melting point machine (SMP 10 Fascia, Stuart Bibby Scientific), FT-IR 
+spectrophotometer (Nicolet iS5-Thermo scientific), electrical conductivity meter (Hanna 
+Instrument, HI 9835, Woon socket R. I USA), heating/drying oven (TT-9053, Techmel and 
+Techmel U.S.A, YLD 2000) and heating mantle among others. 
+2.2 Methodology 
+2.2.1 Syntheses of Schiff base Ligand 
+The method used to synthesize the ligand was adopted from that of Kaczmarek et al. (2009) 
+with minor modifications. Solution of vanillin (0.30g, 2mmol) was mixed with solution of l-
+tryptophan (0.4g, 2mmol) in 15 ml ethanol with 1ml dilute sodium hydroxide solution (0.056g, 
+1mmol) solution. The reaction mixture was stirred for about 10 minutes and refluxed over a water 
+bath for 5hrs at 60 oC. The coloured precipitate formed was filtered, wash with 50% cold ethanol 
+and rinse with diethyl ether. It was allowed to dry in a vacuum desiccator to constant weight which 
+
+4 
+ 
+yield = 72.0% schiff base ligand (SL). The yellowish product was recrystallize in methanol for 
+further purification. 
+2.2.3 Syntheses of the Complexes 
+The complexes were prepared using standard methods of (metal: ligands ratio) adopted 
+from Haddad (2016); Asgharpour et al., (2017) and Hossain et al. (2018) with some minor 
+modifications. An ethanolic solution of schiff base (20 mmol) was prepared and added drop wise 
+to the solutions of metal (II) chloride (10.0 mmol) FeCl2, CoCl2 and NiCl2 stirred for about 20 
+minutes. Same concentrations were prepared and refluxed in water bath at 60oC for 6 hours. The 
+pH of the mixture was adjusted to about 7.5 with few drops of potassium hydroxide solution and 
+the coloured solid formed were filtered, wash with 50% cold ethanol, and finally dried in a vacuum 
+desiccator to constant weight. The synthetic routes is summarised in equations (3) and (4): 
+Fig. 1: A flowchart of Syntheses of Schiff base Ligand and the Metal (II) Complexes 
+ 
+
+5 
+ 
+Fig. 2: A flowchart of Syntheses of Metal (II) Complexes  
+ 
+2.4 Physical Measurements  
+Solubilities of the schiff base and metal complexes were checked in water, 50% ethanol 
+and DMSO.  Melting points of the complexes were also determined to established  their purity 
+using electrically heating melting point apparatus (SMP10, Fascia, Stuart; Bibby scientific), molar 
+conductivity were assessed to test the electrolytic nature of the complexes (with an Hann scientific 
+Electrical conductivity meter). The fourier transform infrared spectra were determined with FT-IR 
+spectrometer (Nicolet iS5, Thermo scientific), wave length of maximum absorption (λmax) of the 
+schiff base and the complexes were measured using an aqueous solution of the compounds in 
+uv/visible spectrophotometer (Jenway, 7315) 1cm cuvette at (200 – 780 nm). Elemental analyses 
+was performed using the match phase crystals analysis software and the report obtained from the 
+PXRD data. Magnetic susceptibility followed by x-rays diffraction technique were used to 
+determine geometries, degree of crystalinity,  and the morphological properties of the complexes. 
+2.5 Assessment of Antimicrobial Activities 
+ 
+ 
+
+6 
+ 
+(i) Minimum Inhibitory Concentration (MIC) 
+The MIC values was determined in accordance with standard method proposed by clinical 
+and laboratory standard institute (CLSI, 2019) using a broth dilution techniques.  Mc Farlands’ 
+turbidity standard was prepared and used for the study. 1ml of sterilized MH- media were added 
+to each of clean, dried and sterilized test-tubes, followed by 1mLof 0.5 MCF bacterial suspension 
+which contained about (1-2×108 CFU/mL bacterial load). 1mL of 2000 µg / mL solution of the 
+test compounds was transferred into each test tube and serially diluted to give concentrations of 
+(1000, 500, 250,  125, 62.5, 31.25, 15.625 µg/mL). 
+The solution was added to the mixture followed by 0.1 mL of the bacterial suspension in a 
+normal saline solution. The test tubes were incubated at 37 oC for 24 hrs to enhance bacterial 
+growth, test-tubes with lowest and the highest turbidity were examined, compared with standard 
+and the results were carefully recorded. 
+(ii) Minimum Bactericidal Concentration (MBC) 
+ 
+              The MBC was assessed using the method developed by Hadariah et al. (2019) to ascertain 
+whether the test pathogens were completely killed or just inhibited their growth. Mueller Hinton 
+agar was sterilized, poured into petri dishes and the media was allowed to cool/solidify. An aliquots 
+from samples which shows sensitive to the MIC test were carefully selected and subjected for 
+MBC evaluation. They were sub-cultured and incubated at 28o C for 7 and 14 days in some cases. 
+To determined CFU, the plates of the media was taken for microscopy and observed for bacterial 
+growth. 
+ 
+ 
+
+7 
+ 
+(iii) Minimum Fungicidal Concentration (MFC) 
+Sarah et al. (2016) and CLSI (2019) methods was adopted to carry out this test using micro 
+broth dilution technique with little modifications. Two fungal spp (Aspergillus Niger and 
+Candida albicans) isolates were used, grown for 7 days at 28 oC incubation period in potato 
+dextrose agar. The stock suspension was diluted with normal saline solution to corresponding 
+concentration of 1×105 cfu /mL and viability of the fungi spp was confirmed using the sabouraud 
+dextrose agar plates and counting the number of cfu/mL 
+Two fold serially diluted solution of the test compounds were prepared to obtained 
+200µg/ml, 100µg/ml, 50µg/ml, 25µg/ml and 12.5µg/ml respectively. A micro pipette was used to 
+transfer 0.1ml of the test fungal strain solution into the test-tubes containing the mixture with 
+varied concentrations of the compounds. The mixture was incubated at 38 oC for 72 hours after 
+which the turbidity was observed and compared with standard for the growth of the pathogens. 
+The turbidity (growth) between the test-tube which contained lowest and the highest concentration 
+of the compound were examined and compared with standard. The results were carefully recorded 
+in accordance with clinical and laboratory standard institute CLSI M100, (2019).  
+ 
+ 
+ 
+ 
+ 
+ 
+
+8 
+ 
+3.1 Results and Discussion 
+Table 1: Shows some Physical Parameters test Results for Schiff base and Metal Complexes 
+ 
+Products 
+ 
+Colour 
+ 
+%Yield 
+ 
+Solubility 
+Mpt 
+(o C) 
+Λm (Sm² mol⁻1) 
+µeff 
+(B.M) 
+Vanillin – Tryptophan SBL 
+Golden 
+Yellow 
+72.0 
+SS (H2O) 
+SL ( EtOH) 
+SL DMSO) 
+84 
+10300 
+- 
+Vanillin – Tryptophan - Fe2+ 
+Greenish 
+yellow 
+85.0 
+SL (H2O) 
+SS (EtOH) 
+SL (H2O) 
+SS (EtOH) 
+245 
+5000 
+5.73 
+Vanillin – Tryptophan - Co2+ 
+Brownish 
+yellow 
+81.0 
+SL (H2O) 
+SL(EtOH) 
+  SL(DMSO) 
+271 
+17300 
+4.52 
+Vanillin – Tryptophan - Ni2+ 
+Yellowish- 
+green, 
+59.4 
+SS (H2O) 
+SL (EtOH) 
+  SL(DMSO) 
+>350 
+52900 
+3.46 
+ 
+Keys: SL = soluble, SS = sparingly soluble 
+ 
+The results in Table 1, indicates colour of the schiff base and the metal complexes, 
+percentage yields and solubilities in some solvents. It revealed that, schiff base ligand is sparingly 
+soluble in water, but it readily dissolved in DMSO and 50% ethanol solutions solution while the 
+complexes are soluble in water, DMSO respectively. It also indicated that, schiff base has lower 
+melting point (84 oC) than the complexes, but Ni2+ complex has the highest melting temperature 
+of (> 350 oC) followed by Cobalt (II) complex (271 oC) while Iron (II) complex has (245 oC).    
+Molar conductivity measurement shows that, both the ligand and the complexes were non 
+electrolytic in nature and magnetic susceptibility measurement obtained at room temperature 
+revealed that, complexes are paramagnetic with observed magnetic moment as: Fe2+ = 5.73 B.M, 
+Co2+ = 4.52 B.M while for Ni2+ = 3.46 B.M respectively which are slightly higher than the spin 
+
+9 
+ 
+only values due to unpaired electrons suggesting an octahedral geometry for all the complexes. 
+This is similar to the work of Zurowska and Bauskey (2011). 
+Table 2: UV-visible Data for the Schiff base Ligand and the Complexes 
+S/N0. Compounds 
+Wave length λmax  (nm) Transition Assigned 
+Geometries 
+1. 
+Schiff base Ligand (SL) 255 – 350 
+330 – 380 
+π   - π⁎ (C =C in benzene) 
+n   - π⁎  (C = N in imine) 
+---- 
+2. 
+SL – Fe2+ Complex 
+460 – 480  
+π   - π⁎ (C =C in benzene) 
+Octahedral 
+3. 
+SL – Co 2+ Complex 
+500 – 520  
+n  - π⁎ (C = N in imine) 
+ 
+Octahedral  
+4. 
+SL – Ni2+ Complex 
+550 – 570 
+369 – 480   
+π   - π⁎ (C=N in imine) 
+π  - π⁎  (C=C in benzene) 
+Octahedral 
+ 
+Results in Table 2, shows electronic transitions within the schiff base and the complexes scanned 
+from 300 – 750 nm in ethanolic solutions (Fig. 2-4) which indicates some bands as  n - π⁎ and         
+π - π⁎ transitions from to non-bonding to anti bonding molecular orbitals and delocalization of pi- 
+electrons within the benzene rings. The absorption bands assigned and the proposed geometries of 
+the complexes are as follows: Fe2+ 460 – 480 nm, Co2+ shows only one bands at 500 – 520nm 
+while Ni2+ shows two absorption bands at 369 – 480 and 550 –570nm respectively. This result is 
+in agreement with that of Suesh and Pracash (2010) and (Crystal, Y. et –al., 2015). 
+Table 3: Elemental Analysis of the Ligand and the Complexes 
+ 
+ Products 
+% of Elements Found (Calculated) 
+ 
+Formulae 
+ C  
+ H 
+O 
+N 
+Fe 
+Co 
+Ni 
+SBL 
+52.35 
+(52.79) 
+5.12 
+(4.63) 
+17. 73 
+(16.90) 
+24.8 
+(24.01) 
+- 
+- 
+- 
+C19H18N2O4 
+SBL – Fe2+   
+39.34 
+(39.72) 
+4.18 
+(3.96) 
+23.03 
+(22.89) 
+3.37  
+(2.65) 
+26.88 
+(26.14) 
+- 
+- 
+C38 H36N4O8Fe  
+SBL– Co2+   
+47.04 
+(46.31) 
+9.71 
+(8.99) 
+10.36  
+(10.20) 
+12.01 
+(11.98) 
+- 
+20.88 
+(20.09) 
+- 
+C38 H36N4O8Co 
+SBL – Ni2+   
+45.23 
+(45.18) 
+4.74 
+(4.98) 
+25.11 
+(25.01) 
+4.40 
+(4.85) 
+- 
+- 
+20.52 
+(20.90) 
+C38 H36N4O8Ni 
+
+10 
+ 
+ 
+Table 3 shows result of elemental analysis and confirmed molecular formula of the schiff 
+base and the metal complexes, it revealed that, the complexes are mononuclear with 2moles of the 
+ligand bonded to 1mole of the metal ion. This implies that, the metal – Ligand ration is 2:1 with 
+molecular formula as [M-(SBL)2. H2O] where SBL = Schiff base ligand, M = Co2+ and Ni+2+ 
+respectively.  
+Table 4: FT-IR Spectral mode (350 – 4000 cm-1) of the Schiff base and the Metal complexes  
+ 
+Complexes 
+V(HC═N) 
+V(M⸺N) 
+V(M⸺O) 
+V(COO) 
+V(C⸺O) 
+V(O⸺H) 
+V(C ⸺N) 
+V(C-O-C) 
+V (C═C) 
+SBL  
+1663 sh 
+- 
+ 
+- 
+1362 sh 
+1282 sh 
+3326 m 
+1373 sh 
+1625 m 
+1521 sh 
+SBL – Fe2+   
+1583.36 sh 
+401.64 sh 
+535.30 sh 
+535.30  br 
+1026.32 sh 
+3208.51 br 
+1509.46 sh 
+- 
+- 
+ 
+SBL– Co2+   
+1583.45 sh 
+463.69 sh 
+511.36 sh 
+588.73 sh 
+1025.14 
+3211.86 br 
+1506.96 sh 
+- 
+- 
+ 
+SBL – Ni2+   
+1586.13 sh 
+419.78 sh 
+385.20 sh 
+410.36 sh 
+1017.85 sh 
+3362.07 br 
+1630.99 sh 
+- 
+- 
+ 
+ 
+Keys: SBL = Vanillin – Tryptophan Schiff base Ligand,  
+SBL – Fe2+= Vanillin – Tryptophan Schiff base – Fe2+ complex, SBL– Co2+  = Vanillin – Tryptophan 
+Schiff base – Co 2+  complex, SBL – Ni2+  = Vanillin – Tryptophan Schiff base – Ni2+ complex 
+Results in Table 4 above, shows FT-IR spectra bands of the ligand and the complexes at 
+1628 v (HCN, str), 382.47 v(O-H str) and 362.12 cm-1 respectively. The FT-IR analysis revealed 
+that, the bonding of the schiff base to the metal centres has occur as expected. In the schiff base 
+ligand, a strong sharp band observed at 1663 cm-1 can be assigned to V (HC═N) azomethine 
+stretching vibration. When reacted to form complex with the metal ions, the band shifted to lower 
+frequencies in 401.64 - 419.78 cm-1 range for v(M⸺N) and  385.2 - 535.30 cm-1 range for  
+v(M⸺O) indicates that, new bonds has formed that, azomethine coordinated to the metal ions 
+through Oxygen and Nitrogen donor atoms in the schiff base ligand.  
+
+11 
+ 
+The symmetric carbonyl stretching v(COO-) is also shifted to a higher frequency from 
+1362cm-1  to 535.30, 588.73 and 410.36cm-1 region for Fe2+, Co2+ and Ni2+ complexes respectively.  
+Similarly the v (C – O) phenolic band in the complexes shows a significant shift to a lower 
+frequencies of 1282 cm-1 in the ligand, 1026.32, 1025.14 and 1017.85 cm-1 for the metal ions 
+respectively. The spectral of the complexes shows broad bands between 3208.51, 3211.86 and 
+3362.07 cm-1 range which attributed to O–H stretching vibration from phenolic group 
+(Fig.5,6and7). Information obtained from the FT-IR results of some of the major functional groups 
+in the schiff base and the complexes revealed that, the schiff base is bidentate ligand and it bound 
+to the metal ions through the phenolic oxygen and azomethine nitrogen atoms to form an 
+octahedral complexes.   
+Table 5: Powder X-rays Diffractions Analysis 
+Complexes 
+Peak position 2θ (o) 
+FWHM  
+β (o) 
+% Crystalinity 
+Index (CI) 
+Crystallite size  
+D (nm) 
+Average  
+D (nm) 
+ Fe2+– SL 
+53.40 
+0.654 
+75 
+19.41 
+ 
+21.63 
+Co2+ - SL 
+58.62 
+0.678 
+80 
+21.32 
+Ni+2+ – SL 
+62.45 
+0.545 
+84 
+22.16 
+                      K = 0.94,                   λ = 0.5418 (nm) 
+ 
+Table 5 shows results of PXRD pattern of the complexes scanned at laboratory temperature 
+(303 K) with an XRD machine (RIGAKU). The degree of crystalinity and crystallite sizes and 
+average crystallite size of the complexes were obtained within the range of 2θ values from 10 – 
+120o at wavelength of λCuKα1 radiation = 1.5418 nm. The XRD results further supported by the 
+inter planar distance and the crystallite sizes (d) of 19.41, 21.32 and 22.16 corresponding to Fe2+– 
+SL, Co2+ - SL and Ni+2+ – SL respectively. 
+
+12 
+ 
+Table 6: Anti-microbial and Antifungal Sensitivity Test for the Schiff base and Complexes 
+ 
+S/N0. 
+Products 
+ 
+MRSA 
+ 
+S. 
+Pnemoniae 
+E. Klebsiella  
+Pneumoniae 
+ESBL 
+E. Coli 
+ 
+F.Shigella 
+ 
+S. Typi 
+ 
+C. Alicans 
+ 
+A. Nigger 
+1. 
+SBL  
+R 
+R 
+MS 
+MS 
+MS 
+R 
+MS 
+MS 
+2. 
+SBL – Fe2+   
+R 
+R 
+S 
+R 
+S 
+R 
+S 
+R 
+3. 
+SBL – Co2+   
+MS 
+S 
+S 
+S 
+R 
+R 
+MS 
+S 
+4. 
+SBL – Ni2+   
+MS 
+S 
+S 
+S 
+R 
+MS 
+R 
+S 
+6. 
+Standard 1 
+* 
+- 
+- 
+- 
+MS 
+- 
+MS 
+MS 
+- 
+7. 
+Standard 2 
+* 
+- 
+- 
+- 
+- 
+- 
+- 
+MS 
+- 
+8. 
+Control  
+- 
+- 
+- 
+- 
+- 
+- 
+- 
+- 
+Key: S = sensitive, MS = mild sensitive,   R = resistance, - = negative (no reaction), C = control, Standard 
+1* = Augmentin,   Standard 2* = Terbinafine 
+Table 5 shows sensitivity test results of the products against some selected pathogens used 
+for the study. It indicates that, one gram +ve with two gram –ve bacteria and the fungal spp shows 
+mild sensitive reactions to the schiff base ligand, Klebsiela P., Shigella and a fungi sp (Candida 
+A.) are very sensitive to Fe2+ complex, Co2+ complex shows its effectiveness to streptococcus P., 
+Klebsiela P., E. coli and Aspergillus N. While Ni2+ complex revealed to be very effective and 
+positive sensitivity test to streptococcus, Klebsiela,  E. coli and Aspergillus N. fungal sp. 
+ 
+ 
+ 
+ 
+
+13 
+ 
+Table 7: Shows Zone of Inhibition (Z.I) Test Results of the Schiff base Ligands and their    
+               Metal (II) Complexes in (mm) 
+ 
+S/N0. 
+Compound 
+ 
+MRSA 
+ 
+S. Pnemoniae 
+ 
+E. Klebsiella  
+Pneumoniae 
+ 
+ESBL 
+E. Coli 
+ 
+Shigella 
+spp 
+ 
+Salmonella 
+spp 
+ 
+C. 
+Albicans 
+ 
+A. Nigger 
+1. 
+SBL  
+- 
+- 
+5 ± 0.2 
+7 ± 0.2 
+10 ± 0.4 
+- 
+8 ± 0.3 
+7 ± 0.1 
+2. 
+SBL – Fe2+   
+- 
+- 
+22 ± 0.1 
+- 
+25 ± 0.3 
+- 
+12 ± 0.2 
+- 
+3. 
+SBL – Co2+   
+20.0 
+23 ± 0.5 
+26.0 
+18.0 
+- 
+- 
+20 ± 0.2 
+10 ± 0.5 
+4. 
+SBL – Ni2+   21 ± 0.5 
+- 
+- 
+23 ± 0.5 
+- 
+13.0 
+23 ± 0.2 
+16 ± 0.4 
+5. 
+Control 
+- 
+- 
+- 
+- 
+- 
+- 
+- 
+- 
+6. 
+Standard 1 * 
+11 
+- 
+15 
+13 
+- 
+17 
+- 
+- 
+7. 
+Standard 2 * 
+- 
+- 
+- 
+- 
+- 
+- 
+8.5 
+10 
+ 
+Key: S = sensitive, MS = mild sensitive,   R = resistance, - = negative (no reaction), C = control, Standard 
+1 * = Augmentin,   Standard 2* = Tarbinafin 
+ 
+The data in Table 6 above, shows results for zone of inhibition test of the products against 
+some those pathogens that shows positive sensitivity (MIC) test from Table 5. It indicates similarly 
+the regions (in mm) where those pathogens are being killed or their growth are inhibited by the 
+compounds as appropriate where the schiff base shows least inhibition zone of (5, 7, 10, 8 and 
+7mm) followed by Fe(II) complex (12, 22 and 25 mm), Ni2+ (13, 16, 21, and 23 mm)  while Co 
+2+ complex shows the widest inhibition zone (10, 18, 20, 23, and 26 mm) respectively. The control 
+agar plate shows negative test result with less than 0. 1 mm. this indicates that, all the microbes 
+grow freely in the absence of the compounds.  
+ 
+ 
+ 
+ 
+
+14 
+ 
+Table 8: Shows Results of minimum Bactericidal (MBC) and Fungicidal Concentrations  
+               (MFC) (µg/mL) for the Schiff base, metal complexes and Standard Drugs against  
+               the Pathogens used for the study 
+ 
+ 
+S/N0. 
+Products 
+ 
+MRSA 
+ 
+S. 
+Pnemoniae 
+ 
+E. Klebsiella  
+Pneumoniae 
+ 
+ESBL 
+E. Coli 
+ 
+F.Shigella  
+ 
+S. Typi 
+ 
+C.Albicans 
+ 
+A. Nigger 
+1. 
+SBL  
+++ 
++++ 
+-- 
+- 
+-- 
++++ 
+-- 
+-- 
+2. 
+SBL – Fe2+   
+++ 
+++ 
+--- 
+++ 
+--- 
++++ 
+-- 
++ 
+3. 
+SBL – Co2+   
+--- 
+-- 
+-- 
+- 
++ 
++ 
+--- 
+-- 
+4. 
+SBL – Ni2+   
+--- 
+++ 
++ 
+-- 
+++ 
+-- 
+-- 
+-- 
+5. 
+Standard1 
+(mg/mL) 
++++ 
+---- 
+---- 
+---- 
++++ 
+---- 
++  
++++ 
+6. 
+Standard2 
+(mg/mL) 
++ 
++++ 
++ 
++++ 
++ 
+++ 
+---- 
++ 
+7. 
+C 
++++ 
++++ 
++++ 
++++ 
++++ 
++++ 
+++ 
++++ 
+ 
+Keys: ++++ = very intense microbial growth +++ = high microbial growth (100 -250 mg/ml), ++ = 
+moderate microbial growth, ++ = mild microbial growth, - No growth slightly clear at (215 µg/ml), -- No 
+growth very clear at (250 µg/ml), --- No growth, very –very clear at (500 µg/ml), ---- No growth immediate 
+sharp clear at (1000 µg/ml), C = control, Standard1* = Augmentin,   Standard 2* = Terbinafine 
+Table 7 shows MBC and MFC results of the test compounds, from the preliminary in vitro 
+antimicrobial activities of the ligand and the complexes. They were screened for their minimum inhibitions 
+concentrations and in vitro antimicrobial actions against the selected microbes (Table 7). The minimum 
+active concentrations measured in (µg/mL) and two standard were used (Augmentin and Terbinafine for 
+fugal spp). The results revealed that, the complexes showed more significantly active for killing the 
+pathogens than the ligand and the overall activities of all the three complexes were found to be very 
+effectives even at very low concentration of less than (250 µg/mL) is being considered to be the minimum 
+concentrations of the schiff base and complexes required to kill (99.99%) of the pathogens completely.   
+ 
+ 
+
+15 
+ 
+3.2 Summary and Conclusion 
+3.2.1 Elemental Analysis 
+The result of elemental analysis confirmed the proposed molecular formula of the schiff 
+base and the metal complexes (C19H18N2O4 and MC38 H36N4O8) and it revealed that, the complexes 
+are mononuclear with 2moles of the ligand bonded to 1mole of the metal ion. This implies that, 
+the metal – Ligand ration is 2:1 with molecular formula as [M-(SL)2 .H2O] where SBL = Schiff 
+base ligand, M = Co2+ and Ni+2+ respectively. 
+3.2.2 Magnetic Susceptibility Measurement 
+Information regarding geometries of the complexes were obtained from their electronic 
+spectral and magnetic moments and the observed magnetic moment are: Fe2+ = 5.73 B.M, Co2+ = 
+4.52 B.M while for Ni2+ = 3.46 B.M respectively which are slightly higher than the spin only 
+values due to unpaired electrons suggesting an octahedral geometries for all the complexes. 
+3.2.3 Conductivity measurement   
+Molar conductivity of the complex in 10-3 moldm-3 DMSO solution were measured at laboratory 
+temperature. The values indicated that, the complexes are non-electrolytic in nature. The result is 
+in line with the findings of Zahid H., and Hazoor A. Shed (2018) and Amadou G. et- al, (2018). 
+3.2.4 Spectral Analysis 
+The electronic spectra of the ligand and the complex shows there bands 1628 v (HCN, str), 382.47 
+v (O-H str) and 362.12 cm-1 respectively. The FT-IR analysis revealed that, the binding of the 
+schiff base to the metal centres has been obtained by comparing the FT-IR –spectra of the ligand 
+with those of the complexes.  
+
+16 
+ 
+3.2.5 Electronic Spectrum  
+The electronic spectrum of the schiff base (Fig. 2 -4) above, shows a broad band at and 
+their metal complexes were determined from the scanned range of 350 – to – 750 visible region 
+using ethanol as solvent. The absorption regions, bands assigned and the proposed geometries of 
+the complexes are shown in Table 2. The scanned electronic transition for Fe2+ shows two bands 
+at 460 – 480 nm, Co2+ shows only one bands at 500 – 520nm while Ni2+ shows two absorption 
+bands at 369 – 480 and 550 –570nm respectively. 
+3.2.6: Powder X-rays Diffractions Analysis 
+Degree of crystalinity, crystal size of the complexes were obtained and PXRD pattern scanned in 
+the range of 2θ values from 10 – 120o at wavelength λCuKα1 radiation = 1.5418 Å. The results 
+revealed that, the complexes are crystalline with high crystalinity index of 72, 80 and 84% 
+respectively. 
+3.2.7 Uv-visible Spectrum of the Schiff base Ligand and the Complexes 
+The electronic spectrum of the schiff base and the complexes shows broad bands, when 
+scanned from 350 – to – 750 visible region in ethanolic solution as are shown in Table 1.0 above. 
+3.2.8 Assessment of Biological Activities of the Schiff base and the Metal (II) Complexes 
+The antibacterial activities test were carryout to find the efficacy of the novel schiff base 
+and the complexes. It was done according to standard procedures in vitro using some clinical 
+strains of microbes MRSA, ESBL Klebsiella Pneumoniae, Streptococcus Pnemoniae , ESBL E. coli, 
+Shigella spp. and Salmonella sp together with  two fungi spp Aspergillus and Candida albicans. The 
+test compounds shows ability to inhibit growth of the microbes which was compared with some 
+
+17 
+ 
+antibiotics (Augmentin and Terbinafine). The results revealed that, SL – Ni2+ complex shows 
+highest activity followed by SL-Co2+ and the schiff base respectively. Comparatively, the 
+inhibition zones of  the SL and the complexes at the different concentrations indicated that, the 
+complexes has better activities than the schiff base ligand used for this study. 
+ 
+Fig1. FT-IR Spectrum of the Schiff base and the Metal (II) Complexes 
+ 
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+0
+50
+100
+150
+200
+250
+300
+350
+400
+% Transmittance
+Wave number (cm--1)
+ SBL- Ni (II) complex
+ SBL- Co (II) complex
+ SBL-Fe(II) complex
+ SBL
+
+18 
+ 
+ 
+Fig 2: A Staked Absorption Spectra of Pure vanillin, SL and SL – Fe2+ Complex 
+ 
+ 
+Fig 3: A Staked Absorption Spectra of Pure vanillin, SL and SL – Co 2+ Complex 
+ 
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+350
+450
+550
+650
+750
+850
+Absorbance
+wave lenght (nm)
+SLA 4 - Fe (II) Complex
+Schiff base Ligand
+Pure vanillin
+0
+0.2
+0.4
+0.6
+0.8
+1
+350
+450
+550
+650
+750
+Absorbance 
+wave length (nm)
+SLA4 Schiff base
+SLA 4 - Co (II)
+Pure Vanilline
+
+19 
+ 
+ 
+Fig 4: A Staked Absorption Spectra of Pure vanillin, SL and SL – Ni2+ Complex 
+ 
+ Fig. 5: FT-IR Spectra of Vanillin – Tryptophan Schiff base Ligand 
+ 
+300
+400
+500
+600
+700
+800
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+Absorbance
+Wave length (nm)
+ SL - Ni (II) Complex
+ Vanillin-Tryp SL
+ Pure Vanillin
+
+90
+85]
+1663.
+80
+3088.25
+1389.22
+99'890
+1205.64
+801.8
+1461.27
+451.94
+%Transmittance
+65-
+1027.16
+928.32
+740.09
+60-
+1425.84
+1257.75
+1117.63
+488.38
+611.72
+1513.37
+523.02
+1621.77
+1282.22
+1158.48
+631.52
+831.50
+727.74
+3.61
+1580.84
+875.02
+858.
+693.91
+549.01
+45
+40-
+0000
+3500
+3000
+2500
+2000
+1500
+1000
+ 500
+Wavenumbers (cm-1)20 
+ 
+  Fig. 6: FT-IR Spectra of Vanillin – Tryptophan –Cobalt (II) Complex 
+ 
+ 
+ Fig. 7: FT-IR Spectra of Vanillin – Tryptophan – Ni2+ Complex 
+ 
+ 
+
+90
+s8
+803
+3211.86
+FOL
+1663.09
+ 651
+1427.70
+813.62
+60
+1196.78
+1025.14
+588.73
+%Transmittance
+1583.45
+1138.07
+50
+511.36
+TTT
+857.79
+ 45
+1463.69
+1297.58~~
+731.65
+T
+ 40
+1263.61
+631.55
+ 30
+1506.96
+754.53
+382.48
+87'L9E
+20
+4000
+3500
+3000
+2500
+2000
+1500
+1000
+500
+Wavenumbers (cm-1)L06
+FoL
+-09 
+1371.16
+880.40
+1017.85
+50
+1499.61L
+1630.99
+1431.34
+1131.02
+733.92
+40
+%Transmittance
+30-
+3362.07
+1555.48
+4
+1269.55
+1159.93
+760.22
+1465.
+Foz
+1586.13
+1309.38
+817.61
+098
+10
+1175.99
+Fo
+.12,
+TTT
+350.50
+-10
+1196.01
+-20J
+86
+-30
+4000
+3500
+3000
+2500
+2000
+1500
+1000
+500
+Wavenumbers (cm-1)21 
+ 
+0
+10
+20
+30
+40
+50
+60
+70
+80
+0
+500
+1000
+1500
+2000
+2500
+Peak Intensity (a.u)
+Vanillin-Tryptophan-Fe2+ Complex (2 Theta (deg.))
+ 
+Fig. 8: PXRD Diffraction Pattern for Vanillin-Tryptophan-Fe2+ Complex  
+ 
+ 
+Fig. 9: Vanillin – Tryptophan - Co 2+ Complex 
+0
+10
+20
+30
+40
+50
+60
+70
+80
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Peak Count (a. u)
+2 theta (deg)
+Vanillin - Tryp Co (II)
+
+22 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+0
+50
+100
+150
+200
+250
+300
+Peak Count (a.u)
+Vanillin-Tryptophan Nickel (II) Complex (peak count)
+
+23 
+ 
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+             Santos, Luciana Dias Ghiraldi Lopes, Paula Aline Zanetti Campanerut-Sa´, Vera  
+             Lucia Dias Siqueira, Katiany Rizzieri Caleffi-Ferracioli, Jorge Juarez Vieira Teixeira   
+             and Rosilene Fressatti Cardoso (2020). Minimum Bactericidal Concentration Techniques 
+             in Mycobacterium tuberculosis: A Systematic Review. 
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+M. (2018). New complexes of Ni ( II ) and Co ( III ) with a Schiff-base ligand derived from    
+             O- vanillin . Crystal structure , magnetic and catalytic properties of a. Polyhedron, Ii.  
+             https://doi.org/10.1016/j.poly.2018.05.007. 
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+Sani, S., and Hussain, S. Y. (2017). A Convenient Method to Synthesis and Characterization of  
+              Ni(II) and Zn(II) Schiff Base Complexes. International Journal of Innovative Research  
+            and Development, 6(12), 8–13. https://doi.org/10.24940/ijird/2017/v6/i12/DEC17084 
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+Saranraj, P., and Devi, D. (2018). Effect of Potassium Sorbate On the Inhibition Of growth of Fungi 
+Isolated From Spoiled Bakery Products. Life Science Archives ( LSA ) Review. April 2017.https:// 
+doi. org/10.22192/lsa.2017.3.2.6 
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+Sheyin, F. ., Ndukwe, G. I., Iyun, O. R. A., Anyam, J. V., and Habila, J. D. (2018). Phytochemical    
+             and Anti-Microbial Screening of Crude Extracts of Natal Fig (FicusNatalensisKraus).  
+             Journal of Applied Sciences and Environmental Management, 22(9), 1457.https:// doi.      
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+Y Yan, GX Zhang, B Gran, F Fallarino IDO upregulates regulatory Tcells via tryptophan 
+Catabolized and suppresses encephalitogenic T cell responses in experimental autoimmune 
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+Zhang, Y., Wang, X., and Ding, L. (2010). Interaction between Tryptophan-vanillin Schiff base  
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+ 
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+
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new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf,len=845
+page_content='0  SYNTHESES AND EVALUATION OF Fe2+, Co2+ AND Ni2+ COMPLEXES WITH DERIVATIVES OF VANILLIN – TRYPTOPHAN SCHIFF BASE LIGAND SALISU ABUBAKAR 1, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Shallangwa⁎ 2 and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Abdulkadir Ibrahim2 1Department of Chemistry Federal College of Education, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' 1041, Zaria 2 Department of Chemistry, Ahmadu Bello University, Zaria, Nigeria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='  Corresponding Author’s abubakarsalisu@abu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='ng +2348136135156 Abstract This research focused on synthesis, characterization and determination of biological   activities of Fe2+ Co2+ and Ni2+ complexes of a novel L  Tryptophan  Vanillin schiff base ligand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' They were synthesized and characterized by determination of their Purity, Solubility, Elemental analyses using phase match XRD data, FT  IR spectra, Molar conductivities, Magnetic susceptibilities, PXRD and Biological activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Results from the analyses revealed that, mpt of schiff base ligand (SL) = 84 – 85 oC, Fe2+  SL = 245  246 oC, Co2+  SL= 271  272 oC while Ni2+  SL is >350 oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Molar conductivities were found to be 10300, 5000, 17300 and 52900 Sm2 mol  1 for the schiff base and the complexes respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' It indicates that, the complexes are non  electrolytic in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Magnetic susceptibilities of the complexes found to be higher than the spin only values due to unpaired electrons (Fe2+ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='73 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='M, Co2+= 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='52 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='M while for Ni2+ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='46 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='M, suggesting an octahedral geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Electronic absorption (λmax) for Fe2+ = 480, Co2+ = 520nm while Ni2+ shows two bands at 480 and 570 which signifies, n  π  and π  π  transition respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Their crystalinity index was also assessed using pxrd technique, it shows that complexes are 75, 80 and 85 % crystalline with average crystallite size of 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='63 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Antimicrobial test results revealed that, Co2+ and Fe2+ complexes have excellent activities against gram negative bacteria (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' coli, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Shigella .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Typi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' They all shows efficient activities against some gram positive bacteria such as: MRSA, Klebsiella P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Pnemoniae and some fungi spp A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' niger and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' albicans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Therefore based on the findings from the study it was concluded that, schiff base is bidentate ligand and octahedral geometries, paramagnetic susceptibilities were suggested for the complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Keywords: Schiff base, Complexes, FT  IR Spectra, Characterization, Magnetic susceptibility, Molar conductivity, diffraction, Absorption, Biological activities, Vanillin and Mole ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='0 Introduction Schiff bases from amino acids and flavones such as vanillin and quercetin and their complexes have numerous biological activities (Akhter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=', 2017b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Sani and Hussain, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' The high affinity for the chelation of the schiff bases towards the transition metals is utilized in preparing their solid complexes and due to its polyhydroxylated chemical structure, vanillin easily forms complexes with species having free orbitals which may be occupied (Barbosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' The potentials of vanillin schiff base ligands in inorganic chemistry can be understood very well from the properties exhibited by its complexes such as: magnetics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' luminescence, chirality, catalysis, cytotoxicity and ferro electricity among others (Al-jeboori and Al-shimiesawi, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Neacşu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=', 2018 and Barbosa et - al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Vanillin derivatives are found to be  active against both gram-positive and gram-negative bacterial strains and has been shown to be effective against some fungal spp ( Sheyin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' L-Tryptophan (Trp) is an essential amino acid which is required for the biosynthesis of proteins in the body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' It is very significant for nitrogen balance, maintenance of muscle mass and body weight in humans (Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='1 The Chemistry of Schiff base Formation In reactions that yield schiff base ligands, the electrophilic carbon atom of carbonyl compounds will experience a nucleophilic attack by amines to give a compound with C=O being replaced by C=N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='  The reaction is reversible and occurs as shown in eqn (1) or (2) depending whether the carbonyl is an aldehyde or a ketone ( Usman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=', )  2  Condensation products of vanillin with amines confers biological activities as well as having good complexation ability with metal ions (Saranraj and Devi, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' This study focused on synthesis, characterization and test for antimicrobial activities of a schiff base ligand derived from condensation of o-vanillin with L– tryptophan and complexes of their ligands with Fe2+, Co 2+ and Ni2+ ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Transition metal complexes with bidentate schiff base ligands that contains nitrogen, oxygen, or sulphur donor atoms contribute immensely in biological systems (Malik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Maalik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Numerous researches carried out on transition metal complexes derived from schiff base ligands have pointed out their significance due to their properties and wider biological applications (Nagesh and Mruthyunjayaswamy, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Agbese et al, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Fugu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' (2013) reported that schiff base complexes derived from 4-hydroxysalicylaldehyde and amines have strong anticancer activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Schiff bases of ketones, their derivatives, and their metal complexes are active anticancer and antioxidant agents (Dhanaraj and Johnson, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Transition metal ions are amongst the important species needed to keep human body healthy and active because several important biological functions depend upon their presence and their absence may lead to deficiency diseases (Arvind  and Andreas  2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Mustafa and Alsharif  3  (2018) opined that, most notable metals that exist in human body are in the form of ions such as: Fe2+, Co2+, Ni2+, Ca2+, Cu2+, Zn2+, and Cr2+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Transition metal complexes are observed to be generally more biologically active than the uncomplexed ligands, they are formed due to the presence of the metal moieties (Mohamed and Omar, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Abu Bakar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=', 2010a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='0 Experimental All chemicals used for this research are of analytical grade (AR) obtained from Sigma Aldrich and other reliable chemical vendors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' The chemicals were used as purchased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='1 Materials The following materials were used for the study: Mettler Analytical weighing balance (MT 200B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='0001-200G), spectrophotometer (JENWAY-7315), magnetic stirrer (Stuart), fume cupboard (Biobase), melting point machine (SMP 10 Fascia, Stuart Bibby Scientific), FT-IR spectrophotometer (Nicolet iS5-Thermo scientific), electrical conductivity meter (Hanna Instrument, HI 9835, Woon socket R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' I USA), heating/drying oven (TT-9053, Techmel and Techmel U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='A, YLD 2000) and heating mantle among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='2 Methodology 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='1 Syntheses of Schiff base Ligand The method used to synthesize the ligand was adopted from that of Kaczmarek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' (2009) with minor modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Solution of vanillin (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='30g, 2mmol) was mixed with solution of l- tryptophan (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='4g, 2mmol) in 15 ml ethanol with 1ml dilute sodium hydroxide solution (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='056g, 1mmol) solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='  The reaction mixture was stirred for about 10 minutes and refluxed over a water bath for 5hrs at 60 oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' The coloured precipitate formed was filtered, wash with 50% cold ethanol and rinse with diethyl ether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' It was allowed to dry in a vacuum desiccator to constant weight which  4  yield = 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='0% schiff base ligand (SL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' The yellowish product was recrystallize in methanol for further purification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='3 Syntheses of the Complexes The complexes were prepared using standard methods of (metal: ligands ratio) adopted from Haddad (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Asgharpour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=', (2017) and Hossain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' (2018) with some minor modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' An ethanolic solution of schiff base (20 mmol) was prepared and added drop wise to the solutions of metal (II) chloride (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='0 mmol) FeCl2, CoCl2 and NiCl2 stirred for about 20 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Same concentrations were prepared and refluxed in water bath at 60oC for 6 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' The pH of the mixture was adjusted to about 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='5 with few drops of potassium hydroxide solution and the coloured solid formed were filtered, wash with 50% cold ethanol, and finally dried in a vacuum desiccator to constant weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' The synthetic routes is summarised in equations (3) and (4): Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' 1: A flowchart of Syntheses of Schiff base Ligand and the Metal (II) Complexes  5  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' 2: A flowchart of Syntheses of Metal (II) Complexes  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='4 Physical Measurements Solubilities of the schiff base and metal complexes were checked in water, 50% ethanol and DMSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Melting points of the complexes were also determined to established  their purity using electrically heating melting point apparatus (SMP10, Fascia, Stuart;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Bibby scientific), molar conductivity were assessed to test the electrolytic nature of the complexes (with an Hann scientific Electrical conductivity meter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' The fourier transform infrared spectra were determined with FT-IR spectrometer (Nicolet iS5, Thermo scientific), wave length of maximum absorption (λmax) of the schiff base and the complexes were measured using an aqueous solution of the compounds in uv/visible spectrophotometer (Jenway, 7315) 1cm cuvette at (200 – 780 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Elemental analyses was performed using the match phase crystals analysis software and the report obtained from the PXRD data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Magnetic susceptibility followed by x-rays diffraction technique were used to determine geometries, degree of crystalinity,  and the morphological properties of the complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='5 Assessment of Antimicrobial Activities  6  (i) Minimum Inhibitory Concentration (MIC) The MIC values was determined in accordance with standard method proposed by clinical and laboratory standard institute (CLSI, 2019) using a broth dilution techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Mc Farlands’ turbidity standard was prepared and used for the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' 1ml of sterilized MH- media were added to each of clean, dried and sterilized test-tubes, followed by 1mLof 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='5 MCF bacterial suspension which contained about (1-2×108 CFU/mL bacterial load).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' 1mL of 2000 µg / mL solution of the test compounds was transferred into each test tube and serially diluted to give concentrations of (1000, 500, 250,  125, 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='5, 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='25, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='625 µg/mL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' The solution was added to the mixture followed by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='1 mL of the bacterial suspension in a normal saline solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' The test tubes were incubated at 37 oC for 24 hrs to enhance bacterial growth, test-tubes with lowest and the highest turbidity were examined, compared with standard and the results were carefully recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' (ii) Minimum Bactericidal Concentration (MBC)  The MBC was assessed using the method developed by Hadariah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' (2019) to ascertain whether the test pathogens were completely killed or just inhibited their growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Mueller Hinton agar was sterilized, poured into petri dishes and the media was allowed to cool/solidify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' An aliquots from samples which shows sensitive to the MIC test were carefully selected and subjected for MBC evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' They were sub-cultured and incubated at 28o C for 7 and 14 days in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' To determined CFU, the plates of the media was taken for microscopy and observed for bacterial growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='  7  (iii) Minimum Fungicidal Concentration (MFC) Sarah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' (2016) and CLSI (2019) methods was adopted to carry out this test using micro broth dilution technique with little modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Two fungal spp (Aspergillus Niger and Candida albicans) isolates were used, grown for 7 days at 28 oC incubation period in potato dextrose agar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' The stock suspension was diluted with normal saline solution to corresponding concentration of 1×105 cfu /mL and viability of the fungi spp was confirmed using the sabouraud dextrose agar plates and counting the number of cfu/mL Two fold serially diluted solution of the test compounds were prepared to obtained 200µg/ml, 100µg/ml, 50µg/ml, 25µg/ml and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='5µg/ml respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' A micro pipette was used to transfer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='1ml of the test fungal strain solution into the test-tubes containing the mixture with varied concentrations of the compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' The mixture was incubated at 38 oC for 72 hours after which the turbidity was observed and compared with standard for the growth of the pathogens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' The turbidity (growth) between the test-tube which contained lowest and the highest concentration of the compound were examined and compared with standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' The results were carefully recorded in accordance with clinical and laboratory standard institute CLSI M100, (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='  8  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='1 Results and Discussion Table 1: Shows some Physical Parameters test Results for Schiff base and Metal Complexes  Products  Colour  %Yield  Solubility  Mpt  (o C)  Λm (Sm² mol⁻1)  µeff  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='M)  Vanillin – Tryptophan SBL  Golden  Yellow  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='0  SS (H2O)  SL ( EtOH)  SL DMSO)  84  10300 Vanillin – Tryptophan Fe2+  Greenish  yellow  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='0  SL (H2O)  SS (EtOH)  SL (H2O)  SS (EtOH)  245  5000  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='73  Vanillin – Tryptophan Co2+  Brownish  yellow  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='0  SL (H2O)  SL(EtOH)  SL(DMSO)  271  17300  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='52  Vanillin – Tryptophan Ni2+  Yellowish green,  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='4  SS (H2O)  SL (EtOH)  SL(DMSO)  >350  52900  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='46  Keys: SL = soluble, SS = sparingly soluble  The results in Table 1, indicates colour of the schiff base and the metal complexes, percentage yields and solubilities in some solvents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' It revealed that, schiff base ligand is sparingly soluble in water, but it readily dissolved in DMSO and 50% ethanol solutions solution while the complexes are soluble in water, DMSO respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' It also indicated that, schiff base has lower melting point (84 oC) than the complexes, but Ni2+ complex has the highest melting temperature of (> 350 oC) followed by Cobalt (II) complex (271 oC) while Iron (II) complex has (245 oC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Molar conductivity measurement shows that, both the ligand and the complexes were non electrolytic in nature and magnetic susceptibility measurement obtained at room temperature revealed that, complexes are paramagnetic with observed magnetic moment as: Fe2+ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='73 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='M, Co2+ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='52 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='M while for Ni2+ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='46 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='M respectively which are slightly higher than the spin  9  only values due to unpaired electrons suggesting an octahedral geometry for all the complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' This is similar to the work of Zurowska and Bauskey (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Table 2: UV-visible Data for the Schiff base Ligand and the Complexes S/N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Compounds Wave length λmax  (nm) Transition Assigned Geometries 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Schiff base Ligand (SL) 255 – 350 330 – 380 π   - π⁎ (C =C in benzene) n   - π⁎  (C = N in imine) ---- 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' SL – Fe2+ Complex 460 – 480 π   - π⁎ (C =C in benzene) Octahedral 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' SL – Co 2+ Complex 500 – 520 n  - π⁎ (C = N in imine)  Octahedral 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' SL – Ni2+ Complex 550 – 570 369 – 480 π   - π⁎ (C=N in imine) π  - π⁎  (C=C in benzene) Octahedral  Results in Table 2, shows electronic transitions within the schiff base and the complexes scanned from 300 – 750 nm in ethanolic solutions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' 2-4) which indicates some bands as  n - π⁎ and π - π⁎ transitions from to non-bonding to anti bonding molecular orbitals and delocalization of pi- electrons within the benzene rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' The absorption bands assigned and the proposed geometries of the complexes are as follows: Fe2+ 460 – 480 nm, Co2+ shows only one bands at 500 – 520nm while Ni2+ shows two absorption bands at 369 – 480 and 550 –570nm respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' This result is in agreement with that of Suesh and Pracash (2010) and (Crystal, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' et –al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Table 3: Elemental Analysis of the Ligand and the Complexes  Products % of Elements Found (Calculated)  Formulae  C  H  O  N  Fe  Co  Ni  SBL  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='35  (52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='79)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='12  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='63)  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' 73  (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='90)  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='8  (24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='01) C19H18N2O4  SBL – Fe2+  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='34  (39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='72)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='18  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='96)  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='03  (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='89)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='37  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='65)  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='88  (26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='14) C38 H36N4O8Fe  SBL– Co2+  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='04  (46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='31)  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='71  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='99)  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='36  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='20)  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='01  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='98) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='88  (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='09) C38 H36N4O8Co  SBL – Ni2+  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='23  (45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='18)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='74  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='98)  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='11  (25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='01)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='40  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='85) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='52  (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='90)  C38 H36N4O8Ni  10  Table 3 shows result of elemental analysis and confirmed molecular formula of the schiff base and the metal complexes, it revealed that, the complexes are mononuclear with 2moles of the ligand bonded to 1mole of the metal ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' This implies that, the metal – Ligand ration is 2:1 with molecular formula as [M-(SBL)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' H2O] where SBL = Schiff base ligand, M = Co2+ and Ni+2+ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Table 4: FT-IR Spectral mode (350 – 4000 cm-1) of the Schiff base and the Metal complexes  Complexes  V(HC═N)  V(M⸺N)  V(M⸺O)  V(COO)  V(C⸺O)  V(O⸺H)  V(C ⸺N)  V(C O C)  V (C═C)  SBL  1663 sh 1362 sh 1282 sh 3326 m 1373 sh 1625 m 1521 sh SBL – Fe2+ 1583.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='36 sh 401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='64 sh 535.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='30 sh 535.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='30  br 1026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='32 sh 3208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='51 br 1509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='46 sh SBL– Co2+  1583.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='45 sh  463.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='69 sh  511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='36 sh  588.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='73 sh  1025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='14  3211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='86 br  1506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='96 sh SBL – Ni2+  1586.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='13 sh  419.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='78 sh  385.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='20 sh  410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='36 sh  1017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='85 sh  3362.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='07 br  1630.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='99 sh Keys: SBL = Vanillin – Tryptophan Schiff base Ligand, SBL – Fe2+= Vanillin – Tryptophan Schiff base – Fe2+ complex, SBL– Co2+  = Vanillin – Tryptophan Schiff base – Co 2+  complex, SBL – Ni2+  = Vanillin – Tryptophan Schiff base – Ni2+ complex Results in Table 4 above, shows FT-IR spectra bands of the ligand and the complexes at 1628 v (HCN, str), 382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='47 v(O-H str) and 362.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='12 cm-1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' The FT-IR analysis revealed that, the bonding of the schiff base to the metal centres has occur as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' In the schiff base ligand, a strong sharp band observed at 1663 cm-1 can be assigned to V (HC═N) azomethine stretching vibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' When reacted to form complex with the metal ions, the band shifted to lower frequencies in 401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='64 - 419.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='78 cm-1 range for v(M⸺N) and  385.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='2 - 535.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='30 cm-1 range for v(M⸺O) indicates that, new bonds has formed that, azomethine coordinated to the metal ions through Oxygen and Nitrogen donor atoms in the schiff base ligand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='  11  The symmetric carbonyl stretching v(COO-) is also shifted to a higher frequency from 1362cm-1  to 535.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='30, 588.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='73 and 410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='36cm-1 region for Fe2+, Co2+ and Ni2+ complexes respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Similarly the v (C – O) phenolic band in the complexes shows a significant shift to a lower frequencies of 1282 cm-1 in the ligand, 1026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='32, 1025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='14 and 1017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='85 cm-1 for the metal ions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' The spectral of the complexes shows broad bands between 3208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='51, 3211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='86 and 3362.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='07 cm-1 range which attributed to O–H stretching vibration from phenolic group (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='5,6and7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Information obtained from the FT-IR results of some of the major functional groups in the schiff base and the complexes revealed that, the schiff base is bidentate ligand and it bound to the metal ions through the phenolic oxygen and azomethine nitrogen atoms to form an octahedral complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Table 5: Powder X-rays Diffractions Analysis Complexes Peak position 2θ (o) FWHM β (o) % Crystalinity Index (CI) Crystallite size D (nm) Average D (nm) Fe2+– SL 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='654 75 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='41  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='63 Co2+ - SL 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='678 80 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='32 Ni+2+ – SL 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='545 84 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='16 K = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='94,                   λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='5418 (nm)  Table 5 shows results of PXRD pattern of the complexes scanned at laboratory temperature (303 K) with an XRD machine (RIGAKU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' The degree of crystalinity and crystallite sizes and average crystallite size of the complexes were obtained within the range of 2θ values from 10 – 120o at wavelength of λCuKα1 radiation = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='5418 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' The XRD results further supported by the inter planar distance and the crystallite sizes (d) of 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='41, 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='32 and 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='16 corresponding to Fe2+– SL, Co2+ - SL and Ni+2+ – SL respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='  12  Table 6: Anti-microbial and Antifungal Sensitivity Test for the Schiff base and Complexes  S/N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='  Products  MRSA  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='  Pnemoniae  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Klebsiella  Pneumoniae  ESBL  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Coli  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='Shigella  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Typi  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Alicans  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Nigger 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' SBL R R MS MS MS R MS MS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' SBL – Fe2+ R R S R S R S R 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' SBL – Co2+ MS S S S R R MS S 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' SBL – Ni2+ MS S S S R MS R S 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Standard 1 * - - - MS - MS MS - 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Standard 2 * - - - - - - MS - 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Control - - - - - - - - Key: S = sensitive, MS = mild sensitive,   R = resistance, - = negative (no reaction), C = control, Standard 1* = Augmentin,   Standard 2* = Terbinafine Table 5 shows sensitivity test results of the products against some selected pathogens used for the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' It indicates that, one gram +ve with two gram –ve bacteria and the fungal spp shows mild sensitive reactions to the schiff base ligand, Klebsiela P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=', Shigella and a fungi sp (Candida A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=') are very sensitive to Fe2+ complex, Co2+ complex shows its effectiveness to streptococcus P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=', Klebsiela P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' coli and Aspergillus N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' While Ni2+ complex revealed to be very effective and positive sensitivity test to streptococcus, Klebsiela,  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' coli and Aspergillus N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' fungal sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='  13  Table 7: Shows Zone of Inhibition (Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='I) Test Results of the Schiff base Ligands and their Metal (II) Complexes in (mm)  S/N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='  Compound  MRSA  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Pnemoniae  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Klebsiella  Pneumoniae  ESBL  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Coli  Shigella  spp  Salmonella  spp  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='  Albicans  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Nigger 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' SBL - - 5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='2 7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='2 10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='4 - 8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='3 7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' SBL – Fe2+ - - 22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='1 - 25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='3 - 12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='2 - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' SBL – Co2+ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='0 23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='5 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='0 - - 20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='2 10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' SBL – Ni2+   21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='5 - - 23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='5 - 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='0 23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='2 16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Control - - - - - - - - 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Standard 1 * 11 - 15 13 - 17 - - 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Standard 2 * - - - - - - 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='5 10  Key: S = sensitive, MS = mild sensitive,   R = resistance, - = negative (no reaction), C = control, Standard 1 * = Augmentin,   Standard 2* = Tarbinafin  The data in Table 6 above, shows results for zone of inhibition test of the products against some those pathogens that shows positive sensitivity (MIC) test from Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' It indicates similarly the regions (in mm) where those pathogens are being killed or their growth are inhibited by the compounds as appropriate where the schiff base shows least inhibition zone of (5, 7, 10, 8 and 7mm) followed by Fe(II) complex (12, 22 and 25 mm), Ni2+ (13, 16, 21, and 23 mm)  while Co 2+ complex shows the widest inhibition zone (10, 18, 20, 23, and 26 mm) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' The control agar plate shows negative test result with less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' 1 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' this indicates that, all the microbes grow freely in the absence of the compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='  14  Table 8: Shows Results of minimum Bactericidal (MBC) and Fungicidal Concentrations (MFC) (µg/mL) for the Schiff base, metal complexes and Standard Drugs against the Pathogens used for the study  S/N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='  Products  MRSA  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='  Pnemoniae  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Klebsiella  Pneumoniae  ESBL  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Coli  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='Shigella  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Typi  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='Albicans  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Nigger 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' SBL ++ +++ -- - -- +++ -- -- 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' SBL – Fe2+ ++ ++ --- ++ --- +++ -- + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' SBL – Co2+ --- -- -- - + + --- -- 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' SBL – Ni2+ --- ++ + -- ++ -- -- -- 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Standard1 (mg/mL) +++ ---- ---- ---- +++ ---- + +++ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Standard2 (mg/mL) + +++ + +++ + ++ ---- + 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' C +++ +++ +++ +++ +++ +++ ++ +++  Keys: ++++ = very intense microbial growth +++ = high microbial growth (100 -250 mg/ml),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' ++ = moderate microbial growth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' ++ = mild microbial growth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' - No growth slightly clear at (215 µg/ml),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' -- No growth very clear at (250 µg/ml),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' --- No growth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' very –very clear at (500 µg/ml),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' ---- No growth immediate sharp clear at (1000 µg/ml),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' C = control,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Standard1* = Augmentin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='   Standard 2* = Terbinafine Table 7 shows MBC and MFC results of the test compounds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' from the preliminary in vitro antimicrobial activities of the ligand and the complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' They were screened for their minimum inhibitions concentrations and in vitro antimicrobial actions against the selected microbes (Table 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' The minimum active concentrations measured in (µg/mL) and two standard were used (Augmentin and Terbinafine for fugal spp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' The results revealed that, the complexes showed more significantly active for killing the pathogens than the ligand and the overall activities of all the three complexes were found to be very effectives even at very low concentration of less than (250 µg/mL) is being considered to be the minimum concentrations of the schiff base and complexes required to kill (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='99%) of the pathogens completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='  15  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='2 Summary and Conclusion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='1 Elemental Analysis The result of elemental analysis confirmed the proposed molecular formula of the schiff base and the metal complexes (C19H18N2O4 and MC38 H36N4O8) and it revealed that, the complexes are mononuclear with 2moles of the ligand bonded to 1mole of the metal ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' This implies that, the metal – Ligand ration is 2:1 with molecular formula as [M-(SL)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='H2O] where SBL = Schiff base ligand, M = Co2+ and Ni+2+ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='2 Magnetic Susceptibility Measurement Information regarding geometries of the complexes were obtained from their electronic spectral and magnetic moments and the observed magnetic moment are: Fe2+ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='73 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='M, Co2+ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='52 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='M while for Ni2+ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='46 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='M respectively which are slightly higher than the spin only values due to unpaired electrons suggesting an octahedral geometries for all the complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='3 Conductivity measurement Molar conductivity of the complex in 10-3 moldm-3 DMSO solution were measured at laboratory temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' The values indicated that, the complexes are non-electrolytic in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' The result is in line with the findings of Zahid H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=', and Hazoor A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Shed (2018) and Amadou G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' et- al, (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='4 Spectral Analysis The electronic spectra of the ligand and the complex shows there bands 1628 v (HCN, str), 382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='47 v (O-H str) and 362.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='12 cm-1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='  The FT-IR analysis revealed that, the binding of the schiff base to the metal centres has been obtained by comparing the FT-IR –spectra of the ligand with those of the complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='  16  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='5 Electronic Spectrum The electronic spectrum of the schiff base (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' 2 -4) above, shows a broad band at and their metal complexes were determined from the scanned range of 350 – to – 750 visible region using ethanol as solvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' The absorption regions, bands assigned and the proposed geometries of the complexes are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' The scanned electronic transition for Fe2+ shows two bands at 460 – 480 nm, Co2+ shows only one bands at 500 – 520nm while Ni2+ shows two absorption bands at 369 – 480 and 550 –570nm respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='6: Powder X-rays Diffractions Analysis Degree of crystalinity, crystal size of the complexes were obtained and PXRD pattern scanned in the range of 2θ values from 10 – 120o at wavelength λCuKα1 radiation = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='5418 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' The results revealed that, the complexes are crystalline with high crystalinity index of 72, 80 and 84% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='7 Uv-visible Spectrum of the Schiff base Ligand and the Complexes The electronic spectrum of the schiff base and the complexes shows broad bands, when scanned from 350 – to – 750 visible region in ethanolic solution as are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='0 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='8 Assessment of Biological Activities of the Schiff base and the Metal (II) Complexes The antibacterial activities test were carryout to find the efficacy of the novel schiff base and the complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='  It was done according to standard procedures in vitro using some clinical strains of microbes MRSA, ESBL Klebsiella Pneumoniae, Streptococcus Pnemoniae , ESBL E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' coli, Shigella spp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' and Salmonella sp together with  two fungi spp Aspergillus and Candida albicans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' The test compounds shows ability to inhibit growth of the microbes which was compared with some  17  antibiotics (Augmentin and Terbinafine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' The results revealed that, SL – Ni2+ complex shows highest activity followed by SL-Co2+ and the schiff base respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Comparatively, the inhibition zones of  the SL and the complexes at the different concentrations indicated that, the complexes has better activities than the schiff base ligand used for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='  Fig1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' FT-IR Spectrum of the Schiff base and the Metal (II) Complexes  500  1000  1500  2000  2500  3000  3500  4000  0  50  100  150  200  250  300  350  400  % Transmittance  Wave number (cm- 1)  SBL Ni (II) complex  SBL Co (II) complex  SBL Fe(II) complex  SBL  18  Fig 2: A Staked Absorption Spectra of Pure vanillin, SL and SL – Fe2+ Complex  Fig 3: A Staked Absorption Spectra of Pure vanillin, SL and SL – Co 2+ Complex  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
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+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
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+page_content='4  350  450  550  650  750  850  Absorbance  wave lenght (nm)  SLA 4 Fe (II) Complex  Schiff base Ligand  Pure vanillin  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
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+page_content='8  1  350  450  550  650  750  Absorbance  wave length (nm)  SLA4 Schiff base  SLA 4 Co (II)  Pure Vanilline  19  Fig 4: A Staked Absorption Spectra of Pure vanillin, SL and SL – Ni2+ Complex  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' 5: FT-IR Spectra of Vanillin – Tryptophan Schiff base Ligand  300  400  500  600  700  800  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
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+page_content='6  Absorbance  Wave length (nm)  SL Ni (II) Complex  Vanillin Tryp SL  Pure Vanillin  90  85]  1663.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
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+page_content=' 8: PXRD Diffraction Pattern for Vanillin-Tryptophan-Fe2+ Complex  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' 9: Vanillin – Tryptophan - Co 2+ Complex 0 10 20 30 40 50 60 70 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
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+page_content='6 Peak Count (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' u) 2 theta (deg) Vanillin - Tryp Co (II)  22  0 10 20 30 40 50 60 70 80 90 0 50 100 150 200 250 300 Peak Count (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='u) Vanillin-Tryptophan Nickel (II) Complex (peak count)  23  References Abu Bakar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=', Bahron, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
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+page_content=' (2010b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Synthesis and characterization of a novel Schiff base derived from 2,4,6-trimethyl-m-phenylenediamine with o-vanillin and its metal complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' CSSR 2010 - 2010 International Conference on Science and Social Research, 463–466.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
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+page_content='5773821  Agbese, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=', Shallangwa, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=', and Idris, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Synthesis, Characterization and Antimicrobial Evaluation of Co(II), Ni(II) and Cu(II) Schiff base complexes of (Z)-4-(1-pyridin-4- ylimino)propyl)phenol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
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+page_content=' Alkynyl Coinage Metal Clusters and Complexes–Syntheses, Structures, and Strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
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+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
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+page_content=' Synthesis, characterization, density functional theory studies and antibacterial activity of a new Schiff base dioxomolybdenum(VI) complex with tryptophan as epoxidation catalyst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
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+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
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+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
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+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' De, Carinhanha, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
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+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Characterization and Stability of the Antimony-Quercetin Complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
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+page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
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+page_content='  Nagesh, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
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+page_content=' Synthesis, characterization and Biological relevance of some metal (II) complexes with oxygen, nitrogen and oxygen (ONO) donor Schiff base ligand derived from thiazole and 2-hydroxy-1-naphthaldehyde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Journal of Molecular Structure, 1085(Ii), 198–206 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='org/ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='1016/ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='molstruc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
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+page_content='058  Nathally Claudiane de Souza Santos, Regiane Bertin de Lima Scodro, Eloı´sa Gibin Sampiron, Andressa Lorena Ieque,2 Hayalla Correˆa de Carvalho, Thais da Silva Santos, Luciana Dias Ghiraldi Lopes, Paula Aline Zanetti Campanerut-Sa´, Vera Lucia Dias Siqueira, Katiany Rizzieri Caleffi-Ferracioli, Jorge Juarez Vieira Teixeira and Rosilene Fressatti Cardoso (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Minimum Bactericidal Concentration Techniques in Mycobacterium tuberculosis: A Systematic Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='  26  Neacşu, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
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+page_content=' New complexes of Ni ( II ) and Co ( III ) with a Schiff-base ligand derived from O- vanillin .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Crystal structure , magnetic and catalytic properties of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Polyhedron, Ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='poly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
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+page_content='007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='  Sani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=', and Hussain, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' A Convenient Method to Synthesis and Characterization of Ni(II) and Zn(II) Schiff Base Complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' International Journal of Innovative Research and Development, 6(12), 8–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='24940/ijird/2017/v6/i12/DEC17084  Saranraj, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=', and Devi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Effect of Potassium Sorbate On the Inhibition Of growth of Fungi Isolated From Spoiled Bakery Products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Life Science Archives ( LSA ) Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' April 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='https:// doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
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+page_content='22192/lsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='6  Sheyin, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=', Ndukwe, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=', Iyun, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=', Anyam, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=', and Habila, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Phytochemical and Anti-Microbial Screening of Crude Extracts of Natal Fig (FicusNatalensisKraus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Journal of Applied Sciences and Environmental Management, 22(9), 1457.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='https:// doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='4314/jasem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='v22i9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='  Y Yan, GX Zhang, B Gran, F Fallarino IDO upregulates regulatory Tcells via tryptophan Catabolized and suppresses encephalitogenic T cell responses in experimental autoimmune encephalomyelitis  Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=', Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=', and Ding, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Interaction between Tryptophan-vanillin Schiff base and herring sperm DNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' Journal of the Serbian Chemical Society, 75(9), 1191–1201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content=' https:www //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='2298/JSC100128107Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
+page_content='  27  28' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE4T4oBgHgl3EQfVAyt/content/2301.05021v1.pdf'}
diff --git a/rdFAT4oBgHgl3EQffx02/content/tmp_files/2301.08583v1.pdf.txt b/rdFAT4oBgHgl3EQffx02/content/tmp_files/2301.08583v1.pdf.txt
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@@ -0,0 +1,444 @@
+qBounce : Systematic shifts of transition frequencies of gravitational states of
+ultra-cold neutrons using Ramsey gravity resonance spectroscopy
+Jakob Mickoa,b, J. Bosinab, S. S. Cranganoreb, T. Jenkea, M. Pitschmannb, S. Rocciaa, R.I.P. Sedmikb,
+H. Abeleb
+ILL, Grenoble, ATI TU Wien
+qBounce is using quantum states of ultra-cold neutrons in the gravitational field of the Earth
+to investigate gravitation in the micrometre range. We present current measurements taken
+in 2021 at the Institut Laue-Langevin (ILL) to determine energy differences of these states
+by mechanically induced transitions. This allows a determination of the local acceleration
+g using a quantum measurement. The data presented here results in g = 9.8120(18) m/s2.
+The classical local value at the experiment is gc = 9.8049 m/s2.
+We present an analysis
+of systematic effects that induces shifts of the transition frequency of order 100 mHz. The
+inferred value for g at the experiment shows a systematic shift of δg ≈ 3.9σ.
+1
+Introduction
+Neutrons are excellent probes to test gravity at short distances – electrically neutral and only
+hardly polarizable. Very slow, so-called ultra-cold neutrons (UCN) form quantum states bound
+by the gravity potential of the Earth. This allows combining gravity experiments with powerful
+resonance spectroscopy techniques, as well as tests of the interplay between gravity and quan-
+tum mechanics. Recently, the qBounce collaboration has finished commissioning its Ramsey
+spectrometer at the ultra-cold and very cold neutron facility PF2 at the Institut Laue-Langevin.
+Here, we present first precision measurements, its result and an analysis of systematics effects.
+A schematic overview of the experiment can be seen in Fig. 1. The neutrons have a horizontal
+velocity of vx ≈ 8 m s−1 and a vertical velocity of vz ≈ 0 m s−1. They are trapped by the gravity
+potential and form bound states2. In section I neutrons enter the system on top of a flat surface.
+They form bound states in the gravity potential with energies in the order of peV for details see
+contribution by J. Bosina in this volume. The Fermi pseudo potential of the mirror is ≈ 100 neV
+and the ground state extends ≈ 15 µm above the mirror, with higher states extending further
+up. On top of section I a rough glass plate (absorber) is placed at a distance of l ≈ 25 µm which
+causes high states to be scattered. This scattering happens on the random surface structure
+aInstitut Laue-Langevin (ILL), Grenoble, France
+bAtominstitut, TU Wien, Austria
+arXiv:2301.08583v1  [hep-ex]  20 Jan 2023
+
+hFigure 1 – Schematic view of the Ramsey spectrometer. UCNs enter from the left and are detected on the right.
+The probability density function of the neutron in the setup is shown in false colour.
+and is non specular. The scattered neutrons have a random velocity in the x-y plane and do
+not enter the detector at the end of the experiment. The scattering probability of low state
+neutrons is lower and after section I the wavefunction is in a superposition of the first three
+states with (P(1), P(2), P(3))T ≈ (0.70, 0.25, 0.05)T . A detailed discussion can be found in 3.
+In section II the mirror oscillates with a specific frequency, displacement and phase where the
+surface position is given by zII(t) = aII sin (ωIIt + φII). In Fig. 1 the transition |1⟩ → |6⟩
+is depicted. This is the transition (mainly) investigated in 2021. If the lower and the excited
+state have an equal probability at the end of section II the applied excitation is called a π
+2 -flip.
+Section III is stationary and the neutron propagates without perturbation. In section IV the
+mirror is oscillating with zIV (t) = aIV sin (ωIV t + φIV ), where ωIV and aIV are tuned to be the
+same frequency and displacement as in section II. If sections II and IV are in phase, such that
+φIV = φII the excitation to the higher state is completed. If φIV = φII + π the neutron returns
+to the ground state. In section V the same selection as in section I is performed. This enables us
+to perform Ramsey type spectroscopy using Gravity states a technique called Gravity Resonance
+Spectroscopy (GRS). From data on the measured transition probability the local acceleration of
+the neutron can be determined.
+2
+Theory
+The evolution of the system can be described by the Schr¨odinger equation:
+i¯h∂tψ = ˆHψ
+(1)
+For a linear potential the solutions can be given in terms of shifted Airy functions.
+With
+linearised Newtonian gravity the solutions only depend on the local acceleration g and the mass
+of the neutron mn.
+qBounce involves infinitely many states which makes precise numerical calculations neces-
+sary. The dominant contribution for this dataset is |1⟩ → |6⟩ (and |2⟩ → |7⟩). A simplified two
+state system, the lower energy one is called |p⟩ and the higher one |q⟩ can be solved analytically.
+The detailed derivation of the transition probability from one state to the other can be found
+for example in 1. For this as an illustrating example the probability to remain in the lower state
+is given by
+P(|p⟩ → |p⟩) =
+���aIIaIV − bIIbIV ei(δIV (T+τ)−δIIτ−(χII+χIV )+φIV −φII)���
+2
+,
+ΩR,i =
+�
+δ2
+i + (aωi|Vpq|)2, χi = arctan
+� δi
+ΩR,i
+tan
+�
+τ ΩR,i
+2
+��
+, δi = ωi − ωqp = 2π (νi − ν0)
+ai =
+�
+cos2
+�
+τ Ωr,i
+2
+�
++
+δi
+ΩR,i
+sin2
+�
+τ Ωr,i
+2
+�
+, bi = aωi|Vqp|
+ΩR,i
+sin
+�
+τ Ωr,i
+2
+�
+,
+(2)
+
+Abs
+Abs
+IV0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0
+5
+10
+15
+20
+oscillation strength aω [mm/s]
+transmiss. r [mcps]
+|1> → |6>
+Figure 2 – Determination of the optimal π/2-flip by increasing the oscillation strength aω. ∆φ = 0 in blue and
+∆φ = π in orange. The solid lines show the theory function fitted to the measurements.
+where aωi, ωi = 2πνi, νi =
+ωi
+2π are the oscillation velocity, the angular frequency and the
+frequency for the i-th section respectively, Vqp is the overlap integral for the perturbation, ν0 =
+ωqp
+2π is the transition frequency, τ is the interaction time in sections II and IV and T is the free
+propagation time in section III. Simplifying to the case that both sections oscillate with the
+same strength aωII = aωIV = aω and frequency νII = νIV = ν and ν − ν0 is small, gives
+P(|p⟩ → |p⟩) = 1 − sin2 (aω|Vpq|)
+2
+�
+1 + cos
+�
+2π(ν − ν0)
+�
+T + tan( aω|Vpq|
+2
+τ)
+aω|Vpq|
+2
+�
++ ∆φ
+��
+(3)
+where ∆φ = φIV − φII. In the case that section II and IV are not oscillating with the same
+vibration strength roughly at resonance ν = ν0 eq. (2) for two different vibration strengths
+simplifies to
+P =
+����cos
+�
+τ aωIIVpq
+2
+�
+cos
+�
+τ aωIV Vpq
+2
+�
+− sin
+�
+τ aωIIVpq
+2
+�
+sin
+�
+τ aωIV Vpq
+2
+�
+ei∆φ
+����
+2
+,
+⇒P(|p⟩ → |p⟩) = cos2
+�τVpq
+2
+(aωII ± aωIV )
+�
+,
+(4)
+with the upper sign for ∆φ = 0 and the lower sign for ∆φ = π.
+3
+Experiment
+During the second reactor cycle at PF2 at the ILL in 2021 the transition |1⟩ → |6⟩ was studied
+in detail. As an initial guess the transition frequency was calculated using the classical value
+of g ≈ 9.805 m/s2 gives an energy of E1 = 1.407 peV E6 = 5.428 peV which results in the
+transition frequency ν16 = 972.345 Hz. The oscillating sections were set to the same frequency
+and amplitude with a phase difference of ∆φ = 0 in (4), all of which was measured with an
+interferometer. The oscillation strength was increased until a minimum in the transmission rate
+through the setup was observed. As a consistency check the same frequencies were also measured
+using ∆φ = π. These measurements can be seen in Fig. 2 with the rate given in counts per
+1000 seconds [mcps].
+Then the vibration strength was fixed and the frequency was varied. Again both 0 and π
+phase measurements were taken. The π phase provides an inverted transition probability as can
+be seen in orange in Fig. 2 and Fig. 3.
+This complete dataset can then be fitted to the theory to determine the transition frequency
+ν16. The value for the local acceleration g can be determined from the transition frequency. In
+addition the transition |2⟩ → |7⟩ was measured. This also provides a check on possible shifts
+for a single transition. The theory gives ν27 = 865.821 Hz for the transition frequency. Fitting
+
+860
+880
+900
+920
+940
+960
+980
+1000
+0
+5
+10
+15
+oscillation frequency
+ν [Hz]
+transmiss. r [mcps]
+|1>→|6>&|2>→|7>
+Figure 3 – Frequency sweep at the optimised oscillation strength. The transition frequencies of |1⟩ → |6⟩ and
+|2⟩ → |7⟩ are indicated by the black lines. The solid lines show the theory function fitted to both transitions.
+Table 1: Resulting parameters for the fit to the complete dataset.
+Parameter
+Result
+Stat. error (1σ)
+Units
+P(1)
+0.61
+0.02
+[1]
+P(2)
+0.22
+0.03
+[1]
+P(3)
+0.04
+0.04
+[1]
+Pbg
+0.14
+-
+[1]
+g
+9.812
+0.0018
+m/s2
+r0
+17.27
+0.165
+mcps
+χ2
+74.4
+neff
+73
+both of these results in Fig. 3 with parameters for the fit given in Tab. 1. This results in
+g = 9.8120(18) m/s2 which is 3.94σ off the classical measured value of gc = 9.804 925 m/s2. The
+value for gc was determined in 2021 next to the qBounce experiment.
+4
+Systematic effects
+qBounce uses high precision frequency measurements to determine the transition frequency
+between gravity states. To do this a Rb-clock is used as a frequency standard and an interfer-
+ometer monitors the applied oscillation to the mirrors with a high accuracy. Nevertheless, there
+are frequency shifts observable for such measurements. One advantage of qBounce is the fact
+that the energy splitting of states is dependent on the local acceleration g and not a controllable
+quantity such as an applied B-Field as in NMR. Intrinsical shifts still present for qBounce are
+the Bloch-Siegert shift, the spectator state shift, and Coriolis forces arising from the movement
+of the neutron beam.
+4.1
+Spectator state shift
+Equation (2) uses a two state system for illustrative purposes. In reality there are infinitely
+many states contributing to the transition probability. This leads to a shift of the theoretically
+predicted curve when compared to the simple two state case, called Spectator state shift and
+arises from neighbouring transitions that are weakly addressed by a driven transition 4. It has
+to be evaluated numerically and is taken into account by simulating a 30 state system. For
+|1⟩ → |6⟩ this shift is ∆ν ≈ 0.03 Hz when compared to the simple two state analysis.
+
+4.2
+Bloch-Siegert shift
+In deriving (2) the mirror movement z(t) = a sin(ωt + φ) was split into terms containing e+iωt
+and e−iωt terms. In combination with the Eigenfrequencies of the states there is one fast and
+one slow oscillating term. In the so called rotating wave (RWA) or secular approximation the
+fast oscillating term is dropped. This results in a frequency shift when compared to the full
+model 5. Evaluating this shift for the transition |1⟩ → |6⟩ gives ∆ν ≈ 15 mHz. In the numerical
+30-state simulation the excitation is not split and includes this shift.
+4.3
+Rotation of the Earth
+By far the largest contribution to the acceleration is the centrifugal force at the surface of
+the Earth.
+This is the classical correction necessary going from a stationary to a rotating
+frame. In the case of the qBounce experiment the neutrons are moving as well, which leads
+to an additional shift due to Coriolis forces. All apparent effects in the rotating frame can be
+calculated by transforming the Schr¨odinger equation to a rotating frame 6. This results in the
+Hamiltonian
+ˆH = p2
+2mi
+− L · Ω + V (r)
+(5)
+where p is the momentum vector of the neutron, Ω is the angular velocity of the Earth and L
+is the angular momentum of the neutron. The gravity potential V (r) = − mgr2
+Eg0
+r
+≈ −mgrEg0 +
+mgg0(r − rE) + O
+�
+(r − rE)2�
+where rE is the radius of the Earth, mg the gravitational mass
+of the neutron and g0 is the acceleration on the surface of the Earth. −L · Ω is constant for
+a non accelerated neutron. Expanding the contribution from angular momentum to first order
+L2
+2mir2 ≈ −
+L2
+2mir2
+E +
+L2
+2mir3
+E (r − rE) + O
+�
+(r − rE)2�
+, where L2 is the total angular momentum
+squared. Extracting the linear part and plugging in the expression for L2 gives
+g = g0 −
+L2
+mimgr2
+E
+= g0 − mi
+mg
+�
+2ωvy sin(χ) + rEω2 sin2(χ) + v2
+rE
+�
+(6)
+where ω is the angular velocity of the Earth, χ is the angle of the position of the particle to
+the north pole (χ = π − latitude), vy is the velocity of the neutron in western direction and
+v2 is the square of the velocity in the lab frame. The velocity independent terms are included
+in a stationary measurement of the local gravitational acceleration, whereas the Coriolis force
+applies to the neutrons since they are moving. For qBounce at the ILL in Grenoble we have
+χ ≈ π
+2 , vy ≈ − sin(0.33) 8 m s−1 ≈ 2.6 m s−1, ω ≈
+2π
+86400 ≈ 7.3 × 10−5 s−1 and rE ≈ 6.37 × 106 m.
+Now for mi = mg this gives
+g ≈ g0 −
+�
+−2.6 × 10−4 + 1.68 × 10−2 + 1 × 10−5 m/s2�
+≈ g0 − 1.66 × 10−2 m/s2 .
+(7)
+For UCN with v = 8 m s−1 in the lab frame, the additional contributions from the Coriolis and
+centrifugal force attributed to the proper motion is suppressed by two orders of magnitude.
+4.4
+Phase offset
+From equation (3) an uncertainty δφ in the relative phase ∆φ between section II and IV leads
+directly to a shift in the transition frequency
+δν0 = −
+δφ
+2π
+�
+T +
+tan
+� aωVpq
+2
+τ
+�
+aωVpq
+2
+� .
+(8)
+For a perfectly tuned π/2-flip, aωVpqτ = π
+2 . The average neutron has v ≈ 8 m s−1, T = 0.34
+v
+≈
+0.0425 s and τ = 0.152
+v
+≈ 0.019 s. The relative phase is measured with a decoupled interferometer
+
+which reaches an accuracy of |δφ| ≤ 1° which results in
+δν0 ≤ − δφ
+0.419 ≈ ±4.16 × 10−2 Hz
+(9)
+The gravitational acceleration has been measured by a stationary measurement with a falling
+corner cube. This is the effective acceleration also including the rotation of the Earth. From this
+the movement by the neutron and the quantum nature of the measurement introduce additional
+shifts. A summary of the described systematic shifts can be seen in Tab. 2 where in the last
+line the difference to the corner cube measurement is shown.
+Table 2: A summary of the discussed systematic effects. In the last line the expected deviation from reference
+measurement using a corner cube is given. This is valid only when comparing to the two state approximation, in
+an analysis using a multi-state system some of the shifts are already included.
+Effect
+δg [m/s2]
+δf [Hz]
+Rotation of the Earth (centrifugal)
+−1.6 · 10−2
+−1.06
+Phase offset
+±6.2 · 10−4
+±4.16 · 10−2
+Spectator shift
+4.5 · 10−4
+3 · 10−2
+Coriolis (movement of the neutron)
+2.6 · 10−4
+1.7 · 10−2
+Bloch-Siegert shift
+2.3 · 10−4
+1.5 · 10−2
+Centrifugal (movement of the neutron)
+−1 · 10−5
+−6.6 · 10−4
+Total
+−1.5 · 10−2 ± 6.2 · 10−4
+−1 ± 4.16 · 10−2
+Total (∆ to corner cube measurement)
+9.3 · 10−4 ± 6.2 · 10−4
+6.1 · 10−2 ± 4.16 · 10−2
+5
+Conclusion
+The qBounce experiment has used Ramsey GRS to probe the gravity states of neutrons
+bound to the surface of mirrors using mechanical oscillations to induce transitions. The tran-
+sition |1⟩ → |6⟩ was observed in detail together with |2⟩ → |7⟩.
+A transition frequency of
+ν16 = 972.81(12) Hz was determined which corresponds to g = 9.8120(18) m/s2. A selection of
+systematic effects contributing to the transition frequency was described and their magnitude
+is given in Fig. 2. The value found for the local acceleration determined by qBounce for this
+dataset as compared to a classical measurement of gc = 9.8049 m/s2 using a falling corner cube
+deviates by a systematic offset of δg ≈ 3.94σ.
+Acknowledgments
+We acknowledge support from the Austrian Fonds zur F¨orderung der Wissenschaftlichen Forschung
+(FWF-P33279). In addition we would like to thank the technical support of Thomas Brenner
+from the ILL staff for their invaluable assistance during the measurements. The data is available
+following the ILL datapolicy at https://doi.ill.fr/10.5291/ILL-DATA.3-14-412.
+References
+1. T. Rechberger, Ramsey spectroscopy of gravitationally bound quantum states of ultracold
+neutrons Diss. TU Wien (2018).
+2. V. V. Nesvizhevsky et al., Nature 415, 297 (2002).
+3. L. A. Chizhova et al., Phys. Rev. E 89, 032907 (2014).
+4. S. Baeßler et al., Phys. Rev. D 91, 042006 (2015).
+5. J. H. Shirley, JAP 34, 783 (1963).
+6. J. Anandan and J. Suzuki, Quantum Mechanics in a Rotating Frame. in Springer ISBN
+978-90-481-6541-8 , 361 ff. (2004).
+
diff --git a/rdFAT4oBgHgl3EQffx02/content/tmp_files/load_file.txt b/rdFAT4oBgHgl3EQffx02/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,251 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf,len=250
+page_content='qBounce : Systematic shifts of transition frequencies of gravitational states of  ultra-cold neutrons using Ramsey gravity resonance spectroscopy  Jakob Mickoa,b, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' Bosinab, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' Cranganoreb, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' Jenkea, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' Pitschmannb, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' Rocciaa, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' Sedmikb,  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' Abeleb  ILL, Grenoble, ATI TU Wien  qBounce is using quantum states of ultra-cold neutrons in the gravitational field of the Earth  to investigate gravitation in the micrometre range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' We present current measurements taken  in 2021 at the Institut Laue-Langevin (ILL) to determine energy differences of these states  by mechanically induced transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' This allows a determination of the local acceleration  g using a quantum measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' The data presented here results in g = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='8120(18) m/s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  The classical local value at the experiment is gc = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='8049 m/s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  We present an analysis  of systematic effects that induces shifts of the transition frequency of order 100 mHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' The  inferred value for g at the experiment shows a systematic shift of δg ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='9σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  1  Introduction  Neutrons are excellent probes to test gravity at short distances – electrically neutral and only  hardly polarizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' Very slow, so-called ultra-cold neutrons (UCN) form quantum states bound  by the gravity potential of the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' This allows combining gravity experiments with powerful  resonance spectroscopy techniques, as well as tests of the interplay between gravity and quan-  tum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' Recently, the qBounce collaboration has finished commissioning its Ramsey  spectrometer at the ultra-cold and very cold neutron facility PF2 at the Institut Laue-Langevin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  Here, we present first precision measurements, its result and an analysis of systematics effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  A schematic overview of the experiment can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' The neutrons have a horizontal  velocity of vx ≈ 8 m s−1 and a vertical velocity of vz ≈ 0 m s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' They are trapped by the gravity  potential and form bound states2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' In section I neutrons enter the system on top of a flat surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  They form bound states in the gravity potential with energies in the order of peV for details see  contribution by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' Bosina in this volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' The Fermi pseudo potential of the mirror is ≈ 100 neV  and the ground state extends ≈ 15 µm above the mirror, with higher states extending further  up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' On top of section I a rough glass plate (absorber) is placed at a distance of l ≈ 25 µm which  causes high states to be scattered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' This scattering happens on the random surface structure  aInstitut Laue-Langevin (ILL), Grenoble, France  bAtominstitut, TU Wien, Austria  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='08583v1  [hep-ex]  20 Jan 2023  hFigure 1 – Schematic view of the Ramsey spectrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' UCNs enter from the left and are detected on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  The probability density function of the neutron in the setup is shown in false colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  and is non specular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' The scattered neutrons have a random velocity in the x-y plane and do  not enter the detector at the end of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' The scattering probability of low state  neutrons is lower and after section I the wavefunction is in a superposition of the first three  states with (P(1), P(2), P(3))T ≈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='70, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='05)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' A detailed discussion can be found in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  In section II the mirror oscillates with a specific frequency, displacement and phase where the  surface position is given by zII(t) = aII sin (ωIIt + φII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' 1 the transition |1⟩ → |6⟩  is depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' This is the transition (mainly) investigated in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' If the lower and the excited  state have an equal probability at the end of section II the applied excitation is called a π  2 -flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  Section III is stationary and the neutron propagates without perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' In section IV the  mirror is oscillating with zIV (t) = aIV sin (ωIV t + φIV ), where ωIV and aIV are tuned to be the  same frequency and displacement as in section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' If sections II and IV are in phase, such that  φIV = φII the excitation to the higher state is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' If φIV = φII + π the neutron returns  to the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' In section V the same selection as in section I is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' This enables us  to perform Ramsey type spectroscopy using Gravity states a technique called Gravity Resonance  Spectroscopy (GRS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' From data on the measured transition probability the local acceleration of  the neutron can be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  2  Theory  The evolution of the system can be described by the Schr¨odinger equation:  i¯h∂tψ = ˆHψ  (1)  For a linear potential the solutions can be given in terms of shifted Airy functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  With  linearised Newtonian gravity the solutions only depend on the local acceleration g and the mass  of the neutron mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  qBounce involves infinitely many states which makes precise numerical calculations neces-  sary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' The dominant contribution for this dataset is |1⟩ → |6⟩ (and |2⟩ → |7⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' A simplified two  state system, the lower energy one is called |p⟩ and the higher one |q⟩ can be solved analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  The detailed derivation of the transition probability from one state to the other can be found  for example in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' For this as an illustrating example the probability to remain in the lower state  is given by  P(|p⟩ → |p⟩) =  ���aIIaIV − bIIbIV ei(δIV (T+τ)−δIIτ−(χII+χIV )+φIV −φII)���  2  ,  ΩR,i =  �  δ2  i + (aωi|Vpq|)2, χi = arctan  � δi  ΩR,i  tan  �  τ ΩR,i  2  ��  , δi = ωi − ωqp = 2π (νi − ν0)  ai =  �  cos2  �  τ Ωr,i  2  �  +  δi  ΩR,i  sin2  �  τ Ωr,i  2  �  , bi = aωi|Vqp|  ΩR,i  sin  �  τ Ωr,i  2  �  ,  (2)  Abs  Abs  IV0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='0  0  5  10  15  20  oscillation strength aω [mm/s]  transmiss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' r [mcps]  |1> → |6>  Figure 2 – Determination of the optimal π/2-flip by increasing the oscillation strength aω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' ∆φ = 0 in blue and  ∆φ = π in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' The solid lines show the theory function fitted to the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  where aωi, ωi = 2πνi, νi =  ωi  2π are the oscillation velocity, the angular frequency and the  frequency for the i-th section respectively, Vqp is the overlap integral for the perturbation, ν0 =  ωqp  2π is the transition frequency, τ is the interaction time in sections II and IV and T is the free  propagation time in section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' Simplifying to the case that both sections oscillate with the  same strength aωII = aωIV = aω and frequency νII = νIV = ν and ν − ν0 is small, gives  P(|p⟩ → |p⟩) = 1 − sin2 (aω|Vpq|)  2  �  1 + cos  �  2π(ν − ν0)  �  T + tan( aω|Vpq|  2  τ)  aω|Vpq|  2  �  + ∆φ  ��  (3)  where ∆φ = φIV − φII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' In the case that section II and IV are not oscillating with the same  vibration strength roughly at resonance ν = ν0 eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' (2) for two different vibration strengths  simplifies to  P =  ����cos  �  τ aωIIVpq  2  �  cos  �  τ aωIV Vpq  2  �  − sin  �  τ aωIIVpq  2  �  sin  �  τ aωIV Vpq  2  �  ei∆φ  ����  2  ,  ⇒P(|p⟩ → |p⟩) = cos2  �τVpq  2  (aωII ± aωIV )  �  ,  (4)  with the upper sign for ∆φ = 0 and the lower sign for ∆φ = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  3  Experiment  During the second reactor cycle at PF2 at the ILL in 2021 the transition |1⟩ → |6⟩ was studied  in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' As an initial guess the transition frequency was calculated using the classical value  of g ≈ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='805 m/s2 gives an energy of E1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='407 peV E6 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='428 peV which results in the  transition frequency ν16 = 972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='345 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' The oscillating sections were set to the same frequency  and amplitude with a phase difference of ∆φ = 0 in (4), all of which was measured with an  interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' The oscillation strength was increased until a minimum in the transmission rate  through the setup was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' As a consistency check the same frequencies were also measured  using ∆φ = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' These measurements can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' 2 with the rate given in counts per  1000 seconds [mcps].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  Then the vibration strength was fixed and the frequency was varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' Again both 0 and π  phase measurements were taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' The π phase provides an inverted transition probability as can  be seen in orange in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' 2 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  This complete dataset can then be fitted to the theory to determine the transition frequency  ν16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' The value for the local acceleration g can be determined from the transition frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' In  addition the transition |2⟩ → |7⟩ was measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' This also provides a check on possible shifts  for a single transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' The theory gives ν27 = 865.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='821 Hz for the transition frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' Fitting  860  880  900  920  940  960  980  1000  0  5  10  15  oscillation frequency  ν [Hz]  transmiss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' r [mcps]  |1>→|6>&|2>→|7>  Figure 3 – Frequency sweep at the optimised oscillation strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' The transition frequencies of |1⟩ → |6⟩ and  |2⟩ → |7⟩ are indicated by the black lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' The solid lines show the theory function fitted to both transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  Table 1: Resulting parameters for the fit to the complete dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  Parameter  Result  Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' error (1σ)  Units  P(1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='61  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='02  [1]  P(2)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='22  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='03  [1]  P(3)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='04  [1]  Pbg  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='14 [1]  g  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='812  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='0018  m/s2  r0  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='165  mcps  χ2  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='4  neff  73  both of these results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' 3 with parameters for the fit given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' This results in  g = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='8120(18) m/s2 which is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='94σ off the classical measured value of gc = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='804 925 m/s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' The  value for gc was determined in 2021 next to the qBounce experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  4  Systematic effects  qBounce uses high precision frequency measurements to determine the transition frequency  between gravity states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' To do this a Rb-clock is used as a frequency standard and an interfer-  ometer monitors the applied oscillation to the mirrors with a high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' Nevertheless, there  are frequency shifts observable for such measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' One advantage of qBounce is the fact  that the energy splitting of states is dependent on the local acceleration g and not a controllable  quantity such as an applied B-Field as in NMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' Intrinsical shifts still present for qBounce are  the Bloch-Siegert shift, the spectator state shift, and Coriolis forces arising from the movement  of the neutron beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='1  Spectator state shift  Equation (2) uses a two state system for illustrative purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' In reality there are infinitely  many states contributing to the transition probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' This leads to a shift of the theoretically  predicted curve when compared to the simple two state case, called Spectator state shift and  arises from neighbouring transitions that are weakly addressed by a driven transition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' It has  to be evaluated numerically and is taken into account by simulating a 30 state system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' For  |1⟩ → |6⟩ this shift is ∆ν ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='03 Hz when compared to the simple two state analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='2  Bloch-Siegert shift  In deriving (2) the mirror movement z(t) = a sin(ωt + φ) was split into terms containing e+iωt  and e−iωt terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' In combination with the Eigenfrequencies of the states there is one fast and  one slow oscillating term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' In the so called rotating wave (RWA) or secular approximation the  fast oscillating term is dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' This results in a frequency shift when compared to the full  model 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' Evaluating this shift for the transition |1⟩ → |6⟩ gives ∆ν ≈ 15 mHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' In the numerical  30-state simulation the excitation is not split and includes this shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='3  Rotation of the Earth  By far the largest contribution to the acceleration is the centrifugal force at the surface of  the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  This is the classical correction necessary going from a stationary to a rotating  frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' In the case of the qBounce experiment the neutrons are moving as well, which leads  to an additional shift due to Coriolis forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' All apparent effects in the rotating frame can be  calculated by transforming the Schr¨odinger equation to a rotating frame 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' This results in the  Hamiltonian  ˆH = p2  2mi  − L · Ω + V (r)  (5)  where p is the momentum vector of the neutron, Ω is the angular velocity of the Earth and L  is the angular momentum of the neutron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' The gravity potential V (r) = − mgr2  Eg0  r  ≈ −mgrEg0 +  mgg0(r − rE) + O  �  (r − rE)2�  where rE is the radius of the Earth, mg the gravitational mass  of the neutron and g0 is the acceleration on the surface of the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' −L · Ω is constant for  a non accelerated neutron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' Expanding the contribution from angular momentum to first order  L2  2mir2 ≈ −  L2  2mir2  E +  L2  2mir3  E (r − rE) + O  �  (r − rE)2�  , where L2 is the total angular momentum  squared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' Extracting the linear part and plugging in the expression for L2 gives  g = g0 −  L2  mimgr2  E  = g0 − mi  mg  �  2ωvy sin(χ) + rEω2 sin2(χ) + v2  rE  �  (6)  where ω is the angular velocity of the Earth, χ is the angle of the position of the particle to  the north pole (χ = π − latitude), vy is the velocity of the neutron in western direction and  v2 is the square of the velocity in the lab frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' The velocity independent terms are included  in a stationary measurement of the local gravitational acceleration, whereas the Coriolis force  applies to the neutrons since they are moving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' For qBounce at the ILL in Grenoble we have  χ ≈ π  2 , vy ≈ − sin(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='33) 8 m s−1 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='6 m s−1, ω ≈  2π  86400 ≈ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='3 × 10−5 s−1 and rE ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='37 × 106 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  Now for mi = mg this gives  g ≈ g0 −  �  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='6 × 10−4 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='68 × 10−2 + 1 × 10−5 m/s2�  ≈ g0 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='66 × 10−2 m/s2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  (7)  For UCN with v = 8 m s−1 in the lab frame, the additional contributions from the Coriolis and  centrifugal force attributed to the proper motion is suppressed by two orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='4  Phase offset  From equation (3) an uncertainty δφ in the relative phase ∆φ between section II and IV leads  directly to a shift in the transition frequency  δν0 = −  δφ  2π  �  T +  tan  � aωVpq  2  τ  �  aωVpq  2  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  (8)  For a perfectly tuned π/2-flip, aωVpqτ = π  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' The average neutron has v ≈ 8 m s−1, T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='34  v  ≈  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='0425 s and τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='152  v  ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='019 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' The relative phase is measured with a decoupled interferometer  which reaches an accuracy of |δφ| ≤ 1° which results in  δν0 ≤ − δφ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='419 ≈ ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='16 × 10−2 Hz  (9)  The gravitational acceleration has been measured by a stationary measurement with a falling  corner cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' This is the effective acceleration also including the rotation of the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' From this  the movement by the neutron and the quantum nature of the measurement introduce additional  shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' A summary of the described systematic shifts can be seen in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' 2 where in the last  line the difference to the corner cube measurement is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  Table 2: A summary of the discussed systematic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' In the last line the expected deviation from reference  measurement using a corner cube is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' This is valid only when comparing to the two state approximation, in  an analysis using a multi-state system some of the shifts are already included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  Effect  δg [m/s2]  δf [Hz]  Rotation of the Earth (centrifugal)  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='6 · 10−2  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='06  Phase offset  ±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='2 · 10−4  ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='16 · 10−2  Spectator shift  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='5 · 10−4  3 · 10−2  Coriolis (movement of the neutron)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='6 · 10−4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='7 · 10−2  Bloch-Siegert shift  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='3 · 10−4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='5 · 10−2  Centrifugal (movement of the neutron)  −1 · 10−5  −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='6 · 10−4  Total  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='5 · 10−2 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='2 · 10−4  −1 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='16 · 10−2  Total (∆ to corner cube measurement)  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='3 · 10−4 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='2 · 10−4  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='1 · 10−2 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='16 · 10−2  5  Conclusion  The qBounce experiment has used Ramsey GRS to probe the gravity states of neutrons  bound to the surface of mirrors using mechanical oscillations to induce transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' The tran-  sition |1⟩ → |6⟩ was observed in detail together with |2⟩ → |7⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  A transition frequency of  ν16 = 972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='81(12) Hz was determined which corresponds to g = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='8120(18) m/s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' A selection of  systematic effects contributing to the transition frequency was described and their magnitude  is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' The value found for the local acceleration determined by qBounce for this  dataset as compared to a classical measurement of gc = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='8049 m/s2 using a falling corner cube  deviates by a systematic offset of δg ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='94σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  Acknowledgments  We acknowledge support from the Austrian Fonds zur F¨orderung der Wissenschaftlichen Forschung  (FWF-P33279).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' In addition we would like to thank the technical support of Thomas Brenner  from the ILL staff for their invaluable assistance during the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' The data is available  following the ILL datapolicy at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='ill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='fr/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='5291/ILL-DATA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='3-14-412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' Rechberger, Ramsey spectroscopy of gravitationally bound quantum states of ultracold  neutrons Diss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' TU Wien (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' Nesvizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=', Nature 415, 297 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' Chizhova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' E 89, 032907 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' Baeßler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' D 91, 042006 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' Shirley, JAP 34, 783 (1963).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' Anandan and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' Suzuki, Quantum Mechanics in a Rotating Frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' in Springer ISBN  978-90-481-6541-8 , 361 ff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
+page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFAT4oBgHgl3EQffx02/content/2301.08583v1.pdf'}
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+eVAE: Evolutionary Variational Autoencoder
+Zhangkai Wu,1 Longbing Cao, 1 Lei Qi 2
+1 University of Technology Sydney
+2 Southeast University
+berenwu1938@gmail.com, Longbing.Cao@uts.edu.au, qilei@seu.edu.cn
+Abstract
+The surrogate loss of variational autoencoders (VAEs) poses
+various challenges to their training, inducing the imbalance
+between task fitting and representation inference. To avert
+this, the existing strategies for VAEs focus on adjusting the
+tradeoff by introducing hyperparameters, deriving a tighter
+bound under some mild assumptions, or decomposing the
+loss components per certain neural settings. VAEs still suffer
+from uncertain tradeoff learning.We propose a novel evolu-
+tionary variational autoencoder (eVAE) building on the vari-
+ational information bottleneck (VIB) theory and integrative
+evolutionary neural learning. eVAE integrates a variational
+genetic algorithm into VAE with variational evolutionary op-
+erators including variational mutation, crossover, and evolu-
+tion. Its inner-outer-joint training mechanism synergistically
+and dynamically generates and updates the uncertain tradeoff
+learning in the evidence lower bound (ELBO) without addi-
+tional constraints. Apart from learning a lossy compression
+and representation of data under the VIB assumption, eVAE
+presents an evolutionary paradigm to tune critical factors of
+VAEs and deep neural networks and addresses the premature
+convergence and random search problem by integrating evo-
+lutionary optimization into deep learning. Experiments show
+that eVAE addresses the KL-vanishing problem for text gen-
+eration with low reconstruction loss, generates all disentan-
+gled factors with sharp images, and improves the image gen-
+eration quality,respectively. eVAE achieves better reconstruc-
+tion loss, disentanglement, and generation-inference balance
+than its competitors.
+Introduction
+Variational autoencoders (Kingma and Welling 2013)
+(VAEs) have attracted significant interest for their capability
+of learning continuous and smooth distributions from obser-
+vations by integrating probabilistic and deep neural learning
+principles. However, VAEs still face some significant issues,
+including dynamic uncertainty learning during variational
+inference and the tradeoff between representation compres-
+sion and task fitting, despite various recent VAE mecha-
+nisms and variants. Below, we briefly analyze these issues
+and the gaps of existing solutions. A novel evolutionary VAE
+(eVAE) framework is then introduced to address some of
+these issues and gaps.
+VAEs and Gap Analysis
+VAEs have demonstrated sig-
+nificant advantages of incorporating prior knowledge, map-
+ping inputs to probabilistic representations, and approximat-
+ing the likelihood of outputs. The integration of stochastic
+gradient variational Bayes (SGVB) estimator (Kingma and
+Welling 2013) with neural settings learns a narrow proba-
+bilistic latent space to infer more representative attributes
+in the hidden space. VAEs have been applied in various
+domains, including time series forecasting (Fortuin et al.
+2019), out-of-domain detection in images (Nalisnick et al.
+2019; Ran et al. 2022; Havtorn et al. 2021; Xiao, Yan, and
+Amit 2020), generating images with spiking signals (Ka-
+mata, Mukuta, and Harada 2022), and generating text by
+language modeling (Zhang et al. 2019).
+Theoretically, the bound optimization of variational infer-
+ence was introduced to substitute the log-likelihood function
+with a surrogate function to make it optimized by gradient
+descent. In practice, the evidence lower bound (ELBO) can-
+not approach the maximum of conditional likelihood with a
+close gap between posterior and prior. It causes dynamic un-
+certainty during inference and failed tradeoff between repre-
+sentation robustness and reconstruction effectiveness. Weak
+KL divergence could incur KL vanishing while strong KL
+divergence could result in bad likelihood. The tradeoff is
+sensitive to posterior distribution disjoint, data characteris-
+tics, and network architectures (Bozkurt et al. 2021).
+Recently, various techniques have been proposed to ad-
+dress these issues. The first set aims to adjust the balance
+in objective functions. Examples are β-VAE incorporating
+a hyperparameter β (Higgins et al. 2017), InfoVAE adding
+a scaling parameter to the KL term (Zhao, Song, and Er-
+mon 2019), SA-VAE having a cyclical annealing schedule
+to progressively increase β for reducing KL vanishing (Fu
+et al. 2019), and ControlVAE introducing the proportional-
+integral-derivative (PID) control to tune the hyperparame-
+ter (Shao et al. 2020). They are partial solutions only ad-
+justing one part of the objectives, failing to weigh and re-
+solve the balance issue in dynamic settings. Another set is
+to tune network architectures or VAE settings, e.g., VQ-
+VAE and their variants (Razavi, Van den Oord, and Vinyals
+2019; Berliner et al. 2022), spiking VAE (Kamata, Mukuta,
+and Harada 2022), multimodal VAE (Sutter, Daunhawer,
+and Vogt 2021), mixture-of-experts multimodal VAE (Shi
+et al. 2019) and tuning of their ELBO (Kviman et al. 2022;
+Jankowiak and Phan 2022) building on the product of ex-
+perts structure. However, these models are subject to specific
+arXiv:2301.00011v1  [cs.NE]  1 Jan 2023
+
+tasks or data settings.
+eVAE From the information bottleneck perspective, VAEs
+can be treated as a lossy information compression process.
+Adjusting the KL within a range controls the information
+bottleneck flowing from representing latent variables to re-
+constructing samples, thus benefiting the trade-off between
+compression and reconstruction (Alemi et al. 2017; Burgess
+et al. 2018). This variational information bottleneck inspires
+us to improve the VAE mechanism and objective function.
+To directly address the generation-inference balance in an
+evolving dynamic setting without constraints on networks
+and inputs, we introduce a novel evolutionary VAE (eVAE).
+eVAE incorporates and integrates variational evolutionary
+learning and variational information bottleneck into VAE to
+achieve better optimum exploration and the tradeoff between
+representation compression and generation fitting. However,
+integrating evolutionary learning into VAE is an open topic.
+We propose the variational genetic algorithm into eVAE to
+optimize VAE objectives and its modeling exploration in an
+evolving manner.
+Consequently, eVAE dynamically optimizes the VAE in-
+ference (the KL terms) in an evolutionary and probabilistic
+manner, which further tunes the VAE generation toward bal-
+ancing generation and inference. To avert premature conver-
+gence, eVAE introduces probabilistic chromosome selection
+for smooth search. To avoid exhaustive random search, we
+apply the simulated binary crossover (Deb and Beyer 2001)
+and Cauchy distributional mutation to guide the training to-
+ward a stable convergence. The main contributions of this
+work are as follows:
+• We propose the eVAE model to balance the tradeoff be-
+tween generation and inference. Specifically, a Cauchy-
+based variational mutation operator and an SBX-based
+variational crossover operator are proposed to train a
+model with an evolving inner-outer-joint training algo-
+rithm, tackling the premature convergence and random
+search problem when optimizing deep models assisted
+with a variational genetic algorithm.
+• We illustrate eVAE by deriving the lower bound on β-
+VAE related models under the information bottleneck
+theory. We analyze the information flow in the cor-
+responding lower bound per the rate-distortion theory
+to show that only the iteration-specific lower bound in
+eVAE can capture the optimization trend to train effec-
+tively.
+• We evaluate eVAE on three tasks: disentangled represen-
+tation learning on dSprites, image generation on CelebA,
+and text generation on PTB corpus, respectively.
+eVAE forms the first evolutionary VAE framework with-
+out empowering constraints on VAE architectures, input set-
+tings, and objective function (ELBO). It integrates varia-
+tional encoding and decoding, information bottleneck, and
+evolutionary learning into the deep neural framework. The
+evaluation against the state-of-the-art VAEs shows that
+eVAE achieves better disentanglement, reconstruction loss,
+and solve the KL vanishing effectively.
+VAE Background and Issues
+Here, we briefly introduce the background of VAEs by fo-
+cusing on their objectives and approaches to addressing the
+inference-generation balance since they are mostly relevant
+to this work. Other developments on VAEs such as adjusting
+VAE network structures, input settings, and learning con-
+straints are excluded due to their irrelevance to this work.
+VAE
+VAEs mitigate autoencoder issues like sparse repre-
+sentation (Tolstikhin et al. 2017) by learning continuous and
+smooth representation distribution p(x), x ∈ X from obser-
+vations X over latent variables z. After learning an encoding
+distribution qφ(z | x) in encoding neural networks, VAEs
+apply variational inference to approximate the posterior dis-
+tribution pθ(x | z). Learning tasks such as reconstructed and
+generated outputs can then be sampled from this learned dis-
+tribution in a generative process. With the SGVB estimator
+and reparameterization trick, the gradients become tractable,
+and the generative parameters θ and inference parameters φ
+are learnable. The objectives of VAEs can be converted to
+ELBO with the expectation over empirical distribution pdata
+of the data towards both reconstruction LR and inference LI
+(Esmaeili et al. 2019; Ghosh et al. 2019).
+LELBO = Ex∼pdata
+�
+Eqφ(z|x) [log pθ(x | z)]
+−KL (qφ(z | x)∥p(z))]
+= Ex∼pdata LR + LI
+(1)
+VAE Issues
+To solve ELBO, the inference model qφ(z|x)
+can be trained jointly by maximizing the ELBO to acquire
+reasonable compression for task fitting. However, a weak ca-
+pacity of the decoder and the variety of data could make
+the expressive posterior favor task fitting rather than opti-
+mal inference (Zhao, Song, and Ermon 2019). For exam-
+ple, in variational language generation, the decoder built on
+autoregressive models such as LSTM and PixelCNN can
+generate language samples by the autoregressive property
+rather than the posterior-based latent variables (Wu, Wang,
+and Zhang 2019). The VAE degenerates to an autoregres-
+sive model where the KL divergence between posterior and
+prior reaches zero quickly during training. This results in
+KL vanishing and poor generalization in test for the lack of
+diversity. The approaches of learning orthogonal transfor-
+mation of priors with the same distribution by the decoder
+(Khemakhem et al. 2020; Mita, Filippone, and Michiardi
+2021) may sacrifice accurate inference in optimal represen-
+tation and generalization of fitting the data. Other research
+attributes the training conflict to the inherent property of
+bound optimization. For example, under a solid factorial as-
+sumption about the posterior distribution (Locatello et al.
+2019), i.e.,
+p(x, z) = p(x | z)
+�
+i
+p (zi) ,
+(2)
+the ELBO constraining the variational samples favors the
+data fitting (Burda, Grosse, and Salakhutdinov 2016) but
+fails to maximize the probability mass on log-likelihood. In
+addition, the vanilla VAE optimizer strengthens the disjoint-
+ness between qφ(z|xi), i.e., µi → ∞, σi → 0+, to separate
+
+the log-likelihood concentrated on each sample, resulting in
+maximizing the mass of joint distribution (Zhao, Song, and
+Ermon 2019).
+VAE Enhancement
+One way to address the above issues
+is to tighten the log-likelihood lower bound for correct vari-
+ational approximation in posterior (Alemi et al. 2018; Wang
+and Wang 2019; Domke and Sheldon 2019; Jankowiak and
+Phan 2022; Ruiz et al. 2021; Kviman et al. 2022), prior
+(Tomczak and Welling 2018; Klushyn et al. 2019) and de-
+composition of ELBO (Zhao, Song, and Ermon 2019; Es-
+maeili et al. 2019) under some mild assumptions. For exam-
+ple, β-VAE (Higgins et al. 2017) adds the hyper-parameter
+β to weigh the LKL term. Then, ELBO minimizes LR to the
+convergence of data fitting collectively with the regulariza-
+tion LI by varying β:
+LELBO =Ex∼pdata
+�
+Eqφ(z|x) [log pθ(x | z)]
+−βDKL (qφ(z | x)∥p(z))]
+=Ex∼pdata [LR + βLI]
+(3)
+β-VAE introduces some fundamental limitations, which
+trigger various follow-up research. InfoVAE introduces a
+scaling parameter λ on the KL-term and converts the ob-
+jective to (Zhao, Song, and Ermon 2019):
+LELBO =αIq(x; z) − DKL (qφ(z | x)∥p(z))
+− Eq(z) [DKL (qφ(z | x)∥pθ(x∥z))]
+(4)
+where Iq is the mutual information with weight α and α +
+λ − 1 = 0. This linear tuning on the KL shows limitation to
+dynamic uncertainty.
+Further, conditional VAE (CVAE) (Sohn, Lee, and Yan
+2015) introduces an initial guess as a conditional variable
+into the objective function for multimodal data. The SA-
+VAE involves a cyclical annealing schedule to split the train-
+ing to multiple cycles starting at β = 0 and progressively
+increases β until β = 1 to reduce the KL vanishing (Fu et al.
+2019). In ControlVAE, the PID control compares the KL di-
+vergence with a set point, with their difference as feedback
+to the controller to tune the hyperparameter β(t) (Shao et al.
+2020). ControlVAE thus optimizes KL dynamically but is
+constrained by the PID controller which follows a separate
+tuning mechanism from the VAE itself.
+The existing work leaves gaps for building an approx-
+imate weight allocation between reconstruction and infer-
+ence, tuning external hypberparameters within the VAE
+working mechanism and handling these issues in a dynamic
+manner over an evolutionary learning process. Our eVAE
+addresses these gaps by incorporating the variational genetic
+learning into balancing inference and generation and evolu-
+tionarily involving their effect into adjusting the VAE learn-
+ing behaviors toward better uncertain tradeoff learning.
+The eVAE Model
+eVAE also tunes the representation-generation balance in
+Eq. (1) by jointly addressing the following issues in Eqs.
+(3) and (4): tuning hyperparameter in an outer circle irrele-
+vant to VAE behaviors, non-dynamic optimization, and con-
+strained settings on data and networks. Figure 1 illustrates
+the eVAE framework of variational evolutionary learning to
+improve the VAE balance.
+eVAE - Evolving Inner-outer-joint Training
+eVAE implements an evolving learning process of optimiz-
+ing both VAE and its evolutionary parameters through an
+inner-outer-joint iterative training process, as shown in Fig-
+ure 2. At time t, given the input xt ∈ X, a VAE inner
+training process trains a VAE model V (we do not constrain
+the VAE framework, for generality, we use β-VAE for case
+study in this paper) with some initialization β0 to learn the
+encoder qφ(zt | xt), the posterior pθ(xt | zt) with prior
+p(z). We obtain the VAE inner objective function with ini-
+tial weight β0:
+LELBOt = Ext∼pX
+�
+Eqφ(zt|xt) [log pθ (xt | zt)] −
+β0DKL (qφ (zt | xt) ∥p(z))]
+(5)
+By applying SGVB and reparameterization trick, the gen-
+erative process of V estimates the posterior as
+pθ(x | z) =
+�
+t
+pθ(xt | zt)
+(6)
+where xt | zt ∼ N(µ(zt; θ), Σ(zt; θ)) with zt = µxt+ϵΣxt
+and ϵ ∼ N(0, I).
+Then, we introduce an outer variational evolutionary
+learner E (illustrated by a variational genetic algorithm in-
+troduced below on Variational Evolution) to optimize the ob-
+jective in Eq. (5) in an outer process. E with the fitness func-
+tion f samples a chromosome βt from an evolving distribu-
+tion βt ∼ R and evolves it by taking variational crossover
+and mutation to generate and improve parameters βt+1.
+The evolved parameter is then incorporated into Vt for
+the next iteration t + 1 training. With the relevant ELBO
+information back-propagated to tune the VAE network and
+optimize the relevant parameters, we obtain the updated re-
+construction and inference performance. Consequently, Eq.
+(5) is evolved to:
+LELBOt+1 = Ext+1∼pX
+�
+Eqφ(zt+1|xt+1) [log pθ (xt+1
+| zt+1)] − βt+1DKL (qφ (zt+1 | xt+1) ∥p(z))]
+(7)
+If the fitness Eq. (17) is satisfied, we update β by βt+1.
+This then results in updated Vt+1 and its parameters φt+1,
+θt+1 and βt+1 for iteration t+1. We further repeat the above
+VGA E to obtain another set of parameters and retrain model
+Vt+1 until it converges.
+The above inner-outer iterative training progresses over
+time and samples in the generative process to iteratively op-
+timize the balance between reconstruction (i.e., minimizing
+LR = ∥xt − ˆxt∥ with the reconstructed ˆxt) and inference
+(i.e., seeking appropriate LI) and the VAE learning behav-
+iors. The VAE V is then optimized until converges.
+Accordingly, the eVAE generative process optimizes the
+following objective function:
+LeV AE = minθt,φt
+�N
+t=1 LELBOt+1 (θt, φt ;
+f (E (φt, θt, {β}, LELBOt)) , xt)
+(8)
+to approach β∗ for the optimal balance between representa-
+tion and construction:
+β∗ ∼ f (E ({β} | φt, θt, LELBOt))
+(9)
+
+N  iterations
+Chromosomes t
+Converge
+Fit
+V-crossover
+Diversity
+Diversity
+𝜷𝑡
+𝑡+1
+𝑡
+Chromosomes t+1
+V-mutation
+・
+・
+・
+・・・
+・ ・
+・
+・
+𝑞∅ ・
+𝑝# ・
+・
+・
+・・・・
+・
+・
+・
+・
+𝑞∅ ・
+𝑝# ・
+𝑓!"#(·)
+ℳ
+𝒞
+𝜷𝑡+1
+{𝛽𝑙}
+Diversity
+Diversity
+Multiple sampling
+Sampling
+V-crossover
+𝛽l,t
+𝛽t-1
+𝛽t+1
+Sampling
+Diversity
+Multiple sampling
+Cauchy/Normal
+Distribution
+𝛽t+1
+V-mutation
+Fitness
+Gene
+Mutate
+Crossover
+More disentangled
+Less disentangled
+𝒞
+ℳ
+𝑡+1 iteration
+Resample
+Sample
+𝜷𝑡-1
+Figure 1: The framework of eVAE. The VAE results inform the chromosome sampling. The genes are then updated by varia-
+tional V-crossover and V-mutation. The evolved results are checked per fitness for t + 1 retraining, giving up, or converging.
+The overall ELBO of eVAE can then be represented by:
+LeV AE = Ez∼qφ(zt|xt) log pθ(xt | zt)+
+fβt∼R(βt, E)DKL(qφ (zt | xt) ∥p(z))
+= Ext+1∼pX [LR + f(β)LI + E(∆LELBO)]
+(10)
+Variational Evolution in eVAE
+Here, we introduce the variational evolutionary learner E op-
+timizing the parameters. We instantiate E by a variational
+genetic algorithm (VGA) with its variational crossover and
+mutation operations. This avoids typical issues including
+premature convergence and random search during discor-
+dant exploitation and exploration.
+Variational Genetic Algorithm
+eVAE integrates the
+VGA-based outer parameter optimization combined with
+the VAE-internal gradient descent-based optimization. Ac-
+cordingly, VGA consists of a few variational steps and op-
+erations, initialization, variational crossover (V-crossover),
+variational mutation (V-mutation), and variational evalua-
+tion (V-evaluation). As shown in the bottom part of Figure
+2, below, we introduce them respectively.
+Chromosome selection: To enable a larger and more
+smooth GA search space compatible with VAEs, chromo-
+somes are embedded by a continuous variable β, sampled
+from an evolving distribution R (e.g., in the crossover). As-
+sume L individual chromosomes forming a candidate group
+{βl} = {β1, . . . , βL}, a chromosome is chosen from this
+candidate group to train the VAE, which is associated with
+a fitness value f after the VGA operations and VAE retrain-
+ing. We thus have the chromosome-fitness pairs embedded
+for the VGA evolution and offspring selection over genera-
+tions, such as:
+{β, f} = ((β1, f1), . . . , (βL, fL))
+(11)
+for optimizing the inner-outer joint training.
+V-crossover: Paired chromosomes at time t − 1 and t are
+crossovered to generate new genes for improving the ge-
+netic variety of offsprings. Following the simulated binary
+crossover (SBX) (Deb and Beyer 2001) method, we imple-
+ment the following crossover operations C to induce a large
+search space evolving over chromosome βt−1 at time t − 1
+and chromosome βt selected from the chromosome candi-
+date group {βl} at time t to obtain the offspring candidate
+chromosome βt+1 for the next time t + 1 in C strategy:
+C :
+�
+βt+1 = 1
+2[(1 + rc)βl,t + (1 − rc)βt−1], or
+βt+1 = 1
+2[(1 − rc)βl,t + (1 + rc)βt−1]
+(12)
+where rc is the crossover rate, which is drawn from a proba-
+bility density function Pc(rc):
+Pc (rc) =
+�
+0.5(η + 1)rn
+c ,
+if rc ≤ 1
+0.5(η + 1)
+1
+rη+2
+c
+,
+otherwise
+(13)
+by concentrating their corresponding parents βt−1 and βl,t.
+By sampling rc from the following function:
+rc =
+�
+�
+�
+(2u)
+1
+η+1 ,
+if u ≤ 0.5
+�
+1
+2(1−u)
+�
+1
+η+1 ,
+otherwise
+(14)
+where u is a random variable, and η is a hyper-parameter for
+scaling, the offsprings βt+1 can then be created by selecting
+the better C strategy (i.e., better satisfying Eq. (17)).
+V-mutation: The crossover-generated offsprings can be
+further mutated to improve their genomic diversity and evo-
+lutionary capacity. Aligning with the V-crossover, a varia-
+tional mutation strategy M is taken to diversify the offspring
+to update βt+1 inherited from crossover or βt from the cur-
+rent moment. With a relatively small probability pm, a chro-
+mosome βt+1 selected from group is mutated as:
+M : βt+1 = βt+1 + rm
+(15)
+where rm is a random variable sampled from the Cauchy
+distribution pm:
+Pm = 1
+π (
+1
+1 + β2
+t+1
+)
+(16)
+
+𝑥
+𝑞∅ 𝑧𝑡|𝑥𝑡
+𝛴!"|#$
+𝜇!"|#$
+𝜖
+z𝑡
+𝑝# 𝑥𝑡|𝑧𝑡
+𝐷%&(𝑞∅,"||𝑝#,")
+𝑥.
+𝑥 − 𝑥. '
+Inner training
+𝛽0
+Outer
+training
+Encoder
+Decoder
+𝛽t, 𝜃𝑡, ∅t, ℒ(&)*"
+𝛽$+,
+∗
+, 𝑓$+,
+∗
+ℒ./01
+=
+𝑓𝑡(·)
+𝑓(𝜀
+𝛽 ∅𝑡, 𝜃𝑡,ℒ(&)*" = ∆ℒ(&)*" +∥ 𝐾𝐿𝑡 − 𝑐 ∥'
+Initialization
+V-evaluation
+V-crossover
+V-mutation
+Variational GA
++
+Figure 2: eVAE - inner-outer joint evolutionary training pro-
+cess. The upper part shows the VAE training at time t,
+the objectives are then incorporated into the lower part -
+the outer training by variational genetic algorithm, whose
+fitness-based optimized results are fed to the VAE for fur-
+ther training.
+V-evaluation & VGA fitness function
+The chromosomes
+updated by the V-crossover and V-mutation operations are
+evaluated in alignment with the VAE objectives to determine
+whether they are transferable to the next generation. Within
+the VAE framework, we take the following heuristic fitness
+function f to guide the evolution of chromesomes and their
+fitness to VAE objectives. The fitness function integrates the
+direction of VAE-oriented stochastic gradient descent and
+the distance to the optimal KL divergence to achieve multi-
+objectives.
+ft+1 = ∆LELBOt+1 + ||KLt+1(βt+1) − c||
+(17)
+where ∆LELBOt+1 is:
+∆LELBOt+1 = LELBOt+1(βt+1) − LELBOt(βt)
+(18)
+denoting the VAE’s ELBO difference after applying the
+evolved β. c is the task-specific information bottleneck,
+which ensures the bound optimization of eVAE over the evo-
+lutionary optimization.
+This V-evaluation and the fitness f guide eVAE to con-
+verge and balance the reconstruction and inference with en-
+hanced evolutionary parameterization. The larger the fitness
+value, the stronger the chromosome in its candidate group,
+which forms a strong pair {β∗, f ∗}.
+The whole training processes of the eVAE model is illus-
+trated in Algorithm 1 in the Supplementary Material A.
+Theoretical Analysis
+In this paper, eVAE takes β-VAE as a case study of evolu-
+tionary VAE. Hence, we analyze the effects of eVAE in ad-
+justing β-VAE parameters, tradeoff between reconstruction
+and regularization, and training performance below.
+β-VAE directly tunes hyperparameter β setting a task-
+relevant ratio between the inference loss LI and the genera-
+tion loss LR. In essence, incrementally regularizing LI by β
+is equivalent to maximizing the likelihood on an experiment-
+specific threshold, i.e., a constant C, (Higgins et al. 2017):
+max
+φ,θ Ex∼pdata LR
+s.t. βLI < C
+(19)
+This learning process can be interpreted in information bot-
+tleneck (IB) theory (Alemi et al. 2018). The βLI constraint
+operates as the bottleneck, compressing the capacity of la-
+tent variable Z organized from input X to expressively rep-
+resent the target ˆX (Alemi et al. 2017). Accordingly, IB
+maximizes I(Z, ˆX) to acquire a concise representation:
+max I(Z, ˆX)
+s.t. I(X, Z) ≤ Ic
+(20)
+Further, the variational information bottleneck (VIB) intro-
+duces an experiment-specific lower bound for optimization:
+Lβ−V AE(θ, φ) = −D − βCR ≤ −βCI − D
+(21)
+where R refers to the compression rate, and D refers to the
+distortion measuring the representation relevance to a task,
+and βC is an experiment-specific constant. VAEs can thus be
+viewed as a process of achieving the rate-distortion tradeoff.
+ControlVAE is a variant of β-VAE, tuning the KL-specific
+βKL with a PID control to a desired KL point:
+LControlV AE(θ, φ) = −D − βKLR ≤ −βKLI − D (22)
+eVAE extends β-VAE tuning the information bottleneck
+βI(X, Z) by a variational evolutionary learner E. After iter-
+ation t of outer training, βt evolves to fit the task and train-
+ing phase, guided by the fitness function f. eVAE generates
+a lower bound to optimize iteration t+1 of the inner training
+phase:
+LeV AE ≤ −E(βt)It − E(βt)Dt
+(23)
+where an evolving iteration-specific lower bound can strike a
+compromise between task fitting in −E(βt)Dt and inference
+quality in E(βt)It. The derivations of Eq. (21) and Eq. (23)
+can be found in the Supplementary Materials B and C.
+We further compare the VIB effects of eVAE against
+VAE, β-VAE and ControlVAE by the rate-distortion (R-D)
+curve in Figure 3. VAE (yellow) tends to sacrifice empirical
+error minimization for representation learning in the early
+stage, where the rate is optimized to decrease in a quarter of
+the iterations. As for β-VAE (green), due to the experiment-
+specific lower bound (−βCI − D), the distortion cannot
+be minimized. ControlVAE (pink) concentrates on optimiz-
+ing the distortion initially to fit data and optimizing the rate
+eventually to acquire smooth representation, resorting to the
+KL-specific lower bound (−βKLI − D). However, directly
+modifying β by the given PID controller leads to fluctuations
+between inference capacity and reconstruction quality dur-
+ing the iterations. Rather than monitoring the optimization
+process, eVAE generates an iteration-specific lower bound
+(−E(βt)It − E(βt)Dt) to achieve a balance between mini-
+mizing distortion and controlling rate.
+Experiments
+We evaluate eVAE in three typical tasks: disentangled rep-
+resentation, image generation, and language modeling.
+
+4
+8
+12
+16
+20
+24
+28
+R
+0
+20
+40
+60
+80
+100
+120
+140
+D
+VAE
+-VAE
+ControlVAE
+eVAE
+Figure 3: The information plane with the R − D curves of
+VAE, β-VAE, ControlVAE and eVAE on dSprites.
+Dataset and Baselines
+Three different learning tasks and their datasets are listed
+below to evaluate eVAE against the VAE baselines.
+Disentangled representation learns independent latent
+variables to generate an image. We validate eVAE on
+dSprites, a 2D-shape dataset with 737,280 binary images
+generated by six ground-truth factors: shape (square, ellipse,
+heart), scale, orientation, position x, and position y. To ex-
+amine the disentanglement performance and image genera-
+tion equally and comprehensively, β-VAE and ControlVAE
+are compared.
+Image generation reconstructs an imagery sample based
+on given datapoints. The CelebA dataset (cropped version)
+(Liu et al. 2015) is used, which is a real-world celebrity face
+dataset with 202,599 RGB images. The baselines are β-VAE
+and ControlVAE.
+Language modeling conducts the word-level text genera-
+tion. The Penn Treebank dataset (PTB) with English corpus
+(Marcinkiewicz 1994) is used. To reveal the performance
+of restraining KL vanishing, we compare eVAE with three
+heuristic methods: cost annealing (noted as Cost-10k) (Bow-
+man et al. 2016), cyclical annealing (Cyc-8) (Fu et al. 2019),
+and PID control (PID-3) (Shao et al. 2020).
+For a fair comparison, we adopt the same embedding net-
+work as β-VAE and tune eVAE to achieve the same KL set
+point as ControlVAE. The VAE architectures and hyperpa-
+rameter settings are given in Supplementary Material D.
+Performance Evaluation
+Performance of disentangled representation learning
+Here, we validate the performance of reconstructing a sharp
+image with disentangled features. Figure 5(a) shows that
+eVAE achieves a lower reconstruction loss of 9.2, compared
+with ControlVAE under the same expected KL divergence
+(KL = 19). Moreover, Figure 5(a) and (b) show that eVAE
+presents a more stable training curve than other VAEs be-
+cause of its dynamic weighting guided by the VGA fitness
+function, rather than a direct modification in β-VAE and
+ControlVAE. The variation of each disentangled factor is il-
+lustrated in Figure 5(c). Figure 6 further demonstrates the
+reliable disentanglement of all factors and sharp reconstruc-
+tion quality of eVAE under a set point KL = 19. In contrast,
+ControlVAE captures the generative factors of scale, x and
+y while β-VAE entangles other four factors except scale.
+Performance of image generation We evaluate eVAE on
+CelebA for image generation. For a fair and concise com-
+parison, the KL set point is set to 200 achieving the best
+reconstruction quality in (Shao et al. 2020). eVAE is tuned
+to achieve the desired KL point similar to ControlVAE. Fig-
+ure 4(a) shows that eVAE has the lowest reconstruction error
+compared with vanilla VAE and ControlVAE. Guided by the
+evolving β, the KL divergence in eVAE can achieve a de-
+sired point dynamically, as illustrated in Figure 4(b).
+(b) KL
+(a) Reconstruction Loss
+Figure 4: Performance comparison of different VAEs for im-
+age generation on CelebA.
+Performance of language modeling To show the valid-
+ity of eVAE in preventing the variational language model
+from degenerating to a standard language model, we draw
+the learning curves of KL, reconstruction loss, and weight
+of KL over iterations. The evolution of β is shown in Figure
+7(a), demonstrating that eVAE can learn to tune β over itera-
+tions automatically rather than by a heuristic setting. Figure
+7(b) shows that the Cost annealing (Cost-10k) model still
+suffers from KL vanishing, with KL approaching 0 at the end
+of iterations. By contrast, the Cyclical annealing (Cyc-8),
+PID, and GA can converge to a nonzero divergence, while
+eVAE achieves the lowest reconstruction loss to 70 in word
+generation, illustrated in Figure 7(c). Further, compared to
+the PID control, VGA can find a path toward a more stable
+stage, without undergoing reconstruction fluctuations during
+training even if the non-zero learned β is fixed.
+Conclusion
+We propose evolutionary VAE (eVAE) to dynamically learn
+the tradeoff between reconstruction effectiveness and infer-
+ence robustness in VAEs. eVAE involves variational evolv-
+ing operators to solve the premature convergence and ran-
+dom search problem. SGD optimization and genetic al-
+gorithm synergistically follow an inner-outer-joint training
+mechanism in eVAE. Following the variational information
+bottleneck theory, eVAE derives an iteration-specific lower
+bound balancing the capacity of compression and decom-
+pression over iterations. Multiple task experiments show that
+eVAE outperforms SOTA VAEs in traversing samples with
+more disentangled features, generating sharp images with
+lower reconstruction loss, and resolving the KL vanishing
+automatically in variational language modeling. We will ex-
+tend eVAE to other VAE models and tasks.
+
+(a) Reconstruction Loss
+(b) KL
+(c) Element-wise KL Divergence
+Figure 5: Learning curves on dSprites1. (a,b) indicate that eVAE has the lowest reconstruction loss compared with VAE (β = 1),
+β-VAE (β = 4), and ControlVAE (KL = 19) under a fixed KL point KL=19. (c) is the element-wise KL divergence as a function
+of iterations in eVAE. We can observe that eVAE retains a stable and reasonable KL divergence of each generator dimension
+(factor): position-y (z2), scale (z3), shape (z4), position-x (z6), orientation (z7). More comparisons in terms of generated KL
+divergence can be found in Supplementary Material E.More comparisons in terms of generated KL divergence can be found in
+Supplementary Material E.
+y
+Scale
+Shape
+x
+Orientation
+eVAE (KL = 19)
+ControlVAE (KL = 19)
+𝜷-VAE (𝜷 = 4)
+Figure 6: Latent traversal on dSprites. To distinguish the latent factors visually, we take the ellipse for illustration. Each row
+represents a latent factor in the traversal, while keeping others fixed. The first column refers to the seed image for initialization,
+and we then manipulate the latent dimension z across the range [-3, 3].
+(c) Reconstruction Loss
+(b) KL
+(a) Beta
+Figure 7: Learning curves on PTB for language modeling. The suffix of PID and VGA refer to a KL set point i.e., KL = 3; Cyc-8
+indicates 8 iteration cycles where the weight increases linearly from 0 to 1; Cost-10k refers to 10k iterations during which the
+weight increases from 0 to 1 by a sigmoid function iteratively.
+
+.References
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+continuous space. In CONLL.
+Bozkurt, A.; Esmaeili, B.; Tristan, J.-B.; Brooks, D. H.; Dy,
+J. G.; and van de Meent, J.-W. 2021. Rate-regularization and
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+Liu, Z.; Luo, P.; Wang, X.; and Tang, X. 2015. Deep learning
+face attributes in the wild. In ICCV.
+Locatello,
+F.;
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+G.;
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+T.;
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+S.;
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+disentangled representations. In NeurIPS.
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+pus of English: The penn treebank. Using Large Corpora.
+Mita, G.; Filippone, M.; and Michiardi, P. 2021. An identifi-
+able double VAE for disentangled representations. In ICML.
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+Lakshminarayanan, B. 2019.
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+Ran, X.; Xu, M.; Mei, L.; Xu, Q.; and Liu, Q. 2022. Detect-
+ing out-of-distribution samples via variational autoencoder
+with reliable uncertainty estimation. Neural Networks.
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+erating diverse high-fidelity images with VQ-VAE-2.
+In
+NeurIPS.
+Ruiz, F. J.; Titsias, M. K.; Cemgil, T.; and Doucet, A. 2021.
+Unbiased gradient estimation for variational autoencoders
+using coupled markov chains. In UAI.
+Shao, H.; Yao, S.; Sun, D.; Zhang, A.; Liu, S.; Liu, D.;
+Wang, J.; and Abdelzaher, T. 2020. Controlvae: Controllable
+variational autoencoder. In ICML.
+Shi, Y.; Paige, B.; Torr, P.; et al. 2019. Variational mixture-
+of-experts autoencoders for multi-modal deep generative
+models. In NeurIPS.
+Sohn, K.; Lee, H.; and Yan, X. 2015.
+Learning struc-
+tured output representation using deep conditional genera-
+tive models. In NeurIPS.
+Sutter, T. M.; Daunhawer, I.; and Vogt, J. E. 2021. General-
+ized multimodal ELBO. arXiv.
+Tolstikhin, I.; Bousquet, O.; Gelly, S.; and Schoelkopf, B.
+2017. Wasserstein autoencoders. arXiv.
+Tomczak, J.; and Welling, M. 2018. VAE with a vampprior.
+In AISTATS.
+Wang, P. Z.; and Wang, W. Y. 2019. Neural gaussian copula
+for variational autoencoder. In EMNLP.
+Wu, D.; Wang, L.; and Zhang, P. 2019. Solving statistical
+mechanics using variational autoregressive networks. Phys-
+ical Review Letters, 122(8): 080602.
+Xiao, Z.; Yan, Q.; and Amit, Y. 2020. Likelihood regret:
+An out-of-distribution detection score for variational auto-
+encoder. NeurIPS.
+Zhang, Y.; Wang, Y.; Zhang, L.; Zhang, Z.; and Gai, K.
+2019. Improve diverse text generation by self labeling con-
+ditional variational auto encoder. In ICASSP.
+Zhao, S.; Song, J.; and Ermon, S. 2019. InfoVAE: Balancing
+learning and inference in variational autoencoders. In AAAI.
+
+References
+Alemi, A.; Poole, B.; Fischer, I.; Dillon, J.; Saurous, R. A.;
+and Murphy, K. 2018. Fixing a broken ELBO. In ICML.
+Alemi, A. A.; Fischer, I.; Dillon, J. V.; and Murphy, K. 2017.
+Deep variational information bottleneck. In ICLR.
+Berliner, A.; Rotman, G.; Adi, Y.; Reichart, R.; and Hazan,
+T. 2022.
+Learning discrete structured variational auto-
+encoder using natural evolution strategies. In ICLR.
+Bowman, S. R.; Vilnis, L.; Vinyals, O.; Dai, A. M.; Jozefow-
+icz, R.; and Bengio, S. 2016. Generating sentences from a
+continuous space. In CONLL.
+Bozkurt, A.; Esmaeili, B.; Tristan, J.-B.; Brooks, D. H.; Dy,
+J. G.; and van de Meent, J.-W. 2021. Rate-regularization and
+generalization in VAEs. AISTATS.
+Burda, Y.; Grosse, R.; and Salakhutdinov, R. 2016. Impor-
+tance weighted autoencoders. In ICLR.
+Burgess, C. P.; Higgins, I.; Pal, A.; Matthey, L.; Watters, N.;
+Desjardins, G.; and Lerchner, A. 2018. Understanding dis-
+entangling in β-VAE. In NeurIPS.
+Deb, K.; and Beyer, H.-G. 2001. Self-adaptive genetic algo-
+rithms with simulated binary crossover. Evolutionary Com-
+putation, 9(2): 197–221.
+Domke, J.; and Sheldon, D. R. 2019. Divide and couple: Us-
+ing monte carlo variational objectives for posterior approxi-
+mation. In NeurIPS.
+Esmaeili, B.; Wu, H.; Jain, S.; Bozkurt, A.; Siddharth, N.;
+Paige, B.; Brooks, D. H.; Dy, J.; and Meent, J.-W. 2019.
+Structured disentangled representations. In AISTATS.
+Fortuin, V.; H¨user, M.; Locatello, F.; Strathmann, H.; and
+R¨atsch, G. 2019. Som-vae: Interpretable discrete represen-
+tation learning on time series. In ICLR.
+Fu, H.; Li, C.; Liu, X.; Gao, J.; Celikyilmaz, A.; and Carin,
+L. 2019. Cyclical annealing schedule: A simple approach to
+mitigating KL vanishing. In NAACL.
+Ghosh, P.; Sajjadi, M. S.; Vergari, A.; Black, M.; and
+Sch¨olkopf, B. 2019. From variational to deterministic au-
+toencoders. In ICLR.
+Havtorn, J. D. D.; Frellsen, J.; Hauberg, S.; and Maaløe, L.
+2021. Hierarchical VAEs know what they don’t know. In
+ICML.
+Higgins, I.; Matthey, L.; Pal, A.; Burgess, C.; Glorot, X.;
+Botvinick, M.; Mohamed, S.; and Lerchner, A. 2017. β-
+VAE: Learning basic visual concepts with a constrained
+variational framework. In ICLR.
+Jankowiak, M.; and Phan, D. 2022. Surrogate likelihoods
+for variational annealed importance sampling. In ICML.
+Kamata, H.; Mukuta, Y.; and Harada, T. 2022. Fully spiking
+variational autoencoder. AAAI.
+Khemakhem, I.; Kingma, D.; Monti, R.; and Hyvarinen, A.
+2020. Variational autoencoders and nonlinear ICA: A unify-
+ing framework. In AISTATS.
+Kingma, D. P.; and Welling, M. 2013. Auto-encoding varia-
+tional bayes. arXiv.
+Klushyn, A.; Chen, N.; Kurle, R.; Cseke, B.; and van der
+Smagt, P. 2019. Learning hierarchical priors in VAEs. In
+NeurIPS.
+Kviman, O.; Melin, H.; Koptagel, H.; Elvira, V.; and Lager-
+gren, J. 2022.
+Multiple importance sampling ELBO and
+deep ensembles of variational approximations. In AISTATS.
+Liu, Z.; Luo, P.; Wang, X.; and Tang, X. 2015. Deep learning
+face attributes in the wild. In ICCV.
+Locatello,
+F.;
+Abbati,
+G.;
+Rainforth,
+T.;
+Bauer,
+S.;
+Sch¨olkopf, B.; and Bachem, O. 2019. On the fairness of
+disentangled representations. In NeurIPS.
+Marcinkiewicz, M. A. 1994. Building a large annotated cor-
+pus of English: The penn treebank. Using Large Corpora.
+Mita, G.; Filippone, M.; and Michiardi, P. 2021. An identifi-
+able double VAE for disentangled representations. In ICML.
+Nalisnick, E.; Matsukawa, A.; Teh, Y. W.; Gorur, D.; and
+Lakshminarayanan, B. 2019.
+Do deep generative models
+know what they don’t know? In ICLR.
+Ran, X.; Xu, M.; Mei, L.; Xu, Q.; and Liu, Q. 2022. Detect-
+ing out-of-distribution samples via variational autoencoder
+with reliable uncertainty estimation. Neural Networks.
+Razavi, A.; Van den Oord, A.; and Vinyals, O. 2019. Gen-
+erating diverse high-fidelity images with VQ-VAE-2.
+In
+NeurIPS.
+Ruiz, F. J.; Titsias, M. K.; Cemgil, T.; and Doucet, A. 2021.
+Unbiased gradient estimation for variational autoencoders
+using coupled markov chains. In UAI.
+Shao, H.; Yao, S.; Sun, D.; Zhang, A.; Liu, S.; Liu, D.;
+Wang, J.; and Abdelzaher, T. 2020. Controlvae: Controllable
+variational autoencoder. In ICML.
+Shi, Y.; Paige, B.; Torr, P.; et al. 2019. Variational mixture-
+of-experts autoencoders for multi-modal deep generative
+models. In NeurIPS.
+Sohn, K.; Lee, H.; and Yan, X. 2015.
+Learning struc-
+tured output representation using deep conditional genera-
+tive models. In NeurIPS.
+Sutter, T. M.; Daunhawer, I.; and Vogt, J. E. 2021. General-
+ized multimodal ELBO. arXiv.
+Tolstikhin, I.; Bousquet, O.; Gelly, S.; and Schoelkopf, B.
+2017. Wasserstein autoencoders. arXiv.
+Tomczak, J.; and Welling, M. 2018. VAE with a vampprior.
+In AISTATS.
+Wang, P. Z.; and Wang, W. Y. 2019. Neural gaussian copula
+for variational autoencoder. In EMNLP.
+Wu, D.; Wang, L.; and Zhang, P. 2019. Solving statistical
+mechanics using variational autoregressive networks. Phys-
+ical Review Letters, 122(8): 080602.
+Xiao, Z.; Yan, Q.; and Amit, Y. 2020. Likelihood regret:
+An out-of-distribution detection score for variational auto-
+encoder. NeurIPS.
+Zhang, Y.; Wang, Y.; Zhang, L.; Zhang, Z.; and Gai, K.
+2019. Improve diverse text generation by self labeling con-
+ditional variational auto encoder. In ICASSP.
+Zhao, S.; Song, J.; and Ermon, S. 2019. InfoVAE: Balancing
+learning and inference in variational autoencoders. In AAAI.
+
+Supplementary Materials -
+eVAE: Evolutionary Variational Autoencoder
+Here, we present some supplemental materials to the main
+content, including the eVAE algorithm, the experimental set-
+tings, the latent traversal on square and heart, and derivation
+of the variational lower and upper bounds.
+A
+The eVAE algorithm
+Below, Algorithm 1 illustrates the modeling process of
+eVAE, which enables an inner-outer-joint training of VAE
+and evolutionary learning iteratively.
+Algorithm 1: Evolving Inner-outer-joint Training
+1: Input: crossover rate Prc, mutation rate Prm, batch
+data x.
+2: Output: {β∗, f ∗}
+3: Initialization: parameters of decoder θ, parameters of
+encoder φ, {βl} = {β1, . . . , βL} //initialize
+group
+4: while t < T do
+5:
+Prt ← sample from N (0, 1) //probability to
+evolve
+6:
+if Prt < Prm then
+7:
+θt,
+φt
+//save current variational
+parameters
+8:
+generate
+βt+1
+by
+V-crossover
+operation
+C(βt−1, βt) by Eqs. (12), (13) and (14)
+9:
+evaluate βt+1 by f(E(φt, θt, {β}, LELBOt)) by
+Eqs. (17) and (18)
+10:
+else if Prt < Prc then
+11:
+θt,
+φt
+//save current variational
+parameters
+12:
+generate βt+1 by V-mutation operation M(βt) by
+Eqs. (15) and (16)
+13:
+evaluate
+βt+1
+by
+fitness
+function
+f(E(φt, θt, {β}, LELBOt))
+by
+Eqs.
+(17)
+and
+(18)
+14:
+else
+15:
+{β∗, f ∗} ← βl //select the strongest
+pair
+16:
+θt+1, φt+1 ← LeV AE(θt, φt, β∗|xt) // update
+variational parameters
+17:
+end if
+18:
+Return: {β∗, f ∗}
+19: end while
+B
+Derivation of β-VAE lower bound
+According to the VIB theory, elbo can be bounded by dis-
+tortion D = −Eqφ(x,z)[log pθ(x | z)] and rate R =
+Eqφ(x,z)[DKL(qφ(z|x)||p(z))], where I(X, Z) has the up-
+per bound I ≤ R:
+R =
+�
+q(x)
+�
+qθ(z | x) log qθ(z | x)
+p(z)
+= I(X, Z) + TC(Z)
+≥ I(X, Z)
+(24)
+and the lower bound I ≥ H − D:
+I =
+�
+dxdzp(x, z) log p(x | z)
+p(x)
+≥
+�
+dxdzp(x, z) log q(x | z)
+p(x)
+=
+�
+dxdzp(x, z) log q(x | z)
+−
+�
+dxp(x) log p(x)
+=
+�
+dxdzp(x, z) log q(x | z)
++ H(X)
+= H − D
+(25)
+All in all, the variational information bottleneck can be re-
+garded as:
+H − D ≤ I ≤ R
+(26)
+where H is the entropy of the data.
+In β-VAE framework, βC impose a experiment-specific
+constraint on compressor and decompressor, giving the vari-
+ational lower bound:
+H − D ≤ I ≤ R
+H − D ≤ βI ≤ βR
+H ≤ βI + D ≤ βR + D
+(27)
+In summary, the lower bound can be denoted as:
+Lβ−V AE(θ, φ) = −D − βCR ≤ −βCI − D
+(28)
+C
+Derivation of eVAE lower bound
+According to Eq. (10) and Eq. (17) , we can obtain the loss
+function of eVAE in iteration t as follows:
+LeV AE = Ez∼qφ(zt|xt) log pθ(xt | zt)+
+fβt∼R(βt, E)DKL(qφ(zt | xt)∥p(z))
+= Ext+1∼pX [LR + f(βt)LI + E(∆LELBO)]
+= Ext+1∼pX [LR + f(βt)LI
++ E(∆LELBOt+1 + ||KLt+1(βt+1) − c||)]
+(29)
+According to the VIB theory, Rt sets the upper bound on
+information bottleneck It(Xt, Zt) in t-th iteration :
+Rt =
+�
+q(xt)
+�
+qθ(zt | xt) log qθ(zt | xt)
+p(zt)
+= It(Xt, Zt) + TC(Zt)
+≥ It(Xt, Zt)
+(30)
+
+and Dt sets the lower bound:
+It =
+�
+dxdzp(xt, zt) log p(xt | zt)
+p(xt)
+≥
+�
+dxdzp(xt, zt) log q(xt | zt)
+p(xt)
+=
+�
+dxdzp(xt, zt) log q(xt | zt)
+−
+�
+dxp(xt) log p(xt)
+=
+�
+dxdzp(xt, zt) log q(xt | zt)
++ H(X)
+= H − Dt
+(31)
+After optimized by a variational evolutionary learner E,
+the loss function of eVAE constrained by a dynamic lower
+bound is as follows:
+LeV AE = Ez∼qφ(zt|xt) log pθ(xt | zt)+
+fβt∼R (βt, E) DKL(qφ(zt | xt)∥p(z))
+= Ext+1∼pX [LR + f(βt)LI + E(∆LELBO)]
+= Ext+1∼pX [LR + f(βt)LI
++ E(∆LELBOt+1 + ||KLt+1(βt+1) − c||)]
+= −Dt − βtRt − E(−Dt+1 − βt+1Rt+1 + Dt + c)
+≤ −E(βt)Dt − E(βt)It
+(32)
+where c is the desired KL point and βt is a chromosome
+sample in t-th iteration.
+D
+Experimental settings
+Table 1 shows the experimental settings of the three learn-
+ing tasks for evaluating eVAE. We give the settings of the
+encoder-decoder network architectures and the parameters
+for the evolutionary learning on three datasets dSprites,
+CelebA, and PTB and their corresponding optimizers.
+E
+Analysis of element-wise KL divergence in
+disentangled representation learning
+Figure 8 further shows the element-wise KL divergence
+evolving over iterations of three VAE models: β-VAE and
+ControlAVE in comparison with our eVAE on the dataset
+dSprites.
+
+Dataset
+Optimiser
+Architecture
+eVAE initialization
+dSprites
+Adam(1e-4)
+Input
+Encoder
+Latent
+Decoder
+(64,64,1)
+Conv(32, 32, 32), ReLu, Conv(32,16,16),
+ReLu, Conv(32,8,8), ReLu,Conv(32,4,4),
+Relu, Linear(256), Relu, Linear(256),
+Relu, Linear(20)
+10
+Linear(256), Relu, Linear(256),
+Relu, Linear(512), Relu,
+Conv(32,4,4), Relu, Conv(32,8,8),
+Relu, Conv(32,16,16), Relu,
+Conv(32,32,32), Relu, (1, 64,64)
+Mutation Rate
+Crossover Rate
+Individual Number
+0.001
+0.04
+20
+CelebA
+(KL = 200)
+Adam(1e-4)
+Input
+Encoder
+Latent
+Decoder
+(128,128,3)
+Conv(32, 32, 32), ReLu,
+Conv(32,16,16), ReLu,
+Conv(32,8,8),ReLu,
+Conv(32,4,4), Relu, Linear(256),
+Relu, Linear(256), Relu, Linear(20)
+10
+Linear(256), Relu, Linear(256),
+Relu, Linear(512), Relu,
+Conv(32,4,4), Relu,
+Conv(32,8,8), Relu,
+Conv(32,16,16), Relu, Conv(32,32,32),
+Relu, (3, 64,64)
+Mutation Rate
+Crossover Rate
+Individual Number
+0.001
+0.03
+20
+PTB
+(KL = 3)
+init lr: 0.001
+threshold:100
+decay factor: 0.5
+max decay: 5
+Input
+Encoder
+Latent
+Decoder
+Mutation Rate
+Crossover Rate
+Individual Number
+0.006
+0.07
+10
+Table 1: Experimental settings for disentangled representation learning on dSprites, image generation on CelebA, and language
+generation on PTB.
+(c) eVAE
+(b) ControlVAE
+(a) 𝛽-VAE
+Figure 8: The element-wise KL divergence over iterations for β-VAE, ControlAVE, and eVAE on dSprites. It shows that each
+KL divergence of eVAE is larger than that of β-VAE, providing an enough space for generating disentangled representation. In
+addition, eVAE generates more stable KL divergence compared to ControlVAE.
+
diff --git a/xtAyT4oBgHgl3EQfOvaI/content/tmp_files/load_file.txt b/xtAyT4oBgHgl3EQfOvaI/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..a403ea8c4f2a7417a63e4a361bc0463c32dc86c5
--- /dev/null
+++ b/xtAyT4oBgHgl3EQfOvaI/content/tmp_files/load_file.txt
@@ -0,0 +1,1247 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf,len=1246
+page_content='eVAE: Evolutionary Variational Autoencoder  Zhangkai Wu,1 Longbing Cao, 1 Lei Qi 2  1 University of Technology Sydney  2 Southeast University  berenwu1938@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='com, Longbing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='Cao@uts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='au, qilei@seu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='cn  Abstract  The surrogate loss of variational autoencoders (VAEs) poses  various challenges to their training, inducing the imbalance  between task fitting and representation inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' To avert  this, the existing strategies for VAEs focus on adjusting the  tradeoff by introducing hyperparameters, deriving a tighter  bound under some mild assumptions, or decomposing the  loss components per certain neural settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' VAEs still suffer  from uncertain tradeoff learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='We propose a novel evolu-  tionary variational autoencoder (eVAE) building on the vari-  ational information bottleneck (VIB) theory and integrative  evolutionary neural learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' eVAE integrates a variational  genetic algorithm into VAE with variational evolutionary op-  erators including variational mutation, crossover, and evolu-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Its inner-outer-joint training mechanism synergistically  and dynamically generates and updates the uncertain tradeoff  learning in the evidence lower bound (ELBO) without addi-  tional constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Apart from learning a lossy compression  and representation of data under the VIB assumption, eVAE  presents an evolutionary paradigm to tune critical factors of  VAEs and deep neural networks and addresses the premature  convergence and random search problem by integrating evo-  lutionary optimization into deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Experiments show  that eVAE addresses the KL-vanishing problem for text gen-  eration with low reconstruction loss, generates all disentan-  gled factors with sharp images, and improves the image gen-  eration quality,respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' eVAE achieves better reconstruc-  tion loss, disentanglement, and generation-inference balance  than its competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Introduction  Variational autoencoders (Kingma and Welling 2013)  (VAEs) have attracted significant interest for their capability  of learning continuous and smooth distributions from obser-  vations by integrating probabilistic and deep neural learning  principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' However, VAEs still face some significant issues,  including dynamic uncertainty learning during variational  inference and the tradeoff between representation compres-  sion and task fitting, despite various recent VAE mecha-  nisms and variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Below, we briefly analyze these issues  and the gaps of existing solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' A novel evolutionary VAE  (eVAE) framework is then introduced to address some of  these issues and gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  VAEs and Gap Analysis  VAEs have demonstrated sig-  nificant advantages of incorporating prior knowledge, map-  ping inputs to probabilistic representations, and approximat-  ing the likelihood of outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' The integration of stochastic  gradient variational Bayes (SGVB) estimator (Kingma and  Welling 2013) with neural settings learns a narrow proba-  bilistic latent space to infer more representative attributes  in the hidden space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' VAEs have been applied in various  domains, including time series forecasting (Fortuin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  2019), out-of-domain detection in images (Nalisnick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Ran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Havtorn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Xiao, Yan, and  Amit 2020), generating images with spiking signals (Ka-  mata, Mukuta, and Harada 2022), and generating text by  language modeling (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Theoretically, the bound optimization of variational infer-  ence was introduced to substitute the log-likelihood function  with a surrogate function to make it optimized by gradient  descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In practice, the evidence lower bound (ELBO) can-  not approach the maximum of conditional likelihood with a  close gap between posterior and prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' It causes dynamic un-  certainty during inference and failed tradeoff between repre-  sentation robustness and reconstruction effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Weak  KL divergence could incur KL vanishing while strong KL  divergence could result in bad likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' The tradeoff is  sensitive to posterior distribution disjoint, data characteris-  tics, and network architectures (Bozkurt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Recently, various techniques have been proposed to ad-  dress these issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' The first set aims to adjust the balance  in objective functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Examples are β-VAE incorporating  a hyperparameter β (Higgins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2017), InfoVAE adding  a scaling parameter to the KL term (Zhao, Song, and Er-  mon 2019), SA-VAE having a cyclical annealing schedule  to progressively increase β for reducing KL vanishing (Fu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2019), and ControlVAE introducing the proportional-  integral-derivative (PID) control to tune the hyperparame-  ter (Shao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' They are partial solutions only ad-  justing one part of the objectives, failing to weigh and re-  solve the balance issue in dynamic settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Another set is  to tune network architectures or VAE settings, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=', VQ-  VAE and their variants (Razavi, Van den Oord, and Vinyals  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Berliner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2022), spiking VAE (Kamata, Mukuta,  and Harada 2022), multimodal VAE (Sutter, Daunhawer,  and Vogt 2021), mixture-of-experts multimodal VAE (Shi  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2019) and tuning of their ELBO (Kviman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Jankowiak and Phan 2022) building on the product of ex-  perts structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' However, these models are subject to specific  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='00011v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='NE]  1 Jan 2023  tasks or data settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  eVAE From the information bottleneck perspective, VAEs  can be treated as a lossy information compression process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Adjusting the KL within a range controls the information  bottleneck flowing from representing latent variables to re-  constructing samples, thus benefiting the trade-off between  compression and reconstruction (Alemi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Burgess  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' This variational information bottleneck inspires  us to improve the VAE mechanism and objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  To directly address the generation-inference balance in an  evolving dynamic setting without constraints on networks  and inputs, we introduce a novel evolutionary VAE (eVAE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  eVAE incorporates and integrates variational evolutionary  learning and variational information bottleneck into VAE to  achieve better optimum exploration and the tradeoff between  representation compression and generation fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' However,  integrating evolutionary learning into VAE is an open topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  We propose the variational genetic algorithm into eVAE to  optimize VAE objectives and its modeling exploration in an  evolving manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Consequently, eVAE dynamically optimizes the VAE in-  ference (the KL terms) in an evolutionary and probabilistic  manner, which further tunes the VAE generation toward bal-  ancing generation and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' To avert premature conver-  gence, eVAE introduces probabilistic chromosome selection  for smooth search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' To avoid exhaustive random search, we  apply the simulated binary crossover (Deb and Beyer 2001)  and Cauchy distributional mutation to guide the training to-  ward a stable convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' The main contributions of this  work are as follows:  We propose the eVAE model to balance the tradeoff be-  tween generation and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Specifically, a Cauchy-  based variational mutation operator and an SBX-based  variational crossover operator are proposed to train a  model with an evolving inner-outer-joint training algo-  rithm, tackling the premature convergence and random  search problem when optimizing deep models assisted  with a variational genetic algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  We illustrate eVAE by deriving the lower bound on β-  VAE related models under the information bottleneck  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' We analyze the information flow in the cor-  responding lower bound per the rate-distortion theory  to show that only the iteration-specific lower bound in  eVAE can capture the optimization trend to train effec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  We evaluate eVAE on three tasks: disentangled represen-  tation learning on dSprites, image generation on CelebA,  and text generation on PTB corpus, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  eVAE forms the first evolutionary VAE framework with-  out empowering constraints on VAE architectures, input set-  tings, and objective function (ELBO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' It integrates varia-  tional encoding and decoding, information bottleneck, and  evolutionary learning into the deep neural framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' The  evaluation against the state-of-the-art VAEs shows that  eVAE achieves better disentanglement, reconstruction loss,  and solve the KL vanishing effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  VAE Background and Issues  Here, we briefly introduce the background of VAEs by fo-  cusing on their objectives and approaches to addressing the  inference-generation balance since they are mostly relevant  to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Other developments on VAEs such as adjusting  VAE network structures, input settings, and learning con-  straints are excluded due to their irrelevance to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  VAE  VAEs mitigate autoencoder issues like sparse repre-  sentation (Tolstikhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2017) by learning continuous and  smooth representation distribution p(x), x ∈ X from obser-  vations X over latent variables z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' After learning an encoding  distribution qφ(z | x) in encoding neural networks, VAEs  apply variational inference to approximate the posterior dis-  tribution pθ(x | z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Learning tasks such as reconstructed and  generated outputs can then be sampled from this learned dis-  tribution in a generative process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' With the SGVB estimator  and reparameterization trick, the gradients become tractable,  and the generative parameters θ and inference parameters φ  are learnable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' The objectives of VAEs can be converted to  ELBO with the expectation over empirical distribution pdata  of the data towards both reconstruction LR and inference LI  (Esmaeili et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  LELBO = Ex∼pdata  �  Eqφ(z|x) [log pθ(x | z)]  −KL (qφ(z | x)∥p(z))]  = Ex∼pdata LR + LI  (1)  VAE Issues  To solve ELBO, the inference model qφ(z|x)  can be trained jointly by maximizing the ELBO to acquire  reasonable compression for task fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' However, a weak ca-  pacity of the decoder and the variety of data could make  the expressive posterior favor task fitting rather than opti-  mal inference (Zhao, Song, and Ermon 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' For exam-  ple, in variational language generation, the decoder built on  autoregressive models such as LSTM and PixelCNN can  generate language samples by the autoregressive property  rather than the posterior-based latent variables (Wu, Wang,  and Zhang 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' The VAE degenerates to an autoregres-  sive model where the KL divergence between posterior and  prior reaches zero quickly during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' This results in  KL vanishing and poor generalization in test for the lack of  diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' The approaches of learning orthogonal transfor-  mation of priors with the same distribution by the decoder  (Khemakhem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Mita, Filippone, and Michiardi  2021) may sacrifice accurate inference in optimal represen-  tation and generalization of fitting the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Other research  attributes the training conflict to the inherent property of  bound optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' For example, under a solid factorial as-  sumption about the posterior distribution (Locatello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  2019), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=',  p(x, z) = p(x | z)  �  i  p (zi) ,  (2)  the ELBO constraining the variational samples favors the  data fitting (Burda, Grosse, and Salakhutdinov 2016) but  fails to maximize the probability mass on log-likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In  addition, the vanilla VAE optimizer strengthens the disjoint-  ness between qφ(z|xi), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=', µi → ∞, σi → 0+, to separate  the log-likelihood concentrated on each sample, resulting in  maximizing the mass of joint distribution (Zhao, Song, and  Ermon 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  VAE Enhancement  One way to address the above issues  is to tighten the log-likelihood lower bound for correct vari-  ational approximation in posterior (Alemi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Wang  and Wang 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Domke and Sheldon 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Jankowiak and  Phan 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Ruiz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Kviman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2022), prior  (Tomczak and Welling 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Klushyn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2019) and de-  composition of ELBO (Zhao, Song, and Ermon 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Es-  maeili et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2019) under some mild assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' For exam-  ple, β-VAE (Higgins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2017) adds the hyper-parameter  β to weigh the LKL term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Then, ELBO minimizes LR to the  convergence of data fitting collectively with the regulariza-  tion LI by varying β:  LELBO =Ex∼pdata  �  Eqφ(z|x) [log pθ(x | z)]  −βDKL (qφ(z | x)∥p(z))]  =Ex∼pdata [LR + βLI]  (3)  β-VAE introduces some fundamental limitations, which  trigger various follow-up research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' InfoVAE introduces a  scaling parameter λ on the KL-term and converts the ob-  jective to (Zhao, Song, and Ermon 2019):  LELBO =αIq(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' z) − DKL (qφ(z | x)∥p(z))  − Eq(z) [DKL (qφ(z | x)∥pθ(x∥z))]  (4)  where Iq is the mutual information with weight α and α +  λ − 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' This linear tuning on the KL shows limitation to  dynamic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Further, conditional VAE (CVAE) (Sohn, Lee, and Yan  2015) introduces an initial guess as a conditional variable  into the objective function for multimodal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' The SA-  VAE involves a cyclical annealing schedule to split the train-  ing to multiple cycles starting at β = 0 and progressively  increases β until β = 1 to reduce the KL vanishing (Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In ControlVAE, the PID control compares the KL di-  vergence with a set point, with their difference as feedback  to the controller to tune the hyperparameter β(t) (Shao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' ControlVAE thus optimizes KL dynamically but is  constrained by the PID controller which follows a separate  tuning mechanism from the VAE itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  The existing work leaves gaps for building an approx-  imate weight allocation between reconstruction and infer-  ence, tuning external hypberparameters within the VAE  working mechanism and handling these issues in a dynamic  manner over an evolutionary learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Our eVAE  addresses these gaps by incorporating the variational genetic  learning into balancing inference and generation and evolu-  tionarily involving their effect into adjusting the VAE learn-  ing behaviors toward better uncertain tradeoff learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  The eVAE Model  eVAE also tunes the representation-generation balance in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' (1) by jointly addressing the following issues in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  (3) and (4): tuning hyperparameter in an outer circle irrele-  vant to VAE behaviors, non-dynamic optimization, and con-  strained settings on data and networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Figure 1 illustrates  the eVAE framework of variational evolutionary learning to  improve the VAE balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  eVAE - Evolving Inner-outer-joint Training  eVAE implements an evolving learning process of optimiz-  ing both VAE and its evolutionary parameters through an  inner-outer-joint iterative training process, as shown in Fig-  ure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' At time t, given the input xt ∈ X, a VAE inner  training process trains a VAE model V (we do not constrain  the VAE framework, for generality, we use β-VAE for case  study in this paper) with some initialization β0 to learn the  encoder qφ(zt | xt), the posterior pθ(xt | zt) with prior  p(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' We obtain the VAE inner objective function with ini-  tial weight β0:  LELBOt = Ext∼pX  �  Eqφ(zt|xt) [log pθ (xt | zt)] −  β0DKL (qφ (zt | xt) ∥p(z))]  (5)  By applying SGVB and reparameterization trick, the gen-  erative process of V estimates the posterior as  pθ(x | z) =  �  t  pθ(xt | zt)  (6)  where xt | zt ∼ N(µ(zt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' θ), Σ(zt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' θ)) with zt = µxt+ϵΣxt  and ϵ ∼ N(0, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Then, we introduce an outer variational evolutionary  learner E (illustrated by a variational genetic algorithm in-  troduced below on Variational Evolution) to optimize the ob-  jective in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' (5) in an outer process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' E with the fitness func-  tion f samples a chromosome βt from an evolving distribu-  tion βt ∼ R and evolves it by taking variational crossover  and mutation to generate and improve parameters βt+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  The evolved parameter is then incorporated into Vt for  the next iteration t + 1 training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' With the relevant ELBO  information back-propagated to tune the VAE network and  optimize the relevant parameters, we obtain the updated re-  construction and inference performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Consequently, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  (5) is evolved to:  LELBOt+1 = Ext+1∼pX  �  Eqφ(zt+1|xt+1) [log pθ (xt+1  | zt+1)] − βt+1DKL (qφ (zt+1 | xt+1) ∥p(z))]  (7)  If the fitness Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' (17) is satisfied, we update β by βt+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  This then results in updated Vt+1 and its parameters φt+1,  θt+1 and βt+1 for iteration t+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' We further repeat the above  VGA E to obtain another set of parameters and retrain model  Vt+1 until it converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  The above inner-outer iterative training progresses over  time and samples in the generative process to iteratively op-  timize the balance between reconstruction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=', minimizing  LR = ∥xt − ˆxt∥ with the reconstructed ˆxt) and inference  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=', seeking appropriate LI) and the VAE learning behav-  iors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' The VAE V is then optimized until converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Accordingly, the eVAE generative process optimizes the  following objective function:  LeV AE = minθt,φt  �N  t=1 LELBOt+1 (θt, φt ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  f (E (φt, θt, {β}, LELBOt)) , xt)  (8)  to approach β∗ for the optimal balance between representa-  tion and construction:  β∗ ∼ f (E ({β} | φt, θt, LELBOt))  (9)  N  iterations  Chromosomes t  Converge  Fit  V-crossover  Diversity  Diversity  𝜷𝑡  𝑡+1  𝑡  Chromosomes t+1  V-mutation  ・  ・  ・  ・・・  ・ ・  ・  ・  𝑞∅ ・  𝑝# ・  ・  ・  ・・・・  ・  ・  ・  ・  𝑞∅ ・  𝑝# ・  𝑓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' "#(·)  ℳ  𝒞  𝜷𝑡+1  {𝛽𝑙}  Diversity  Diversity  Multiple sampling  Sampling  V-crossover  𝛽l,t  𝛽t-1  𝛽t+1  Sampling  Diversity  Multiple sampling  Cauchy/Normal  Distribution  𝛽t+1  V-mutation  Fitness  Gene  Mutate  Crossover  More disentangled  Less disentangled  𝒞  ℳ  𝑡+1 iteration  Resample  Sample  𝜷𝑡-1  Figure 1: The framework of eVAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' The VAE results inform the chromosome sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' The genes are then updated by varia-  tional V-crossover and V-mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' The evolved results are checked per fitness for t + 1 retraining, giving up, or converging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  The overall ELBO of eVAE can then be represented by:  LeV AE = Ez∼qφ(zt|xt) log pθ(xt | zt)+  fβt∼R(βt, E)DKL(qφ (zt | xt) ∥p(z))  = Ext+1∼pX [LR + f(β)LI + E(∆LELBO)]  (10)  Variational Evolution in eVAE  Here, we introduce the variational evolutionary learner E op-  timizing the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' We instantiate E by a variational  genetic algorithm (VGA) with its variational crossover and  mutation operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' This avoids typical issues including  premature convergence and random search during discor-  dant exploitation and exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Variational Genetic Algorithm  eVAE integrates the  VGA-based outer parameter optimization combined with  the VAE-internal gradient descent-based optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Ac-  cordingly, VGA consists of a few variational steps and op-  erations, initialization, variational crossover (V-crossover),  variational mutation (V-mutation), and variational evalua-  tion (V-evaluation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' As shown in the bottom part of Figure  2, below, we introduce them respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Chromosome selection: To enable a larger and more  smooth GA search space compatible with VAEs, chromo-  somes are embedded by a continuous variable β, sampled  from an evolving distribution R (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=', in the crossover).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' As-  sume L individual chromosomes forming a candidate group  {βl} = {β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' , βL}, a chromosome is chosen from this  candidate group to train the VAE, which is associated with  a fitness value f after the VGA operations and VAE retrain-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' We thus have the chromosome-fitness pairs embedded  for the VGA evolution and offspring selection over genera-  tions, such as:  {β, f} = ((β1, f1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' , (βL, fL))  (11)  for optimizing the inner-outer joint training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  V-crossover: Paired chromosomes at time t − 1 and t are  crossovered to generate new genes for improving the ge-  netic variety of offsprings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Following the simulated binary  crossover (SBX) (Deb and Beyer 2001) method,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' we imple-  ment the following crossover operations C to induce a large  search space evolving over chromosome βt−1 at time t − 1  and chromosome βt selected from the chromosome candi-  date group {βl} at time t to obtain the offspring candidate  chromosome βt+1 for the next time t + 1 in C strategy:  C :  �  βt+1 = 1  2[(1 + rc)βl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='t + (1 − rc)βt−1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' or  βt+1 = 1  2[(1 − rc)βl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='t + (1 + rc)βt−1]  (12)  where rc is the crossover rate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' which is drawn from a proba-  bility density function Pc(rc):  Pc (rc) =  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='5(η + 1)rn  c ,  if rc ≤ 1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='5(η + 1)  1  rη+2  c  ,  otherwise  (13)  by concentrating their corresponding parents βt−1 and βl,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  By sampling rc from the following function:  rc =  �  �  �  (2u)  1  η+1 ,  if u ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='5  �  1  2(1−u)  �  1  η+1 ,  otherwise  (14)  where u is a random variable, and η is a hyper-parameter for  scaling, the offsprings βt+1 can then be created by selecting  the better C strategy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=', better satisfying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' (17)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  V-mutation: The crossover-generated offsprings can be  further mutated to improve their genomic diversity and evo-  lutionary capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Aligning with the V-crossover, a varia-  tional mutation strategy M is taken to diversify the offspring  to update βt+1 inherited from crossover or βt from the cur-  rent moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' With a relatively small probability pm, a chro-  mosome βt+1 selected from group is mutated as:  M : βt+1 = βt+1 + rm  (15)  where rm is a random variable sampled from the Cauchy  distribution pm:  Pm = 1  π (  1  1 + β2  t+1  )  (16)  𝑥  𝑞∅ 𝑧𝑡|𝑥𝑡  𝛴!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' "|#$  𝜇!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' "|#$  𝜖  z𝑡  𝑝# 𝑥𝑡|𝑧𝑡  𝐷%&(𝑞∅,"||𝑝#,")  𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  𝑥 − 𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' \'  Inner training  𝛽0  Outer  training  Encoder  Decoder  𝛽t, 𝜃𝑡, ∅t, ℒ(&)*"  𝛽$+,  ∗  , 𝑓$+,  ∗  ℒ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='/01  =  𝑓𝑡(·)  𝑓(𝜀  𝛽 ∅𝑡, 𝜃𝑡,ℒ(&)*" = ∆ℒ(&)*" +∥ 𝐾𝐿𝑡 − 𝑐 ∥\'  Initialization  V-evaluation  V-crossover  V-mutation  Variational GA  +  Figure 2: eVAE - inner-outer joint evolutionary training pro-  cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' The upper part shows the VAE training at time t,  the objectives are then incorporated into the lower part -  the outer training by variational genetic algorithm, whose  fitness-based optimized results are fed to the VAE for fur-  ther training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  V-evaluation & VGA fitness function  The chromosomes  updated by the V-crossover and V-mutation operations are  evaluated in alignment with the VAE objectives to determine  whether they are transferable to the next generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Within  the VAE framework, we take the following heuristic fitness  function f to guide the evolution of chromesomes and their  fitness to VAE objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' The fitness function integrates the  direction of VAE-oriented stochastic gradient descent and  the distance to the optimal KL divergence to achieve multi-  objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  ft+1 = ∆LELBOt+1 + ||KLt+1(βt+1) − c||  (17)  where ∆LELBOt+1 is:  ∆LELBOt+1 = LELBOt+1(βt+1) − LELBOt(βt)  (18)  denoting the VAE’s ELBO difference after applying the  evolved β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' c is the task-specific information bottleneck,  which ensures the bound optimization of eVAE over the evo-  lutionary optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  This V-evaluation and the fitness f guide eVAE to con-  verge and balance the reconstruction and inference with en-  hanced evolutionary parameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' The larger the fitness  value, the stronger the chromosome in its candidate group,  which forms a strong pair {β∗, f ∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  The whole training processes of the eVAE model is illus-  trated in Algorithm 1 in the Supplementary Material A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Theoretical Analysis  In this paper, eVAE takes β-VAE as a case study of evolu-  tionary VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Hence, we analyze the effects of eVAE in ad-  justing β-VAE parameters, tradeoff between reconstruction  and regularization, and training performance below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  β-VAE directly tunes hyperparameter β setting a task-  relevant ratio between the inference loss LI and the genera-  tion loss LR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In essence, incrementally regularizing LI by β  is equivalent to maximizing the likelihood on an experiment-  specific threshold, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=', a constant C, (Higgins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2017):  max  φ,θ Ex∼pdata LR  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' βLI < C  (19)  This learning process can be interpreted in information bot-  tleneck (IB) theory (Alemi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' The βLI constraint  operates as the bottleneck, compressing the capacity of la-  tent variable Z organized from input X to expressively rep-  resent the target ˆX (Alemi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Accordingly, IB  maximizes I(Z, ˆX) to acquire a concise representation:  max I(Z, ˆX)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' I(X, Z) ≤ Ic  (20)  Further, the variational information bottleneck (VIB) intro-  duces an experiment-specific lower bound for optimization:  Lβ−V AE(θ, φ) = −D − βCR ≤ −βCI − D  (21)  where R refers to the compression rate, and D refers to the  distortion measuring the representation relevance to a task,  and βC is an experiment-specific constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' VAEs can thus be  viewed as a process of achieving the rate-distortion tradeoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  ControlVAE is a variant of β-VAE, tuning the KL-specific  βKL with a PID control to a desired KL point:  LControlV AE(θ, φ) = −D − βKLR ≤ −βKLI − D (22)  eVAE extends β-VAE tuning the information bottleneck  βI(X, Z) by a variational evolutionary learner E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' After iter-  ation t of outer training, βt evolves to fit the task and train-  ing phase, guided by the fitness function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' eVAE generates  a lower bound to optimize iteration t+1 of the inner training  phase:  LeV AE ≤ −E(βt)It − E(βt)Dt  (23)  where an evolving iteration-specific lower bound can strike a  compromise between task fitting in −E(βt)Dt and inference  quality in E(βt)It.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' The derivations of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' (21) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' (23)  can be found in the Supplementary Materials B and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  We further compare the VIB effects of eVAE against  VAE, β-VAE and ControlVAE by the rate-distortion (R-D)  curve in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' VAE (yellow) tends to sacrifice empirical  error minimization for representation learning in the early  stage, where the rate is optimized to decrease in a quarter of  the iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' As for β-VAE (green), due to the experiment-  specific lower bound (−βCI − D), the distortion cannot  be minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' ControlVAE (pink) concentrates on optimiz-  ing the distortion initially to fit data and optimizing the rate  eventually to acquire smooth representation, resorting to the  KL-specific lower bound (−βKLI − D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' However, directly  modifying β by the given PID controller leads to fluctuations  between inference capacity and reconstruction quality dur-  ing the iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Rather than monitoring the optimization  process, eVAE generates an iteration-specific lower bound  (−E(βt)It − E(βt)Dt) to achieve a balance between mini-  mizing distortion and controlling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Experiments  We evaluate eVAE in three typical tasks: disentangled rep-  resentation, image generation, and language modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  4  8  12  16  20  24  28  R  0  20  40  60  80  100  120  140  D  VAE  VAE  ControlVAE  eVAE  Figure 3: The information plane with the R − D curves of  VAE, β-VAE, ControlVAE and eVAE on dSprites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Dataset and Baselines  Three different learning tasks and their datasets are listed  below to evaluate eVAE against the VAE baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Disentangled representation learns independent latent  variables to generate an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' We validate eVAE on  dSprites, a 2D-shape dataset with 737,280 binary images  generated by six ground-truth factors: shape (square, ellipse,  heart), scale, orientation, position x, and position y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' To ex-  amine the disentanglement performance and image genera-  tion equally and comprehensively, β-VAE and ControlVAE  are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Image generation reconstructs an imagery sample based  on given datapoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' The CelebA dataset (cropped version)  (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2015) is used, which is a real-world celebrity face  dataset with 202,599 RGB images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' The baselines are β-VAE  and ControlVAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Language modeling conducts the word-level text genera-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' The Penn Treebank dataset (PTB) with English corpus  (Marcinkiewicz 1994) is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' To reveal the performance  of restraining KL vanishing, we compare eVAE with three  heuristic methods: cost annealing (noted as Cost-10k) (Bow-  man et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2016), cyclical annealing (Cyc-8) (Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2019),  and PID control (PID-3) (Shao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  For a fair comparison, we adopt the same embedding net-  work as β-VAE and tune eVAE to achieve the same KL set  point as ControlVAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' The VAE architectures and hyperpa-  rameter settings are given in Supplementary Material D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Performance Evaluation  Performance of disentangled representation learning  Here, we validate the performance of reconstructing a sharp  image with disentangled features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Figure 5(a) shows that  eVAE achieves a lower reconstruction loss of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='2, compared  with ControlVAE under the same expected KL divergence  (KL = 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Moreover, Figure 5(a) and (b) show that eVAE  presents a more stable training curve than other VAEs be-  cause of its dynamic weighting guided by the VGA fitness  function, rather than a direct modification in β-VAE and  ControlVAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' The variation of each disentangled factor is il-  lustrated in Figure 5(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Figure 6 further demonstrates the  reliable disentanglement of all factors and sharp reconstruc-  tion quality of eVAE under a set point KL = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In contrast,  ControlVAE captures the generative factors of scale, x and  y while β-VAE entangles other four factors except scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Performance of image generation We evaluate eVAE on  CelebA for image generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' For a fair and concise com-  parison, the KL set point is set to 200 achieving the best  reconstruction quality in (Shao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' eVAE is tuned  to achieve the desired KL point similar to ControlVAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Fig-  ure 4(a) shows that eVAE has the lowest reconstruction error  compared with vanilla VAE and ControlVAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Guided by the  evolving β, the KL divergence in eVAE can achieve a de-  sired point dynamically, as illustrated in Figure 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  (b) KL  (a) Reconstruction Loss  Figure 4: Performance comparison of different VAEs for im-  age generation on CelebA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Performance of language modeling To show the valid-  ity of eVAE in preventing the variational language model  from degenerating to a standard language model, we draw  the learning curves of KL, reconstruction loss, and weight  of KL over iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' The evolution of β is shown in Figure  7(a), demonstrating that eVAE can learn to tune β over itera-  tions automatically rather than by a heuristic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Figure  7(b) shows that the Cost annealing (Cost-10k) model still  suffers from KL vanishing, with KL approaching 0 at the end  of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' By contrast, the Cyclical annealing (Cyc-8),  PID, and GA can converge to a nonzero divergence, while  eVAE achieves the lowest reconstruction loss to 70 in word  generation, illustrated in Figure 7(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Further, compared to  the PID control, VGA can find a path toward a more stable  stage, without undergoing reconstruction fluctuations during  training even if the non-zero learned β is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Conclusion  We propose evolutionary VAE (eVAE) to dynamically learn  the tradeoff between reconstruction effectiveness and infer-  ence robustness in VAEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' eVAE involves variational evolv-  ing operators to solve the premature convergence and ran-  dom search problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' SGD optimization and genetic al-  gorithm synergistically follow an inner-outer-joint training  mechanism in eVAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Following the variational information  bottleneck theory, eVAE derives an iteration-specific lower  bound balancing the capacity of compression and decom-  pression over iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Multiple task experiments show that  eVAE outperforms SOTA VAEs in traversing samples with  more disentangled features, generating sharp images with  lower reconstruction loss, and resolving the KL vanishing  automatically in variational language modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' We will ex-  tend eVAE to other VAE models and tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  (a) Reconstruction Loss  (b) KL  (c) Element-wise KL Divergence  Figure 5: Learning curves on dSprites1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' (a,b) indicate that eVAE has the lowest reconstruction loss compared with VAE (β = 1),  β-VAE (β = 4), and ControlVAE (KL = 19) under a fixed KL point KL=19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' (c) is the element-wise KL divergence as a function  of iterations in eVAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' We can observe that eVAE retains a stable and reasonable KL divergence of each generator dimension  (factor): position-y (z2), scale (z3), shape (z4), position-x (z6), orientation (z7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' More comparisons in terms of generated KL  divergence can be found in Supplementary Material E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='More comparisons in terms of generated KL divergence can be found in  Supplementary Material E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  y  Scale  Shape  x  Orientation  eVAE (KL = 19)  ControlVAE (KL = 19)  𝜷-VAE (𝜷 = 4)  Figure 6: Latent traversal on dSprites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' To distinguish the latent factors visually, we take the ellipse for illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Each row  represents a latent factor in the traversal, while keeping others fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' The first column refers to the seed image for initialization,  and we then manipulate the latent dimension z across the range [-3, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  (c) Reconstruction Loss  (b) KL  (a) Beta  Figure 7: Learning curves on PTB for language modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' The suffix of PID and VGA refer to a KL set point i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=', KL = 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Cyc-8  indicates 8 iteration cycles where the weight increases linearly from 0 to 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Cost-10k refers to 10k iterations during which the  weight increases from 0 to 1 by a sigmoid function iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='References  Alemi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Poole, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Fischer, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Dillon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Saurous, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  and Murphy, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Fixing a broken ELBO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In ICML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Alemi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Fischer, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Dillon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Murphy, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Deep variational information bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Berliner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Rotman, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Adi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Reichart, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Hazan,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Learning discrete structured variational auto-  encoder using natural evolution strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Bowman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Vilnis, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Vinyals, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Dai, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Jozefow-  icz, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Bengio, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Generating sentences from a  continuous space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In CONLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Bozkurt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Esmaeili, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Tristan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Brooks, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Dy,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and van de Meent, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Rate-regularization and  generalization in VAEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' AISTATS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Burda, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Grosse, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Salakhutdinov, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Impor-  tance weighted autoencoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Burgess, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Higgins, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Pal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Matthey, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Watters, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Desjardins, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Lerchner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Understanding dis-  entangling in β-VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In NeurIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Deb, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Beyer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Self-adaptive genetic algo-  rithms with simulated binary crossover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Evolutionary Com-  putation, 9(2): 197–221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Domke, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Sheldon, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Divide and couple: Us-  ing monte carlo variational objectives for posterior approxi-  mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In NeurIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Esmaeili, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Wu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Jain, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Bozkurt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Siddharth, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Paige, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Brooks, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Dy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Meent, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Structured disentangled representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In AISTATS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Fortuin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' H¨user, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Locatello, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Strathmann, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and  R¨atsch, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Som-vae: Interpretable discrete represen-  tation learning on time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Fu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Gao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Celikyilmaz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Carin,  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Cyclical annealing schedule: A simple approach to  mitigating KL vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In NAACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Ghosh, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Sajjadi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Vergari, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Black, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and  Sch¨olkopf, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' From variational to deterministic au-  toencoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Havtorn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Frellsen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Hauberg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Maaløe, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Hierarchical VAEs know what they don’t know.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In  ICML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Higgins, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=' Matthey, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Pal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Burgess, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Glorot, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Botvinick, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=' Mohamed, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Lerchner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' β-  VAE: Learning basic visual concepts with a constrained  variational framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Jankowiak, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=' and Phan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Surrogate likelihoods  for variational annealed importance sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In ICML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Kamata, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Mukuta, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Harada, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Fully spiking  variational autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' AAAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Khemakhem, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content='  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Variational autoencoders and nonlinear ICA: A unify-  ing framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In AISTATS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Kingma, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Auto-encoding varia-  tional bayes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Klushyn, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=' Chen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Kurle, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Cseke, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content='  Kviman, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=' Melin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=' In AISTATS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Luo, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Tang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Deep learning  face attributes in the wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In ICCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Locatello,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content='  Sch¨olkopf, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' On the fairness of  disentangled representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In NeurIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Marcinkiewicz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' An identifi-  able double VAE for disentangled representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In ICML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Nalisnick, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Gorur, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=' In ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Ran, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Xu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Mei, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Xu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Liu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Detect-  ing out-of-distribution samples via variational autoencoder  with reliable uncertainty estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Razavi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Van den Oord, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Vinyals, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Gen-  erating diverse high-fidelity images with VQ-VAE-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  In  NeurIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Ruiz, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Titsias, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Cemgil, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Doucet, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Unbiased gradient estimation for variational autoencoders  using coupled markov chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In UAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Shao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Yao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Sun, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Zhang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Liu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Abdelzaher, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Controlvae: Controllable  variational autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In ICML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Shi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Paige, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Torr, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=' Variational mixture-  of-experts autoencoders for multi-modal deep generative  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In NeurIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Lee, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Yan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Learning struc-  tured output representation using deep conditional genera-  tive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In NeurIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Sutter, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=' Daunhawer, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=' and Vogt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' General-  ized multimodal ELBO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Bousquet, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=' Gelly, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Schoelkopf, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Wasserstein autoencoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Welling, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content='  In AISTATS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Wang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=' and Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=' Neural gaussian copula  for variational autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=' and Zhang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Solving statistical  mechanics using variational autoregressive networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Phys-  ical Review Letters, 122(8): 080602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Yan, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Amit, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Likelihood regret:  An out-of-distribution detection score for variational auto-  encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' NeurIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Gai, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Improve diverse text generation by self labeling con-  ditional variational auto encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In ICASSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Zhao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Song, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Ermon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' InfoVAE: Balancing  learning and inference in variational autoencoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In AAAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  References  Alemi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Poole, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Fischer, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Dillon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Saurous, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  and Murphy, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Fixing a broken ELBO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In ICML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Alemi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Fischer, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Dillon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Murphy, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Deep variational information bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Berliner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Rotman, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Adi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Reichart, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Hazan,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Learning discrete structured variational auto-  encoder using natural evolution strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Bowman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Vilnis, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Vinyals, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Dai, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Jozefow-  icz, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Bengio, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Generating sentences from a  continuous space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In CONLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Bozkurt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Esmaeili, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Tristan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Brooks, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Dy,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and van de Meent, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Rate-regularization and  generalization in VAEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' AISTATS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Burda, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Grosse, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Salakhutdinov, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Impor-  tance weighted autoencoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Burgess, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Higgins, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Pal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Matthey, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Watters, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Desjardins, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Lerchner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Understanding dis-  entangling in β-VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In NeurIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Deb, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Beyer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Self-adaptive genetic algo-  rithms with simulated binary crossover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Evolutionary Com-  putation, 9(2): 197–221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Domke, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Sheldon, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Divide and couple: Us-  ing monte carlo variational objectives for posterior approxi-  mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In NeurIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Esmaeili, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Wu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Jain, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Siddharth, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Dy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Meent, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Structured disentangled representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In AISTATS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Fortuin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' H¨user, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Locatello, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Strathmann, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Som-vae: Interpretable discrete represen-  tation learning on time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Fu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Gao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Celikyilmaz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Carin,  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Cyclical annealing schedule: A simple approach to  mitigating KL vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In NAACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Ghosh, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Sajjadi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Vergari, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Black, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and  Sch¨olkopf, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' From variational to deterministic au-  toencoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Havtorn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Frellsen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Hauberg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Maaløe, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Hierarchical VAEs know what they don’t know.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In  ICML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Higgins, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Matthey, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Pal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Burgess, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Glorot, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Botvinick, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Mohamed, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Lerchner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' β-  VAE: Learning basic visual concepts with a constrained  variational framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Jankowiak, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Phan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Surrogate likelihoods  for variational annealed importance sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In ICML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Kamata, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Mukuta, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Harada, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Fully spiking  variational autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' AAAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Khemakhem, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Kingma, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Monti, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Hyvarinen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Variational autoencoders and nonlinear ICA: A unify-  ing framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In AISTATS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Kingma, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Welling, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Auto-encoding varia-  tional bayes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Klushyn, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Chen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Kurle, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Cseke, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and van der  Smagt, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=' In  NeurIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Kviman, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=' Melin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Koptagel, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=' Elvira, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content=' and Lager-  gren, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Multiple importance sampling ELBO and  deep ensembles of variational approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In AISTATS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Luo, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Tang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Deep learning  face attributes in the wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In ICCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Locatello,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+page_content='  Abbati,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Rainforth,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Bauer,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Sch¨olkopf, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Bachem, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' On the fairness of  disentangled representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In NeurIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Marcinkiewicz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Building a large annotated cor-  pus of English: The penn treebank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Using Large Corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Mita, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Filippone, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Michiardi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' An identifi-  able double VAE for disentangled representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In ICML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Nalisnick, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Matsukawa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Teh, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Gorur, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and  Lakshminarayanan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Do deep generative models  know what they don’t know?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Ran, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Xu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Mei, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Xu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Liu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Detect-  ing out-of-distribution samples via variational autoencoder  with reliable uncertainty estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Razavi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Van den Oord, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Vinyals, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Gen-  erating diverse high-fidelity images with VQ-VAE-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  In  NeurIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Ruiz, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Titsias, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Cemgil, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Doucet, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Unbiased gradient estimation for variational autoencoders  using coupled markov chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In UAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Shao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Yao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Sun, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Zhang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Liu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Abdelzaher, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Controlvae: Controllable  variational autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In ICML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Shi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Paige, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Torr, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Variational mixture-  of-experts autoencoders for multi-modal deep generative  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In NeurIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Sohn, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Lee, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Yan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Learning struc-  tured output representation using deep conditional genera-  tive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In NeurIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Sutter, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Daunhawer, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Vogt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' General-  ized multimodal ELBO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Tolstikhin, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Bousquet, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Gelly, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Schoelkopf, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Wasserstein autoencoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Tomczak, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Welling, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' VAE with a vampprior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  In AISTATS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Wang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Neural gaussian copula  for variational autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In EMNLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Wu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Zhang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Solving statistical  mechanics using variational autoregressive networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Phys-  ical Review Letters, 122(8): 080602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Xiao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Yan, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Amit, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Likelihood regret:  An out-of-distribution detection score for variational auto-  encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' NeurIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Gai, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Improve diverse text generation by self labeling con-  ditional variational auto encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In ICASSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Zhao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Song, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' and Ermon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' InfoVAE: Balancing  learning and inference in variational autoencoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In AAAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Supplementary Materials -  eVAE: Evolutionary Variational Autoencoder  Here, we present some supplemental materials to the main  content, including the eVAE algorithm, the experimental set-  tings, the latent traversal on square and heart, and derivation  of the variational lower and upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  A  The eVAE algorithm  Below, Algorithm 1 illustrates the modeling process of  eVAE, which enables an inner-outer-joint training of VAE  and evolutionary learning iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Algorithm 1: Evolving Inner-outer-joint Training  1: Input: crossover rate Prc, mutation rate Prm, batch  data x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  2: Output: {β∗, f ∗}  3: Initialization: parameters of decoder θ, parameters of  encoder φ, {βl} = {β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' , βL} //initialize  group  4: while t < T do  5:  Prt ← sample from N (0, 1) //probability to  evolve  6:  if Prt < Prm then  7:  θt,  φt  //save current variational  parameters  8:  generate  βt+1  by  V-crossover  operation  C(βt−1, βt) by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' (12), (13) and (14)  9:  evaluate βt+1 by f(E(φt, θt, {β}, LELBOt)) by  Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' (17) and (18)  10:  else if Prt < Prc then  11:  θt,  φt  //save current variational  parameters  12:  generate βt+1 by V-mutation operation M(βt) by  Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' (15) and (16)  13:  evaluate  βt+1  by  fitness  function  f(E(φt, θt, {β}, LELBOt))  by  Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  (17)  and  (18)  14:  else  15:  {β∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' f ∗} ← βl //select the strongest  pair  16:  θt+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' φt+1 ← LeV AE(θt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' φt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' β∗|xt) // update  variational parameters  17:  end if  18:  Return: {β∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' f ∗}  19: end while  B  Derivation of β-VAE lower bound  According to the VIB theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' elbo can be bounded by dis-  tortion D = −Eqφ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='z)[log pθ(x | z)] and rate R =  Eqφ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='z)[DKL(qφ(z|x)||p(z))],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' where I(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Z) has the up-  per bound I ≤ R:  R =  �  q(x)  �  qθ(z | x) log qθ(z | x)  p(z)  = I(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Z) + TC(Z)  ≥ I(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Z)  (24)  and the lower bound I ≥ H − D:  I =  �  dxdzp(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' z) log p(x | z)  p(x)  ≥  �  dxdzp(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' z) log q(x | z)  p(x)  =  �  dxdzp(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' z) log q(x | z)  −  �  dxp(x) log p(x)  =  �  dxdzp(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' z) log q(x | z)  + H(X)  = H − D  (25)  All in all,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' the variational information bottleneck can be re-  garded as:  H − D ≤ I ≤ R  (26)  where H is the entropy of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  In β-VAE framework, βC impose a experiment-specific  constraint on compressor and decompressor, giving the vari-  ational lower bound:  H − D ≤ I ≤ R  H − D ≤ βI ≤ βR  H ≤ βI + D ≤ βR + D  (27)  In summary, the lower bound can be denoted as:  Lβ−V AE(θ, φ) = −D − βCR ≤ −βCI − D  (28)  C  Derivation of eVAE lower bound  According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' (10) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' (17) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' we can obtain the loss  function of eVAE in iteration t as follows:  LeV AE = Ez∼qφ(zt|xt) log pθ(xt | zt)+  fβt∼R(βt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' E)DKL(qφ(zt | xt)∥p(z))  = Ext+1∼pX [LR + f(βt)LI + E(∆LELBO)]  = Ext+1∼pX [LR + f(βt)LI  + E(∆LELBOt+1 + ||KLt+1(βt+1) − c||)]  (29)  According to the VIB theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Rt sets the upper bound on  information bottleneck It(Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Zt) in t-th iteration :  Rt =  �  q(xt)  �  qθ(zt | xt) log qθ(zt | xt)  p(zt)  = It(Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Zt) + TC(Zt)  ≥ It(Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' Zt)  (30)  and Dt sets the lower bound:  It =  �  dxdzp(xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' zt) log p(xt | zt)  p(xt)  ≥  �  dxdzp(xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' zt) log q(xt | zt)  p(xt)  =  �  dxdzp(xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' zt) log q(xt | zt)  −  �  dxp(xt) log p(xt)  =  �  dxdzp(xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' zt) log q(xt | zt)  + H(X)  = H − Dt  (31)  After optimized by a variational evolutionary learner E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  the loss function of eVAE constrained by a dynamic lower  bound is as follows:  LeV AE = Ez∼qφ(zt|xt) log pθ(xt | zt)+  fβt∼R (βt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' E) DKL(qφ(zt | xt)∥p(z))  = Ext+1∼pX [LR + f(βt)LI + E(∆LELBO)]  = Ext+1∼pX [LR + f(βt)LI  + E(∆LELBOt+1 + ||KLt+1(βt+1) − c||)]  = −Dt − βtRt − E(−Dt+1 − βt+1Rt+1 + Dt + c)  ≤ −E(βt)Dt − E(βt)It  (32)  where c is the desired KL point and βt is a chromosome  sample in t-th iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  D  Experimental settings  Table 1 shows the experimental settings of the three learn-  ing tasks for evaluating eVAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' We give the settings of the  encoder-decoder network architectures and the parameters  for the evolutionary learning on three datasets dSprites,  CelebA, and PTB and their corresponding optimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  E  Analysis of element-wise KL divergence in  disentangled representation learning  Figure 8 further shows the element-wise KL divergence  evolving over iterations of three VAE models: β-VAE and  ControlAVE in comparison with our eVAE on the dataset  dSprites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  Dataset  Optimiser  Architecture  eVAE initialization  dSprites  Adam(1e-4)  Input  Encoder  Latent  Decoder  (64,64,1)  Conv(32, 32, 32), ReLu, Conv(32,16,16),  ReLu, Conv(32,8,8), ReLu,Conv(32,4,4),  Relu, Linear(256), Relu, Linear(256),  Relu, Linear(20)  10  Linear(256), Relu, Linear(256),  Relu, Linear(512), Relu,  Conv(32,4,4), Relu, Conv(32,8,8),  Relu, Conv(32,16,16), Relu,  Conv(32,32,32), Relu, (1, 64,64)  Mutation Rate  Crossover Rate  Individual Number  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='04  20  CelebA  (KL = 200)  Adam(1e-4)  Input  Encoder  Latent  Decoder  (128,128,3)  Conv(32, 32, 32), ReLu,  Conv(32,16,16), ReLu,  Conv(32,8,8),ReLu,  Conv(32,4,4), Relu, Linear(256),  Relu, Linear(256), Relu, Linear(20)  10  Linear(256), Relu, Linear(256),  Relu, Linear(512), Relu,  Conv(32,4,4), Relu,  Conv(32,8,8), Relu,  Conv(32,16,16), Relu, Conv(32,32,32),  Relu, (3, 64,64)  Mutation Rate  Crossover Rate  Individual Number  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='03  20  PTB  (KL = 3)  init lr: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='001  threshold:100  decay factor: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='5  max decay: 5  Input  Encoder  Latent  Decoder  Mutation Rate  Crossover Rate  Individual Number  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='07  10  Table 1: Experimental settings for disentangled representation learning on dSprites, image generation on CelebA, and language  generation on PTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content='  (c) eVAE  (b) ControlVAE  (a) 𝛽-VAE  Figure 8: The element-wise KL divergence over iterations for β-VAE, ControlAVE, and eVAE on dSprites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' It shows that each  KL divergence of eVAE is larger than that of β-VAE, providing an enough space for generating disentangled representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
+page_content=' In  addition, eVAE generates more stable KL divergence compared to ControlVAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfOvaI/content/2301.00011v1.pdf'}
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+arXiv:2301.01433v1  [math.DG]  4 Jan 2023
+SEMI-STABLE TWISTED HOLOMORPHIC VECTOR BUNDLES OVER
+GAUDUCHON MANIFOLDS
+ZHENGHAN SHEN
+Abstract. In this paper, we study the semi-stable twisted holomorphic vector bun-
+dles over compact Gauduchon manifolds. By using Uhlenbeck–Yau’s continuity method,
+we show that the existence of approximate Hermitian–Einstein structure and the semi-
+stability of twisted holomorphic vector bundles are equivalent over compact Gauduchon
+manifolds. As its application, we show that the Bogomolov type inequality is also valid
+for a semi-stable twisted holomorphic vector bundle.
+1.
+Introduction
+Let M be a complex manifold of dimension n and g a Hermitian metric with the
+associated K¨ahler form ω. The Hermitian metric g is called Gauduchon if
+∂ ¯∂ωn−1 = 0.
+It has been proved by Gauduchon ([18]) that every Hermitian metric is conformal to a
+Gauduchon metric (uniquely up to scaling when n ≥ 2). The Hermitian metric ω is said
+to be Astheno-K¨ahler if ∂ ¯∂ωn−2 = 0. The concept of Astheno-K¨ahler was introduced by
+Jost and Yau in [23]. In this case, the second Chern number of a twisted vector bundle
+can be well defined. Throughout the paper, we will assume ω is Gauduchon if there is no
+additional emphasis.
+Twisted sheaves were introduced by Giraud in [19], and can be defined in several equiv-
+alent ways. It can be regarded as family of sheaves on an open covering of M together with
+a twisted gluing, as sheaves of modules over an Azumaya algebra on M ([10]), as sheaves
+over a gerb on M ([14, 19, 42]), etc. These objects have found many applications in
+physics for the description of the so-called B-fields and the K-theory. We should mention
+that the twisted K-theory has appeared earlier in the mathematical physics literature, in
+the study of the quantum Hall effect ([11, 12]).
+In recent years, the existence of Hermitian–Einstein has been extensively studied. The
+classical Donaldson–Uhlenbeck–Yau theorem says that a holomorphic vector bundle over
+compact K¨ahler manifolds admit Hermitian–Einstein metrics if it is stable. This result
+was first proved by Narasimhan and Seshadri ([34]) for compact Riemannian surface by
+using the methods of algebraic geometry. Subsequently, Donaldson ([15, 16, 17]) gave a
+new proof of the Narasimhan-Seshadri theorem, and generalized to algebraic surfaces and
+2020 Mathematics Subject Classification. 53C07, 57N16.
+Key words and phrases. approximate Hermitian-Einstein metric, twisted vector bundle, semi-stability,
+Gauduchon manifold.
+1
+
+2
+Z. Shen
+algebraic manifolds by using Hermitian–Yang–Mills flow. Uhlenbeck and Yau ([40, 41])
+proved it for general compact K¨ahler manifold by using continuity method.
+The inverse of the problem which states that a holomorphic bundle carrying a Hermitian–
+Einstein metric must be polystable is also valid. It was proved by Kobayashi ([29]) and
+L¨ubke ([31]) independently. So there is a one-to-one correspondence bewteen the algebraic
+notion of stability and the existence of Hermitian–Einstein on holomorphic vector bundle.
+This is usually referred as to Hitchin–Kobayashi correspondence (see [1, 4, 21, 32, 38] and
+references therein). There are several interesting generalizations for this coorespondence.
+For example, Li–Yau ([26]) studied the coorespondence for Hermitian manifolds with
+Gauduchon metric; Hitchin ([20]) and Simpson ([39]) studied the Higgs bundle case which
+plays an important role in non-abelian Hodge theory; Mochizuki ([33]) studied such corre-
+spondence on the non-compact K¨ahler manifolds; Biswas–Loftin–Stemmler ([7]) studied
+for affine manifolds; Biswas–Kasuya ([5, 6]) studied for Sasakian manifolds; Zhang–Zhang–
+Zhang ([43]) studied for some non-compact non-K¨ahler manifolds.
+If one considers the weaker stability condition, there is an analog of such correspondence
+between the semi-stability and the approximate Hermitian-Einstein metric. Kobayashi
+([29]) introduced the notion of approximate Hermitian–Einstein metric for holomorphic
+vector bundles. He proved a holomorphic vector bundle admitting an approximate Hermitian–
+Einstein metric must be semi-stable over a compact K¨ahler manifold. In [8], Bruzzo and
+Gra˜na Otero generalized above result to the Higgs bundle case. When the base manifold
+is projective, Kobayashi proved the inverse part and conjectured that it should be true
+for general K¨ahler manifolds. It was solved by Li–Zhang ([25]) and Jacob ([22]) indepen-
+dently for the general K¨ahler manifold case. In the principal bundle case, it was studied
+by Biswas–Jacob–Stemmler ([2, 3]). Recently, Nie–Zhang ([35]) proved the existence of
+approximate Hermitian–Einstein structures is equivalent to the semi-stability on Higgs
+bundles over compact Gauduchon manifolds. Subsequently, Zhang–Zhang–Zhang ([43])
+also studied the semi-stable Higgs bundles on some non-compact Gauducon manifolds.
+In the present paper, our considerations focus on the twisted vector bundles over com-
+pact Gauduchon manifolds. The twisted vector bundles are local vector bundles on a
+open covering with a twisted gluing, which is the point of view of C˘ald˘araru ([10, 13]).
+Let U := {Ui}i∈I be a fixed open covering of M. A B-field on M with respect to U is a
+family B = {Bi}i∈I, where each Bi is a 2-form on Ui such that
+(1.1)
+Bi − Bj = dβij,
+for some 1-forms βij on Uij := Ui ∩ Uj. Notice that d(βij + βjk + βki) = 0, we may find
+U(1)-valued functions αijk on Uijk := Ui ∩ Uj ∩ Uk such that
+(1.2)
+βij + βjk + βki = −α−1
+ijkdαijk,
+when the open covering U is sufficiently fine. Then αB := {αijk} is a 2-cocycle whose
+cohomology class lies in H2(M, O∗
+M), which is called the twist induced by B. We will omit
+the subscript B when there is no confusion. An α-twisted holomorphic vector bundle is
+a pair E := ({Ei}i∈I, {ϕij}i,j∈I), where Ei is a holomorphic vector bundle on Ui and
+ϕij : Ej|Uij → Ei|Uij
+
+Semi-stable twisted holomorphic vector bundles over Gauduchon manifolds
+3
+is an isomorphism of holomorphic vector bundles on Uij, such that ϕii = IdEi, ϕ−1
+ij = ϕji
+and ϕki◦ϕjk◦ϕij = αijk·IdE|Uijk for each i, j, k ∈ I. Here we usually omit the holomporphic
+structure ¯∂Ei on Ei when there is no confusion.
+It can be easily see that 1-twisted
+holomorphic vector bundle is the usual holomorphic vector bundle.
+In [42], S. Wang proved the Hitchin–Kobayashi coorespondence for twisted holomorphic
+vector bundles over a gerbe on compact K¨ahler manifolds. All the definitions in [42] are
+in the category of twisted holomorphic vector bundles as holomorphic vector bundles over
+a gerb. Recently, A. Perego ([37]) generalized Wang’s result to complex manifolds with
+a Gauduchon metric. And he also proved the twisted version of approximate Hitchin–
+Kobayashi coorespondence on compact K¨ahler manifolds. Inspired by Perego’s work, we
+consider the case that the approximate Hermitian–Einstein metric for twisted holomor-
+phic vector bundle over compact Gauduchon manifolds. In fact, we prove the following
+theorem.
+Theorem 1.1. Let E = {Ei, ϕij}i,j∈I be an α-twisted holomorphic vector bundle of rank
+r over a compact Gauduchon manifold (M, ω). Then E is ω-semi-stable if and only if it
+admits an approximate Hermitian–Einstein structure.
+We should remark that Perego ([37]) only proved the above theorem when the base man-
+ifold is a K¨ahler manifold. He mainly follow the methods of Jacob ([22]) and Kobayashi
+([29]) by using Donaldson’s heat flow. The proof relies on the properties of the Donald-
+son’s Lagrangian. However, the Donaldson’s Lagrangian is not well-defined when the base
+manifold is only Gauduchon. So his argument can not be generalized to Gauduchon case
+directly. In this paper, we will adapt Nie–Zhang’s arguments ([35]) to our case, but there
+are some differences in the treatment of certain details.
+As an application, we obtain the following Bogomolov type inequality.
+Theorem 1.2. Let (M, ˜ω) be a compact Astheno-K¨ahler manifold of dimension n, and
+ω a Gauduchon metric which is conformal to ˜ω. Let E = {Ei, ϕij}i,j∈I be an α-twisted
+holomorohic vector bundle of rank r over M. If E is ω-semi-stable, then we have the
+following Bogomolov type inequality
+(1.3)
+�
+M
+�
+2c2(E) − r − 1
+r
+c1(E) ∧ c1(E)
+�
+∧
+˜ωn−2
+(n − 2)! ≥ 0.
+The Bogomolov type inequality was first proved by Bogomolov ([9]) for semi-stable
+holomorphic vector bundles over complex algrebraic surfaces. Subsequently, it was gen-
+eralized to Hermitian–Einstein vector bundles over compact K¨ahler manifolds by L¨ubke (
+[30]). Recently, Li–Nie–Zhang ([24]) proved the Bogomolov type inequality over Gaudu-
+chon Astheno-K¨ahler manifolds. In twisted case, Perego ([37]) obtained the Bogomolov
+type inequality for semi-stable α-twisted holomorphic vector bundle over compact K¨ahler
+manifolds.
+In our paper, we will proved the Bogomolov type inequality by following
+Li–Nie–Zhang’s arguments.
+This article is organized as follows. In Section 2, we briefly present some notations for
+α-twisted holomorphic vector bundle, such as connection, curvature and semi-stability.
+In Section 3, we give the detailed proof of Theorem 1.1.
+In Section 4, we prove the
+Bogomolov type inequality for semi-stable α-twisted holomorphic vector bundles over
+Gauduchon Astheno-K¨ahler manifolds.
+
+4
+Z. Shen
+2. Preliminary
+In this section, we will introduce the basic notations of α-twisted vector bundles that
+will be used throughout the paper. We will follow the notations of [37].
+2.1. Hermitian metrics, connections and curvatures. Let E = {Ei, ϕij}i,j∈I be
+an α-twisted holomorphic vector bundle over compact Gauduchon manifold (M, ω). A
+Hermitian metric on E is a connection H = {Hi}i∈I, where Hi is a Hermitian metric on
+Ei and Hi = ϕT
+ijHj ¯ϕij for each i, j ∈ I, i.e., for any ξ, η ∈ Γ(Ei),
+Hi(ξ, η) = Hj(ϕij(ξ), ϕij(η)).
+Since αijk is U(1)-valued, it can be easily seen that the definition is well-defined. And
+there always exists a Hermitian metric H = {Hi}i∈I on an α-twisted vector bundle (see
+[37] for details).
+We should remark that Hi also denotes the matrix of smooth functions with respect to
+a given local frame and ϕij denotes the matrix of smooth functions with respect to the
+chosen local frames of Ei and Ej when there is no confusion.
+A connection on α-twisted vector bundle is a family D = {Di}i∈I, where Di is a
+connection on Ei and locally the connection 1-form Ai satisfies
+Ai = ϕ−1
+ij Ajϕij + ϕ−1
+ij dϕij + βij · IdE.
+The notion of connection on a twisted bundle can be found in [28].
+We say that a
+connection D = {Di}i∈I is compatible with the Hermitian metric H = {Hi}i∈I, if each
+Di is compatible with Hi, i.e., for any ξ, η ∈ Γ(Ei),
+d(Hi(ξ, η)) = Hi(Di(ξ), η) + Hi(ξ, Di(η)).
+In a local given frame of Ei, it usually represents as
+dHi = AT
+i Hi + Hi ¯Ai.
+Let FD ∈ Ω2(End(E)) be curvature form of the connection D = {Di}i∈I. According to
+[37], we have
+FD|Ui = FDi − Bi · IdEi,
+where FDi is the curvature of Di for every i ∈ I. Thoughtout the paper, we assume
+that each Bi of the B-filed B is a (1, 1)-form on Ui and βij is a (1, 0)-form on Uij. Then
+given a Hermitian metric H on the α-twisted holomorphic vector bundle E, there is a
+unique Chern connection DH which is compatible with the Hermitian metric H and the
+holomorphic structure of E. In fact, we have the following lemma.
+Lemma 2.1 (Lemma 2.25 in [37]). Let E be an α-twisted holomorphic vector bundle and
+H be a Hermitian metric on E.
+(1) There is a unique connection DH which is compatible both with the holomorphic
+structure of E and the Hermitian metric H.
+(2) If each Bi of the B-field is a (1, 1)-form for every i ∈ I, then FD ∈ Ω1,1(End(E)).
+Let FH ∈ Ω1,1(End(E)) be the curvature form of the Chern connection DH, then we
+have
+FH|Ui = FHi − Bi · IdEi,
+
+Semi-stable twisted holomorphic vector bundles over Gauduchon manifolds
+5
+where FHi is the curvature form of the Chern connection DHi on Ei. A Hermitian metric
+H = {Hi}i∈I in E = {Ei, ϕij}i,j∈I is called a Hermitian–Einstein metric if the curvature
+FH of the Chern connection DH satisfies
+(2.1)
+√
+−1ΛωFH = λ · IdE,
+where Λω denotes the contraction of differential forms by ω, and λ is a real constant.
+An α-twisted holomorphic vector bundle E = {Ei, ϕij}i,j∈I is said to be admitting an
+approximate Hermitian–Einstein structure if for every ε > 0, there is a Hermitian metric
+Hε = {Hi,ε}i∈I such that
+(2.2)
+max
+X
+|
+√
+−1ΛωFHε − λ · IdE|Hε < ε.
+2.2. Stability. Let E = {Ei, ϕij}i,j∈I be an α-twisted holomorphic vector bundle of rank
+r and H = {Hi}i∈I be a Hermitian metric on E. Then the first Chern form of (E, H) is
+(2.3)
+c1(E, H) =
+√−1
+2π tr(FH) ∈ Ω1,1(M).
+The second Chern form of (E, H) is
+(2.4)
+c2(E, H) = − 1
+8π2
+�
+(trFH)2 − tr(FH ∧ FH)
+�
+.
+The degree of (E, H) with respect to the Gauduchon metric ω on M is defined to be
+(2.5)
+degω(E) :=
+�
+M
+c1(E, H) ∧
+ωn−1
+(n − 1)!.
+Since ω is a Gauduchon metric, the definition of the degree is well-defined and independent
+of the choice of the Hermitian metric H.
+Let F = {Fi, ϕij}i,j∈I be an α-twisted coherent sheaf of rank s on M. The definition of
+stability for twisted coherent sheaves was first introduced by Lieblich ([27]) and A. Perego
+([36, 37]) studied the stability of twisted coherent sheaves in the language of twisted
+gluing of coherent sheaves. Given a rank s α-twisted coherent sheaves, the determinant
+of F = {det(Fi), det(ϕij)}i,j∈I is a locally free αs-twisted sheaf of rank 1. The ω-degree
+of F is defined by
+(2.6)
+degω(F) := degω(det(F)).
+If F is non-trivial, the slope of F with respect to ω is defined by
+(2.7)
+µω(F) = degω(F)
+rank(F).
+Let E = {Ei, ϕij}i,j∈I be an α-twisted holomorphic vector bundle on M. We say an α-
+twisted holomorphic vector bundle E is ω-stable (resp. ω-semi-stable) if for every proper
+α-twisted coherent subsheaf F of E, it holds that
+(2.8)
+µω(F) < µω(E)
+(resp.
+µω(F) ≤ µω(E)).
+
+6
+Z. Shen
+2.3. Endomorphisms of twisted vector bundles. In the rest of this section, we will
+introduce the endomorphisms of twisted vector bundles. Let E = {Ei, ϕij}i,j∈I be an
+α-twisted vector bundle on M. An endomorphism f : E → E is a family f = {fi}i∈I,
+where fi : Ei → Ei is an endomorphism of Ei on Ui and for every i, j ∈ I,
+(2.9)
+ϕij ◦ fi = fj ◦ ϕij.
+It is easy to check that the eigenvalues of f are smooth functions on the whole M. In
+fact, if λi is a smooth function on Ui and ξ ∈ Γ(Ei) is a nowhere vanishing smooth section
+such that fi(ξ) = λiξ, then
+fj(ϕij(ξ)) = ϕij(fj(ξ)) = ϕij(λiξ) = λiϕij(ξ).
+So λi is an eigenvalue for fj with a nowhere vanishing smooth section ϕij(ξ). Hence the
+eigenvalues of fi glue together to give global functions on M.
+Given a Hermitian metric H = {Hi}i∈I on E = {Ei, ϕij}i,j∈I, we say that an endo-
+morphism f is self-adjoint with respect to H, if for every i ∈ I and ξ, η ∈ Γ(Ei), it
+holds
+(2.10)
+Hi(fi(ξ), η) = Hi(ξ, fi(η)).
+We denote
+Herm(E, H) = {f ∈ End(E)|f ∗H = f},
+and
+Herm+(E, H) = {f ∈ Herm(E, H)|f > 0},
+where f > 0 means that all the eigenvalues of f are positive.
+If H = {Hi}i∈I and
+K = {Ki}i∈I are two Hermitian metrics on α-twisted vector bundles, then for every i ∈ I
+and ξ, η ∈ Γ(Ei), the endomorphism h = {hi}i∈I is defined by
+Hi(ξ, η) = Ki(hiξ, η).
+It is easy to see that h = K−1H is self-adjoint with respect to H and K.
+For any h ∈ Herm+(E, K), we can choose an open dense subset W ⊂ M satisfying
+that, at each x0 ∈ W there exists an open neighborhood Ui of x0, a local unitary basis
+{ej}r
+j=1 with respect to Ki and functions {λj ∈ C∞(Ui, R)}r
+i=1 such that
+h(x) =
+r
+�
+j=1
+eλj(x)ej(x) ⊗ ej(x)
+for all x ∈ Ui, where {ej}r
+j=1 is the dual basis corresponding to {ej}r
+j=1. Then according
+to [37, Lemma 2.56], we have
+log h(x) =
+r
+�
+j=1
+λj(x)ej(x) ⊗ ej(x).
+Suppose Ψ : R × R → R is a smooth function, A = �r
+i,j=1 Aijei ⊗ ej ∈ End(E), and
+s ∈ Herm(E, K). Then we define Ψ(s)(A) by
+Ψ(s)(A) =
+n
+�
+i,j=1
+Ψ(λi, λj)Aijei ⊗ ej.
+
+Semi-stable twisted holomorphic vector bundles over Gauduchon manifolds
+7
+3. Proof of Theorem 1.1
+Let E = {Ei, ϕij}i,j∈I be an α-twisted holomorphic vector bundle of rank r over a
+compact Gauduchon manifold (M, ω). Suppose H = {Hi}i∈I and K = {Ki}i∈I are two
+Hermitian metrics on E. Let DH = {DHi}i∈I (resp.
+DK = {DKi}i∈I) be the Chern
+connection of (E, H) (resp. (E, K)). Then for every i ∈ I, we have
+(3.1)
+∂Hi = ∂Ki + h−1
+i ∂Kihi,
+where ∂Ki denotes the (1, 0)-part of the connection induced by DKi on End(Ei). The
+relation between FHi and FKi is given by
+FHi = FKi + ¯∂i(h−1
+i ∂Kihi).
+(3.2)
+Then we have
+FH|Ui =FHi − Bi · IdEi = FKi + ¯∂i(h−1
+i ∂Kihi) − Bi · IdEi
+=FK|Ui + ¯∂i(h−1
+i ∂Kihi).
+(3.3)
+So if we glue together, it follows that
+(3.4)
+FH = FK + ¯∂(h−1∂Kh).
+Therefore H solves the Hermitian–Einstein equation on α-twisted holomorphic bundle E
+if and only if there exist Hermitian metric K and h ∈ Herm+(E, K) such that
+(3.5)
+√
+−1ΛωFK − λ · IdE +
+√
+−1Λω ¯∂(h−1∂Kh) = 0.
+Now, fixing a proper background Hermitian metric K = {Ki}i∈I on α-twisted holomor-
+phic vector bundle E = {Ei, ϕij}i,j∈I, we consider the following perturbed equation
+Lε(hε) : =
+√
+−1ΛωFHε − λ · IdE + ε log(hε)
+=
+√
+−1ΛωFK − λ · IdE +
+√
+−1Λω ¯∂(h−1
+ε ∂Khε) + ε log(hε),
+(3.6)
+where hε = K−1Hε ∈ Herm+(E, K) and ε ∈ (0, 1]. It is obvious that hε and log hε are
+self-adjoint with respect to K and Hε in the sense of (2.10). In [37], Perego proved that
+the equation (3.6) is solvable for all ε ∈ (0, 1] by using the continuity method. If the
+α-twisted holomorphic vector bundle E is semi-stbale, we can show that
+(3.7)
+lim
+ε→0 ε max
+M | log hε|K = 0.
+This implies that maxM |√−1ΛωFHε − λ · IdE| → 0 as ε → 0. So we obtain the existence
+of approximate Hermitian–Einstein structure on E.
+Before giving the detailed proof of (3.7), we need the following two lemmas.
+Lemma 3.1 (Lemma 5.7 in [37]). Let E = {Ei, ϕij}i,j∈I be an α-twisted holomorphic
+vector bundle on M, and K be a Hermitian metric on E. If h ∈ Herm+(E, K) satisfies
+Lε(h) = 0 for some ε > 0.
+(1) We have
+(3.8)
+1
+2
+√
+−1Λω ¯∂∂(| log(h)|2) + ε| log(h)|2
+K ≤ |Φ(K)|K · | log(h)|K,
+where Φ(K) := √−1ΛωFK − λ · IdE.
+
+8
+Z. Shen
+(2) If m := maxM | log(h)|K(x), then we have
+(3.9)
+m ≤ 1
+ε max
+x∈M |Φ(K)|K(x).
+(3) There is a real number C (depending only on ω and K) such that
+(3.10)
+m ≤ C(|| log h||L2 + max
+M |Φ(K)|K).
+Lemma 3.2. Let E = {Ei, ϕij}i,j∈I be an α-twisted holomorphic vector bundle on M,
+and K = {Ki}i∈I be a Hermitian metric on E. If hε = {hi,ε}i∈I ∈ Herm+(E, K) solves
+(3.6) for some ε, then it holds that
+(3.11)
+�
+M
+tr(Φ(K)sε)ωn
+n! +
+�
+M
+⟨Ψ(sε(¯∂sε), ¯∂sε)⟩K
+ωn
+n! = −ε||sε||2
+L2,
+where sε = {si,ε}i∈I = log hε, Φ(K) = √−1ΛωFK − λ · IdE and
+Ψ(x, y) =
+�
+ey−x−1
+y−x ,
+x ̸= y;
+1,
+x = y.
+Proof. Since hε solves (3.6), then we have
+(3.12)
+⟨Φ(K), sε⟩K + ⟨
+√
+−1Λω ¯∂(h−1
+ε ∂Khε), sε⟩K = −⟨sε, sε⟩.
+Integrating over M gives
+(3.13)
+�
+M
+tr(Φ(K)sε)ωn
+n! +
+�
+M
+⟨
+√
+−1Λω ¯∂(h−1
+ε ∂Khε), sε⟩K
+ωn
+n! = −||sε||L2.
+Then comparing (3.13) with (3.11), it suffices to prove
+(3.14)
+�
+M
+⟨
+√
+−1Λω ¯∂(h−1
+ε ∂Khε), sε⟩K
+ωn
+n! =
+�
+M
+⟨Ψ(sε(¯∂sε), ¯∂sε)⟩K
+ωn
+n! .
+An easy calculation (see Proposition 3.1 in [35] for details) shows that
+(3.15)
+�
+M
+⟨
+√
+−1Λω ¯∂(h−1
+ε ∂Khε), sε⟩K
+ωn
+n! =
+�
+M
+√
+−1tr(h−1
+ε ∂Khε ∧ ¯∂sε) ωn−1
+(n − 1)!.
+On the other hand, following the methods in [32], we can choose an open dense subset
+W ⊂ M satisfying that at each x ∈ W there exists an open neighborhood Ui ∈ U of x, a
+local unitary basis {ej}r
+j=1 of Ei with respect to Ki and eigenvalues {λj ∈ C∞(Ui, R)}r
+j=1
+such that
+si,ε(y) =
+r
+�
+j=1
+λj(y)ej(y) ⊗ ej(y),
+for all y ∈ Ui, where {ej}r
+j=1 is the dual basis of E∗
+i . Here λ1, · · · , λr are eigenvalues of sε
+(and hence of si,ε). Therefore, we have
+∂Kihi,ε(x) = eλj∂λjej ⊗ ej + (eλk − eλj)Aj
+kej ⊗ ek,
+and
+¯∂si,ε(x) = ¯∂λjej ⊗ ej + (λk − λj)(−Ak
+j)ej ⊗ ek,
+
+Semi-stable twisted holomorphic vector bundles over Gauduchon manifolds
+9
+where {Ak
+j}r
+j,k=1 are the (1, 0)-forms defined defined by ∂Kiej = Ak
+jek. Then on each Ui,
+it follows that
+tr
+√
+−1Λω(h−1
+i,ε ∂Kihi,ε ∧ ¯∂si,ε)
+=
+r
+�
+j=1
+|¯∂λj|2 +
+�
+j̸=k
+(eλk − eλj)(λj − λk)
+√
+−1ΛωAj
+k ∧ (−Ak
+j)
+=
+r
+�
+j=1
+|¯∂λj|2 +
+�
+j̸=k
+eλk−λj − 1
+λk − λj
+(λk − λj)2| − Aj
+k|2
+=
+r
+�
+j,k=1
+Ψ(λj, λk)|(¯∂si,ε)k
+j|2.
+As both sides glue together, we have
+tr
+√
+−1Λω(h−1
+ε ∂Khε ∧ ¯∂sε) =
+r
+�
+j,k=1
+Ψ(λj, λk)|(¯∂s)k
+j|2.
+Using the similar argument, we can also prove
+⟨Ψ(s)(¯∂s), ¯∂s⟩K =
+r
+�
+j,k=1
+Ψ(λj, λk)|(¯∂s)k
+j|2.
+Therefore, we have
+(3.16)
+⟨Ψ(s)(¯∂s), ¯∂s⟩K = tr
+√
+−1Λω(h−1
+ε ∂Khε ∧ ¯∂sε).
+Combining (3.15) and (3.16), we complete the proof.
+□
+In the following, we are ready to prove Theorem 1.1. First, we show that the semi-
+stability implies the existence of approximate Hermitian–Einstein structure on α-twisted
+holomorphic vector bundles. In fact, we have the following theorem.
+Theorem 3.3. Let E = {Ei, ϕij}i,j∈I be an α-twisted holomorphic vector bundle over
+(M, ω). If E is ω-semi-stable, then there is an approximate Hermitian–Einstein structure
+on E. i.e.,
+(3.17)
+max
+M |Φ(Hε)|Hε → 0,
+as
+ε → 0.
+Proof. Let K be a background Hermitian metric on E. Without loss of generality, we can
+assume that the Hermitian K satisfies
+tr(
+√
+−1ΛωFK − λ · IdE) = 0.
+In fact, by an approximate conformal change K = eφ ˜K, we have
+(3.18)
+√
+−1Λω ¯∂∂φ = −1
+rtr(
+√
+−1ΛωF ˜
+K − λ · IdE),
+where ˜K is an arbitrary Hermitian metric on E. Since
+�
+M tr(√−1ΛωF ˜
+K − λ · IdE) ωn
+n! = 0,
+then there is a smooth function φ satisfying (3.18).
+
+10
+Z. Shen
+Let {hε}0<ε≤1 be the solutions of the perturbed equation (3.6) with the background
+Hermitian metric K. Then we have
+|| log hε||2
+L2 = −1
+ε
+�
+M
+⟨Φ(Hε), log hε⟩Hε
+ωn
+n! ,
+where Φ(Hε) = √−1ΛωFHε − λ · IdE. By [37, Lemma 5.2], we also have
+(3.19)
+det(hε) = 1.
+We will consider the limit behavior of || log hε||L2 in two cases.
+Case 1: || log hε||L2 is bounded. i.e., there exists a constant C1 such that
+|| log hε||L2 < C1 < +∞.
+In this case, from Lemma 3.1, we have
+max
+M |Φ(Hε)|Hε = ε · max
+M | log hε|Hε < εC · (|| log hε||L2 + max
+M |Φ(K)|K)
+≤ εC(C1 + max
+M |Φ(K)|K) → 0
+as ε → 0.
+Case 2: lim supε→0 || log hε||L2 = +∞. In this case, if E = {Ei, ϕij}i,j∈I is ω-semi-
+stable, then we also have
+(3.20)
+lim
+ε→0 max
+M |Φ(Hε)|Hε = 0.
+We will follow Simpson’s argument ([39]) to show that if (3.19) is not valid, there exists
+a coherent α-twisted subsheaf contradicting the ω-semi-stability of E.
+We will prove (3.20) by contradiction. If (3.20) does not hold, then there exist δ > 0
+and a subsequence εi → 0, as i → +∞, such that
+|| log hεi||L2 → +∞
+and
+(3.21)
+max
+M |Φ(Hεi)|Hεi = εi max
+M | log hεi|Hεi ≥ δ.
+Set
+sεi = log hεi, li = ||sεi||L2, uεi = sεi
+li
+.
+Then we have ||uεi||L2 = 1. From (3.19), it follows that tr(uεi) = 0. Then combining
+(3.21) with Lemma 3.1, we have
+(3.22)
+li ≥
+δ
+Cεi
+− max
+M |Φ(K)|K
+and
+(3.23)
+max
+M |uεi| < C
+li
+�
+li + max
+M |Φ(K)|K
+�
+< C2 < +∞.
+In the following, we divide our proof into two steps.
+Step 1 We will show that uεi converge to u∞ weakly as i → +∞. We need to show
+that ||uεi||L2
+1 are uniformly bounded. Since ||uεi||L2 = 1, it enough to prove ||¯∂uεi||L2 are
+uniformly bounded.
+
+Semi-stable twisted holomorphic vector bundles over Gauduchon manifolds
+11
+By Lemma 3.11, for each hεi, it holds
+(3.24)
+�
+M
+tr(Φ(K)uεi)ωn
+n! + li
+�
+M
+⟨Ψ(liuεi)(¯∂uεi, ¯∂uεi)⟩K
+ωn
+n! = −εili.
+Combining (3.22) and (3.24), we obtain
+(3.25)
+δ
+C +
+�
+M
+tr(Φ(K)uεi)ωn
+n! + li
+�
+M
+⟨Ψ(liuεi)(¯∂uεi, ¯∂uεi)⟩K
+ωn
+n! ≤ εi max
+M |Φ(K)|K.
+Next, we consider the function
+(3.26)
+lΨ(lx, ly) =
+�
+l,
+x = y;
+el(y−x)−1
+y−x
+,
+x ̸= y.
+Because of (3.23), we may assume that (x, y) ∈ [−C2, C2] × [−C2, C2]. It is easy to check
+that
+(3.27)
+lΨ(lx, ly) →
+�
+(x − y)−1,
+x > y;
++∞,
+x ≤ y,
+increases monotonically as l → +∞. Let ζ ∈ C∞(R×R, R+) satisfying ζ(x, y) < (x−y)−1
+whenever x > y. From (3.25), (3.27) and the same arguments in [39, Lemma 5.4], for i
+large enough, we have
+(3.28)
+δ
+C +
+�
+M
+tr(Φ(K)uεi)ωn
+n! +
+�
+M
+⟨ζ(uεi)(¯∂uεi, ¯∂uεi)⟩K
+ωn
+n! ≤ εi max
+M |Φ(K)|K.
+In particular, we take ζ =
+1
+3C2.
+It can be easy check that
+1
+3C2 <
+1
+x−y when (x, y) ∈
+[−C2, C2] × [−C2, C2] and x > y. This implies that
+δ
+C +
+�
+M
+tr(Φ(K)uεi)ωn
+n! +
+�
+M
+1
+3C2
+|¯∂uεi|2
+K
+ωn
+n! ≤ εi max
+M |Φ(K)|K,
+for i >> 0. Then we obtain
+�
+M
+|¯∂uεi|2
+K ≤ 3C2
+2 max
+M |Φ(K)|KVol(M).
+Therefore, uεi are bounded in L2
+1. Then we can choose a subsequence {uεij} (still denoted
+by uεi for simplicity) such that uεi ⇀ u∞ weakly in L2
+1. Since L2
+1 ֒→ L2. it follows that
+1 =
+�
+M
+|uεi|2
+H0 →
+�
+M
+|u∞|2
+K.
+This indicates that ||u∞||L2 = 1 and u∞ is non-trivial. Then using (3.28) and following a
+similar argument as in [39, Lemma 5.4], we have
+(3.29)
+δ
+C +
+�
+M
+tr(Φ(K)u∞)ωn
+n! +
+�
+M
+⟨ζ(u∞)(¯∂u∞, ¯∂u∞)⟩K
+ωn
+n! ≤ 0.
+Step 2 We will use Uhlenbeck and Yau’s trick from [40] to construct an α-twisted
+coherent subsheaf which contradicts the ω-semi-stability of E. Before going on, we need
+the following lemma.
+
+12
+Z. Shen
+Lemma 3.4 (Lemma 5.12 in [37]). Let E = {Ei, ϕij} be an α-twisted holomorphic vector
+bundle on M whose associated locally free α-twisted sheaf is E . If π ∈ L2
+1(End(E)) is a
+weakly holomorphic α-twisted subbundle of E, there is a coherent α-twisted subsheaf F of
+E and an analytic subset S of M such that:
+(1) S has codimension at least 2 in M,
+(2) π|M\S ∈ A0(E|M\S) and have π∗
+|M\S = π|M\S = π2
+|M\S and (IdE|M\S − π|M\S) ◦
+¯∂(π|M\S),
+(3) F|M\S is the image of π|M\S and an α-twisted holomorphic subbundle of E.
+From (3.29) and the technique in [39, Lemma 5.5], we conclude that the eigenvalues of
+u∞ are constant almost everywhere. Let λ1 < λ2 < · · · < λl be the distinct eigenvalues
+of u∞. Since tr(u∞) = tr(uεi) = 0 and ||u∞||L2 = 1, we conclude that 2 ≤ l ≤ r be the
+distinct eigenvalues of u∞. Then for each eigenvalue λj (1 ≤ l − 1), which is a global
+function on M, we can construct a function
+Pj : R → R
+such that
+Pj =
+�
+1,
+x ≤ λj,
+0,
+x ≥ λj.
+We denote πj = {πi,j}i∈I = Pj(u∞). Following the same arguments as in [39], we have
+(i) πj ∈ L2
+1(End(E));
+(ii) π2
+j = πj = π∗K
+j ;
+(iii) (IdE − πj) ◦ ¯∂πj = 0.
+By Lemma 3.4, we know that the weakly holomorphic α-twisted vector bundles {πj}l−1
+j=1
+determine l − 1 coherent α-twisted subsheaves of E. Denote Ej = {Ei,j}i∈I = πj(E).
+Since tr(u∞) = 0 and u∞ = λl · IdE − �l−1
+j=1(λj+1 − λj)πj, it holds that
+(3.30)
+λlrank(E) =
+l−1
+�
+j=1
+(λj+1 − λj)rank(Ej).
+Set
+(3.31)
+ν = λl deg(E) −
+l−1
+�
+j=1
+(λj+1 − λj) deg(Ej).
+On the one hand, substituting (3.30) into (3.31) directly, we have
+(3.32)
+ν =
+l−1
+�
+j=1
+(λj+1 − λj)rank(Ej)
+� deg(E)
+rank(E) − deg(Ej)
+rank(Ej)
+�
+.
+On the other hand, from the twisted Gauss–Codazzi equation ([37]), we have
+(3.33)
+deg(Ej) =
+�
+M
+tr(πj
+√
+−1ΛωFK) − |¯∂πj|2
+K
+ωn
+n!
+
+Semi-stable twisted holomorphic vector bundles over Gauduchon manifolds
+13
+Then substituting (3.33) into (3.31), we have
+2πν = λl
+�
+M
+tr(
+√
+−1ΛωFK)ωn
+n!
+−
+l−1
+�
+j=1
+(λj+1 − λj)
+�
+M
+tr(πj
+√
+−1ΛωFK) − |¯∂πj|2
+K
+ωn
+n!
+=
+�
+M
+tr
+��
+λl · IdE −
+l−1
+�
+j=1
+(λj+1 − λj)πj
+�√
+−1ΛωFK
+�ωn
+n!
++
+l−1
+�
+j=1
+(λj+1 − λj)
+�
+M
+|¯∂πj|2
+K
+ωn
+n!
+=
+�
+M
+tr(u∞
+√
+−1ΛωFK)ωn
+n! +
+�
+M
+� l−1
+�
+j=1
+(λj+1 − λj)(dPα)2(u∞)(¯∂u∞), ¯∂u∞
+�
+K
+ωn
+n! ,
+where the functions dPj : R × R → R are defined by
+dPj(x, y) =
+�Pj(x)−Pj(y)
+x−y
+,
+x ̸= y;
+P ′
+j(x),
+x = y.
+By simple calculation, if λa ̸= λb, we have
+l−1
+�
+j=1
+(λj+1 − λj)(dPj)2(λa, λb) = |λa − λb|−1.
+Since tr(u∞) = 0, according to (3.29) and the same arguments in [25], it follows that
+2πv =
+�
+M
+tr(u∞
+√
+−1ΛωFK)ωn
+n! +
+�
+M
+� l−1
+�
+j=1
+(λj+1 − λj)(dPα)2(u∞)(¯∂u∞), ¯∂u∞
+�
+K
+ωn
+n!
+< − δ
+C .
+(3.34)
+Combining (3.32) with (3.34), we obtain
+l−1
+�
+j=1
+(λj+1 − λj)rank(Ej)
+� deg(E)
+rank(E) − deg(Ej)
+rank(Ej)
+�
+< 0.
+This indicates that there must exist a term µ(E) − µ(Ej0) < 0, which contradicts the
+ω-semi-stability of E.
+□
+Finally, we prove the other direction of the theorem 1.1. In fact, we get the following
+theorem. The proof of the following theorem is standard and we present the proof for the
+readers’ convenience.
+
+14
+Z. Shen
+Theorem 3.5. Let (M, ω) be a compact Gauduchon manifold and E = {Ei, ϕij}i,j∈I be
+an α-twisted holomorphic vector bundle of rank r over M. If E admits an approximate
+Hermitian–Einstein structure, then E is ω-semi-stable.
+Before we giving the detailed proof, we need the following vanishing theorem.
+Proposition 3.6 (Proposition 3.31 in [37]). Let E1 and E2 be two α-twisted holomorphic
+vector bundles on M of respective ranks r1 and r2, and suppose that
+degω(E1)
+r1
+> degω(E2)
+r2
+.
+If E1 and E2 admit approximate Hermitian–Einstein structure, then every morphism f ∈
+Hom(E1, E2) is zero.
+Proof of theorem 3.5. Let E be the locally free α-twisted coherent sheaf associated to E
+and F be any α-twisted coherent subsheaf of E with rank 0 < p < r. Let L be the
+αp-twisted holomorphic line bundle corresponding to det(F). We consider the following
+untwisted holomorphic vector bundle
+G = ∧pE ⊗ L−1,
+where L−1 is the dual α−p-twisted line bundle of L. Following the same arguments as in
+[37], we know that G admits an approximate Hermitian–Einstein structure. Since there
+is a nontrivial holomorphic section of G, then by Proposition 3.6, we have
+degω(G) ≥ 0.
+Then
+0 ≤ degω(G) = degω(∧pE) − rank(∧pE) degω(L)
+= degω(∧pE) − rank(∧pE) degω(F).
+So we have
+0 ≤ µω(G) = µω(∧pE) − pµω(F) = p(µω(E) − µω(F)).
+This implies that µω(F) ≤ µω(E), i.e., E is ω-semi-stable.
+□
+4. Bogomolov type inequality for semi-stable α-twisted holomorphic
+vector bundles
+In this section, we will prove the Bogomolov type inequality for semi-stable α-twisted
+holomorphic vector bundles over compact Gauduchon Astheno-K¨ahler manifolds. Before
+giving the detailed proof of Theorem 1.2, we need the following lemma.
+Lemma 4.1. Let (M, ˜ω) be a compact Hermitian manifold of dimension n, and ω a
+Gauduchon metric which is conformal to ˜ω. Let E = {Ei, ϕij}i,j∈I be an α-twisted holo-
+morphic vector bundle of rank r over M. If E is ω-semi-stable, then for any ε > 0, there
+exists a Hermitian metric Hε = {Hi,ε}i∈I on E such that
+(4.1)
+sup
+M
+|
+√
+−1ΛωF ⊥
+Hε|Hε < ε,
+where F ⊥
+Hε = FHε − 1
+rtrFHε ⊗ IdE is the trace free part of FHε.
+
+Semi-stable twisted holomorphic vector bundles over Gauduchon manifolds
+15
+Proof. Since ω is conformal to ˜ω, then there exists a smooth function φ such that
+ω = eφ˜ω.
+Then by direct calculation, for any Hermitian metric H, we have
+sup
+M
+|
+√
+−1Λ˜ωF ⊥
+H |H = sup
+M
+����
+√
+−1Λ˜ωFH −
+�1
+r
+√
+−1Λ˜ωtrFH
+�
+⊗ IdE
+����
+= sup
+M
+eφ
+����
+√
+−1ΛωFH −
+�1
+r
+√
+−1ΛωtrFH
+�
+⊗ IdE
+����
+(4.2)
+Due to the fact that √−1ΛωFH is self-adjoint with respect to H, we know that the
+eigenvalues of √−1ΛωFH are real values. Then for any λ1, · · · , λr ∈ R, C ∈ R, we have
+the inequality
+r
+�
+j=1
+(λj − ¯λ) ≤
+r
+�
+j=1
+(λj − C)2,
+where ¯λ = 1
+r
+�r
+j=1 λj. So we have
+(4.3)
+����
+√
+−1ΛωFH −
+�1
+r
+√
+−1ΛωtrFH
+�
+⊗ IdE
+���� ≤ |
+√
+−1ΛωFH − λ · IdE|H,
+where λ = 2π degω(E)
+rVol(M,ω) . Combining (4.2) and (4.3) yields that
+(4.4)
+sup
+M
+|
+√
+−1Λ˜ωF ⊥
+H |H ≤ esupM φ sup
+M
+|
+√
+−1ΛωFH − λ · IdE|H.
+Since E is ω-semi-stable, by Theorem 1.1, it admits an approximate Hermitian–Einstein
+structure on E. So for any ε > 0, there exists a Hermitian metric Hε such that
+(4.5)
+sup
+M
+|
+√
+−1ΛωFHε − λ · IdE|Hε < εe− supM φ.
+Combining (4.5) with (4.4) gives (4.1).
+□
+Proof of theorem 1.2. Let (M, ˜ω) be a compact Astheno-K¨ahler manifold of dimension n,
+and ω a Gauduchon metric which is conformal to ˜ω. Let E = {Ei, ϕij}i,j∈I be an α-twisted
+holomorohic vector bundle of rank r over M. Given any Hermitian metric H = {Hi}i∈I,
+recall that
+c1(E, H) =
+√−1
+2π trFH, c2(E, H) = − 1
+8π2
+�
+(trFH)2 − tr(F 2
+H)
+�
+.
+Since F ⊥
+H = FH − 1
+rtrFH ⊗ IdE, one can easily get
+tr(F 2
+H) = 1
+r(trFH)2 + tr(F ⊥
+H ∧ F ⊥
+H).
+Then we have
+4π2
+�
+2c2(E, H) − r − 1
+r
+c1(E, H) ∧ c1(E, H)
+�
+= −
+�
+(trFH)2 − tr(F 2
+H)
+�
++ r − 1
+r
+(trFH)2 = tr(F 2
+H) − 1
+r(trFH)2
+= tr(F ⊥
+H ∧ F ⊥
+H).
+
+16
+Z. Shen
+By the Riemann bilinear relations, we have
+�
+M
+�
+2c2(E) − r − 1
+r
+c1(E) ∧ c1(E)
+�
+∧
+˜ωn−2
+(n − 2)!
+=
+�
+M
+�
+2c2(E, H) − r − 1
+r
+c1(E, H) ∧ c1(E, H)
+�
+∧
+˜ωn−2
+(n − 2)!
+=
+1
+4π2
+�
+M
+tr(F ⊥
+H ∧ F ⊥
+H ) ∧
+˜ωn−2
+(n − 2)!
+=
+1
+4π2
+�
+M
+|F ⊥
+H|H,˜ω − |
+√
+−1Λ˜ωF ⊥
+H |2
+H
+˜
+ωn
+n! .
+(4.6)
+Since E is ω-semi-stable, by Lemma 4.1, for every ε > 0, there exists a Hermitian metric
+Hε such that
+(4.7)
+sup
+M
+|
+√
+−1Λ˜ωF ⊥
+H |Hε ≤ ε.
+Combining (4.6) and (4.7), and letting ε → 0, we obtain
+�
+M
+�
+2c2(E) − r − 1
+r
+c1(E) ∧ c1(E)
+�
+∧
+˜ωn−2
+(n − 2)! ≥ 0.
+□
+Data Availability Statement: Not applicable.
+Acknowledgements: The author would like to thank Prof. Xi Zhang and Dr. Pan
+Zhang for their useful discussions and helpful comments. The research was supported by
+the National Key R and D Program of China 2020YFA0713100. The author is partially
+supported by NSF in China No.12141104 and 11721101.
+Conflicts of Interest: The authors declare no conflicts of interest.
+References
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+Zhenghan Shen, School of Mathematics and Statistics, Nanjing University of Science
+and Technology, Nanjing, 210094,P.R. China,
+Email address: mathszh@njust.edu.cn
+
diff --git a/y9AzT4oBgHgl3EQfePxr/content/tmp_files/load_file.txt b/y9AzT4oBgHgl3EQfePxr/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..9509e701ce1a68498fc3ef4369def4644a5ecb6d
--- /dev/null
+++ b/y9AzT4oBgHgl3EQfePxr/content/tmp_files/load_file.txt
@@ -0,0 +1,786 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf,len=785
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='01433v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='DG]  4 Jan 2023  SEMI-STABLE TWISTED HOLOMORPHIC VECTOR BUNDLES OVER  GAUDUCHON MANIFOLDS  ZHENGHAN SHEN  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' In this paper, we study the semi-stable twisted holomorphic vector bun-  dles over compact Gauduchon manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' By using Uhlenbeck–Yau’s continuity method,  we show that the existence of approximate Hermitian–Einstein structure and the semi-  stability of twisted holomorphic vector bundles are equivalent over compact Gauduchon  manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' As its application, we show that the Bogomolov type inequality is also valid  for a semi-stable twisted holomorphic vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Introduction  Let M be a complex manifold of dimension n and g a Hermitian metric with the  associated K¨ahler form ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' The Hermitian metric g is called Gauduchon if  ∂ ¯∂ωn−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  It has been proved by Gauduchon ([18]) that every Hermitian metric is conformal to a  Gauduchon metric (uniquely up to scaling when n ≥ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' The Hermitian metric ω is said  to be Astheno-K¨ahler if ∂ ¯∂ωn−2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' The concept of Astheno-K¨ahler was introduced by  Jost and Yau in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' In this case, the second Chern number of a twisted vector bundle  can be well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Throughout the paper, we will assume ω is Gauduchon if there is no  additional emphasis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Twisted sheaves were introduced by Giraud in [19], and can be defined in several equiv-  alent ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' It can be regarded as family of sheaves on an open covering of M together with  a twisted gluing, as sheaves of modules over an Azumaya algebra on M ([10]), as sheaves  over a gerb on M ([14, 19, 42]), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' These objects have found many applications in  physics for the description of the so-called B-fields and the K-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' We should mention  that the twisted K-theory has appeared earlier in the mathematical physics literature, in  the study of the quantum Hall effect ([11, 12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  In recent years, the existence of Hermitian–Einstein has been extensively studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' The  classical Donaldson–Uhlenbeck–Yau theorem says that a holomorphic vector bundle over  compact K¨ahler manifolds admit Hermitian–Einstein metrics if it is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' This result  was first proved by Narasimhan and Seshadri ([34]) for compact Riemannian surface by  using the methods of algebraic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Subsequently, Donaldson ([15, 16, 17]) gave a  new proof of the Narasimhan-Seshadri theorem, and generalized to algebraic surfaces and  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' 53C07, 57N16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' approximate Hermitian-Einstein metric, twisted vector bundle, semi-stability,  Gauduchon manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  1  2  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Shen  algebraic manifolds by using Hermitian–Yang–Mills flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Uhlenbeck and Yau ([40, 41])  proved it for general compact K¨ahler manifold by using continuity method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  The inverse of the problem which states that a holomorphic bundle carrying a Hermitian–  Einstein metric must be polystable is also valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' It was proved by Kobayashi ([29]) and  L¨ubke ([31]) independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' So there is a one-to-one correspondence bewteen the algebraic  notion of stability and the existence of Hermitian–Einstein on holomorphic vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  This is usually referred as to Hitchin–Kobayashi correspondence (see [1, 4, 21, 32, 38] and  references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' There are several interesting generalizations for this coorespondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  For example, Li–Yau ([26]) studied the coorespondence for Hermitian manifolds with  Gauduchon metric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Hitchin ([20]) and Simpson ([39]) studied the Higgs bundle case which  plays an important role in non-abelian Hodge theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Mochizuki ([33]) studied such corre-  spondence on the non-compact K¨ahler manifolds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Biswas–Loftin–Stemmler ([7]) studied  for affine manifolds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Biswas–Kasuya ([5, 6]) studied for Sasakian manifolds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Zhang–Zhang–  Zhang ([43]) studied for some non-compact non-K¨ahler manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  If one considers the weaker stability condition, there is an analog of such correspondence  between the semi-stability and the approximate Hermitian-Einstein metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Kobayashi  ([29]) introduced the notion of approximate Hermitian–Einstein metric for holomorphic  vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' He proved a holomorphic vector bundle admitting an approximate Hermitian–  Einstein metric must be semi-stable over a compact K¨ahler manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' In [8], Bruzzo and  Gra˜na Otero generalized above result to the Higgs bundle case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' When the base manifold  is projective, Kobayashi proved the inverse part and conjectured that it should be true  for general K¨ahler manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' It was solved by Li–Zhang ([25]) and Jacob ([22]) indepen-  dently for the general K¨ahler manifold case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' In the principal bundle case, it was studied  by Biswas–Jacob–Stemmler ([2, 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Recently, Nie–Zhang ([35]) proved the existence of  approximate Hermitian–Einstein structures is equivalent to the semi-stability on Higgs  bundles over compact Gauduchon manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Subsequently, Zhang–Zhang–Zhang ([43])  also studied the semi-stable Higgs bundles on some non-compact Gauducon manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  In the present paper, our considerations focus on the twisted vector bundles over com-  pact Gauduchon manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' The twisted vector bundles are local vector bundles on a  open covering with a twisted gluing, which is the point of view of C˘ald˘araru ([10, 13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Let U := {Ui}i∈I be a fixed open covering of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' A B-field on M with respect to U is a  family B = {Bi}i∈I, where each Bi is a 2-form on Ui such that  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='1)  Bi − Bj = dβij,  for some 1-forms βij on Uij := Ui ∩ Uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Notice that d(βij + βjk + βki) = 0, we may find  U(1)-valued functions αijk on Uijk := Ui ∩ Uj ∩ Uk such that  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='2)  βij + βjk + βki = −α−1  ijkdαijk,  when the open covering U is sufficiently fine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Then αB := {αijk} is a 2-cocycle whose  cohomology class lies in H2(M, O∗  M), which is called the twist induced by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' We will omit  the subscript B when there is no confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' An α-twisted holomorphic vector bundle is  a pair E := ({Ei}i∈I, {ϕij}i,j∈I), where Ei is a holomorphic vector bundle on Ui and  ϕij : Ej|Uij → Ei|Uij  Semi-stable twisted holomorphic vector bundles over Gauduchon manifolds  3  is an isomorphism of holomorphic vector bundles on Uij, such that ϕii = IdEi, ϕ−1  ij = ϕji  and ϕki◦ϕjk◦ϕij = αijk·IdE|Uijk for each i, j, k ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Here we usually omit the holomporphic  structure ¯∂Ei on Ei when there is no confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  It can be easily see that 1-twisted  holomorphic vector bundle is the usual holomorphic vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  In [42], S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Wang proved the Hitchin–Kobayashi coorespondence for twisted holomorphic  vector bundles over a gerbe on compact K¨ahler manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' All the definitions in [42] are  in the category of twisted holomorphic vector bundles as holomorphic vector bundles over  a gerb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Recently, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Perego ([37]) generalized Wang’s result to complex manifolds with  a Gauduchon metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' And he also proved the twisted version of approximate Hitchin–  Kobayashi coorespondence on compact K¨ahler manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Inspired by Perego’s work, we  consider the case that the approximate Hermitian–Einstein metric for twisted holomor-  phic vector bundle over compact Gauduchon manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' In fact, we prove the following  theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Let E = {Ei, ϕij}i,j∈I be an α-twisted holomorphic vector bundle of rank  r over a compact Gauduchon manifold (M, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Then E is ω-semi-stable if and only if it  admits an approximate Hermitian–Einstein structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  We should remark that Perego ([37]) only proved the above theorem when the base man-  ifold is a K¨ahler manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' He mainly follow the methods of Jacob ([22]) and Kobayashi  ([29]) by using Donaldson’s heat flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' The proof relies on the properties of the Donald-  son’s Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' However, the Donaldson’s Lagrangian is not well-defined when the base  manifold is only Gauduchon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' So his argument can not be generalized to Gauduchon case  directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' In this paper, we will adapt Nie–Zhang’s arguments ([35]) to our case, but there  are some differences in the treatment of certain details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  As an application, we obtain the following Bogomolov type inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Let (M, ˜ω) be a compact Astheno-K¨ahler manifold of dimension n, and  ω a Gauduchon metric which is conformal to ˜ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Let E = {Ei, ϕij}i,j∈I be an α-twisted  holomorohic vector bundle of rank r over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' If E is ω-semi-stable, then we have the  following Bogomolov type inequality  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='3)  �  M  �  2c2(E) − r − 1  r  c1(E) ∧ c1(E)  �  ∧  ˜ωn−2  (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  The Bogomolov type inequality was first proved by Bogomolov ([9]) for semi-stable  holomorphic vector bundles over complex algrebraic surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Subsequently, it was gen-  eralized to Hermitian–Einstein vector bundles over compact K¨ahler manifolds by L¨ubke (  [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Recently, Li–Nie–Zhang ([24]) proved the Bogomolov type inequality over Gaudu-  chon Astheno-K¨ahler manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' In twisted case, Perego ([37]) obtained the Bogomolov  type inequality for semi-stable α-twisted holomorphic vector bundle over compact K¨ahler  manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  In our paper, we will proved the Bogomolov type inequality by following  Li–Nie–Zhang’s arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  This article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' In Section 2, we briefly present some notations for  α-twisted holomorphic vector bundle, such as connection, curvature and semi-stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  In Section 3, we give the detailed proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  In Section 4, we prove the  Bogomolov type inequality for semi-stable α-twisted holomorphic vector bundles over  Gauduchon Astheno-K¨ahler manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  4  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Shen  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Preliminary  In this section, we will introduce the basic notations of α-twisted vector bundles that  will be used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' We will follow the notations of [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Hermitian metrics, connections and curvatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Let E = {Ei, ϕij}i,j∈I be  an α-twisted holomorphic vector bundle over compact Gauduchon manifold (M, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' A  Hermitian metric on E is a connection H = {Hi}i∈I, where Hi is a Hermitian metric on  Ei and Hi = ϕT  ijHj ¯ϕij for each i, j ∈ I, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=', for any ξ, η ∈ Γ(Ei),  Hi(ξ, η) = Hj(ϕij(ξ), ϕij(η)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Since αijk is U(1)-valued, it can be easily seen that the definition is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' And  there always exists a Hermitian metric H = {Hi}i∈I on an α-twisted vector bundle (see  [37] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  We should remark that Hi also denotes the matrix of smooth functions with respect to  a given local frame and ϕij denotes the matrix of smooth functions with respect to the  chosen local frames of Ei and Ej when there is no confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  A connection on α-twisted vector bundle is a family D = {Di}i∈I, where Di is a  connection on Ei and locally the connection 1-form Ai satisfies  Ai = ϕ−1  ij Ajϕij + ϕ−1  ij dϕij + βij · IdE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  The notion of connection on a twisted bundle can be found in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  We say that a  connection D = {Di}i∈I is compatible with the Hermitian metric H = {Hi}i∈I, if each  Di is compatible with Hi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=', for any ξ, η ∈ Γ(Ei),  d(Hi(ξ, η)) = Hi(Di(ξ), η) + Hi(ξ, Di(η)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  In a local given frame of Ei, it usually represents as  dHi = AT  i Hi + Hi ¯Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Let FD ∈ Ω2(End(E)) be curvature form of the connection D = {Di}i∈I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' According to  [37], we have  FD|Ui = FDi − Bi · IdEi,  where FDi is the curvature of Di for every i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Thoughtout the paper, we assume  that each Bi of the B-filed B is a (1, 1)-form on Ui and βij is a (1, 0)-form on Uij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Then  given a Hermitian metric H on the α-twisted holomorphic vector bundle E, there is a  unique Chern connection DH which is compatible with the Hermitian metric H and the  holomorphic structure of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' In fact, we have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='1 (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='25 in [37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Let E be an α-twisted holomorphic vector bundle and  H be a Hermitian metric on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  (1) There is a unique connection DH which is compatible both with the holomorphic  structure of E and the Hermitian metric H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  (2) If each Bi of the B-field is a (1, 1)-form for every i ∈ I, then FD ∈ Ω1,1(End(E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Let FH ∈ Ω1,1(End(E)) be the curvature form of the Chern connection DH, then we  have  FH|Ui = FHi − Bi · IdEi,  Semi-stable twisted holomorphic vector bundles over Gauduchon manifolds  5  where FHi is the curvature form of the Chern connection DHi on Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' A Hermitian metric  H = {Hi}i∈I in E = {Ei, ϕij}i,j∈I is called a Hermitian–Einstein metric if the curvature  FH of the Chern connection DH satisfies  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='1)  √  −1ΛωFH = λ · IdE,  where Λω denotes the contraction of differential forms by ω, and λ is a real constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  An α-twisted holomorphic vector bundle E = {Ei, ϕij}i,j∈I is said to be admitting an  approximate Hermitian–Einstein structure if for every ε > 0, there is a Hermitian metric  Hε = {Hi,ε}i∈I such that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='2)  max  X  |  √  −1ΛωFHε − λ · IdE|Hε < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Let E = {Ei, ϕij}i,j∈I be an α-twisted holomorphic vector bundle of rank  r and H = {Hi}i∈I be a Hermitian metric on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Then the first Chern form of (E, H) is  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='3)  c1(E, H) =  √−1  2π tr(FH) ∈ Ω1,1(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  The second Chern form of (E, H) is  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='4)  c2(E, H) = − 1  8π2  �  (trFH)2 − tr(FH ∧ FH)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  The degree of (E, H) with respect to the Gauduchon metric ω on M is defined to be  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='5)  degω(E) :=  �  M  c1(E, H) ∧  ωn−1  (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='.  Since ω is a Gauduchon metric, the definition of the degree is well-defined and independent  of the choice of the Hermitian metric H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Let F = {Fi, ϕij}i,j∈I be an α-twisted coherent sheaf of rank s on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' The definition of  stability for twisted coherent sheaves was first introduced by Lieblich ([27]) and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Perego  ([36, 37]) studied the stability of twisted coherent sheaves in the language of twisted  gluing of coherent sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Given a rank s α-twisted coherent sheaves, the determinant  of F = {det(Fi), det(ϕij)}i,j∈I is a locally free αs-twisted sheaf of rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' The ω-degree  of F is defined by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='6)  degω(F) := degω(det(F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  If F is non-trivial, the slope of F with respect to ω is defined by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='7)  µω(F) = degω(F)  rank(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Let E = {Ei, ϕij}i,j∈I be an α-twisted holomorphic vector bundle on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' We say an α-  twisted holomorphic vector bundle E is ω-stable (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' ω-semi-stable) if for every proper  α-twisted coherent subsheaf F of E, it holds that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='8)  µω(F) < µω(E)  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  µω(F) ≤ µω(E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  6  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Shen  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Endomorphisms of twisted vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' In the rest of this section, we will  introduce the endomorphisms of twisted vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Let E = {Ei, ϕij}i,j∈I be an  α-twisted vector bundle on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' An endomorphism f : E → E is a family f = {fi}i∈I,  where fi : Ei → Ei is an endomorphism of Ei on Ui and for every i, j ∈ I,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='9)  ϕij ◦ fi = fj ◦ ϕij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  It is easy to check that the eigenvalues of f are smooth functions on the whole M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' In  fact, if λi is a smooth function on Ui and ξ ∈ Γ(Ei) is a nowhere vanishing smooth section  such that fi(ξ) = λiξ, then  fj(ϕij(ξ)) = ϕij(fj(ξ)) = ϕij(λiξ) = λiϕij(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  So λi is an eigenvalue for fj with a nowhere vanishing smooth section ϕij(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Hence the  eigenvalues of fi glue together to give global functions on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Given a Hermitian metric H = {Hi}i∈I on E = {Ei, ϕij}i,j∈I, we say that an endo-  morphism f is self-adjoint with respect to H, if for every i ∈ I and ξ, η ∈ Γ(Ei), it  holds  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='10)  Hi(fi(ξ), η) = Hi(ξ, fi(η)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  We denote  Herm(E, H) = {f ∈ End(E)|f ∗H = f},  and  Herm+(E, H) = {f ∈ Herm(E, H)|f > 0},  where f > 0 means that all the eigenvalues of f are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  If H = {Hi}i∈I and  K = {Ki}i∈I are two Hermitian metrics on α-twisted vector bundles, then for every i ∈ I  and ξ, η ∈ Γ(Ei), the endomorphism h = {hi}i∈I is defined by  Hi(ξ, η) = Ki(hiξ, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  It is easy to see that h = K−1H is self-adjoint with respect to H and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  For any h ∈ Herm+(E, K), we can choose an open dense subset W ⊂ M satisfying  that, at each x0 ∈ W there exists an open neighborhood Ui of x0, a local unitary basis  {ej}r  j=1 with respect to Ki and functions {λj ∈ C∞(Ui, R)}r  i=1 such that  h(x) =  r  �  j=1  eλj(x)ej(x) ⊗ ej(x)  for all x ∈ Ui, where {ej}r  j=1 is the dual basis corresponding to {ej}r  j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Then according  to [37, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='56], we have  log h(x) =  r  �  j=1  λj(x)ej(x) ⊗ ej(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Suppose Ψ : R × R → R is a smooth function, A = �r  i,j=1 Aijei ⊗ ej ∈ End(E), and  s ∈ Herm(E, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Then we define Ψ(s)(A) by  Ψ(s)(A) =  n  �  i,j=1  Ψ(λi, λj)Aijei ⊗ ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Semi-stable twisted holomorphic vector bundles over Gauduchon manifolds  7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='1  Let E = {Ei, ϕij}i,j∈I be an α-twisted holomorphic vector bundle of rank r over a  compact Gauduchon manifold (M, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Suppose H = {Hi}i∈I and K = {Ki}i∈I are two  Hermitian metrics on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Let DH = {DHi}i∈I (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  DK = {DKi}i∈I) be the Chern  connection of (E, H) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' (E, K)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Then for every i ∈ I, we have  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='1)  ∂Hi = ∂Ki + h−1  i ∂Kihi,  where ∂Ki denotes the (1, 0)-part of the connection induced by DKi on End(Ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' The  relation between FHi and FKi is given by  FHi = FKi + ¯∂i(h−1  i ∂Kihi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='2)  Then we have  FH|Ui =FHi − Bi · IdEi = FKi + ¯∂i(h−1  i ∂Kihi) − Bi · IdEi  =FK|Ui + ¯∂i(h−1  i ∂Kihi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='3)  So if we glue together, it follows that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='4)  FH = FK + ¯∂(h−1∂Kh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Therefore H solves the Hermitian–Einstein equation on α-twisted holomorphic bundle E  if and only if there exist Hermitian metric K and h ∈ Herm+(E, K) such that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='5)  √  −1ΛωFK − λ · IdE +  √  −1Λω ¯∂(h−1∂Kh) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Now, fixing a proper background Hermitian metric K = {Ki}i∈I on α-twisted holomor-  phic vector bundle E = {Ei, ϕij}i,j∈I, we consider the following perturbed equation  Lε(hε) : =  √  −1ΛωFHε − λ · IdE + ε log(hε)  =  √  −1ΛωFK − λ · IdE +  √  −1Λω ¯∂(h−1  ε ∂Khε) + ε log(hε),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='6)  where hε = K−1Hε ∈ Herm+(E, K) and ε ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' It is obvious that hε and log hε are  self-adjoint with respect to K and Hε in the sense of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' In [37], Perego proved that  the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='6) is solvable for all ε ∈ (0, 1] by using the continuity method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' If the  α-twisted holomorphic vector bundle E is semi-stbale, we can show that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='7)  lim  ε→0 ε max  M | log hε|K = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  This implies that maxM |√−1ΛωFHε − λ · IdE| → 0 as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' So we obtain the existence  of approximate Hermitian–Einstein structure on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Before giving the detailed proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='7), we need the following two lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='1 (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='7 in [37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Let E = {Ei, ϕij}i,j∈I be an α-twisted holomorphic  vector bundle on M, and K be a Hermitian metric on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' If h ∈ Herm+(E, K) satisfies  Lε(h) = 0 for some ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  (1) We have  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='8)  1  2  √  −1Λω ¯∂∂(| log(h)|2) + ε| log(h)|2  K ≤ |Φ(K)|K · | log(h)|K,  where Φ(K) := √−1ΛωFK − λ · IdE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  8  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Shen  (2) If m := maxM | log(h)|K(x), then we have  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='9)  m ≤ 1  ε max  x∈M |Φ(K)|K(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  (3) There is a real number C (depending only on ω and K) such that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='10)  m ≤ C(|| log h||L2 + max  M |Φ(K)|K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Let E = {Ei, ϕij}i,j∈I be an α-twisted holomorphic vector bundle on M,  and K = {Ki}i∈I be a Hermitian metric on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' If hε = {hi,ε}i∈I ∈ Herm+(E, K) solves  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='6) for some ε, then it holds that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='11)  �  M  tr(Φ(K)sε)ωn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' +  �  M  ⟨Ψ(sε(¯∂sε), ¯∂sε)⟩K  ωn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' = −ε||sε||2  L2,  where sε = {si,ε}i∈I = log hε, Φ(K) = √−1ΛωFK − λ · IdE and  Ψ(x, y) =  �  ey−x−1  y−x ,  x ̸= y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  1,  x = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Since hε solves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='6), then we have  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='12)  ⟨Φ(K), sε⟩K + ⟨  √  −1Λω ¯∂(h−1  ε ∂Khε), sε⟩K = −⟨sε, sε⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Integrating over M gives  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='13)  �  M  tr(Φ(K)sε)ωn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' +  �  M  ⟨  √  −1Λω ¯∂(h−1  ε ∂Khε), sε⟩K  ωn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' = −||sε||L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Then comparing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='13) with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='11), it suffices to prove  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='14)  �  M  ⟨  √  −1Λω ¯∂(h−1  ε ∂Khε), sε⟩K  ωn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' =  �  M  ⟨Ψ(sε(¯∂sε), ¯∂sε)⟩K  ωn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  An easy calculation (see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='1 in [35] for details) shows that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='15)  �  M  ⟨  √  −1Λω ¯∂(h−1  ε ∂Khε), sε⟩K  ωn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' =  �  M  √  −1tr(h−1  ε ∂Khε ∧ ¯∂sε) ωn−1  (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='.  On the other hand, following the methods in [32], we can choose an open dense subset  W ⊂ M satisfying that at each x ∈ W there exists an open neighborhood Ui ∈ U of x, a  local unitary basis {ej}r  j=1 of Ei with respect to Ki and eigenvalues {λj ∈ C∞(Ui, R)}r  j=1  such that  si,ε(y) =  r  �  j=1  λj(y)ej(y) ⊗ ej(y),  for all y ∈ Ui, where {ej}r  j=1 is the dual basis of E∗  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Here λ1, · · · , λr are eigenvalues of sε  (and hence of si,ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Therefore, we have  ∂Kihi,ε(x) = eλj∂λjej ⊗ ej + (eλk − eλj)Aj  kej ⊗ ek,  and  ¯∂si,ε(x) = ¯∂λjej ⊗ ej + (λk − λj)(−Ak  j)ej ⊗ ek,  Semi-stable twisted holomorphic vector bundles over Gauduchon manifolds  9  where {Ak  j}r  j,k=1 are the (1, 0)-forms defined defined by ∂Kiej = Ak  jek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Then on each Ui,  it follows that  tr  √  −1Λω(h−1  i,ε ∂Kihi,ε ∧ ¯∂si,ε)  =  r  �  j=1  |¯∂λj|2 +  �  j̸=k  (eλk − eλj)(λj − λk)  √  −1ΛωAj  k ∧ (−Ak  j)  =  r  �  j=1  |¯∂λj|2 +  �  j̸=k  eλk−λj − 1  λk − λj  (λk − λj)2| − Aj  k|2  =  r  �  j,k=1  Ψ(λj, λk)|(¯∂si,ε)k  j|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  As both sides glue together, we have  tr  √  −1Λω(h−1  ε ∂Khε ∧ ¯∂sε) =  r  �  j,k=1  Ψ(λj, λk)|(¯∂s)k  j|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Using the similar argument, we can also prove  ⟨Ψ(s)(¯∂s), ¯∂s⟩K =  r  �  j,k=1  Ψ(λj, λk)|(¯∂s)k  j|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Therefore, we have  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='16)  ⟨Ψ(s)(¯∂s), ¯∂s⟩K = tr  √  −1Λω(h−1  ε ∂Khε ∧ ¯∂sε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='15) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='16), we complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  □  In the following, we are ready to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' First, we show that the semi-  stability implies the existence of approximate Hermitian–Einstein structure on α-twisted  holomorphic vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' In fact, we have the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Let E = {Ei, ϕij}i,j∈I be an α-twisted holomorphic vector bundle over  (M, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' If E is ω-semi-stable, then there is an approximate Hermitian–Einstein structure  on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=',  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='17)  max  M |Φ(Hε)|Hε → 0,  as  ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Let K be a background Hermitian metric on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Without loss of generality, we can  assume that the Hermitian K satisfies  tr(  √  −1ΛωFK − λ · IdE) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  In fact, by an approximate conformal change K = eφ ˜K, we have  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='18)  √  −1Λω ¯∂∂φ = −1  rtr(  √  −1ΛωF ˜  K − λ · IdE),  where ˜K is an arbitrary Hermitian metric on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Since  �  M tr(√−1ΛωF ˜  K − λ · IdE) ωn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' = 0,  then there is a smooth function φ satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  10  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Shen  Let {hε}0<ε≤1 be the solutions of the perturbed equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='6) with the background  Hermitian metric K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Then we have  || log hε||2  L2 = −1  ε  �  M  ⟨Φ(Hε), log hε⟩Hε  ωn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' ,  where Φ(Hε) = √−1ΛωFHε − λ · IdE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' By [37, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='2], we also have  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='19)  det(hε) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  We will consider the limit behavior of || log hε||L2 in two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Case 1: || log hε||L2 is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=', there exists a constant C1 such that  || log hε||L2 < C1 < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  In this case, from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='1, we have  max  M |Φ(Hε)|Hε = ε · max  M | log hε|Hε < εC · (|| log hε||L2 + max  M |Φ(K)|K)  ≤ εC(C1 + max  M |Φ(K)|K) → 0  as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Case 2: lim supε→0 || log hε||L2 = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' In this case, if E = {Ei, ϕij}i,j∈I is ω-semi-  stable, then we also have  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='20)  lim  ε→0 max  M |Φ(Hε)|Hε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  We will follow Simpson’s argument ([39]) to show that if (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='19) is not valid, there exists  a coherent α-twisted subsheaf contradicting the ω-semi-stability of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  We will prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='20) by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' If (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='20) does not hold, then there exist δ > 0  and a subsequence εi → 0, as i → +∞, such that  || log hεi||L2 → +∞  and  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='21)  max  M |Φ(Hεi)|Hεi = εi max  M | log hεi|Hεi ≥ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Set  sεi = log hεi, li = ||sεi||L2, uεi = sεi  li  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Then we have ||uεi||L2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='19), it follows that tr(uεi) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Then combining  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='21) with Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='1, we have  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='22)  li ≥  δ  Cεi  − max  M |Φ(K)|K  and  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='23)  max  M |uεi| < C  li  �  li + max  M |Φ(K)|K  �  < C2 < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  In the following, we divide our proof into two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Step 1 We will show that uεi converge to u∞ weakly as i → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' We need to show  that ||uεi||L2  1 are uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Since ||uεi||L2 = 1, it enough to prove ||¯∂uεi||L2 are  uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Semi-stable twisted holomorphic vector bundles over Gauduchon manifolds  11  By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='11, for each hεi, it holds  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='24)  �  M  tr(Φ(K)uεi)ωn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' + li  �  M  ⟨Ψ(liuεi)(¯∂uεi, ¯∂uεi)⟩K  ωn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' = −εili.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='22) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='24), we obtain  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='25)  δ  C +  �  M  tr(Φ(K)uεi)ωn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' + li  �  M  ⟨Ψ(liuεi)(¯∂uεi, ¯∂uεi)⟩K  ωn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' ≤ εi max  M |Φ(K)|K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Next, we consider the function  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='26)  lΨ(lx, ly) =  �  l,  x = y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  el(y−x)−1  y−x  ,  x ̸= y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Because of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='23), we may assume that (x, y) ∈ [−C2, C2] × [−C2, C2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' It is easy to check  that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='27)  lΨ(lx, ly) →  �  (x − y)−1,  x > y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  +∞,  x ≤ y,  increases monotonically as l → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Let ζ ∈ C∞(R×R, R+) satisfying ζ(x, y) < (x−y)−1  whenever x > y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='25), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='27) and the same arguments in [39, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='4], for i  large enough, we have  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='28)  δ  C +  �  M  tr(Φ(K)uεi)ωn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' +  �  M  ⟨ζ(uεi)(¯∂uεi, ¯∂uεi)⟩K  ωn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' ≤ εi max  M |Φ(K)|K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  In particular, we take ζ =  1  3C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  It can be easy check that  1  3C2 <  1  x−y when (x, y) ∈  [−C2, C2] × [−C2, C2] and x > y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' This implies that  δ  C +  �  M  tr(Φ(K)uεi)ωn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' +  �  M  1  3C2  |¯∂uεi|2  K  ωn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' ≤ εi max  M |Φ(K)|K,  for i >> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Then we obtain  �  M  |¯∂uεi|2  K ≤ 3C2  2 max  M |Φ(K)|KVol(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Therefore, uεi are bounded in L2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Then we can choose a subsequence {uεij} (still denoted  by uεi for simplicity) such that uεi ⇀ u∞ weakly in L2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Since L2  1 ֒→ L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' it follows that  1 =  �  M  |uεi|2  H0 →  �  M  |u∞|2  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  This indicates that ||u∞||L2 = 1 and u∞ is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Then using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='28) and following a  similar argument as in [39, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='4], we have  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='29)  δ  C +  �  M  tr(Φ(K)u∞)ωn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' +  �  M  ⟨ζ(u∞)(¯∂u∞, ¯∂u∞)⟩K  ωn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Step 2 We will use Uhlenbeck and Yau’s trick from [40] to construct an α-twisted  coherent subsheaf which contradicts the ω-semi-stability of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Before going on, we need  the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  12  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Shen  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='4 (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='12 in [37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Let E = {Ei, ϕij} be an α-twisted holomorphic vector  bundle on M whose associated locally free α-twisted sheaf is E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' If π ∈ L2  1(End(E)) is a  weakly holomorphic α-twisted subbundle of E, there is a coherent α-twisted subsheaf F of  E and an analytic subset S of M such that:  (1) S has codimension at least 2 in M,  (2) π|M\\S ∈ A0(E|M\\S) and have π∗  |M\\S = π|M\\S = π2  |M\\S and (IdE|M\\S − π|M\\S) ◦  ¯∂(π|M\\S),  (3) F|M\\S is the image of π|M\\S and an α-twisted holomorphic subbundle of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='29) and the technique in [39, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='5], we conclude that the eigenvalues of  u∞ are constant almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Let λ1 < λ2 < · · · < λl be the distinct eigenvalues  of u∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Since tr(u∞) = tr(uεi) = 0 and ||u∞||L2 = 1, we conclude that 2 ≤ l ≤ r be the  distinct eigenvalues of u∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Then for each eigenvalue λj (1 ≤ l − 1), which is a global  function on M, we can construct a function  Pj : R → R  such that  Pj =  �  1,  x ≤ λj,  0,  x ≥ λj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  We denote πj = {πi,j}i∈I = Pj(u∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Following the same arguments as in [39], we have  (i) πj ∈ L2  1(End(E));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  (ii) π2  j = πj = π∗K  j ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  (iii) (IdE − πj) ◦ ¯∂πj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='4, we know that the weakly holomorphic α-twisted vector bundles {πj}l−1  j=1  determine l − 1 coherent α-twisted subsheaves of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Denote Ej = {Ei,j}i∈I = πj(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Since tr(u∞) = 0 and u∞ = λl · IdE − �l−1  j=1(λj+1 − λj)πj, it holds that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='30)  λlrank(E) =  l−1  �  j=1  (λj+1 − λj)rank(Ej).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Set  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='31)  ν = λl deg(E) −  l−1  �  j=1  (λj+1 − λj) deg(Ej).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  On the one hand, substituting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='30) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='31) directly, we have  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='32)  ν =  l−1  �  j=1  (λj+1 − λj)rank(Ej)  � deg(E)  rank(E) − deg(Ej)  rank(Ej)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  On the other hand, from the twisted Gauss–Codazzi equation ([37]), we have  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='33)  deg(Ej) =  �  M  tr(πj  √  −1ΛωFK) − |¯∂πj|2  K  ωn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Semi-stable twisted holomorphic vector bundles over Gauduchon manifolds  13  Then substituting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='33) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='31), we have  2πν = λl  �  M  tr(  √  −1ΛωFK)ωn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  −  l−1  �  j=1  (λj+1 − λj)  �  M  tr(πj  √  −1ΛωFK) − |¯∂πj|2  K  ωn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  =  �  M  tr  ��  λl · IdE −  l−1  �  j=1  (λj+1 − λj)πj  �√  −1ΛωFK  �ωn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  +  l−1  �  j=1  (λj+1 − λj)  �  M  |¯∂πj|2  K  ωn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  =  �  M  tr(u∞  √  −1ΛωFK)ωn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' +  �  M  � l−1  �  j=1  (λj+1 − λj)(dPα)2(u∞)(¯∂u∞), ¯∂u∞  �  K  ωn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' ,  where the functions dPj : R × R → R are defined by  dPj(x, y) =  �Pj(x)−Pj(y)  x−y  ,  x ̸= y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  P ′  j(x),  x = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  By simple calculation, if λa ̸= λb, we have  l−1  �  j=1  (λj+1 − λj)(dPj)2(λa, λb) = |λa − λb|−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Since tr(u∞) = 0, according to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='29) and the same arguments in [25], it follows that  2πv =  �  M  tr(u∞  √  −1ΛωFK)ωn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' +  �  M  � l−1  �  j=1  (λj+1 − λj)(dPα)2(u∞)(¯∂u∞), ¯∂u∞  �  K  ωn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  < − δ  C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='34)  Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='32) with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='34), we obtain  l−1  �  j=1  (λj+1 − λj)rank(Ej)  � deg(E)  rank(E) − deg(Ej)  rank(Ej)  �  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  This indicates that there must exist a term µ(E) − µ(Ej0) < 0, which contradicts the  ω-semi-stability of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  □  Finally, we prove the other direction of the theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' In fact, we get the following  theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' The proof of the following theorem is standard and we present the proof for the  readers’ convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  14  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Shen  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Let (M, ω) be a compact Gauduchon manifold and E = {Ei, ϕij}i,j∈I be  an α-twisted holomorphic vector bundle of rank r over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' If E admits an approximate  Hermitian–Einstein structure, then E is ω-semi-stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Before we giving the detailed proof, we need the following vanishing theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='6 (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='31 in [37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Let E1 and E2 be two α-twisted holomorphic  vector bundles on M of respective ranks r1 and r2, and suppose that  degω(E1)  r1  > degω(E2)  r2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  If E1 and E2 admit approximate Hermitian–Einstein structure, then every morphism f ∈  Hom(E1, E2) is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Proof of theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Let E be the locally free α-twisted coherent sheaf associated to E  and F be any α-twisted coherent subsheaf of E with rank 0 < p < r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Let L be the  αp-twisted holomorphic line bundle corresponding to det(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' We consider the following  untwisted holomorphic vector bundle  G = ∧pE ⊗ L−1,  where L−1 is the dual α−p-twisted line bundle of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Following the same arguments as in  [37], we know that G admits an approximate Hermitian–Einstein structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Since there  is a nontrivial holomorphic section of G, then by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='6, we have  degω(G) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Then  0 ≤ degω(G) = degω(∧pE) − rank(∧pE) degω(L)  = degω(∧pE) − rank(∧pE) degω(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  So we have  0 ≤ µω(G) = µω(∧pE) − pµω(F) = p(µω(E) − µω(F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  This implies that µω(F) ≤ µω(E), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=', E is ω-semi-stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Bogomolov type inequality for semi-stable α-twisted holomorphic  vector bundles  In this section, we will prove the Bogomolov type inequality for semi-stable α-twisted  holomorphic vector bundles over compact Gauduchon Astheno-K¨ahler manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Before  giving the detailed proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='2, we need the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Let (M, ˜ω) be a compact Hermitian manifold of dimension n, and ω a  Gauduchon metric which is conformal to ˜ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Let E = {Ei, ϕij}i,j∈I be an α-twisted holo-  morphic vector bundle of rank r over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' If E is ω-semi-stable, then for any ε > 0, there  exists a Hermitian metric Hε = {Hi,ε}i∈I on E such that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='1)  sup  M  |  √  −1ΛωF ⊥  Hε|Hε < ε,  where F ⊥  Hε = FHε − 1  rtrFHε ⊗ IdE is the trace free part of FHε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Semi-stable twisted holomorphic vector bundles over Gauduchon manifolds  15  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Since ω is conformal to ˜ω, then there exists a smooth function φ such that  ω = eφ˜ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Then by direct calculation, for any Hermitian metric H, we have  sup  M  |  √  −1Λ˜ωF ⊥  H |H = sup  M  ����  √  −1Λ˜ωFH −  �1  r  √  −1Λ˜ωtrFH  �  ⊗ IdE  ����  = sup  M  eφ  ����  √  −1ΛωFH −  �1  r  √  −1ΛωtrFH  �  ⊗ IdE  ����  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='2)  Due to the fact that √−1ΛωFH is self-adjoint with respect to H, we know that the  eigenvalues of √−1ΛωFH are real values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Then for any λ1, · · · , λr ∈ R, C ∈ R, we have  the inequality  r  �  j=1  (λj − ¯λ) ≤  r  �  j=1  (λj − C)2,  where ¯λ = 1  r  �r  j=1 λj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' So we have  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='3)  ����  √  −1ΛωFH −  �1  r  √  −1ΛωtrFH  �  ⊗ IdE  ���� ≤ |  √  −1ΛωFH − λ · IdE|H,  where λ = 2π degω(E)  rVol(M,ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='2) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='3) yields that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='4)  sup  M  |  √  −1Λ˜ωF ⊥  H |H ≤ esupM φ sup  M  |  √  −1ΛωFH − λ · IdE|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Since E is ω-semi-stable, by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='1, it admits an approximate Hermitian–Einstein  structure on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' So for any ε > 0, there exists a Hermitian metric Hε such that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='5)  sup  M  |  √  −1ΛωFHε − λ · IdE|Hε < εe− supM φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='5) with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='4) gives (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  □  Proof of theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Let (M, ˜ω) be a compact Astheno-K¨ahler manifold of dimension n,  and ω a Gauduchon metric which is conformal to ˜ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Let E = {Ei, ϕij}i,j∈I be an α-twisted  holomorohic vector bundle of rank r over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Given any Hermitian metric H = {Hi}i∈I,  recall that  c1(E, H) =  √−1  2π trFH, c2(E, H) = − 1  8π2  �  (trFH)2 − tr(F 2  H)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Since F ⊥  H = FH − 1  rtrFH ⊗ IdE, one can easily get  tr(F 2  H) = 1  r(trFH)2 + tr(F ⊥  H ∧ F ⊥  H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Then we have  4π2  �  2c2(E, H) − r − 1  r  c1(E, H) ∧ c1(E, H)  �  = −  �  (trFH)2 − tr(F 2  H)  �  + r − 1  r  (trFH)2 = tr(F 2  H) − 1  r(trFH)2  = tr(F ⊥  H ∧ F ⊥  H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  16  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Shen  By the Riemann bilinear relations, we have  �  M  �  2c2(E) − r − 1  r  c1(E) ∧ c1(E)  �  ∧  ˜ωn−2  (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  =  �  M  �  2c2(E, H) − r − 1  r  c1(E, H) ∧ c1(E, H)  �  ∧  ˜ωn−2  (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  =  1  4π2  �  M  tr(F ⊥  H ∧ F ⊥  H ) ∧  ˜ωn−2  (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  =  1  4π2  �  M  |F ⊥  H|H,˜ω − |  √  −1Λ˜ωF ⊥  H |2  H  ˜  ωn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='6)  Since E is ω-semi-stable, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='1, for every ε > 0, there exists a Hermitian metric  Hε such that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='7)  sup  M  |  √  −1Λ˜ωF ⊥  H |Hε ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='6) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='7), and letting ε → 0, we obtain  �  M  �  2c2(E) − r − 1  r  c1(E) ∧ c1(E)  �  ∧  ˜ωn−2  (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  □  Data Availability Statement: Not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Acknowledgements: The author would like to thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Xi Zhang and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' Pan  Zhang for their useful discussions and helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' The research was supported by  the National Key R and D Program of China 2020YFA0713100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content=' The author is partially  supported by NSF in China No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='12141104 and 11721101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
+page_content='  Conflicts of Interest: The authors declare no conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfePxr/content/2301.01433v1.pdf'}
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diff --git a/ytAzT4oBgHgl3EQf7_7g/content/tmp_files/2301.01899v1.pdf.txt b/ytAzT4oBgHgl3EQf7_7g/content/tmp_files/2301.01899v1.pdf.txt
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+arXiv:2301.01899v1  [cs.IT]  5 Jan 2023
+On Sparse Regression LDPC Codes
+Jamison R. Ebert, Jean-Francois Chamberland, Krishna R. Narayanan
+Department of Electrical and Computer Engineering, Texas A&M University
+Abstract—Belief propagation applied to iterative decoding and
+sparse recovery through approximate message passing (AMP) are
+two research areas that have seen monumental progress in recent
+decades. Inspired by these advances, this article introduces sparse
+regression LDPC codes and their decoding. Sparse regression
+codes (SPARCs) are a class of error correcting codes that
+build on ideas from compressed sensing and can be decoded
+using AMP. In certain settings, SPARCs are known to achieve
+capacity; yet, their performance suffers at finite block lengths.
+Likewise, LDPC codes can be decoded efficiently using belief
+propagation and can also be capacity achieving. This article
+introduces a novel concatenated coding structure that combines
+an LDPC outer code with a SPARC-inspired inner code. Efficient
+decoding for such a code can be achieved using AMP with a
+denoiser that performs belief propagation on the factor graph
+of the outer LDPC code. The proposed framework exhibits
+performance improvements over SPARCs and standard LDPC
+codes for finite block lengths and results in a steep waterfall
+in error performance, a phenomenon not observed in uncoded
+SPARCs. Findings are supported by numerical results.
+Index Terms—LDPC codes, sparse regression codes, approxi-
+mate message passing, belief propagation.
+I. INTRODUCTION AND BACKGROUND
+Low-density parity check (LDPC) codes have been the
+subject of many scientific inquiries in the past, with landmark
+contributions ranging from theoretical advances to practical
+implementations [1]–[8]. LDPC ensembles are amenable to
+analysis and, under certain conditions, encoded messages can
+be recovered efficiently using iterative belief propagation (BP)
+decoding. Interestingly, the complexity of BP decoding grows
+linearly with block length and, as such, this paradigm offers a
+pragmatic solution for long block lengths [9]. Some spatially
+coupled LDPC constructions feature capacity approaching
+iterative decoding thresholds while also avoiding the pitfall
+of error floors [10]–[15]. Unfortunately, systems operating at
+shorter block lengths may not be conducive to the application
+of spatial coupling. In such situations, non-binary LDPC
+codes have been leveraged as a means to provide adequate
+performance [6], [7], [16]–[18].
+In a seemingly unrelated research direction, Joseph and
+Barron introduced the concept of a sparse regression code
+(SPARC), which establishes a connection between code design
+and sparse recovery in high dimensions [19]–[21]. The code-
+words contained in a SPARC codebook consist of sparse su-
+perpositions of columns from a design matrix, also known as a
+sensing matrix in the compressed sensing (CS) literature [22],
+[23]. In some sense, the SPARC decoding process is equivalent
+This material is based upon work supported, in part, by the National
+Science Foundation (NSF) under Grant CCF-2131106, and by Qualcomm
+Technologies, Inc., through their University Relations Program.
+to recovering the non-zero entries in a suitably designed index
+vector. Low complexity algorithms have been proposed for
+the efficient decoding of SPARC codewords. Notably, Rush et
+al. discuss an approximate message passing (AMP) decoder
+for SPARCs in [24], [25]. It is also worth mentioning that
+AMP has independently gained much research momentum in
+recent years owing to its low implementation cost, mathemat-
+ical tractability, and excellent performance [26]–[30]. Sparse
+regression codes paired with AMP decoding are known to
+achieve asymptotic single-user AWGN channel capacity under
+an appropriately chosen power allocation. Concurrently, there
+have been efforts to improve the finite block-length perfor-
+mance of SPARCs [31], [32].
+A popular strategy to improve the performance of codes
+in practical settings is to adopt a concatenated structure. For
+example, SPARCs and LDPC codes have been paired in
+the past to create concatenated schemes that improve per-
+formance [31]–[34]. Concatenated structures with a SPARC-
+like inner code have also been proposed in unsourced random
+access [35]–[37]. In [36], it is shown that the structure of
+a judiciously designed outer code can be integrated into the
+denoising function of the inner AMP recovery algorithm.
+Surprisingly, despite being mentioned by Liu et al. in [34] as
+a possible future research direction, such an approach has not
+been considered for the single-user scenario. This article seeks
+to address this defficiency by introducing sparse regression
+LDPC codes.
+Sparse regression LDPC codes feature a concatenated inner
+SPARC-like code and an outer non-binary LDPC code. This
+concatenated structure is amenable to a decoder that exchanges
+information between inner and outer codes via a dynamic
+AMP denoiser. In this setting, the denoiser performs BP on
+the factor graph of the outer LDPC code to improve the
+AMP state estimates. This in turn improves the effective
+observations which in turn translate into better local belief
+vectors for the BP LDPC decoder. Such a denoiser improves
+AMP’s convergence and lowers the residual mean square
+error (MSE) in the estimated SPARC codeword. Once the
+MSE of the iterative AMP-BP decoding levels off, the outer
+LDPC code can be leveraged once more to further improve
+performance. The goal is to use AMP-BP decoding to increase
+the SNR of the effective observation to a level that exceeds
+the BP threshold of the graph and then to apply message
+passing to the outer LDPC code to recover the message.
+Numerical results demonstrate the performance advantages of
+the proposed model over related architectures at reasonably
+short block lengths.
+
+II. SYSTEM MODEL
+We consider a point-to-point memoryless additive white
+Gaussian noise (AWGN) channel with one transmitter and one
+receiver, each equipped with a single antenna. The received
+signal y ∈ Rn is given by
+y = x + z,
+(1)
+where x ∈ Rn denotes the transmitted signal and z ∼
+N
+�
+0, σ2I
+�
+represents AWGN. As mentioned above, we wish
+to study a coding architecture composed of a sparse regression
+inner code [19]–[21], and a non-binary LDPC outer code [6]–
+[8]. We elaborate on the encoding process and the decoding
+scheme below.
+A. Sparse Regression LDPC Encoding
+The proposed encoding process features a sequence of
+three distinct steps: q-ary LDPC encoding, indexing LDPC
+symbols, and inner CS encoding. First, the information bits
+are encoded into a q-ary LDPC codeword via well-established
+operations [6]–[8]. The second step transforms the q-ary LDPC
+codeword into a suitable sparse vector. The last step is the
+matrix multiplication emblematic of a sparse regression code;
+the output is sometimes referred to as a large random matrix
+system [19]–[21], [34]. We summarize the notions pertaining
+to this encoding process below while concurrently introducing
+necessary notation.
+1) q-ary LDPC Encoding: The LDPC encoder takes a
+binary sequence w ∈ FB
+2 as its input and maps it to a q-
+ary codeword v ∈ FL
+q . We represent the resultant codeword in
+concatenated form as
+v = (v1, v2, . . . , vL)
+(2)
+where the ℓth element vℓ lies in finite field Fq and L is the
+length of the codeword.
+Note that there exists a bijection Φ : Fq → [q] between
+the elements of Fq and the integers [q] = (0, 1, . . . , q − 1),
+where 0 ∈ Z maps to 0 ∈ Fq and 1 ∈ Z maps to 1 ∈ Fq.
+Throughout, we adopt such an arbitrary, but fixed bijection
+and exploit this relation by employing the same variable for
+a field element g ∈ Fq and for its corresponding integer
+Φ (g) ∈ [q]. This slight abuse of notation greatly simplifies
+exposition, and its use should not lead to confusion because
+one can unambiguously infer from context whether an instance
+refers to the field element or to its integer representation.
+2) LDPC Symbol Indexing:
+Coded symbol sparsifica-
+tion/indexing consists of mapping vℓ ∈ Fq to standard basis
+vector evℓ ∈ Rq and subsequently stacking the L basis vectors
+together. We seize this opportunity to reinforce the notion that
+entry ℓ of v is an element of Fq, but vℓ in evℓ refers to an
+integer in [q] under our overloaded notation. With that, the
+output of the indexing process becomes
+s =
+
+
+ev1
+...
+evL
+
+ ,
+(3)
+where s is an L-sparse vector of length qL. Vector s has
+a structure akin to that of a sparse regression code prior
+to multiplication by a large random matrix. This structured
+sparsity can be exploited during decoding.
+3) Inner CS Encoding: The last phase of the encoding
+process consists in pre-multiplying vector s by matrix A to
+get x = As, where A ∈ Rn×qL and Ai,j ∼ N
+�
+0, 1
+n
+�
+. Note
+that n ≪ qL to conform to the CS framework. Equation (1)
+may thus be rewritten as:
+y = As + z.
+(4)
+Having specified the code structure, we are now ready to
+discuss the process of recovering s from y.
+B. Sparse Regression LDPC Decoding
+Paralleling the development of AMP for sparse regression
+codes [24] and drawing inspiration from concatenated AMP
+systems [33], [34], we wish to create a composite iterative
+process to recover state vector s from y that exploits the
+sparsity in s and the parity structure of the LDPC code. This
+can be accomplished by incorporating message passing on the
+factor graph of the LDPC code into the AMP denoiser. A
+similar approach can be found in [36]; however, the denoiser
+we wish to utilize below differs in the fact that only one
+codeword is present within y.
+Our AMP composite algorithm is as follows,
+r(t) = ATz(t) + s(t)
+(5)
+z(t) = y − As(t) + z(t−1)
+n
+div ηt−1
+�
+r(t−1)�
+(6)
+s(t+1) = ηt
+�
+r(t)�
+(7)
+where the superscript t denotes the iteration count. The al-
+gorithm is initialized with conditions r(0) = s(0) = 0 and
+z(0) = y.
+Equation (5) specifies the effective observation, which is
+characteristic of AMP. Equation (6) computes the residual
+error enhanced with an Onsager correction term, whereas (7)
+provides a state estimate. The denoising functions (ηt(·))t≥0
+seek to exploit the structure of s while computing the state
+updates.
+Generally speaking, a choice denoiser is the conditional
+expectation of s given r(t), Unfortunately, this approach is
+computationally intractable in the present context because
+it entails summing over all the possible codewords. As an
+alternative, we know that BP can be efficiently applied to
+q-ary LDPC codes. Furthermore, at any point during BP, a
+belief on individual LDPC symbols can be formed based on
+incoming messages from neighboring factor nodes and local
+observations. Thus, we can potentially apply a few iterations
+of BP as a means to get an estimate for the state vector by
+leveraging the connection between sections of s and LDPC
+symbols. In this sense, iterative message passing can act as a
+foundation for pragmatic denoising functions. We elaborate on
+this connection below and, concurrently, we review pertinent
+notions of BP applied to q-ary LDPC codes.
+
+III. CREATING A BP DEONOISER FOR AMP
+In this section, we introduce the denoising function we wish
+to employ within AMP. To begin, we emphasize that r admits
+a sectionized representation akin to that of the state vector s
+in (3). That is, we can view both the state estimate and the
+effective observation as concatenations of L vectors, each of
+length q. Mathematically, we have
+r =
+
+
+r1
+...
+rL
+
+
+ˆs =
+
+
+ˆs1
+...
+ˆsL
+
+ .
+This point is crucially important because the denoiser is
+constructed in a block-wise fashion. As a side note, we neglect
+the superscript (t) for most of the discussion below to lighten
+notation. Furthermore, we emphasize that the presence of the
+hat in ˆs distinguishes the estimate from the true state vector s
+in the absence of an iteration count (t).
+Each section rℓ in r acts as a vector observation about
+the value of sℓ ∈ {eg : g ∈ [q]}. An astounding and enabling
+property of AMP is that, under certain technical conditions,
+the effective observation r is asymptotically distributed as
+s+ τζ, where ζ is a random vector with independent N(0, 1)
+components and τ is a deterministic quantity. This fact hinges
+on the presence of the Onsager term in (6) and on some
+smoothness conditions for the denoising functions. We delay
+the treatment of these technical conditions for the time being;
+rather, we posit the requirements and formally introduce them
+as a condition.
+Condition 1. The effective observation r(t) is asymptotically
+distributed as s + τtζt, where ζt is a random vector with
+independent N(0, 1) components and τt is specified by a set
+of deterministic equations. This asymptotic characterization
+takes place in the dimensions of the system, as opposed to
+time or iteration count.
+We describe below our rationale behind the denoising
+function assuming Condition 1 holds; we provide a rigorous
+foundation for this condition in Appendix B, but this can
+only be done once the structure of the denoiser is established.
+Consider the effective observation restricted to section ℓ.
+Under Condition 1, the distribution of random observation
+vector Rℓ given section Sℓ = eg is given by
+fRℓ|Sℓ (rℓ|eg) =
+1
+(2π)
+q
+2 τ q exp
+�
+−∥rℓ − eg∥2
+2τ 2
+�
+.
+Under a uniform input distribution, the conditional probability
+of Sℓ = eg becomes
+αℓ(g) = Pr (Sℓ = eg|Rℓ = rℓ) = Pr (Vℓ = g|Rℓ = rℓ)
+=
+fRℓ|Vℓ (rℓ|g)
+�
+h∈Fq fRℓ|Vℓ (rℓ|h) =
+e− ∥rℓ−eg∥2
+2τ2
+�
+h∈Fq e− ∥rℓ−eh∥2
+2τ2
+=
+e
+rℓ(g)
+τ2
+�
+h∈Fq e
+rℓ(h)
+τ2
+.
+(8)
+A possible estimate for Sℓ can be computed by taking its
+conditional expectation, given observation Rℓ = rℓ, with
+E [Sℓ|Rℓ = rℓ] =
+�
+g∈Fq
+egPr (Sℓ = eg|Rℓ = rℓ) .
+(9)
+A variant of this approach can be found in [24] for a system
+without an outer code. Ideally, we would like to take advantage
+of the outer code with the more precise MMSE estimate of
+the form E [Sℓ|R = r]. Unfortunately, as mentioned above,
+computing this conditional expectation is far too complex to
+be implemented in practice. A viable alternative that trades
+off performance and complexity is to execute BP on the
+factor graph of the outer LDPC code. Implicitly, this approach
+computes an estimate for every Sℓ based on the computation
+tree of the code, up to a certain depth [8], [38].
+Frameworks to perform BP on factor graphs are well-
+established [39]; thus, we assume some familiarity with such
+iterative procedures. We proceed by first considering the
+nuances of non-binary LDPC factor graphs, then presenting
+a BP algorithm, and finally proposing a dynamic denoiser for
+SR-LDPC codes.
+A. Non-Binary LDPC Graphs
+The factor graph for an Fq LDPC code features L variable
+(left) nodes and L(1−R) check (right) nodes enforcing parity
+constraints, where R is the rate of the LDPC code [6]–[8]. An
+important distinction between binary and non-binary LDPC
+codes is that a factor graph for the latter code typically includes
+edge labels. These labels take values in Fq \ {0}. A vector
+v ∈ FL
+q is a valid codeword if it satisfies the parity equations
+�
+vℓ∈N(cp)
+ωℓ,p ⊗ vℓ = 0
+∀p ∈ [L(1 − R)],
+(10)
+where N(cp) is the collection of variable nodes adjacent
+to parity check cp. The summation and the multiplication
+operator ⊗ in (10) take place over finite field Fq. Parameter
+ωℓ,p represents the label or weight assigned with the edge
+connecting variable node vℓ and check node cp. Adopting
+common factor graph concepts [39], [40], we denote the graph
+neighbors of variable node vℓ by N(vℓ). The factor associated
+with cp and derived from parity equation (10) can be expressed
+as an indicator function
+Gp(vp) = 1
+
+
+�
+vℓ∈N(cp)
+ωℓ,p ⊗ vℓ = 0
+
+
+(11)
+where vp = (vℓ ∈ N(cp)) is a shorthand notation for the
+restriction of v to entries associated with graph neighborhood
+N(cp). With these definitions, the factor function associated
+with our LDPC code assumes the product decomposition given
+by
+G(v) =
+�
+p∈[L(1−R)]
+Gp(vp).
+(12)
+In words, G(v) is an indicator function that assesses whether
+its argument is a valid codeword.
+
+B. Belief Propagation
+We view messages for a non-binary LDPC code as multi-
+dimensional belief vectors over Fq. Messages from variable
+nodes to check nodes are denoted by µv→c, and messages
+in the reverse direction are represented by µc→v. Formally,
+a message going from check node cp to variable node vℓ ∈
+N(cp) is computed component-wise through
+µcp→vℓ(g) =
+�
+vp:vℓ=g
+Gp (vp)
+�
+vj∈N(cp)\vℓ
+µvj→cp(gj).
+(13)
+While (13) is shown in compact form, the actual summation
+operation is cumbersome. Finding the set of summands entails
+identifying sequences of the form (gj ∈ Fq : vj ∈ N(cp) \ vℓ)
+that fulfill local condition (10) or, equivalently,
+�
+vj∈N(cp)\vℓ
+ωj,p ⊗ gj = −ωℓ,p ⊗ g.
+(14)
+A belief vector passed from variable node vℓ to check node
+cp, where p ∈ N(vℓ), is calculated component-wise via
+µvℓ→cp(g) ∝ αℓ(g)
+�
+cξ∈N(vℓ)\cp
+µcξ→vℓ(g).
+(15)
+The ‘∝’ symbol indicates that the positive measure can be
+normalized before being sent out as a message. Vector αℓ
+in (15) can be viewed as a collection of beliefs based on
+local observations, as in (8). That is, entry αℓ(g) captures
+the probability that symbol g is the true field element within
+section ℓ. Other BP messages are initialized with µv→c = 1
+and µc→v = 1. The traditional parallel sum-product algorithm
+iterates between (13) and (15), alternating between updated
+rightbound messages and leftbound messages.
+One of the key advantages of indexing vectors using field
+elements in Fq, as pointed out in [7], is the ensuing ability
+to define pertinent operators on these vectors. Paralleling
+established literature, we define two actions.
+Definition 2 (Vector +g Operator [7]). For field element g ∈
+Fq, the vector +g operator acting on b and denoted by b+g
+is defined as
+b+g =
+�
+bg, bg⊕1, . . . , bg⊕(q−1)
+�
+= (bh⊕g : h ∈ Fq)
+where subscript addition ⊕ is performed in Fq.
+Definition 3 (Vector ×g Operator [7]). For field element g ∈
+Fg \ {0}, we define the vector ×g operator acting on b and
+denoted by b×g by
+b×g =
+�
+b0, bg, b2⊗g, . . . , b(q−1)⊗g
+�
+= (bh⊗g : h ∈ Fq)
+where subscript product ⊗ takes place in Fq.
+Under these operations, we can rewrite (13) in a concise
+manner:
+µcp→vℓ =
+
+
+�
+vj∈N(cp)\vℓ
+�
+µvj→cp
+�×ω−1
+j,p
+
+
+×(−ωℓ,p)
+(16)
+where ωj,p is the label on the edge between variable node vj
+and factor node cp. The operator ⊙ denotes the Fq convolution
+between two vectors,
+[µ ⊙ ν]g =
+�
+h∈Fq
+µh · νg−h
+g ∈ Fq.
+The exposition can be simplified further if we absorb the edge
+labels within the messages themselves. Specifically, we adopt
+the definitions
+µvj→cp =
+�
+µvj→cp
+�×ω−1
+j,p
+(17)
+µcp→vℓ =
+�
+µcp→vℓ
+�×(−ω−1
+ℓ,p)
+.
+(18)
+Then (16) morphs into the simpler expression
+µcp→vℓ =
+�
+vj∈N(cp)\vℓ
+µvj→cp.
+(19)
+This equation highlights the role of the Fq convolution within
+BP for non-binary LDPC codes. The message from variable
+node vℓ to check node cp found in (15) also admits a
+more compact form. For p ∈ N(vℓ), the traditional outgoing
+message from a variable node can be written as
+µvℓ→cp =
+αℓ ◦
+�
+⃝cξ∈N(vℓ)\cp µcξ→vℓ
+�
+���αℓ ◦
+�
+⃝cξ∈N(vℓ)\cp µcξ→vℓ
+����
+1
+(20)
+where ◦ represents the Hadamard product.
+A natural estimate for the distribution associated with vari-
+able node vℓ, including intrinsic information, is
+ˆSℓ =
+αℓ ◦
+�
+⃝cp∈N(vℓ) µcp→vℓ
+�
+���αℓ ◦
+�
+⃝cp∈N(vℓ) µcp→vℓ
+����
+1
+.
+(21)
+As we will shortly see, (21) is the output of our proposed
+denoiser.
+Remark 4. In our construction, q = 2m, m ≥ 1 because
+indexing is derived from sequences of bits. This invites the
+application of fast techniques to implement message passing
+over the corresponding factor graph. Specifically, the fast
+Walsh-Hadamard transform (FWHT) can be utilized to rapidly
+and efficiently compute (19), with
+µcp→vℓ ∝ fwht−1
+
+
+�
+vj∈N(cp)\vℓ
+fwht
+�
+µvj→cp
+�
+
+ .
+Alternatively, one could adopt a different finite field convolu-
+tion or a ring structure amenable to the circular convolution
+to create local factor functions conducive to the fast Fourier
+transform [6], [41], [42].
+With these tools in mind, we are ready to formally define
+our proposed denoiser.
+
+C. Sparse Regression LDPC Denoiser
+Conceptually, one can initiate the state of the LDPC factor
+graph using the effective observation r, run BP, and then
+gain an estimate for the state based on (21). As mentioned
+before, in the absence of BP iterations, local estimates reduce
+to the conditional expectation E [Sℓ|Rℓ = rℓ] found in (9).
+Yet, as more iterations of the BP algorithm are performed, the
+estimate for Sℓ is refined based on the computation tree of the
+outer code, up to a certain depth.
+Definition 5 (BP-N Denoiser). Let N denote the number of
+BP iterations to perform, where N is less than the girth of the
+LDPC factor graph. The BP-N denoiser
+1) initializes the LDPC factor graph with estimates αℓ
+computed from rℓ for ℓ ∈ [L] according to (8);
+2) computes and passes variable to check messages (see
+(20)) and check to variable messages (see (17), (19), (18)
+and Remark 4) along the edges of the factor graph in an
+alternating fashion N times;
+3) computes updated state estimates according to (21).
+The output of this denoiser can then be passed to the AMP
+composite algorithm for the computation of the next residual,
+enhanced with the Onsager term.
+The derivation of the Onsager correction term corresponding
+to this denoiser is included in Appendix A. At this point, we
+turn to a performance assessment of sparse regression LDPC
+codes via numerical simulations.
+IV. SIMULATION RESULTS
+In this section, we seek to characterize the performance of
+SR-LDPC codes.1 As benchmarks, we compare our results to
+the concatenated SPARC/LDPC scheme presented in [31] as
+well as 5G-NR binary LDPC encoded BPSK. We note that
+the latter comparison provides limited insight because the SR-
+LDPC code is allowed real channel inputs whereas the 5G-NR
+code is constrained to binary inputs; nevertheless, we make the
+comparison to see whether SR-LDPC codes are competitive
+with state of the art, commonly used codes.
+The scenario of interest is one in which 5888 information
+bits are to be transmitted over 7350 real channel uses. We
+define the SR-LDPC rate to be the number of information
+bits over the number of channel uses; thus, we have that
+RSRLDPC ≈ 0.80. The non-binary LDPC code employed is a
+(751, 736) code (RLDPC ≈ 0.98) over GF(256) whose edges
+are generated via progressive edge growth (PEG) and whose
+weights were chosen uniformly at random from the elements
+of F256 \ 0. The fraction of degree-2 variable nodes is chosen
+to be higher than usual so that the AMP-BP iterations can
+bootstrap in high-noise environments. The AMP-BP decoder
+is run for up to 100 iterations and then the non-binary LDPC
+BP decoder is run for another 100 iterations, or until a valid
+codeword is obtained. Figure 1 highlights our results.
+1The source code used to generate these results is available online at
+https://github.com/EngProjects/mMTC/tree/code.
+1
+1.5
+2
+2.5
+3
+3.5
+10−4
+10−3
+10−2
+10−1
+100
+Eb/N0 (dB)
+BER
+SPARC + LDPC [31]
+5G-NR LDPC + BPSK
+SR-LDPC
+Fig. 1: BER performance comparison of a sparse regression
+LDPC code, the concatenated SPARC/LDPC construction
+from [31], and 5G-NR binary LDPC encoded BPSK as a
+function of Eb/N0. The SR-LDPC code requires a lower
+Eb/N0 to achieve a target BER than LDPC encoded BPSK
+and the scheme in [31].
+From Fig. 1, it is clear that sparse regression LDPC codes
+provide a roughly 0.75 dB improvement at a BER of 10−3 over
+other SPARC/LDPC concatenated coding structures. Further-
+more, the SR-LDPC code outperforms 5G-NR LDPC encoded
+BPSK in some regimes; we note however, that 5G-NR LDPC
+encoded BPSK has a lower error floor than the SR-LDPC
+code.
+V. CONCLUSION
+In conclusion, this article presents sparse regression LDPC
+codes and their decoding. A sparse regression code is a
+concatenated structure with an inner SPARC-like code and
+an outer non-binary LDPC code. The inner code is decoded
+using AMP with a dynamic denoiser which runs BP on the
+factor graph of the outer LDPC code to improve the state
+estimate at every iteration. After running many iterations of
+the AMP-BP decoder, the final effective observation is fed
+into a standard LDPC BP decoder, which seeks to correct
+any residual errors that may be present in the received signal.
+Numerical results show that sparse regression codes exhibit
+performance improvements over standard LDPC codes and
+other concatenated SPARC/LDPC constructions.
+As mentioned in the introduction, both SPARCs and LDPC
+codes are known to achieve capacity in certain circumstances,
+yet both codes generally suffer at short block lengths. The
+success of sparse regression LDPC codes is evidence that
+SPARCs and LDPC codes can be seamlessly combined to
+significantly improve performance. Directions for future work
+include optimizing the outer LDPC code and applying coded
+demixing [37] to SR-LDPC codes to accommodate multiple
+users.
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+APPENDIX A
+DERIVATION OF ONSAGER TERM
+Intuitively, the role of the Onsager term is to (asymptoti-
+cally) cancel the first-order correlations between ATz(t) and
+s(t) and thereby maintain a structure conducive to prompt
+convergence and analysis. This factor, emblematic of AMP
+algorithms, appears in (6) and is given by
+z(t−1)
+n
+div ηt−1
+�
+ATz(t−1) + s(t−1)�
+= z(t−1)
+n
+div ηt−1
+�
+r(t−1)�
+where the div operator can be expanded into
+div η (r) =
+�
+ℓ∈[L]
+div ˆsℓ(r, τ) =
+�
+ℓ∈[L]
+�
+g∈Fq
+∂ˆsℓ (g, r, τ)
+∂rℓ(g)
+. (22)
+Thus, as an intermediate step, we must calculate the partial
+derivative of ˆsℓ (g, r, τ) with respect to rℓ(g). This computa-
+tion is rendered much simpler under the following condition.
+Condition 6 (Sub-Girth AMP-BP). The BP-N denoiser is
+said to possess the sub-girth AMP-BP condition when fewer
+message passing iterations are performed on the factor graph
+of the LDPC code than the shortest cycle of this same graph,
+per AMP denoising step.
+This condition ensures that the message passing operations
+employed during denoising yield valid computation trees with-
+out cycles. Fortunately, this is the regime we are primarily
+interested in.
+Lemma 7. Under Condition 6, the partial derivative of
+ˆsℓ (g, r, τ) with respect to rℓ(g) is equal to
+∂ˆsℓ (g, r, τ)
+∂rℓ(g)
+= 1
+τ 2 ˆsℓ (g, r, τ) (1 − ˆsℓ (g, r, τ))
+(23)
+where g ∈ Fq.
+Proof: Recall that the output of the BP denoiser defined
+in (21) can be expressed as
+ˆsℓ (g, r, τ) =
+αℓ(g) �
+p∈N(vℓ) µcp→vℓ(g)
+�
+h∈Fq αℓ(h) �
+p∈N(vℓ) µcp→vℓ(h)
+=
+αℓ(g)µvℓ→c0(g)
+�
+h∈Fq αℓ(h)µvℓ→c0(h).
+(24)
+Under Condition 6, belief vector µvℓ→c0 is based solely on
+extrinsic information and, hence, it is determined based on
+{rj : j ∈ [L] \ {ℓ}}. Consequence, we gather that
+∂µvℓ→c0(g)
+∂rℓ(g)
+= 0.
+Under such circumstances, we can calculate the desired deriva-
+tive in a straightforward manner, with
+∂ˆsℓ (g, r, τ)
+∂rℓ(g)
+=
+∂
+∂rℓ(g)
+e
+rℓ(g)
+τ2 µvℓ→c0(g)
+�
+h∈Fq e
+rℓ(h)
+τ2 µvℓ→c0(h)
+= 1
+τ 2
+e
+rℓ(g)
+τ2 µvℓ→c0(g)
+�
+h∈Fq e
+rℓ(g)
+τ2 µvℓ→c0(g)
+− 1
+τ 2
+�
+e
+rℓ(g)
+τ2 µvℓ→c0(g)
+�2
+��
+h∈Fq e
+rℓ(g)
+τ2 µvℓ→c0(g)
+�2
+= 1
+τ 2 ˆsℓ (g, r, τ) (1 − ˆsℓ (g, r, τ)) .
+This last line corresponds to the statement of the lemma.
+It is worth emphasizing that the derivative in (23) remains
+unchanged irrespective of the number of BP rounds computed
+on the factor graph, so long as Condition 6 is satisfied. The
+divergence of (22) assumes the same simple form under such
+circumstances.
+Proposition 8. The divergence of η (r) with respect to r is
+equal to
+div η (r) = 1
+τ 2
+�
+∥η (r)∥1 − ∥η (r)∥2�
+.
+(25)
+Proof: We expand the div operator as
+div η (r) =
+�
+ℓ∈[L]
+div ˆsℓ (r, τ) =
+�
+ℓ∈[L]
+�
+g∈Fq
+∂ˆsℓ (g, r, τ)
+∂rℓ(g)
+=
+�
+ℓ∈[L]
+�
+g∈Fq
+1
+τ 2 ˆsℓ (g, r, τ) (1 − ˆsℓ (g, r, τ))
+= 1
+τ 2
+�
+∥η (r)∥1 − ∥η (r)∥2�
+.
+(26)
+The last equality follows from the fact that, since ˆsℓ (g, r, τ)
+lies between zero and one, the corresponding partial derivative
+with respect to rℓ(g) found in Lemma 7 is always non-
+negative.
+APPENDIX B
+DENOISER IS LIPSCHITZ CONTINUOUS
+The mathematical underpinnings for the rigorous appli-
+cation of AMP in this article are presented by Berthier,
+Montanari, and Nguyen in [30]. One of the conditions for the
+state evolution to hold for non-separable functions is that the
+denoiser must be pseudo-Lipschitz of a certain order. For the
+problem at hand, we are able to show the stronger Lipschitz
+condition, which is sufficient. Thus, the main objective of
+this section is to demonstrate that the denoiser introduced
+in Definition 5 is Lipschitz continuous under Condition 6.
+To achieve this goal, our strategy is to demonstrate that the
+magnitudes of the entries in the Jacobian matrix of η(r) with
+respect to r are uniformly bounded.
+
+Recall that the denoiser assumes a sectional form, as
+described in Definition 5. The vector estimate for section ℓ
+becomes
+ˆsℓ(r) =
+�
+g∈Fq
+Pr (Vℓ = g|Rtree = rtree) eg,
+where Rtree denotes the measurements associated with the
+computational tree of the LDPC code rooted at section ℓ. The
+(realized) scaling factors found in (21) are given by
+ˆsℓ(r) =
+⃝p∈N0(vℓ) µcp→vℓ
+���⃝p∈N0(vℓ) µcp→vℓ
+���
+1
+.
+(27)
+where N0(vℓ) denotes the neighborhood of vℓ including the
+local observation and µc0→vℓ
+= αℓ. We are ultimately
+interested in Jacobian entries of the form
+∂ˆsℓ (r, g)
+∂rj(h)
+=
+∂
+∂rj(h)
+�
+p∈N0(vℓ) µcp→vℓ(g)
+���⃝p∈N0(vℓ) µcp→vℓ
+���
+1
+=
+∂
+∂rj(h)
+αℓ(g) �
+p∈N(vℓ) µcp→vℓ(g)
+�
+k∈Fq αℓ(k) �
+p∈N(vℓ) µcp→vℓ(k)
+(28)
+for g, h ∈ Fq and ℓ, j ∈ [L]. We adopt a divide-and-conquer
+approach to identify and bound these derivatives. Specifically,
+we focus on the rooted tree obtained by taking the factor graph
+of the outer LDPC code, setting vℓ as the root, and retaining
+only the nodes involved in the computation of ˆsℓ(r, g). Under
+Condition 6, this sub-graph must form a proper tree; Fig. 2
+offers a notional diagram to illustrate the outcome of this
+process.
+We seek to bound the magnitude of the derivatives in (28)
+based on the distance between vℓ and vj in this rooted tree.
+We begin with local observations.
+Proposition 9. (Local Observations) The partial derivatives
+of αj with respect to rk(h) are given by
+∂αj
+∂rk(h) =
+�
+1
+τ 2 αj(h) (eh − αj)
+j = k
+0
+j ̸= k
+(29)
+∀h ∈ Fq and where τ > 0 is the standard deviation of the
+effective observation.
+Proof: As defined in (8), the vector αj is given by
+αj(g) =
+e
+rj (g)
+τ2
+�
+k∈Fq e
+rj (k)
+τ2
+∀g ∈ Fq.
+When j ̸= k, it immediately follows that ∂αj/∂rk(h) = 0
+as αj does not depend on rk(h). We thus consider the case
+where k = j. When g = h, we have that
+∂αj(h)
+∂rj(h) = 1
+τ 2
+e
+rj(h)
+τ2
+�
+k∈Fq e
+rj(k)
+τ2
+− 1
+τ 2
+e
+rj(h)
+τ2 e
+rj(h)
+τ2
+��
+k∈Fq e
+rj(k)
+τ2
+�2
+= 1
+τ 2 αj(h) (1 − αj(h)) .
+vℓ
+cp
+µcp→vℓ
+vk
+µc0→vk
+cρ
+vj
+µvj→cρ
+µc0→vj
+Fig. 2: Computation tree for variable node vℓ obtained by
+taking the factor graph of the outer LDPC code, setting vℓ
+as the root, and retaining only the nodes involved in the
+computation of ˆsℓ (r, g). We use this data structure to compute
+the derivatives in (28).
+When g ̸= h, we get
+∂αj(g)
+∂rj(h) = − 1
+τ 2
+e
+rj (g)
+τ2 e
+rj(h)
+τ2
+��
+κ∈Fq e
+rj (κ)
+τ2
+�2 = − 1
+τ 2 αj(g)αj(h).
+Collecting these findings and condensing them into a more
+compact form, we arrive at (29), which is the desired expres-
+sion.
+Corollary 10. The absolute value of the partial derivatives of
+αj with respect to rk(h) are bounded by
+����
+∂αj
+∂rk(h)
+���� ≤
+1
+4τ 2
+(30)
+where τ
+> 0 is the standard deviation of the effective
+observation.
+The proof of this corollary is trivial when αj is a valid
+probability vector, as is the case in this article. We also note
+that, based on the state evolution of AMP, τ 2 ≥ σ2 at every
+iteration irrespective of the iteration number. We can therefore
+establish a uniform bound across iterations. We are now ready
+to show that the absolute value of (28) is bounded whenever
+ℓ = j, i.e., at the root level of the computation tree.
+Proposition 11 (Root Derivatives). The partial derivatives of
+ˆsℓ (r, g) with respect to rℓ(h) are given by
+∂ˆsℓ (r)
+∂rℓ(h) = 1
+τ 2 ˆsℓ (r, h) (eh − ˆsℓ (r))
+∀h ∈ Fq
+(31)
+where τ
+> 0 is the standard deviation of the effective
+observation.
+
+Proof: Leveraging Proposition 9 and denoting the stan-
+dard inner product by ⟨·, ·⟩, we have
+∂ˆsℓ (r)
+∂rℓ(h) =
+∂
+∂rℓ(h)
+αℓ ◦ ⃝p∈N(vℓ) µcp→vℓ
+���αℓ ◦ ⃝p∈N(vℓ) µcp→vℓ
+���
+1
+= αℓ(h)
+τ 2
+(eh − αℓ) ◦
+�
+⃝cp∈N(vℓ) µcp→vℓ
+�
+���αℓ ◦
+�
+⃝cp∈N(vℓ) µcp→vℓ
+����
+1
+− αℓ(h)
+τ 2
+ˆsℓ (r)
+�
+eh − αℓ, ⃝cp∈N(vℓ) µcp→vℓ
+�
+���αℓ ◦
+�
+⃝cp∈N(vℓ) µcp→vℓ
+����
+1
+= αℓ(h)
+τ 2
+eh ◦
+�
+⃝cp∈N(vℓ) µcp→vℓ
+�
+���αℓ ◦
+�
+⃝cp∈N(vℓ) µcp→vℓ
+����
+1
+− αℓ(h)
+τ 2
+ˆsℓ (r)
+�
+eh, ⃝cp∈N(vℓ) µcp→vℓ
+�
+���αℓ ◦
+�
+⃝cp∈N(vℓ) µcp→vℓ
+����
+1
+= 1
+τ 2 ˆsℓ (r, h) (eh − ˆsℓ (r)) ,
+which is the desired expression.
+Corollary 12. The absolute value of the partial derivatives of
+ˆsℓ (r) with respect to rℓ(h) are bounded by
+����
+∂ˆsℓ (r)
+∂rℓ(h)
+���� ≤
+1
+4τ 2
+(32)
+The proof of this corollary follows that of Corollary 10
+because like αj, ˆsℓ (r) forms a valid probability vector.
+Proposition 11 offers a blueprint for the general result we
+wish to establish. Yet, the situation gets more complicated
+when ℓ ̸= j because we have to involve the message passing
+rules. In doing so, we obtain a key intermediate result using
+mathematical induction. We start with the variable node closest
+to the root node, and then progress outward step by step.
+To circumvent a notational nightmare, we restrict the proof
+to cases where all edge weights are equal to 1 ∈ Fq.
+Conceptually, the edge can be interpreted as permutations on
+the belief vectors. From the point of view of bounding partial
+derivatives, this is a benign operation, yet the accounting
+that comes with permutations is dreadful, hence our focus on
+the simpler case. Moving forward, we assume the following
+condition.
+Condition 13. All edge weights within the factor graph of the
+LDPC outer code are equal to 1 ∈ Fq.
+The extension of the following propositions to the case
+with arbitrary edge weights (i.e., beyond Condition 13) is
+conceptually straightforward.
+Proposition 14. Suppose Condition 6 holds and let vj be a
+descendant of root node vℓ in the computation tree. Moreover,
+let cp ∈ N(vℓ) be the unique check neighbor of vℓ on the path
+from vℓ to vj within the tree. Then, there exists vector ν, with
+0 ⪯ ν ⪯ µcp→vℓ, such that the partial derivative of µcp→vℓ
+with respect to rj(h) is given by
+∂µcp→vℓ
+∂rj(h)
+= 1
+τ 2
+�
+ν − ∥ν∥1 µcp→vℓ
+�
+,
+(33)
+where τ
+> 0 is the standard deviation of the effective
+observation. Here, ⪯ denotes elementwise comparison of the
+entries in the vector.
+Proof: Under condition 6, we know that vj appears at
+most once within the computation tree rooted at vℓ. Thus,
+we establish (33) via mathematical induction on the distance
+between vℓ and its descendant vj on the computation tree. The
+distance that we are interested in only considers the number of
+variable nodes between vℓ and vj. Before beginning, we point
+out that if vj is not a descendant of vℓ, then the corresponding
+partial derivatives vanish and the claim is immediate.
+We begin with generic results that are useful for both the
+base case and the inductive step. Let vj be a descendant of vℓ
+and let vk be the variable child of vℓ on the path from vℓ to
+vj. Let cp be the unique check node in N(vℓ) ∩ N(vk) and
+let c̺ be the unique check node child of vk on the path from
+vk to vj; if vk = vj, let ̺ = 0. Then, we have that
+∂µvk→cp
+∂rj(h)
+=
+∂
+∂rj(h)
+⃝cξ∈N0(vk)\cp µcξ→vk
+���⃝cξ∈N0(vk)\cp µcξ→vk
+���
+1
+.
+(34)
+We can also examine the partial derivatives of the probability
+vector µc̺→vk. Suppose vk ̸= vj and let vo be the unique
+variable child of vk on the path from vℓ to vj. Using the Fq
+convolution, we have
+∂µc̺→vk
+∂rj(h)
+=
+∂
+∂rj(h)
+
+
+�
+vl∈N(c̺)\vk
+µvl→c̺
+
+
+=
+∂µvo→c̺
+∂rj(h) ⊙
+
+
+�
+vl∈N(c̺)\{vk,vo}
+µvl→c̺
+
+
+=
+∂µvo→c̺
+∂rj(h) ⊙ νc̺\vk,vo.
+(35)
+We emphasize that νc̺\vk,vo, as defined implicitly above, is a
+probability distribution.
+Having established these results, we turn our attention to
+the base case where vj is a variable child of vℓ (vk = vj,
+
+̺ = 0). Applying (34) and Proposition 9, we obtain
+∂µvj→cp
+∂rj(h)
+=
+∂
+∂rj(h)
+αj ◦
+�
+⃝cξ∈N(vj)\cp µcξ→vj
+�
+���αj ◦
+�
+⃝cξ∈N(vj)\cp µcξ→vj
+����
+1
+=
+∂αj
+∂rj(h) ◦
+�
+⃝cξ∈N(vj)\cp µcξ→vj
+�
+�
+αj, ⃝cξ∈N(vj)\cp µcξ→vj
+�
+− µvj→cp
+�
+∂αj
+∂rj(h), ⃝cξ∈N(vj)\cp µcξ→vj
+�
+�
+αj, ⃝cξ∈N(vj)\cp µcξ→vj
+�
+= 1
+τ 2
+αj(h)eh ◦
+�
+⃝cξ∈N(vj)\cp µcξ→vj
+�
+�
+αj, ⃝cξ∈N(vj)\cp µcξ→vj
+�
+− 1
+τ 2 µvj→cp
+�
+αj(h)eh, ⃝cξ∈N(vj)\cp µcξ→vj
+�
+�
+αj, ⃝cξ∈N(vj)\cp µcξ→vj
+�
+= 1
+τ 2
+�
+νvj→cp −
+��νvj→cp
+��
+1 µvj→cp
+�
+,
+(36)
+where
+νvj→cp =
+αj(h)eh ◦
+�
+⃝cξ∈N(vj)\cp µcξ→vj
+�
+�
+αj, ⃝cξ∈N(vj)\cp µcξ→vj
+�
+.
+By construction, we have 0 ⪯ νvj→cp ⪯ µvj→cp. We turn to
+the second graph operation and apply (35), which yields
+∂µcp→vℓ
+∂rj(h)
+=
+∂µvj→cp
+∂rj(h) ⊙ νcp\vℓ,vj
+= 1
+τ 2
+�
+νvj→cp −
+��νvj→cp
+��
+1 µvj→cp
+�
+⊙ νcp\vℓ,vj
+= 1
+τ 2
+�
+νvj→cp ⊙ νcp\vℓ,vj −
+��νvj→cp
+��
+1 µcp→vℓ
+�
+.
+(37)
+Thus, in this case, we take ν = νvj→cp⊙νcp\vℓ,vj as a suitable
+vector. Based on the fact that νcp\vℓ,vj is a probability vector,
+together with the aforementioned component-wise ordering,
+we gather that
+0 ⪯ ν = νvj→cp ⊙ νcp\vℓ,vj
+⪯ µvj→cp ⊙ νcp\vℓ,vj = µcp→vℓ.
+Moreover, leveraging the properties of the Fq convolution for
+non-negative vectors, we get
+∥ν∥1 =
+��νvj→cp
+��
+1
+��νcp\vℓ,vj
+��
+1 =
+��νvj→cp
+��
+1 .
+Thus, for this choice of ν, we arrive at
+∂µcp→vℓ
+∂rj(h)
+= 1
+τ 2
+�
+ν − ∥ν∥1 µcp→vℓ
+�
+,
+(38)
+as claimed. That is, the base case conforms to the structure of
+Proposition 14.
+We now consider the inductive step in our proof. As our
+hypothesis, we assume that (33) holds for all computation trees
+wherein the distance between the root node and vj is less than
+or equal to γ ∈ N. Consider a rooted computation tree and
+suppose the distance between vℓ and its descendant vj in the
+tree is exactly γ+1. Under Condition 6, there is a unique path
+from vℓ to node vj. Let vk be the variable child of vℓ that is
+also an ascendant of vj, and denote the unique check node that
+connects the two by cp ∈ N(vℓ)∩N(vk). Furthermore, let c̺ ∈
+N(vk) be the unique check node within this neighborhood
+that is an ascendant of vj on the computation tree. Finally, let
+vo ∈ N(c̺) be the unique variable child of vk that is also an
+ascendant of vj (or, perhaps, vj itself).
+The sub-tree starting at vk can be viewed as a rooted tree
+containing vj; the graph distance between these two variable
+nodes within the sub-tree is exactly γ. As such, our inductive
+hypothesis applies. That is, there exists vector ν such that
+0 ⪯ ν ⪯ µc̺→vk where the partial derivative of µc̺→vk with
+respect to rj(h) is equal to
+∂µc̺→vk
+∂rj(h)
+= 1
+τ 2
+�
+ν − ∥ν∥1 µc̺→vk
+�
+.
+(39)
+Applying (34) and our inductive hypothesis, we have that
+∂µvk→cp
+∂rj(h)
+=
+∂
+∂rj(h)
+⃝cξ∈N0(vk)\cp µcξ→vk
+���⃝cξ∈N0(vk)\cp µcξ→vk
+���
+1
+=
+∂µc̺→vk
+∂rj(h)
+◦
+�
+⃝cξ∈N0(vk)\cp,c̺ µcξ→vk
+�
+�
+µc̺→vk, ⃝cξ∈N0(vk)\cp,c̺ µcξ→vk
+�
+− µvk→cp
+� ∂µc̺→vk
+∂rj(h) , ⃝cξ∈N0(vk)\cp,c̺ µcξ→vk
+�
+�
+µc̺→vk, ⃝cξ∈N0(vk)\cp,c̺ µcξ→vk
+�
+= 1
+τ 2
+ν ◦
+�
+⃝cξ∈N0(vk)\cp,c̺ µcξ→vk
+�
+�
+µc̺→vk, ⃝cξ∈N0(vk)\cp,c̺ µcξ→vk
+�
+− 1
+τ 2 µvk→cp
+�
+ν, ⃝cξ∈N0(vk)\cp,c̺ µcξ→vk
+�
+�
+µc̺→vk, ⃝cξ∈N0(vk)\cp,c̺ µcξ→vk
+�
+= 1
+τ 2
+�
+νvk→cp −
+��νvk→cp
+��
+1 µvk→cp
+�
+,
+(40)
+where we have utilized the shorthand notation
+νvk→cp =
+ν ◦
+�
+⃝cξ∈N0(vk)\cp,c̺ µcξ→vk
+�
+�
+µc̺→vk, ⃝cξ∈N0(vk)\cp,c̺ µcξ→vk
+�.
+We emphasize that two of the terms in the derivation above
+cancel out, as before. Furthermore, by construction, we im-
+mediately obtain 0 ⪯ νvk→cp ⪯ µvk→cp. These observations
+closely parallel the description for the base case.
+The derivation of the second graph operation for the induc-
+tive step is in complete analogy with the base case, except for
+labeling. Specifically, we apply (35) and obtain
+∂µcp→vℓ
+∂rj(h)
+=
+∂µvk→cp
+∂rj(h)
+⊙ νcp\vℓ,vk
+= 1
+τ 2
+�
+νvk→cp −
+��νvk→cp
+��
+1 µvk→cp
+�
+⊙ νcp\vℓ,vk
+= 1
+τ 2
+�
+νvk→cp ⊙ νcp\vℓ,vk −
+��νvk→cp
+��
+1 µcp→vℓ
+�
+.
+(41)
+
+For the inductive step, we define ν′ = νvk→cp ⊙ νcp\vℓ,vk as
+the candidate vector. Based on component-wise ordering, we
+can write
+0 ⪯ ν′ = νvk→cp ⊙ νcp\vℓ,vk
+⪯ µvk→cp ⊙ νcp\vℓ,vk = µcp→vℓ.
+As before, we have that
+∥ν′∥1 =
+��νvk→cp
+��
+1
+��νcp\vℓ,vk
+��
+1 =
+��νvk→cp
+��
+1 .
+Hence, candidate vector ν′ is such that 0 ⪯ ν′ ⪯ µcp→vℓ and
+∂µcp→vℓ
+∂rj(h)
+= 1
+τ 2
+�
+ν′ − ∥ν′∥1 µcp→vℓ
+�
+.
+(42)
+This completes the proof for Proposition 14.
+We have nearly attained out goal of showing that the
+magnitudes of the entries in the Jacobian matrix of η(r) with
+respect to r are uniformly bounded. To achieve the desired
+result, it sufficies to connect the partial derivative of the
+incoming message with the partial derivative of state estimate
+ˆsℓ (r, g). This is accomplished below.
+Proposition 15. Under Condition 6, the absolute value of the
+entries in the Jacobian are bounded by
+����
+∂ˆsℓ (r, g)
+∂rj(h)
+���� ≤ 1
+τ 2
+∀g, h ∈ Fq, ℓ, j ∈ [L],
+(43)
+where τ
+> 0 is the standard deviation of the effective
+observation.
+Proof: When vj does not appear in the rooted tree
+of vℓ, the partial derivative is equal to zero and the result
+immediately follows. Furthermore, when vℓ = vj, the result
+follows from Corollary 12. Thus, we can focus on the scenario
+wherein vj is a descendant of vℓ.
+Let cp be the unique check node in N(vℓ) that lies on the
+path between vℓ and vj. By Proposition 14, there exists vector
+ν, with 0 ⪯ ν ⪯ µcp→vℓ, such that the partial derivative of
+µcp→vℓ with respect to rj(h) is given by
+∂µcp→vℓ
+∂rj(h)
+= 1
+τ 2
+�
+ν − ∥ν∥1 µcp→vℓ
+�
+.
+(44)
+Drawing an analogy to (40), we have
+∂ˆsℓ (r)
+∂rj(h) =
+∂
+∂rj(h)
+⃝cξ∈N0(vℓ) µcξ→vℓ
+���⃝cξ∈N0(vℓ) µcξ→vℓ
+���
+1
+= 1
+τ 2
+ν ◦
+�
+⃝cξ∈N0(vℓ)\cp µcξ→vℓ
+�
+�
+µcp→vℓ, ⃝cξ∈N0(vℓ)\cp µcξ→vℓ
+�
+− 1
+τ 2 ˆsℓ (r)
+�
+ν, ⃝cξ∈N0(vℓ)\cp µcξ→vℓ
+�
+�
+µcp→vℓ, ⃝cξ∈N0(vℓ)\cp µcξ→vℓ
+�
+= 1
+τ 2
+�
+νvℓ − ∥νvℓ∥1 ˆsℓ (r)
+�
+,
+(45)
+where we have implicitly defined
+νvℓ =
+ν ◦
+�
+⃝cξ∈N0(vℓ)\cp µcξ→vℓ
+�
+�
+µcp→vℓ, ⃝cξ∈N0(vℓ)\cp µcξ→vℓ
+�.
+We note that 0 ⪯ νvℓ ⪯ ˆsℓ (r). Thus, we have that:
+∂ˆsℓ (r)
+∂rj(h) ≤ 1
+τ 2 νvℓ ≤ 1
+τ 2 ˆsℓ (r) .
+(46)
+Since we are interested in bounding the absolute value of the
+partial derivatives, we also consider a lower bound.
+∂ˆsℓ (r)
+∂rj(h) ≥ − 1
+τ 2 ∥νvℓ∥1 ˆsℓ (r) ≥ − 1
+τ 2 ˆsℓ (r) .
+(47)
+Combining these two observations with the properties of
+probability vectors, we obtain the desired expression.
+Theorem 16. Under Condition 6, the BP-N denoiser presented
+in definition 5 is Lipschitz continuous.
+Proof: By Proposition 15, the magnitudes of the entries
+of the Jacobian matrix of η(r) with respect to r are uniformly
+bounded. Thus, the denoiser is Lipschitz continuous.
+
diff --git a/ytAzT4oBgHgl3EQf7_7g/content/tmp_files/load_file.txt b/ytAzT4oBgHgl3EQf7_7g/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..6a3bb03cd28a95d9fc9b1e437f4ba36dd00f7368
--- /dev/null
+++ b/ytAzT4oBgHgl3EQf7_7g/content/tmp_files/load_file.txt
@@ -0,0 +1,806 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf,len=805
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='01899v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='IT]  5 Jan 2023  On Sparse Regression LDPC Codes  Jamison R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Ebert, Jean-Francois Chamberland, Krishna R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Narayanan  Department of Electrical and Computer Engineering, Texas A&M University  Abstract—Belief propagation applied to iterative decoding and  sparse recovery through approximate message passing (AMP) are  two research areas that have seen monumental progress in recent  decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Inspired by these advances, this article introduces sparse  regression LDPC codes and their decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Sparse regression  codes (SPARCs) are a class of error correcting codes that  build on ideas from compressed sensing and can be decoded  using AMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' In certain settings, SPARCs are known to achieve  capacity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' yet, their performance suffers at finite block lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Likewise, LDPC codes can be decoded efficiently using belief  propagation and can also be capacity achieving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' This article  introduces a novel concatenated coding structure that combines  an LDPC outer code with a SPARC-inspired inner code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Efficient  decoding for such a code can be achieved using AMP with a  denoiser that performs belief propagation on the factor graph  of the outer LDPC code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' The proposed framework exhibits  performance improvements over SPARCs and standard LDPC  codes for finite block lengths and results in a steep waterfall  in error performance, a phenomenon not observed in uncoded  SPARCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Findings are supported by numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Index Terms—LDPC codes, sparse regression codes, approxi-  mate message passing, belief propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' INTRODUCTION AND BACKGROUND  Low-density parity check (LDPC) codes have been the  subject of many scientific inquiries in the past, with landmark  contributions ranging from theoretical advances to practical  implementations [1]–[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' LDPC ensembles are amenable to  analysis and, under certain conditions, encoded messages can  be recovered efficiently using iterative belief propagation (BP)  decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Interestingly, the complexity of BP decoding grows  linearly with block length and, as such, this paradigm offers a  pragmatic solution for long block lengths [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Some spatially  coupled LDPC constructions feature capacity approaching  iterative decoding thresholds while also avoiding the pitfall  of error floors [10]–[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Unfortunately, systems operating at  shorter block lengths may not be conducive to the application  of spatial coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' In such situations, non-binary LDPC  codes have been leveraged as a means to provide adequate  performance [6], [7], [16]–[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  In a seemingly unrelated research direction, Joseph and  Barron introduced the concept of a sparse regression code  (SPARC), which establishes a connection between code design  and sparse recovery in high dimensions [19]–[21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' The code-  words contained in a SPARC codebook consist of sparse su-  perpositions of columns from a design matrix, also known as a  sensing matrix in the compressed sensing (CS) literature [22],  [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' In some sense, the SPARC decoding process is equivalent  This material is based upon work supported, in part, by the National  Science Foundation (NSF) under Grant CCF-2131106, and by Qualcomm  Technologies, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=', through their University Relations Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  to recovering the non-zero entries in a suitably designed index  vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Low complexity algorithms have been proposed for  the efficient decoding of SPARC codewords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Notably, Rush et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' discuss an approximate message passing (AMP) decoder  for SPARCs in [24], [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' It is also worth mentioning that  AMP has independently gained much research momentum in  recent years owing to its low implementation cost, mathemat-  ical tractability, and excellent performance [26]–[30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Sparse  regression codes paired with AMP decoding are known to  achieve asymptotic single-user AWGN channel capacity under  an appropriately chosen power allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Concurrently, there  have been efforts to improve the finite block-length perfor-  mance of SPARCs [31], [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  A popular strategy to improve the performance of codes  in practical settings is to adopt a concatenated structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' For  example, SPARCs and LDPC codes have been paired in  the past to create concatenated schemes that improve per-  formance [31]–[34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Concatenated structures with a SPARC-  like inner code have also been proposed in unsourced random  access [35]–[37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' In [36], it is shown that the structure of  a judiciously designed outer code can be integrated into the  denoising function of the inner AMP recovery algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Surprisingly, despite being mentioned by Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' in [34] as  a possible future research direction, such an approach has not  been considered for the single-user scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' This article seeks  to address this defficiency by introducing sparse regression  LDPC codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Sparse regression LDPC codes feature a concatenated inner  SPARC-like code and an outer non-binary LDPC code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' This  concatenated structure is amenable to a decoder that exchanges  information between inner and outer codes via a dynamic  AMP denoiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' In this setting, the denoiser performs BP on  the factor graph of the outer LDPC code to improve the  AMP state estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' This in turn improves the effective  observations which in turn translate into better local belief  vectors for the BP LDPC decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Such a denoiser improves  AMP’s convergence and lowers the residual mean square  error (MSE) in the estimated SPARC codeword.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Once the  MSE of the iterative AMP-BP decoding levels off, the outer  LDPC code can be leveraged once more to further improve  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' The goal is to use AMP-BP decoding to increase  the SNR of the effective observation to a level that exceeds  the BP threshold of the graph and then to apply message  passing to the outer LDPC code to recover the message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Numerical results demonstrate the performance advantages of  the proposed model over related architectures at reasonably  short block lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' SYSTEM MODEL  We consider a point-to-point memoryless additive white  Gaussian noise (AWGN) channel with one transmitter and one  receiver, each equipped with a single antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' The received  signal y ∈ Rn is given by  y = x + z,  (1)  where x ∈ Rn denotes the transmitted signal and z ∼  N  �  0, σ2I  �  represents AWGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' As mentioned above, we wish  to study a coding architecture composed of a sparse regression  inner code [19]–[21], and a non-binary LDPC outer code [6]–  [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' We elaborate on the encoding process and the decoding  scheme below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Sparse Regression LDPC Encoding  The proposed encoding process features a sequence of  three distinct steps: q-ary LDPC encoding, indexing LDPC  symbols, and inner CS encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' First, the information bits  are encoded into a q-ary LDPC codeword via well-established  operations [6]–[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' The second step transforms the q-ary LDPC  codeword into a suitable sparse vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' The last step is the  matrix multiplication emblematic of a sparse regression code;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  the output is sometimes referred to as a large random matrix  system [19]–[21], [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' We summarize the notions pertaining  to this encoding process below while concurrently introducing  necessary notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  1) q-ary LDPC Encoding: The LDPC encoder takes a  binary sequence w ∈ FB  2 as its input and maps it to a q-  ary codeword v ∈ FL  q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' We represent the resultant codeword in  concatenated form as  v = (v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' , vL)  (2)  where the ℓth element vℓ lies in finite field Fq and L is the  length of the codeword.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Note that there exists a bijection Φ : Fq → [q] between  the elements of Fq and the integers [q] = (0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' , q − 1),  where 0 ∈ Z maps to 0 ∈ Fq and 1 ∈ Z maps to 1 ∈ Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Throughout, we adopt such an arbitrary, but fixed bijection  and exploit this relation by employing the same variable for  a field element g ∈ Fq and for its corresponding integer  Φ (g) ∈ [q].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' This slight abuse of notation greatly simplifies  exposition, and its use should not lead to confusion because  one can unambiguously infer from context whether an instance  refers to the field element or to its integer representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  2) LDPC Symbol Indexing:  Coded symbol sparsifica-  tion/indexing consists of mapping vℓ ∈ Fq to standard basis  vector evℓ ∈ Rq and subsequently stacking the L basis vectors  together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' We seize this opportunity to reinforce the notion that  entry ℓ of v is an element of Fq, but vℓ in evℓ refers to an  integer in [q] under our overloaded notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' With that, the  output of the indexing process becomes  s =  \uf8ee  \uf8ef\uf8f0  ev1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  evL  \uf8f9  \uf8fa\uf8fb ,  (3)  where s is an L-sparse vector of length qL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Vector s has  a structure akin to that of a sparse regression code prior  to multiplication by a large random matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' This structured  sparsity can be exploited during decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  3) Inner CS Encoding: The last phase of the encoding  process consists in pre-multiplying vector s by matrix A to  get x = As, where A ∈ Rn×qL and Ai,j ∼ N  �  0, 1  n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Note  that n ≪ qL to conform to the CS framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Equation (1)  may thus be rewritten as:  y = As + z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  (4)  Having specified the code structure, we are now ready to  discuss the process of recovering s from y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Sparse Regression LDPC Decoding  Paralleling the development of AMP for sparse regression  codes [24] and drawing inspiration from concatenated AMP  systems [33], [34], we wish to create a composite iterative  process to recover state vector s from y that exploits the  sparsity in s and the parity structure of the LDPC code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' This  can be accomplished by incorporating message passing on the  factor graph of the LDPC code into the AMP denoiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' A  similar approach can be found in [36];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' however, the denoiser  we wish to utilize below differs in the fact that only one  codeword is present within y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Our AMP composite algorithm is as follows,  r(t) = ATz(t) + s(t)  (5)  z(t) = y − As(t) + z(t−1)  n  div ηt−1  �  r(t−1)�  (6)  s(t+1) = ηt  �  r(t)�  (7)  where the superscript t denotes the iteration count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' The al-  gorithm is initialized with conditions r(0) = s(0) = 0 and  z(0) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Equation (5) specifies the effective observation, which is  characteristic of AMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Equation (6) computes the residual  error enhanced with an Onsager correction term, whereas (7)  provides a state estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' The denoising functions (ηt(·))t≥0  seek to exploit the structure of s while computing the state  updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Generally speaking, a choice denoiser is the conditional  expectation of s given r(t), Unfortunately, this approach is  computationally intractable in the present context because  it entails summing over all the possible codewords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' As an  alternative, we know that BP can be efficiently applied to  q-ary LDPC codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Furthermore, at any point during BP, a  belief on individual LDPC symbols can be formed based on  incoming messages from neighboring factor nodes and local  observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Thus, we can potentially apply a few iterations  of BP as a means to get an estimate for the state vector by  leveraging the connection between sections of s and LDPC  symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' In this sense, iterative message passing can act as a  foundation for pragmatic denoising functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' We elaborate on  this connection below and, concurrently, we review pertinent  notions of BP applied to q-ary LDPC codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' CREATING A BP DEONOISER FOR AMP  In this section, we introduce the denoising function we wish  to employ within AMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' To begin, we emphasize that r admits  a sectionized representation akin to that of the state vector s  in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' That is, we can view both the state estimate and the  effective observation as concatenations of L vectors, each of  length q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Mathematically, we have  r =  \uf8ee  \uf8ef\uf8f0  r1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  rL  \uf8f9  \uf8fa\uf8fb  ˆs =  \uf8ee  \uf8ef\uf8f0  ˆs1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  ˆsL  \uf8f9  \uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  This point is crucially important because the denoiser is  constructed in a block-wise fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' As a side note, we neglect  the superscript (t) for most of the discussion below to lighten  notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Furthermore, we emphasize that the presence of the  hat in ˆs distinguishes the estimate from the true state vector s  in the absence of an iteration count (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Each section rℓ in r acts as a vector observation about  the value of sℓ ∈ {eg : g ∈ [q]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' An astounding and enabling  property of AMP is that, under certain technical conditions,  the effective observation r is asymptotically distributed as  s+ τζ, where ζ is a random vector with independent N(0, 1)  components and τ is a deterministic quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' This fact hinges  on the presence of the Onsager term in (6) and on some  smoothness conditions for the denoising functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' We delay  the treatment of these technical conditions for the time being;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  rather, we posit the requirements and formally introduce them  as a condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Condition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' The effective observation r(t) is asymptotically  distributed as s + τtζt, where ζt is a random vector with  independent N(0, 1) components and τt is specified by a set  of deterministic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' This asymptotic characterization  takes place in the dimensions of the system, as opposed to  time or iteration count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  We describe below our rationale behind the denoising  function assuming Condition 1 holds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' we provide a rigorous  foundation for this condition in Appendix B, but this can  only be done once the structure of the denoiser is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Consider the effective observation restricted to section ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Under Condition 1, the distribution of random observation  vector Rℓ given section Sℓ = eg is given by  fRℓ|Sℓ (rℓ|eg) =  1  (2π)  q  2 τ q exp  �  −∥rℓ − eg∥2  2τ 2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Under a uniform input distribution, the conditional probability  of Sℓ = eg becomes  αℓ(g) = Pr (Sℓ = eg|Rℓ = rℓ) = Pr (Vℓ = g|Rℓ = rℓ)  =  fRℓ|Vℓ (rℓ|g)  �  h∈Fq fRℓ|Vℓ (rℓ|h) =  e− ∥rℓ−eg∥2  2τ2  �  h∈Fq e− ∥rℓ−eh∥2  2τ2  =  e  rℓ(g)  τ2  �  h∈Fq e  rℓ(h)  τ2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  (8)  A possible estimate for Sℓ can be computed by taking its  conditional expectation, given observation Rℓ = rℓ, with  E [Sℓ|Rℓ = rℓ] =  �  g∈Fq  egPr (Sℓ = eg|Rℓ = rℓ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  (9)  A variant of this approach can be found in [24] for a system  without an outer code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Ideally, we would like to take advantage  of the outer code with the more precise MMSE estimate of  the form E [Sℓ|R = r].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Unfortunately, as mentioned above,  computing this conditional expectation is far too complex to  be implemented in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' A viable alternative that trades  off performance and complexity is to execute BP on the  factor graph of the outer LDPC code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Implicitly, this approach  computes an estimate for every Sℓ based on the computation  tree of the code, up to a certain depth [8], [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Frameworks to perform BP on factor graphs are well-  established [39];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' thus, we assume some familiarity with such  iterative procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' We proceed by first considering the  nuances of non-binary LDPC factor graphs, then presenting  a BP algorithm, and finally proposing a dynamic denoiser for  SR-LDPC codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Non-Binary LDPC Graphs  The factor graph for an Fq LDPC code features L variable  (left) nodes and L(1−R) check (right) nodes enforcing parity  constraints, where R is the rate of the LDPC code [6]–[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' An  important distinction between binary and non-binary LDPC  codes is that a factor graph for the latter code typically includes  edge labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' These labels take values in Fq \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' A vector  v ∈ FL  q is a valid codeword if it satisfies the parity equations  �  vℓ∈N(cp)  ωℓ,p ⊗ vℓ = 0  ∀p ∈ [L(1 − R)],  (10)  where N(cp) is the collection of variable nodes adjacent  to parity check cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' The summation and the multiplication  operator ⊗ in (10) take place over finite field Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Parameter  ωℓ,p represents the label or weight assigned with the edge  connecting variable node vℓ and check node cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Adopting  common factor graph concepts [39], [40], we denote the graph  neighbors of variable node vℓ by N(vℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' The factor associated  with cp and derived from parity equation (10) can be expressed  as an indicator function  Gp(vp) = 1  \uf8eb  \uf8ed  �  vℓ∈N(cp)  ωℓ,p ⊗ vℓ = 0  \uf8f6  \uf8f8  (11)  where vp = (vℓ ∈ N(cp)) is a shorthand notation for the  restriction of v to entries associated with graph neighborhood  N(cp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' With these definitions, the factor function associated  with our LDPC code assumes the product decomposition given  by  G(v) =  �  p∈[L(1−R)]  Gp(vp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  (12)  In words, G(v) is an indicator function that assesses whether  its argument is a valid codeword.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Belief Propagation  We view messages for a non-binary LDPC code as multi-  dimensional belief vectors over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Messages from variable  nodes to check nodes are denoted by µv→c, and messages  in the reverse direction are represented by µc→v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Formally,  a message going from check node cp to variable node vℓ ∈  N(cp) is computed component-wise through  µcp→vℓ(g) =  �  vp:vℓ=g  Gp (vp)  �  vj∈N(cp)\\vℓ  µvj→cp(gj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  (13)  While (13) is shown in compact form, the actual summation  operation is cumbersome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Finding the set of summands entails  identifying sequences of the form (gj ∈ Fq : vj ∈ N(cp) \\ vℓ)  that fulfill local condition (10) or, equivalently,  �  vj∈N(cp)\\vℓ  ωj,p ⊗ gj = −ωℓ,p ⊗ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  (14)  A belief vector passed from variable node vℓ to check node  cp, where p ∈ N(vℓ), is calculated component-wise via  µvℓ→cp(g) ∝ αℓ(g)  �  cξ∈N(vℓ)\\cp  µcξ→vℓ(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  (15)  The ‘∝’ symbol indicates that the positive measure can be  normalized before being sent out as a message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Vector αℓ  in (15) can be viewed as a collection of beliefs based on  local observations, as in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' That is, entry αℓ(g) captures  the probability that symbol g is the true field element within  section ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Other BP messages are initialized with µv→c = 1  and µc→v = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' The traditional parallel sum-product algorithm  iterates between (13) and (15), alternating between updated  rightbound messages and leftbound messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  One of the key advantages of indexing vectors using field  elements in Fq, as pointed out in [7], is the ensuing ability  to define pertinent operators on these vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Paralleling  established literature, we define two actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Definition 2 (Vector +g Operator [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' For field element g ∈  Fq, the vector +g operator acting on b and denoted by b+g  is defined as  b+g =  �  bg, bg⊕1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' , bg⊕(q−1)  �  = (bh⊕g : h ∈ Fq)  where subscript addition ⊕ is performed in Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Definition 3 (Vector ×g Operator [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' For field element g ∈  Fg \\ {0}, we define the vector ×g operator acting on b and  denoted by b×g by  b×g =  �  b0, bg, b2⊗g, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' , b(q−1)⊗g  �  = (bh⊗g : h ∈ Fq)  where subscript product ⊗ takes place in Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Under these operations, we can rewrite (13) in a concise  manner:  µcp→vℓ =  \uf8eb  \uf8ed  �  vj∈N(cp)\\vℓ  �  µvj→cp  �×ω−1  j,p  \uf8f6  \uf8f8  ×(−ωℓ,p)  (16)  where ωj,p is the label on the edge between variable node vj  and factor node cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' The operator ⊙ denotes the Fq convolution  between two vectors,  [µ ⊙ ν]g =  �  h∈Fq  µh · νg−h  g ∈ Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  The exposition can be simplified further if we absorb the edge  labels within the messages themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Specifically, we adopt  the definitions  µvj→cp =  �  µvj→cp  �×ω−1  j,p  (17)  µcp→vℓ =  �  µcp→vℓ  �×(−ω−1  ℓ,p)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  (18)  Then (16) morphs into the simpler expression  µcp→vℓ =  �  vj∈N(cp)\\vℓ  µvj→cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  (19)  This equation highlights the role of the Fq convolution within  BP for non-binary LDPC codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' The message from variable  node vℓ to check node cp found in (15) also admits a  more compact form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' For p ∈ N(vℓ), the traditional outgoing  message from a variable node can be written as  µvℓ→cp =  αℓ ◦  �  ⃝cξ∈N(vℓ)\\cp µcξ→vℓ  �  ���αℓ ◦  �  ⃝cξ∈N(vℓ)\\cp µcξ→vℓ  ����  1  (20)  where ◦ represents the Hadamard product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  A natural estimate for the distribution associated with vari-  able node vℓ, including intrinsic information, is  ˆSℓ =  αℓ ◦  �  ⃝cp∈N(vℓ) µcp→vℓ  �  ���αℓ ◦  �  ⃝cp∈N(vℓ) µcp→vℓ  ����  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  (21)  As we will shortly see, (21) is the output of our proposed  denoiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' In our construction, q = 2m, m ≥ 1 because  indexing is derived from sequences of bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' This invites the  application of fast techniques to implement message passing  over the corresponding factor graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Specifically, the fast  Walsh-Hadamard transform (FWHT) can be utilized to rapidly  and efficiently compute (19), with  µcp→vℓ ∝ fwht−1  \uf8eb  \uf8ed  �  vj∈N(cp)\\vℓ  fwht  �  µvj→cp  �  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Alternatively, one could adopt a different finite field convolu-  tion or a ring structure amenable to the circular convolution  to create local factor functions conducive to the fast Fourier  transform [6], [41], [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  With these tools in mind, we are ready to formally define  our proposed denoiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Sparse Regression LDPC Denoiser  Conceptually, one can initiate the state of the LDPC factor  graph using the effective observation r, run BP, and then  gain an estimate for the state based on (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' As mentioned  before, in the absence of BP iterations, local estimates reduce  to the conditional expectation E [Sℓ|Rℓ = rℓ] found in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Yet, as more iterations of the BP algorithm are performed, the  estimate for Sℓ is refined based on the computation tree of the  outer code, up to a certain depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Definition 5 (BP-N Denoiser).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Let N denote the number of  BP iterations to perform, where N is less than the girth of the  LDPC factor graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' The BP-N denoiser  1) initializes the LDPC factor graph with estimates αℓ  computed from rℓ for ℓ ∈ [L] according to (8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  2) computes and passes variable to check messages (see  (20)) and check to variable messages (see (17), (19), (18)  and Remark 4) along the edges of the factor graph in an  alternating fashion N times;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  3) computes updated state estimates according to (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  The output of this denoiser can then be passed to the AMP  composite algorithm for the computation of the next residual,  enhanced with the Onsager term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  The derivation of the Onsager correction term corresponding  to this denoiser is included in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' At this point, we  turn to a performance assessment of sparse regression LDPC  codes via numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' SIMULATION RESULTS  In this section, we seek to characterize the performance of  SR-LDPC codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='1 As benchmarks, we compare our results to  the concatenated SPARC/LDPC scheme presented in [31] as  well as 5G-NR binary LDPC encoded BPSK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' We note that  the latter comparison provides limited insight because the SR-  LDPC code is allowed real channel inputs whereas the 5G-NR  code is constrained to binary inputs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' nevertheless, we make the  comparison to see whether SR-LDPC codes are competitive  with state of the art, commonly used codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  The scenario of interest is one in which 5888 information  bits are to be transmitted over 7350 real channel uses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' We  define the SR-LDPC rate to be the number of information  bits over the number of channel uses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' thus, we have that  RSRLDPC ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' The non-binary LDPC code employed is a  (751, 736) code (RLDPC ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='98) over GF(256) whose edges  are generated via progressive edge growth (PEG) and whose  weights were chosen uniformly at random from the elements  of F256 \\ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' The fraction of degree-2 variable nodes is chosen  to be higher than usual so that the AMP-BP iterations can  bootstrap in high-noise environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' The AMP-BP decoder  is run for up to 100 iterations and then the non-binary LDPC  BP decoder is run for another 100 iterations, or until a valid  codeword is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Figure 1 highlights our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  1The source code used to generate these results is available online at  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='com/EngProjects/mMTC/tree/code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='5  10−4  10−3  10−2  10−1  100  Eb/N0 (dB)  BER  SPARC + LDPC [31]  5G-NR LDPC + BPSK  SR-LDPC  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' 1: BER performance comparison of a sparse regression  LDPC code, the concatenated SPARC/LDPC construction  from [31], and 5G-NR binary LDPC encoded BPSK as a  function of Eb/N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' The SR-LDPC code requires a lower  Eb/N0 to achieve a target BER than LDPC encoded BPSK  and the scheme in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' 1, it is clear that sparse regression LDPC codes  provide a roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='75 dB improvement at a BER of 10−3 over  other SPARC/LDPC concatenated coding structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Further-  more, the SR-LDPC code outperforms 5G-NR LDPC encoded  BPSK in some regimes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' we note however, that 5G-NR LDPC  encoded BPSK has a lower error floor than the SR-LDPC  code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' CONCLUSION  In conclusion, this article presents sparse regression LDPC  codes and their decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' A sparse regression code is a  concatenated structure with an inner SPARC-like code and  an outer non-binary LDPC code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' The inner code is decoded  using AMP with a dynamic denoiser which runs BP on the  factor graph of the outer LDPC code to improve the state  estimate at every iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' After running many iterations of  the AMP-BP decoder, the final effective observation is fed  into a standard LDPC BP decoder, which seeks to correct  any residual errors that may be present in the received signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Numerical results show that sparse regression codes exhibit  performance improvements over standard LDPC codes and  other concatenated SPARC/LDPC constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  As mentioned in the introduction, both SPARCs and LDPC  codes are known to achieve capacity in certain circumstances,  yet both codes generally suffer at short block lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' The  success of sparse regression LDPC codes is evidence that  SPARCs and LDPC codes can be seamlessly combined to  significantly improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Directions for future work  include optimizing the outer LDPC code and applying coded  demixing [37] to SR-LDPC codes to accommodate multiple  users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
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+page_content=' Colas, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Gelle, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Declercq,  “FFT-based BP  decoding of general LDPC codes over Abelian groups,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  on Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' 55, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' 644–649, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  APPENDIX A  DERIVATION OF ONSAGER TERM  Intuitively, the role of the Onsager term is to (asymptoti-  cally) cancel the first-order correlations between ATz(t) and  s(t) and thereby maintain a structure conducive to prompt  convergence and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' This factor, emblematic of AMP  algorithms, appears in (6) and is given by  z(t−1)  n  div ηt−1  �  ATz(t−1) + s(t−1)�  = z(t−1)  n  div ηt−1  �  r(t−1)�  where the div operator can be expanded into  div η (r) =  �  ℓ∈[L]  div ˆsℓ(r, τ) =  �  ℓ∈[L]  �  g∈Fq  ∂ˆsℓ (g, r, τ)  ∂rℓ(g)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' (22)  Thus, as an intermediate step, we must calculate the partial  derivative of ˆsℓ (g, r, τ) with respect to rℓ(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' This computa-  tion is rendered much simpler under the following condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Condition 6 (Sub-Girth AMP-BP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' The BP-N denoiser is  said to possess the sub-girth AMP-BP condition when fewer  message passing iterations are performed on the factor graph  of the LDPC code than the shortest cycle of this same graph,  per AMP denoising step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  This condition ensures that the message passing operations  employed during denoising yield valid computation trees with-  out cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Fortunately, this is the regime we are primarily  interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Under Condition 6, the partial derivative of  ˆsℓ (g, r, τ) with respect to rℓ(g) is equal to  ∂ˆsℓ (g, r, τ)  ∂rℓ(g)  = 1  τ 2 ˆsℓ (g, r, τ) (1 − ˆsℓ (g, r, τ))  (23)  where g ∈ Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Proof: Recall that the output of the BP denoiser defined  in (21) can be expressed as  ˆsℓ (g, r, τ) =  αℓ(g) �  p∈N(vℓ) µcp→vℓ(g)  �  h∈Fq αℓ(h) �  p∈N(vℓ) µcp→vℓ(h)  =  αℓ(g)µvℓ→c0(g)  �  h∈Fq αℓ(h)µvℓ→c0(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  (24)  Under Condition 6, belief vector µvℓ→c0 is based solely on  extrinsic information and, hence, it is determined based on  {rj : j ∈ [L] \\ {ℓ}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Consequence, we gather that  ∂µvℓ→c0(g)  ∂rℓ(g)  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Under such circumstances, we can calculate the desired deriva-  tive in a straightforward manner, with  ∂ˆsℓ (g, r, τ)  ∂rℓ(g)  =  ∂  ∂rℓ(g)  e  rℓ(g)  τ2 µvℓ→c0(g)  �  h∈Fq e  rℓ(h)  τ2 µvℓ→c0(h)  = 1  τ 2  e  rℓ(g)  τ2 µvℓ→c0(g)  �  h∈Fq e  rℓ(g)  τ2 µvℓ→c0(g)  − 1  τ 2  �  e  rℓ(g)  τ2 µvℓ→c0(g)  �2  ��  h∈Fq e  rℓ(g)  τ2 µvℓ→c0(g)  �2  = 1  τ 2 ˆsℓ (g, r, τ) (1 − ˆsℓ (g, r, τ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  This last line corresponds to the statement of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  It is worth emphasizing that the derivative in (23) remains  unchanged irrespective of the number of BP rounds computed  on the factor graph, so long as Condition 6 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' The  divergence of (22) assumes the same simple form under such  circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' The divergence of η (r) with respect to r is  equal to  div η (r) = 1  τ 2  �  ∥η (r)∥1 − ∥η (r)∥2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  (25)  Proof: We expand the div operator as  div η (r) =  �  ℓ∈[L]  div ˆsℓ (r, τ) =  �  ℓ∈[L]  �  g∈Fq  ∂ˆsℓ (g, r, τ)  ∂rℓ(g)  =  �  ℓ∈[L]  �  g∈Fq  1  τ 2 ˆsℓ (g, r, τ) (1 − ˆsℓ (g, r, τ))  = 1  τ 2  �  ∥η (r)∥1 − ∥η (r)∥2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  (26)  The last equality follows from the fact that, since ˆsℓ (g, r, τ)  lies between zero and one, the corresponding partial derivative  with respect to rℓ(g) found in Lemma 7 is always non-  negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  APPENDIX B  DENOISER IS LIPSCHITZ CONTINUOUS  The mathematical underpinnings for the rigorous appli-  cation of AMP in this article are presented by Berthier,  Montanari, and Nguyen in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' One of the conditions for the  state evolution to hold for non-separable functions is that the  denoiser must be pseudo-Lipschitz of a certain order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' For the  problem at hand, we are able to show the stronger Lipschitz  condition, which is sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Thus, the main objective of  this section is to demonstrate that the denoiser introduced  in Definition 5 is Lipschitz continuous under Condition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  To achieve this goal, our strategy is to demonstrate that the  magnitudes of the entries in the Jacobian matrix of η(r) with  respect to r are uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Recall that the denoiser assumes a sectional form, as  described in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' The vector estimate for section ℓ  becomes  ˆsℓ(r) =  �  g∈Fq  Pr (Vℓ = g|Rtree = rtree) eg,  where Rtree denotes the measurements associated with the  computational tree of the LDPC code rooted at section ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' The  (realized) scaling factors found in (21) are given by  ˆsℓ(r) =  ⃝p∈N0(vℓ) µcp→vℓ  ���⃝p∈N0(vℓ) µcp→vℓ  ���  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  (27)  where N0(vℓ) denotes the neighborhood of vℓ including the  local observation and µc0→vℓ  = αℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' We are ultimately  interested in Jacobian entries of the form  ∂ˆsℓ (r, g)  ∂rj(h)  =  ∂  ∂rj(h)  �  p∈N0(vℓ) µcp→vℓ(g)  ���⃝p∈N0(vℓ) µcp→vℓ  ���  1  =  ∂  ∂rj(h)  αℓ(g) �  p∈N(vℓ) µcp→vℓ(g)  �  k∈Fq αℓ(k) �  p∈N(vℓ) µcp→vℓ(k)  (28)  for g, h ∈ Fq and ℓ, j ∈ [L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' We adopt a divide-and-conquer  approach to identify and bound these derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Specifically,  we focus on the rooted tree obtained by taking the factor graph  of the outer LDPC code, setting vℓ as the root, and retaining  only the nodes involved in the computation of ˆsℓ(r, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Under  Condition 6, this sub-graph must form a proper tree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' 2  offers a notional diagram to illustrate the outcome of this  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  We seek to bound the magnitude of the derivatives in (28)  based on the distance between vℓ and vj in this rooted tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  We begin with local observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' (Local Observations) The partial derivatives  of αj with respect to rk(h) are given by  ∂αj  ∂rk(h) =  �  1  τ 2 αj(h) (eh − αj)  j = k  0  j ̸= k  (29)  ∀h ∈ Fq and where τ > 0 is the standard deviation of the  effective observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Proof: As defined in (8), the vector αj is given by  αj(g) =  e  rj (g)  τ2  �  k∈Fq e  rj (k)  τ2  ∀g ∈ Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  When j ̸= k, it immediately follows that ∂αj/∂rk(h) = 0  as αj does not depend on rk(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' We thus consider the case  where k = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' When g = h, we have that  ∂αj(h)  ∂rj(h) = 1  τ 2  e  rj(h)  τ2  �  k∈Fq e  rj(k)  τ2  − 1  τ 2  e  rj(h)  τ2 e  rj(h)  τ2  ��  k∈Fq e  rj(k)  τ2  �2  = 1  τ 2 αj(h) (1 − αj(h)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  vℓ  cp  µcp→vℓ  vk  µc0→vk  cρ  vj  µvj→cρ  µc0→vj  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' 2: Computation tree for variable node vℓ obtained by  taking the factor graph of the outer LDPC code, setting vℓ  as the root, and retaining only the nodes involved in the  computation of ˆsℓ (r, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' We use this data structure to compute  the derivatives in (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  When g ̸= h, we get  ∂αj(g)  ∂rj(h) = − 1  τ 2  e  rj (g)  τ2 e  rj(h)  τ2  ��  κ∈Fq e  rj (κ)  τ2  �2 = − 1  τ 2 αj(g)αj(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Collecting these findings and condensing them into a more  compact form, we arrive at (29), which is the desired expres-  sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' The absolute value of the partial derivatives of  αj with respect to rk(h) are bounded by  ����  ∂αj  ∂rk(h)  ���� ≤  1  4τ 2  (30)  where τ  > 0 is the standard deviation of the effective  observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  The proof of this corollary is trivial when αj is a valid  probability vector, as is the case in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' We also note  that, based on the state evolution of AMP, τ 2 ≥ σ2 at every  iteration irrespective of the iteration number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' We can therefore  establish a uniform bound across iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' We are now ready  to show that the absolute value of (28) is bounded whenever  ℓ = j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=', at the root level of the computation tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Proposition 11 (Root Derivatives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' The partial derivatives of  ˆsℓ (r, g) with respect to rℓ(h) are given by  ∂ˆsℓ (r)  ∂rℓ(h) = 1  τ 2 ˆsℓ (r, h) (eh − ˆsℓ (r))  ∀h ∈ Fq  (31)  where τ  > 0 is the standard deviation of the effective  observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Proof: Leveraging Proposition 9 and denoting the stan-  dard inner product by ⟨·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' ·⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' we have  ∂ˆsℓ (r)  ∂rℓ(h) =  ∂  ∂rℓ(h)  αℓ ◦ ⃝p∈N(vℓ) µcp→vℓ  ���αℓ ◦ ⃝p∈N(vℓ) µcp→vℓ  ���  1  = αℓ(h)  τ 2  (eh − αℓ) ◦  �  ⃝cp∈N(vℓ) µcp→vℓ  �  ���αℓ ◦  �  ⃝cp∈N(vℓ) µcp→vℓ  ����  1  − αℓ(h)  τ 2  ˆsℓ (r)  �  eh − αℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' ⃝cp∈N(vℓ) µcp→vℓ  �  ���αℓ ◦  �  ⃝cp∈N(vℓ) µcp→vℓ  ����  1  = αℓ(h)  τ 2  eh ◦  �  ⃝cp∈N(vℓ) µcp→vℓ  �  ���αℓ ◦  �  ⃝cp∈N(vℓ) µcp→vℓ  ����  1  − αℓ(h)  τ 2  ˆsℓ (r)  �  eh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' ⃝cp∈N(vℓ) µcp→vℓ  �  ���αℓ ◦  �  ⃝cp∈N(vℓ) µcp→vℓ  ����  1  = 1  τ 2 ˆsℓ (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' h) (eh − ˆsℓ (r)) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  which is the desired expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Corollary 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' The absolute value of the partial derivatives of  ˆsℓ (r) with respect to rℓ(h) are bounded by  ����  ∂ˆsℓ (r)  ∂rℓ(h)  ���� ≤  1  4τ 2  (32)  The proof of this corollary follows that of Corollary 10  because like αj, ˆsℓ (r) forms a valid probability vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Proposition 11 offers a blueprint for the general result we  wish to establish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Yet, the situation gets more complicated  when ℓ ̸= j because we have to involve the message passing  rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' In doing so, we obtain a key intermediate result using  mathematical induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' We start with the variable node closest  to the root node, and then progress outward step by step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  To circumvent a notational nightmare, we restrict the proof  to cases where all edge weights are equal to 1 ∈ Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Conceptually, the edge can be interpreted as permutations on  the belief vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' From the point of view of bounding partial  derivatives, this is a benign operation, yet the accounting  that comes with permutations is dreadful, hence our focus on  the simpler case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Moving forward, we assume the following  condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Condition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' All edge weights within the factor graph of the  LDPC outer code are equal to 1 ∈ Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  The extension of the following propositions to the case  with arbitrary edge weights (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=', beyond Condition 13) is  conceptually straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Proposition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Suppose Condition 6 holds and let vj be a  descendant of root node vℓ in the computation tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Moreover,  let cp ∈ N(vℓ) be the unique check neighbor of vℓ on the path  from vℓ to vj within the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Then, there exists vector ν, with  0 ⪯ ν ⪯ µcp→vℓ, such that the partial derivative of µcp→vℓ  with respect to rj(h) is given by  ∂µcp→vℓ  ∂rj(h)  = 1  τ 2  �  ν − ∥ν∥1 µcp→vℓ  �  ,  (33)  where τ  > 0 is the standard deviation of the effective  observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Here, ⪯ denotes elementwise comparison of the  entries in the vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Proof: Under condition 6, we know that vj appears at  most once within the computation tree rooted at vℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Thus,  we establish (33) via mathematical induction on the distance  between vℓ and its descendant vj on the computation tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' The  distance that we are interested in only considers the number of  variable nodes between vℓ and vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Before beginning, we point  out that if vj is not a descendant of vℓ, then the corresponding  partial derivatives vanish and the claim is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  We begin with generic results that are useful for both the  base case and the inductive step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Let vj be a descendant of vℓ  and let vk be the variable child of vℓ on the path from vℓ to  vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Let cp be the unique check node in N(vℓ) ∩ N(vk) and  let c̺ be the unique check node child of vk on the path from  vk to vj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' if vk = vj, let ̺ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Then, we have that  ∂µvk→cp  ∂rj(h)  =  ∂  ∂rj(h)  ⃝cξ∈N0(vk)\\cp µcξ→vk  ���⃝cξ∈N0(vk)\\cp µcξ→vk  ���  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  (34)  We can also examine the partial derivatives of the probability  vector µc̺→vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Suppose vk ̸= vj and let vo be the unique  variable child of vk on the path from vℓ to vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Using the Fq  convolution, we have  ∂µc̺→vk  ∂rj(h)  =  ∂  ∂rj(h)  \uf8eb  \uf8ed  �  vl∈N(c̺)\\vk  µvl→c̺  \uf8f6  \uf8f8  =  ∂µvo→c̺  ∂rj(h) ⊙  \uf8eb  \uf8ed  �  vl∈N(c̺)\\{vk,vo}  µvl→c̺  \uf8f6  \uf8f8  =  ∂µvo→c̺  ∂rj(h) ⊙ νc̺\\vk,vo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  (35)  We emphasize that νc̺\\vk,vo, as defined implicitly above, is a  probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Having established these results, we turn our attention to  the base case where vj is a variable child of vℓ (vk = vj,  ̺ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Applying (34) and Proposition 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' we obtain  ∂µvj→cp  ∂rj(h)  =  ∂  ∂rj(h)  αj ◦  �  ⃝cξ∈N(vj)\\cp µcξ→vj  �  ���αj ◦  �  ⃝cξ∈N(vj)\\cp µcξ→vj  ����  1  =  ∂αj  ∂rj(h) ◦  �  ⃝cξ∈N(vj)\\cp µcξ→vj  �  �  αj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' ⃝cξ∈N(vj)\\cp µcξ→vj  �  − µvj→cp  �  ∂αj  ∂rj(h),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' ⃝cξ∈N(vj)\\cp µcξ→vj  �  �  αj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' ⃝cξ∈N(vj)\\cp µcξ→vj  �  = 1  τ 2  αj(h)eh ◦  �  ⃝cξ∈N(vj)\\cp µcξ→vj  �  �  αj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' ⃝cξ∈N(vj)\\cp µcξ→vj  �  − 1  τ 2 µvj→cp  �  αj(h)eh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' ⃝cξ∈N(vj)\\cp µcξ→vj  �  �  αj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' ⃝cξ∈N(vj)\\cp µcξ→vj  �  = 1  τ 2  �  νvj→cp −  ��νvj→cp  ��  1 µvj→cp  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  (36)  where  νvj→cp =  αj(h)eh ◦  �  ⃝cξ∈N(vj)\\cp µcξ→vj  �  �  αj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' ⃝cξ∈N(vj)\\cp µcξ→vj  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  By construction, we have 0 ⪯ νvj→cp ⪯ µvj→cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' We turn to  the second graph operation and apply (35), which yields  ∂µcp→vℓ  ∂rj(h)  =  ∂µvj→cp  ∂rj(h) ⊙ νcp\\vℓ,vj  = 1  τ 2  �  νvj→cp −  ��νvj→cp  ��  1 µvj→cp  �  ⊙ νcp\\vℓ,vj  = 1  τ 2  �  νvj→cp ⊙ νcp\\vℓ,vj −  ��νvj→cp  ��  1 µcp→vℓ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  (37)  Thus, in this case, we take ν = νvj→cp⊙νcp\\vℓ,vj as a suitable  vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Based on the fact that νcp\\vℓ,vj is a probability vector,  together with the aforementioned component-wise ordering,  we gather that  0 ⪯ ν = νvj→cp ⊙ νcp\\vℓ,vj  ⪯ µvj→cp ⊙ νcp\\vℓ,vj = µcp→vℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Moreover, leveraging the properties of the Fq convolution for  non-negative vectors, we get  ∥ν∥1 =  ��νvj→cp  ��  1  ��νcp\\vℓ,vj  ��  1 =  ��νvj→cp  ��  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Thus, for this choice of ν, we arrive at  ∂µcp→vℓ  ∂rj(h)  = 1  τ 2  �  ν − ∥ν∥1 µcp→vℓ  �  ,  (38)  as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' That is, the base case conforms to the structure of  Proposition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  We now consider the inductive step in our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' As our  hypothesis, we assume that (33) holds for all computation trees  wherein the distance between the root node and vj is less than  or equal to γ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Consider a rooted computation tree and  suppose the distance between vℓ and its descendant vj in the  tree is exactly γ+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Under Condition 6, there is a unique path  from vℓ to node vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Let vk be the variable child of vℓ that is  also an ascendant of vj, and denote the unique check node that  connects the two by cp ∈ N(vℓ)∩N(vk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Furthermore, let c̺ ∈  N(vk) be the unique check node within this neighborhood  that is an ascendant of vj on the computation tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Finally, let  vo ∈ N(c̺) be the unique variable child of vk that is also an  ascendant of vj (or, perhaps, vj itself).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  The sub-tree starting at vk can be viewed as a rooted tree  containing vj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' the graph distance between these two variable  nodes within the sub-tree is exactly γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' As such, our inductive  hypothesis applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' That is, there exists vector ν such that  0 ⪯ ν ⪯ µc̺→vk where the partial derivative of µc̺→vk with  respect to rj(h) is equal to  ∂µc̺→vk  ∂rj(h)  = 1  τ 2  �  ν − ∥ν∥1 µc̺→vk  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  (39)  Applying (34) and our inductive hypothesis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' we have that  ∂µvk→cp  ∂rj(h)  =  ∂  ∂rj(h)  ⃝cξ∈N0(vk)\\cp µcξ→vk  ���⃝cξ∈N0(vk)\\cp µcξ→vk  ���  1  =  ∂µc̺→vk  ∂rj(h) �  ⃝cξ∈N0(vk)\\cp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='c̺ µcξ→vk  �  �  µc̺→vk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' ⃝cξ∈N0(vk)\\cp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='c̺ µcξ→vk  �  − µvk→cp  � ∂µc̺→vk  ∂rj(h) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' ⃝cξ∈N0(vk)\\cp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='c̺ µcξ→vk  �  �  µc̺→vk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' ⃝cξ∈N0(vk)\\cp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='c̺ µcξ→vk  �  = 1  τ 2  ν ◦  �  ⃝cξ∈N0(vk)\\cp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='c̺ µcξ→vk  �  �  µc̺→vk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' ⃝cξ∈N0(vk)\\cp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='c̺ µcξ→vk  �  − 1  τ 2 µvk→cp  �  ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' ⃝cξ∈N0(vk)\\cp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='c̺ µcξ→vk  �  �  µc̺→vk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' ⃝cξ∈N0(vk)\\cp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='c̺ µcξ→vk  �  = 1  τ 2  �  νvk→cp −  ��νvk→cp  ��  1 µvk→cp  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  (40)  where we have utilized the shorthand notation  νvk→cp =  ν ◦  �  ⃝cξ∈N0(vk)\\cp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='c̺ µcξ→vk  �  �  µc̺→vk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' ⃝cξ∈N0(vk)\\cp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='c̺ µcξ→vk  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  We emphasize that two of the terms in the derivation above  cancel out, as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Furthermore, by construction, we im-  mediately obtain 0 ⪯ νvk→cp ⪯ µvk→cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' These observations  closely parallel the description for the base case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  The derivation of the second graph operation for the induc-  tive step is in complete analogy with the base case, except for  labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Specifically, we apply (35) and obtain  ∂µcp→vℓ  ∂rj(h)  =  ∂µvk→cp  ∂rj(h)  ⊙ νcp\\vℓ,vk  = 1  τ 2  �  νvk→cp −  ��νvk→cp  ��  1 µvk→cp  �  ⊙ νcp\\vℓ,vk  = 1  τ 2  �  νvk→cp ⊙ νcp\\vℓ,vk −  ��νvk→cp  ��  1 µcp→vℓ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  (41)  For the inductive step, we define ν′ = νvk→cp ⊙ νcp\\vℓ,vk as  the candidate vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Based on component-wise ordering, we  can write  0 ⪯ ν′ = νvk→cp ⊙ νcp\\vℓ,vk  ⪯ µvk→cp ⊙ νcp\\vℓ,vk = µcp→vℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  As before, we have that  ∥ν′∥1 =  ��νvk→cp  ��  1  ��νcp\\vℓ,vk  ��  1 =  ��νvk→cp  ��  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Hence, candidate vector ν′ is such that 0 ⪯ ν′ ⪯ µcp→vℓ and  ∂µcp→vℓ  ∂rj(h)  = 1  τ 2  �  ν′ − ∥ν′∥1 µcp→vℓ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  (42)  This completes the proof for Proposition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  We have nearly attained out goal of showing that the  magnitudes of the entries in the Jacobian matrix of η(r) with  respect to r are uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' To achieve the desired  result, it sufficies to connect the partial derivative of the  incoming message with the partial derivative of state estimate  ˆsℓ (r, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' This is accomplished below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Proposition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Under Condition 6, the absolute value of the  entries in the Jacobian are bounded by  ����  ∂ˆsℓ (r, g)  ∂rj(h)  ���� ≤ 1  τ 2  ∀g, h ∈ Fq, ℓ, j ∈ [L],  (43)  where τ  > 0 is the standard deviation of the effective  observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Proof: When vj does not appear in the rooted tree  of vℓ, the partial derivative is equal to zero and the result  immediately follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Furthermore, when vℓ = vj, the result  follows from Corollary 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Thus, we can focus on the scenario  wherein vj is a descendant of vℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Let cp be the unique check node in N(vℓ) that lies on the  path between vℓ and vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' By Proposition 14, there exists vector  ν, with 0 ⪯ ν ⪯ µcp→vℓ, such that the partial derivative of  µcp→vℓ with respect to rj(h) is given by  ∂µcp→vℓ  ∂rj(h)  = 1  τ 2  �  ν − ∥ν∥1 µcp→vℓ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  (44)  Drawing an analogy to (40), we have  ∂ˆsℓ (r)  ∂rj(h) =  ∂  ∂rj(h)  ⃝cξ∈N0(vℓ) µcξ→vℓ  ���⃝cξ∈N0(vℓ) µcξ→vℓ  ���  1  = 1  τ 2  ν ◦  �  ⃝cξ∈N0(vℓ)\\cp µcξ→vℓ  �  �  µcp→vℓ, ⃝cξ∈N0(vℓ)\\cp µcξ→vℓ  �  − 1  τ 2 ˆsℓ (r)  �  ν, ⃝cξ∈N0(vℓ)\\cp µcξ→vℓ  �  �  µcp→vℓ, ⃝cξ∈N0(vℓ)\\cp µcξ→vℓ  �  = 1  τ 2  �  νvℓ − ∥νvℓ∥1 ˆsℓ (r)  �  ,  (45)  where we have implicitly defined  νvℓ =  ν ◦  �  ⃝cξ∈N0(vℓ)\\cp µcξ→vℓ  �  �  µcp→vℓ, ⃝cξ∈N0(vℓ)\\cp µcξ→vℓ  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  We note that 0 ⪯ νvℓ ⪯ ˆsℓ (r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Thus, we have that:  ∂ˆsℓ (r)  ∂rj(h) ≤ 1  τ 2 νvℓ ≤ 1  τ 2 ˆsℓ (r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  (46)  Since we are interested in bounding the absolute value of the  partial derivatives, we also consider a lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  ∂ˆsℓ (r)  ∂rj(h) ≥ − 1  τ 2 ∥νvℓ∥1 ˆsℓ (r) ≥ − 1  τ 2 ˆsℓ (r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  (47)  Combining these two observations with the properties of  probability vectors, we obtain the desired expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Theorem 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Under Condition 6, the BP-N denoiser presented  in definition 5 is Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content='  Proof: By Proposition 15, the magnitudes of the entries  of the Jacobian matrix of η(r) with respect to r are uniformly  bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
+page_content=' Thus, the denoiser is Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQf7_7g/content/2301.01899v1.pdf'}
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